Recent developments in meteorological physics

PHYSICS REPORTS (Review Section of Physics Letters) 48, No. 2 (1978) 65-177. NORTH-HOLLAND PUBLISHING COMPANY

RECENT DEVELOPMENTS IN METEOROLOGICAL PHYSICS R.R. Rogers McGill University, Montreal, Canada and

Gabor VALI University of Wyoming, Laramie, U.S.A.

Received April 1978

Contents: I. Introduction 2. Light scattering in the atmosphere 2.1. Introduction and background 2.2. Light scattering by aerosols 2.3. Light scattering by clouds 3. Atmospheric absorption and emission of radiation 3.1. Introduction 3.2. Effects of gases 3.3. Clouds and aerosol layers 3.4. Radiative energy budget 4. Physics of clouds and precipitation 4.1. Introduction 4.2. The formation of clouds

67 68 68 80 95 105 105 107 108 112 115 115 116

4.3. The “warm-rain” process 4.4. The formation of ice in clouds 4.5. Growth and precipitation of ice 4.6. Precipitation modification 5. New methods of observing the atmosphere 5.1. Introduction 5.2. Microwave radar 5.3. Laser applications 5.4. Acoustic radar 5.5. In-situ cloud measurements 5.6. Remote sensing by satellite Appendix 1. List of principal symbols References

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 48, No. 2 (1978) 65—177. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price DII. 43.00, postage included.

124 127 37 141 41 141 141 147 152 55 158 167 169

RECENT DEVELOPMENTS IN METEOROLOGIC AL PHYSIC S

R.R. ROGERS McGill University, Montreal, Canada and

Gabor VALI University of Wyoming, Laramie, U.S.A.

1 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

67

1. Introduction Although much of meteorology is physics, there is within the body of atmospheric science a group of phenomena which, by tradition, are considered to comprise physical meteorology. These are phenomena not directly linked with atmospheric circulation, the domain of dynamical meteorology, and include atmospheric optical, accoustical and electrical effects, radiation and cloud physics, thermodynamics and the study of atmospheric composition. Too much is included in this definition to be covered in a modest review article. To achieve a manageable size we have limited ourselves to certain topics in radiation and atmospheric optics, cloud physics and methods of observing the atmosphere. Although aeronomy, turbulence and convection, and atmospheric electricity are thus omitted, the topics retained are at the core of physical meteorology and have undergone remarkable recent development. Moreover, these are topics to which meteorologists actively contribute whereas, in some of the subjects omitted, research is carried out primarily by chemists, fluid dynamicists, or applied mathematicians. By the choice of material, we are therefore giving a meteorologist’s view of atmospheric physics. The scope of the review has been further limited by confining discussion to the terrestrial atmosphere, in spite of spectacular recent advances in the understanding of atmospheres of other planets. The review is arranged into four main sections. Sections 2 and 3 are on radiation, the emphasis of section 2 being solar radiation and that of section 3 terrestrial radiation. The extended introduction of section 2 prepares the way for much of what follows by introducing the terminology and some of the fundamental concepts of radiative transfer. Section 4 is on the physics of clouds and precipitation. Section 5 is devoted largely to atmospheric measurements by remote sensing. Developments of the past decade are emphasized throughout. Although some attempt was made to have each of the major sections independent and self-contained, there is obviously common ground in sections 2 and 3, and section 5 draws from much of the preceding material. Notable for the past century has been the high degree of international cooperation in observational meteorology. This has become more conspicuous during the last decade as a result of world-wide cooperative research programs, of which the Global Atmospheric Research Program (GARP) is the best example. Considerable recent progress in physical meteorology has been stimulated by the needs and opportunities posed by GARP and its sub-programs. Some research projects, especially those in satellite meteorology and numerical modeling, are strongly interdisciplinary, requiring a team effort and accounting for the prevalence of multiple authorship in the cited publications. References which have been particularly helpful in preparing this review and which we recommend for further reading—are the recent books of McCartney [271]and Paltridge and Platt [300]and the comprehensive reviews of Hansen and Travis [150]and Houghton and Taylor [179].We should also mention the standard references on atmospheric radiation [74,128, 177, 227,409] and cloud physics [111,267].A short book [1761has just appeared, which treats the fundamentals of atmospheric physics and takes a broader view of the subject than that of this review. Wherever possible we have avoided citing unpublished reports and conference papers, limiting the references to papers in refereed journals and readily available books. The nomenclature and units used in atmospheric physics, especially radiation, are far from standard. We have generally used SI units and the symbols and nomenclature recommended by most international commissions on standardization. Exceptions are the units of pressure, for which we have retained milhibars (1 mb = 102 Pa), and the units appearing on some of the illustrations, which were left unchanged from the original sources. A list of frequently used symbols is given in appendix 1. —

68

R.R. Rogers and Gabor Vali. Recent developments in meteorological physics

For assistance in preparing the manuscript we thank Mrs. Olga Cartier, Miss Ursula Seidenfuss and Mrs. Sandra Yip. Dr. Henry Leighton made several good suggestions for improving parts of the manuscript. One of us (R.R.R.) was supported by grants from the National Research Council of Canada and the Atmospheric Environment Service while working on the review. When agreeing in early 1977 to undertake this project we decided on areas for which we were individually responsible within a general outline and anticipated a relatively close collaboration. Under the best of conditions frequent consultation would not have been possible owing to our geographical separation. As it turned out, communication was made more awkward than expected because both of us were away on sabbatical leaves during part of the time available for writing the report. Consequently the final manuscript, though a joint effort, strongly reflects the separate contributions of its authors: section 4 is essentially the work of G.V. only whereas R.R.R. wrote most of the rest. Although it is not clear whether closer collaboration would have produced a manuscript substantially different, we thought that the particular circumstances under which the review was written needed to be mentioned.

2. Light scattering in the atmosphere 2.1. Introduction and background Natural and artificial light may be refracted, absorbed, or scattered by atmospheric constituents. The study of these effects comprises atmospheric optics, the oldest branch of theoretical meteorology. Its early development was related to astronomy because of the limitations imposed by the atmosphere on distant seeing. Over the years contributions were made by a number of outstanding figures. Leonardo da Vinci, for example, in his quest for realism in painting, studied the phenomena of the rainbow and the reduction of visual contrast with range. First to be explained acceptably were refractive effects: the rainbow by Isaac Newton, and later the halo. The celebrated scattering theory published in 1871 by John Strutt (afterwards Lord Rayleigh) was developed to explain the color and polarization of light from the clear sky. Recent progress in atmospheric optics has been inspired by new ways of observing—from satellites, spacecraft and high-altitude airplanes, and by means of laser light. Developing simultaneously with the new observational techniques, the capacity of computers has increased sufficiently to allow scattering calculations on a scale much larger than previously possible. Progress has also been stimulated by the development of global general circulation models and climate models, in which a careful accounting of the disposition of solar radiation in the atmosphere is essential. Of all the atmospheric constituents, clouds have the most profound influence on sunlight. Their effect is predominantly to scatter the incident radiation; they reflect a large fraction of solar energy back to space, making it unavailable to support processes at the earth’s surface or in the atmosphere. Not surprisingly, light scattering by clouds is a subject of long-standing meteorological interest. Recent developments in the theory of multiple scattering and in methods of solving radiative transfer problems have made possible a greatly improved understanding of this subject. The scattering by ever-present aerosol particles accounts for atmospheric turbidity and is treated by the same theory as clouds. Recent activity in this area is related to the mounting concern over atmospheric pollution. On the one hand, light scattering provides a means of remotely measuring

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

69

aerosol properties and monitoring emissions from pollution sources. On the other, there is concern that an increase in atmospheric aerosol content could lead to climatic change. This section is devoted to recent work on meteorological aspects of the scattering of natural light by clouds and aerosols. To provide a background for some of these rather specialized studies, this introductory part continues with a description of the solar spectrum and a short review of scattering theory. The solar spectrum Figure 2.1 gives the approximate form of the spectral irradiance of extraterrestrial solar energy and schematically illustrates the spectrum as observed under clear skies at sea level. The absorption bands of atmospheric gases are indicated. Absorption by ozone occurs in the stratosphere and that by water vapor mainly in the lower troposphere. Molecular scattering accounts for some attenuation across the entire spectrum and predominates in the visible range of wavelengths and into the ultraviolet. During the past decade there has been considerable refinement of the estimates of the extraterrestrial spectrum and of its integral, the solar constant. Based on measurements from high-altitude aircraft, balloons, and spacecraft, the accepted value of the solar constant is 1353 W/m2 [383]. Standard values of the spectral irradiance have been tabulated between 0.3 and 0.61 p~mat intervals of 0.1 A [382]; with coarser resolution the spectrum has been extended to 100 ~m [236].Drummond [97] described the instruments used for these measurements and presented a historical survey of estimates of the solar constant. Extinction of light As a collimated beam of radiation traverses a distance ds of atmosphere, energy may be lost from the direct beam by scattering and absorption. The total depletion is given by the law variously 22

4 \~

20

if

I8

‘I

11

16

\\ \‘ o~

I I I i I

14

E

Loergy carve for bloch body a’ 6000’t< ~ ~ç~k—Solar energy curre ootsde atmosphere H~0\ Solar energy cuooe at sea leoel

010

H 20

\

I

I

-

~6-

\Hy0

~

103 I

-

4.

2

-

0

H50

I

/ 0.2

I 0.4

0.6

0.8

‘

1.0

1.2 1.4 1$ Wovelenqth, j.tm

l.6~

C02-r120

22

2.4

2.62.8~0&?

Fig. 2.1. Solar spectral irradiance at sea level and extrapolated outside the atmosphere. Darkened areas indicate gaseous absorption in the atmosphere [237].

70

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

attributed to Bouguer, Lambert, or Beer, dE/E

(2.1) 2) and y is the total volume extinction coefficient (e.g. in where E denotes the irradiance (e.g. in W/m m2/m3 or m’). This coefficient may conveniently be written as the sum of two extinction coefficients, one due to molecules of the gaseous atmospheric constituents, the other due to particles such as aerosols and cloud droplets. Both of these coefficients may in turn be written as the sum of a scattering coefficient and an absorption coefficient. Over a path of length I the transmittance t is defined by =

—

—yds,

E(l)/E 0 = exp(_

j yds).

where E(l) denotes the irradiance at s = I and E0 is the irradiance at the beginning of the path, s The incremental optical thickness dr along the path is defined by dr

=

yds.

(2.2) =

0.

(2.3)

The total optical thickness of a path of length I is therefore T =

f

yds.

(2.4)

Ordinarily the extinction coefficient depends on the wavelength A of the radiation. Therefore T and t are in general functions of the wavelength and eq. (2.1) itself applies only for radiation that is approximately monochromatic. Goody [128]and Zuev [447]discussed conditions under which eq. (2.1) might not be satisfied, but concluded that it correctly describes the attenuation of natural light in the atmosphere. In this section consideration is given to scattering by molecules and extinction by particles. These are the principal atmospheric attenuating processes in the visible region of the electromagnetic spectrum. Section 3 is devoted to molecular absorption, the remaining process which contributes to total extinction, and an important source of atmospheric attenuation in the infrared region of the spectrum.

Rayleigh scattering Rayleigh’s theory for scattering by air molecules explains the blue color of the clear daytime sky, its brightness, and the degree of polarization of skylight. Excellent accounts of the Rayleigh theory are given by van de Hulst [409], Kerker [206] and McCartney [271]. The main result expresses the angular scattering cross section of a small particle, assumed to scatter as an elementary dipole, as 22 2 + 2) 2 (1 + cos 0), 4 ~~n2 (2.5) 9irVin—l~ UR(O) I(O)/E0 = 2A where V is the volume of the particle, n is its refractive index, A is the wavelength of the radiation and E 2). This equation applies for unpolarized incident radiation; 1(0) is the incident irradiance is the0scattered intensity (W/sr) (W/m in the direction at angle 0 from the direction of propagation of the beam. The units of O~R(O)are m2/sr. The total scattering cross section of a particle of volume V is obtained by integrating eq. (2.5) over

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

71

‘4ir steradians, 3

I

,2

~

IT 4rV ~n2+2)~ 217i_~1\

(2.6)

o-R_J0R(8)dfl—2

If the particle is a sphere of radius r, then 56 2 2 128ITr (n_—1\ ~ 3A~ (\fl2+2) (2.7) The volume scattering coefficient is obtained by summing the scattering cross sections of all the particles in a unit volume. If, for example, the particles are all identical and with concentration N per unit volume, then the total volume scattering coefficient for unpolarized incident light is obtained from eq. (2.6) as .

3

13R =

NUR

IT

=

/ 2 2ifl’

~

24N~V ki~+2

(.)

The refractive index fla of air may be defined by (na— 1) = N(n

—

1)V = p(n

the air density and m0 is the mass of an air molecule. In the limit as n 2 NV

(“~i~i ~)_*~~[N(n

—

1)V]2 =

~(fla

—~

—

l)V/m

0, where p is

1,

1)2.

Then eq. (2.8) may be written in terms of fla as /3R = (32ir313NA4) (fla

—

(2.9)

1)2,

which is the usual form of Rayleigh’s inverse fourth-power scattering law. By employing the relationship between (fla — 1) and p given by Kondratyev [227]for air at 273 K and 1000 mb, eq. (2.9)

may be reduced to 8.4 X h0-~ptA4 where ~R is in km~,p is in kg/rn3, and A is in ~m. This result neglects the weak dependence of ii,, Ofl wavelength, but is accurate to within 3% over the visible portion of the spectrum. The foregoing relations assume the scattering particles to be isotropic, in the sense that the induced dipole moment is exactly aligned with the incident electric vector. Because atmospheric gases are weakly anisotropic eq. (2.9) is only an approximation for molecular scattering. Cabannes derived a correction factor to account for molecular anisotropy, expressible as (6+ 38)/(6 78) where 8 is a depolarization factor. This quantity is applied as a multiplier to any of the scattering coefficients. For example, the angular volume coefficient for molecular scattering of unpolarized radiation is =

—

13R(O) =

2IT~4 1)(~+

(1 + cos2o).

(2.10)

As the depolarization factor for air is estimated to be 0.035, the correction factor amounts to approximately 1.06 [271,448]. Over a path of length I the Rayleigh transmittance may be defined by tR = exp( Tg) where the optical thickness ‘TR is given by —

TR0J

~RdS.

(2.11)

72

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

The normal optical depth r0 of the earth’s atmosphere depends on wavelength and is obtained by integrating over a vertical path from the top of the atmosphere to the surface. Hansen and Travis [1501 explained that for a standard atmosphere and a surface pressure of 1013.25 mb, molecular scattering causes a normal optical depth given by 4(1 + 0.01 13A2 + 0.000l3A4), (2.12) = 0.008569A with A in ~m. For a wavelength of 0.5 ~m this formula gives T 0 0.144, corresponding 4 law because oftothea transmittance of 0.87. The expression for ‘ro deviates slightly from the A dependence of the refractive index fla on wavelength. The optical depth above any pressure level p in the Rayleigh atmosphere is proportional to pressure, ‘r(p) =

(2.13)

Top/po.

Another result from classical Rayleigh theory expresses the degree of linear polarization of scattered light as I /1= pol

1

+

sin20 cos2O + 28/(1

—

6)’

(~14)

where ~ and I denote respectively the intensity of the linearly polarized light and the total intensity of light scattered in direction 0. Unpolarized incident radiation is assumed. If all the light reaching the surface through a Rayleigh-scattering atmosphere were scattered only once, then eq. (2.14) would describe the distribution over the sky of the degree of polarization of skylight. Again assuming only primary scattering, it is possible to solve for the distribution of radiance of the skylight, for a given solar zenith angle, by suitably integrating the angular scattering coefficient over altitude. Such results agree only approximately with observed clear-sky radiances and polarizations.* The two main reasons for the discrepancies are multiple scattering effects and the presence in the atmosphere of dust and haze particles which are too large for the Rayleigh theory to apply. These reasons had long been known but only recently have the effects been incorporated into meteorological optics. The approach which replaces Rayleigh scattering in the case of particles whose sizes are no longer negliglible compared to the wavelength is the Mie scattering theory for spheres, available since 1908 but applied extensively only since the advent of fast electronic computers. The treatment of multiple scattering has been made possible by the mathematical theory of Chandrasekhar [49]published in 1950. Most of the developments in meteorological optics during the past two decades have exploited either the Mie theory or Chandrasekhar’s formulation of the radiative transfer problem. Mie scattering is extensively covered in a number of books [27,90, 206, 271, 397,409] and review articles [41, 150]. Multiple scattering in the earth’s atmosphere has been under study by Sekera and his colleagues [72,81,91,356] and has been included in excellent recent reviews [150,188]. Mie scattering The angular scattering cross section o~,(0)of any particle may be defined as in eq. (2.5) as the proportionality factor between incident irradiance E 0 and the intensity 1(0) scattered in direction 0. That is, 1(0)

=

E0o0(0).

~Ground-basedinstruments for measuring solar and terrestrial radiation, and their history, are described by Coulson [74].

(2.15)

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

73

For a homogeneous spherical particle of radius r, Mie’s solution to the scattering problem gives for the cross section (2.16) o~(0)= (A218IT2) (1S 2 + S 2),

11

where S

21

1 and S2 are dimensionless complex amplitudes expressed as infinite series. These amplitudes depend on the size and composition of the particle and the scattering angle. The size enters through the dimensionless parameter a = 2irr/A. The significant aspect of composition is the complex refractive index m = n in, where n and n~denote the real and imaginary parts. The imaginary part of m is associated with absorption of radiation by the particle, an effect which was disregarded in the —

description of Rayleigh scattering. The quantity 4ITn1/A is referred to as the Lambert absorption coefficient of the material. In the limit as IamI—~0,and for unpolarized incident light, a~~(0) approaches the Rayleigh cross

section given by eq. (2.5). As a increases, the tendency is for o~,(0)to become more irregular in shape, with an increasingly prominent peak for forward scattering (0 = 0). The total scattering cross section o~,is obtained by integrating o-~(O)over all scattering directions, =

J

(2ar

o~,(0)dil

=

21rJ

o0(0)sinOd0.

(2.17)

0

Results are usually presented in the form of the dimensionless scattering efficiency factor Q0, defined by

2.

(2.18)

Q5=u~/irr

Because absorption occurs when the refractive index has a non-zero imaginary part, the total flux removed from the beam exceeds the amount scattered. The total flux removed is described by the extinction cross section, which may be written as the product of the geometrical cross section ITT2 times an extinction efficiency factor Q~.From Mie theory Q~is obtained from either of the complex amplitude functions (S 1 or S2) evaluated at 0 = 0. Because the effects are additive, it follows that

Q~= Qs + Qa,

(2.19)

where Qa denotes the efficiency factor for absorption. Numerous examples of these efficiency factors for different materials and over a wide range of size parameter are given in the standard references [90,206,409].

Scattering efficiency factors for two kinds of nonabsorbing spheres are plotted on logarithmic coordinates in fig. 2.2. The curve for n = 1.50 approximates a dry haze particle; that for n = 1.33 corresponds to a water droplet. For both curves the slope becomes equal to approximately 4 for a <1, consistent with Rayleigh scattering. The flattening of the curves as a increases indicates the transition from Rayleigh to Mie scattering, whether due to increasing particle size or decreasing wavelength. The maximum value of Q8 for the curve n = 1.33 occurs at a = 6.5, indicating that a water droplet scatters most efficiently when its radius is approximately equal to the wavelength. Curves of Q~and Q. are plotted in fig. 2.3 to illustrate the effect of increasing absorption. For all the curves n = 1.525, which is thought to be representative of dry dust particles. Six values of n, were used: 0, 0.005, 0.01, 0.02, 0.05 and 0.1, of which 0.005 is believed to be typical of dust aerosols. As the size parameter increases the Qe values oscillate about a smooth curve that approaches Q~= 2 for large particles, the geometric optics limit [150,206,409].The largest oscillations are associated with nonabsorbing particles (n1 = 0); the amplitude of the oscillations decreases with increasing n1. In fig.

74

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

Fr I

1.50/

is

1 33

I

-

~

~

/

—

0

/

I

~

0

—

I

5

-.

4

20

40 60 80 MIE SIZE PARAMETER

100

(b)

I

-

>-

~

iO~~

3

/I

2

0 0

-~

I 1

2

5

7 1.0

10

100

Size p~ror~eter, ~

Fig. 2.2. Scattering efficiency factors for nonabsorbing spheres haying refractive indices approximating those of water (n = 1.33) and dry aerosols (n = 1.50) [271].

.

20

~ 40 60 MIE SIZE PARAMETER

80

100

.

Fig. 2.3. Extinction efficiency (a) and scattering efficiency (b) for spheres with n = 1.525 and n, = 0, 0.005, 0.01, 0.02, 0.05 and 0.10, plotted against the Mie size parameter a = 2irr/A. In (b) the topmost curve is for nonabsorbing particles (n, 0), and those below are for increasing values of n, [304].

2.3, Qs = Qe for the nonabsorbing particles but the Q. values decrease with increasing absorption, approaching Q. = 1 for large particles. Volume scattering and extinction coefficients are obtained by summing the appropriate cross sections of the individual particles in a unit volume. Accordingly, the volume total coefficients for extinction and scattering are given by

f3~=

2 Qen(r)dr

(2.20)

irr2Q,n(r)dr

(2.21)

ITT

/3e =

f

where n(r) dr is the number of particles per unit volume with radii between r and r + dr. The volume

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

75

angular scattering coefficient is likewise defined by I3s(0)°°J u~(0)n(r)dr. (2.22) The directional dependence of scattering is described in terms of the dimensionless phase function

P(0), defined by P(0) = 4irf3~(0)/I3~.

(2.23)

Thus defined, P(0) is normalized in the sense that ~—JP(0)df1

=

1.

(2.24)

From eqs. (2.9) and (2.10), the phase function for Rayleigh scattering, disregarding the depolarization factor, is seen to be P(0) = ~(l+ cos20). Instead of the complete phase function, it suffices in some treatments of radiative transfer to know only the asymmetry factor (cos0), defined by (cos0) =

J

j~— “iT

P(0)cosOdfl,

4~.

which is the mean value of cos0 over a sphere, weighted by the phase function. For small values of the scattering parameter a, (cos0) approaches zero because of the symmetry of the Rayleigh scattering cross section about the forward and backward directions. Another quantity used to characterize a polydisperse scattering medium is the albedo for single scattering w 0 defined by =

13s/Pe

f~J(fl~ + faa).

A nonabsorbing medium is clearly one for which mounts co~tends to decrease.

(2.25) wo =

1, and as the importance of absorption

The relationships given up to this point that involve total scattering are valid for incident light of any polarization. The relations involving the directional dependence of scattering apply only for unpolarized incident radiation, which is the case for direct sunlight in the atmosphere. In none of the

relations is there any information about the polarization properties of the scattered light, and this light will in general be partially polarized, even for unpolarized incident radiation. To characterize the polarization state of light completely requires specifying four parameters. The particular four that are most useful in scattering problems are the Stokes parameters, I, Q, U and V. The scattered intensity may be regarded as a vector whose four components are the Stokes parameters. The scattering

transformation may then be written as IS

10

g~ V0

~

goV0

(2.26)

where f3. is the total volume scattering coefficient and P is the phase matrix, four rows and four columns of dimensionless numbers expressed in terms of the Mie scattering amplitudes. For homogeneous spheres the elements of P depend only on S1 and S2. When the polarization of the scattered light is of interest it is necessary to employ this full vector formulation. For the purposes of

76

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

computing radiant energy flux and heating rates, the scalar form of the equations is an excellent approximation, even when multiple scattering is taken into account [237]. For a comprehensive exposition of Mie scattering theory employing the Stokes vectors, the reader is referred to the standard sources [27,90, 150, 206,409]. Starting with Deirmendjian’s book [90], a number of calculations have been published on Mie scattering functions for atmospheric polydispersions. Most of these are based on analytic models of the size distribution function n(r) which approximate the particles encountered in fogs, hazes, or clouds. Hansen and Travis [150],for example, employed the two-parameter cloud droplet model r/ab), (2.27) n(r) rt’3”~”exp (—

where n(r)dr is the number of droplets per unit volume with radii in dr, and a and b are the “effective” (area-weighted) mean and variance of the distribution. Although the size-spectra of natural clouds are highly variable and often irregular in shape, this simple model approximates some of the observed distributions. Figure 2.4 shows the dependence of total scattering cross section on mean droplet size for several values of the effective variance b. On the ordinate is plotted f3 2n(r)dr, which may be interpreted as the scattering efficiency factor for the ensemble of drops. On5/firr the abscissa is plotted 2ira/A, which is the effective Mie size parameter for a polydispersion of average radius a. The solid curve (b = 0) is for a distribution of zero width; hence it represents the scattering efficiency factor for a single drop of radius r = a, or for a set of equal-sized drops with radius a. As the width of the droplet spectrum increases, the fine irregularities or ripples are first smoothed out; then the peak scattering efficiency, which occurs between a size parameter of 5 and 10, is reduced. 5

___________

0

5

5.?

0 2iro/~

50

00

Fig. 2.4. Effective scattering efficiency for the droplet model defined by eq. (2.27). as a function of the size parameter 2ira/A. Curves shown for four values of the variance parameter b. The refractive index is m = 1.33—01 [150].

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

0

—rr11~

77

I

9

-

.8

..———

-

-

//~ 7. -

/

/

/

/

,,.—

-

-

-

/ -

/

.5

b~0 b~.0l

-

b~.I

/

4/

.3

-

-

-

I

-

I 5

I

0 2ira/X

50

00

Fig. 2.5. Effective asymmetry parameter (cos 0) for the droplet model defined by (2.27), as a function of the size parameter 2ira/A. Curves shown for four values of the variance parameter b. Refractive index as in fig. 2.4 [150].

The influence of spectrum width on the asymmetry parameter is shown in fig. 2.5. Again the solid for b = 0 corresponds to a single drop of radius r = a. For small values of 2ITafA, ~cos0) approaches zero in accordance with Rayleigh theory. As the size parameter increases beyond about 20 the dependence of (cos 0) on a/A and b becomes weak. For typical cloud droplet sizes and optical and infrared wavelengths, 2ira/A is large enough that (cos 0) may be regarded approximately as independent of a and b. The limiting value of (cos 0) depends upon the refractive index, however. Other results of Hansen and Travis show that for a size parameter of 100 and for n1 = 0, (cos 0) decreases curve

from about 0.91 to 0.70 as the real part n increases from 1.2 to 2.0. When some absorption is permitted (n, 0), the value of (cos 0) approaches 0.97 for large drops with n = 1.33. For the same droplet model, fig. 2.6 illustrates the dependence of the single scattering albedo on n1. All solid curves are for a single drop (b = 0), and are drawn for different values of the size parameter 2ITr/A. The dotted curves correspond to polydispersions with b = 0.1, and are for size parameters of 1 and 5. The albedos approach unity with decreasing n1, corresponding to nonabsorbing spheres. The effect of the spread parameter b on albedo is small, and becomes negligible for size parameters greater than about 10. For large values of the size parameter, and for n1 not too large, w~approaches 0.53, which can be explained by geometric optics [150]. Many results of this kind were presented by Deirmendjian [90] for a variety of cloud and haze models, based on a modified gamma distribution for n(r). Hansen and Travis compared the phase functions for four analytic models of n (r), including their own and Deirmendjian’s, and concluded that the general behavior of P(0) is determined by the mean radius and the standard deviation, regardless of the particular model used. This of course lends support to the use of simple models in estimating

78

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

.8

-..~--.

\“~

\

\~

0.0

-•\~

~

III

\—\~ L)~.\~5 -.

0 -

S

..

---

-.

.4 —

:L1 .0001

.001

.01

0i

I

_ 0

00

Fig. 2.6. Single scattering albedo as a function of n, for the droplet model (2.27). Solid curves are for b = 0, corresponding to a single drop for which 2ira(A = 2irr/A = a. The real part of the refractive index is n = 1.33 [150].

the scattering properties of natural clouds, and it also indicates that scattering measurements might be employed for deducing information about the mean droplet size and the spread of sizes in clouds. Radiative transfer As the simplest form of radiative transfer equation, Bouguer’s law (eq. 2.1) expresses the rate of depletion of energy from a collimated, monochromatic beam as it traverses an absorbing or scattering medium. More general forms of the equation include an additional term to allow for the possibility of sources of radiant energy along the path which contribute to the radiance in the same direction and at the same wavelength as the incident beam. Such sources can arise from either scattering of radiation into the beam or emission of radiation at the same wavelength by material along the path. Owing to the relatively cold temperature of the terrestrial atmosphere, the extra term for wavelengths in the solar spectrum arises entirely from scattering. On the other hand, for the longer wavelengths in the terrestrial radiation spectrum, both emission and scattering sources may be important in cloudy conditions, though emission effects predominate in clear skies. When emission sources are negligible, the radiative transfer equation may be written dL/ds

—

L+~J

L(o)P(o)dcl].

(2.28) 2sr_i) is used instead of irradiance E to allow for diffuse radiation. In this form, L(W gives m The first termthein radiance the brackets the extinction of radiance due to scattering and absorption by molecules and particles. The second term is the scattering source term, which accounts for incident radiances from all directions that intersect the scattering volume at ds and are there scattered through =

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

79

angle 0 into the direction of the main beam. This equation is little more than a formal description of

the scattering process. Analytical solutions are usually impossible owing to the difficulty in evaluating the source term. Chandrasekhar’s theory [49]made possible an exact solution of this equation for the special case of a plane, horizontally homogeneous Rayleigh atmosphere. Up until that time (1950), several efforts had been made to solve the problem, but severe difficulties were encountered when attempting to include the effects beyond first-order (primary) scattering. Chandrasekhar’s solution incorporated all orders of scattering and the effects of ground reflection as well. It was then possible for the first time to evaluate theoretically the amount and spectral distribution of solar radiation reaching the ground under clear skies. Deirmendjian and Sekera [91] were among the first to apply Chandrasekhar’s method. They separated the total solar radiation into the direct-beam “sun radiation” and diffuse “sky radiation” and for a plane-parallel Rayleigh atmosphere solved for the spectral irradiance at the ground for various solar zenith angles. Figure 2.7 illustrates their results for zenith angles of

00,

53.1°, and 84.3°.The

uppermost curve represents the solar irradiance at the top of the atmosphere. Then, for the three zenith angles, the families of curves below show the irradiance at the ground due to the direct beam, the diffuse radiation, and the additional diffuse radiation caused by reflection from the ground of a fraction of the incident radiation. Solution was possible assuming a Lambert surface; in this example the surface albedo is 0.25. Quite evident is the increasing importance of diffuse radiation at short wavelengths. It is also clear that the surface reflectance increases the sky radiation significantly,

especially for the smaller zenith angles. Marggraf and Griggs [265]gave a useful formula for the total monochromatic solar irradiance at the 24 22 20

-

18

11) 14 ~

12 10

~

Sun radiation

~

Sky radiation Reflected radiation (albedo=0.25)

-

-

i~F 5301

2 0 0.10

~

I

~

0.20

~

84~.3

I

_______

I

0.50 0.80 1.00 Wavelength (lim)

2.00

__

5.00

Fig. 2.7. Solar spectral irradiance on a horizontal surface at sea level for three solar zenith angles. The topmost curve represents the extraterrestrial irradiance. Hatched areas represent additional radiation received from sky due to multiple Rayleigh scattering. Blackened areas are component reflected from surface and scattered back to the ground [91].

80

R.R. Rogers and Gabor Vali. Recent developments in meteorological physics

ground (direct beam plus skylight), based on the work of Deirmendjian and Sekera, as E(r,

.i,~,A)

Eoj.~oZ(r,~io)/[l S(r)A] —

where r = r~is the normal optical thickness of a Rayleigh atmosphere (given, for example, by eq. (2.12)), A is the surface (Lambert) albedo, E0 is the extraterrestrial solar irradiance and ~ = cos 0~ with 0~the solar zenith angle. The functions Z and S were presented in graphical form, from which values can be read to solve for E for any combination of wavelength, zenith angle, and surface albedo. A similar expression, involving the same functions, was given for the irradiance emerging from the top of a Rayleigh atmosphere. Within a decade, most of the scattering properties of a Rayleigh atmosphere had been calculated, including polarization effects and radiation scattered back to space, as well as the downward radiation in the atmosphere [81].These results are only a first approximation for the real atmosphere, however, because of the usual preponderance of Mie over Rayleigh scattering and because of the marked horizontal irregularity in the distribution of scatterers. Assuming horizontal homogeneity, exact solutions of eq. (2.28) have been possible for simple forms of the phase function, such as that for Rayleigh scattering. As the forward-scattering lobe of P(0) increases, it even leads to numerical difficulties in approximate solution of the equation [147]. The various methods that have been employed for solving multiple scattering problems are comprehensively reviewed by Irvine [188] and Hansen and Travis [150]. It should be mentioned here that phenomena of the twilight sky, in which the sun is below the horizon, may not be treated by the usual assumption of a plane-parallel atmosphere. The spherical shape of the atmosphere becomes all important and refractive effects, generally negligible at solar elevations above a few degrees, must be taken into account. In his treatise on twilight [333], Rozenberg included approximate formulas for the radiance pattern of the sky, based largely on single scattering theory. More recently, using the Monte Carlo numerical technique, Blättner et al. [26] solved for the radiance and polarization distribution in the twilight sky and found that results which include multiple scattering are in approximate agreement with observations. 2.2. Light scattering by aerosols During the past decade, great impetus has been given to studies of the radiative effects of aerosols primarily for two reasons: (1) increased concern about long-range climatic consequences of presumed increases in the atmosphere’s aerosol content arising from the activities of man; (2) the advent of meteorological satellites, which allow the atmosphere to be observed remotely and investigated by way of its scattering and emitting characteristics. Two technological factors have also contributed to the blossoming activity in this field: (1) readily available fast computers, which make feasible the calculation of Mie scattering functions for nearly any desired combination of particle size, refractive index, and scattering angle; (2) the appearance of the laser, which may be used for observing aerosols

in much the same way as radar is employed for studying rain and snow. Concentration, size and cornposition of aerosol particles According to Fenn the aerosol burden of populated air in the continental lower troposphere ranges from 100— 3in large cities[104] to about 50 ~g/m3 in thinly areas. Estimates by Junge 200 ~ag/m [199]of the clean-air, background aerosol mass are an order of magnitude less. The aerosols consist of (1) inorganic, water-insoluble material, (2) organic, water-insoluble material, and (3) inorganic, water-

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

81

soluble material, in the approximate proportions of 60%, 15% and 25%, respectively. On the global scale, Brock [35] has estimated that about 75% of the total aerosol mass arises from natural and anthropogenic primary sources such as dust stirred up by wind (20%), sea spray (40%), forest fires (10%), combustion and other industrial operations (5%). The remaining 25% is attributed to secondary sources which involve conversion by photochemical and other chemical processes of gaseous atmospheric constituents to particulates. Detailed tabulations by Bach [10]of the magnitudes of these components as reported by various authors indicate a wide latitude of uncertainy. Roughly speaking, it seems that secondary sources may account for up to half the aerosol content of the atmosphere and that the anthropogenic contribution to total global particle production is about 15%. Natural aerosols range in size from about i0~~m radius for the small ions, made up of charged clusters of a few molecules, to more than 10 ~m for the largest salt, soil, and combustion particles. Particles up to 100 ~m in size have been observed near the ground, and even up to the altitude of cloud base during thunderstorm conditions. Swept up by strong winds at the surface, these particles remain in the atmosphere only a short time before settling out. Slinn [363] has given a schematic presentation of aerosol size spectra and their general dependence on height above the ground, distance from sources, and wind speed. Background (clean-air) distributions tend to peak at radii near 0.1 j.tm and to have significant content from about 10_2 ~m to 10 ~m. The aerosol size distributions in polluted urban air are typically much broader, extending from l0~~m to 102 j.~m,with a tendency for two peaks, one near 10 j.~m,the other at 10 ~m. The analytical model distribution most often cited in the literature is the Junge distribution, n(r) = cr~”,

r1

<

r < r2,

(2.29)

where c and v are adjustable parameters and r~and r2 are the minimum and maximum radii for which eq. (2.29) applies. This model is usually assumed to approximate background aerosol populations in the radius interval between 10_i and 10 j.Lm and with v equal to approximately 4. The parameter c obviously determines the aerosol concentration. Twomey [397]has shown that for such distributions more than half the scattering at visible wavelengths is accounted for by particles between 0.1 and 0.6 ~m. Much used in scattering studies is the modified gamma distribution introduced by Deirmend-

jian, defined by

7), -0< r
bimodal tendency. Optical properties of aerosols Reports vary on the value of the refractive index for aerosols, probably owing to real variability among aerosol samples. Fenn [104] gave the value 1.53 for the real part, n, based on measurements of the residue from precipitation. Fischer [106]measured the absorption coefficient of thin films of impaction-collected aerosols in several urban and remote sites and found values of n from 3/g, where p is the density of the aerosol material. His curves show 1/p thatranging for a given i0~to l0_2 cm sample the absorption tends to increase with increasing wavelength and is perhaps twice as strong at

82

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

2 ~tm as at 0.5 ~m. The values quoted for n1/p are for a relative humidity of 35%. There was considerable variability from one sample to the next and a strong dependence on relative humidity, especially at high relative humidities. With increasing humidity the hygroscopic aerosols, which might comprise one-quarter of a typical sample, become wet and undergo changes in both size and refractive index. As they become wetter and larger, the refractive index approaches that of water. These complex effects of humidity on the optical properties of aerosols have been reviewed by Hänel [145]and Tuomi [387]. Lindberg et al. (254) briefly described a program which has been initiated to measure the optical properties of aerosol samples collected at a number of locations around the world over a two-year period. They presented preliminary results which indicate values of n, in the visible part of the spectrum ranging from less than 0.01 for some samples up to about 0.06. Over the range of wavelengths from 0.3 to 1.7 ~m there is a weak tendency for n, to increase with wavelength. Certain types of aerosols are known to be relatively strong absorbers. Reviewing the literature on refractive indices for various carbonaceous materials, such as graphite, coals and soot, Twitty and Weinman [390]found that for visible wavelengths n varies from about 1.5 to 2.6 and n, from 0 to 1.5. The value they decided on as generally appropriate for carbonaceous aerosols is rn = 1.8— 0.5 i (0.25
—

Scattering and extinction Essential for any theoretical calculation of scattering by aerosols are the extinction coefficient /3e, the single scattering albedo w~,and the phase function P(0) which characterize the aerosol distribution. Deirmendjian [90] has tabulated these parameters for a variety of aerosol and cloud populations based on eq. (2.30). McCartney [271] has cited tabulations published prior to Deirmendjian’s. Figure 2.8 illustrates the phase functions for two of Deirmendjian’s model distributions. The Haze M model, designed to approximate marine or coastal aerosol distributions, has a broad maximum in n(r) at about 0.1 ~m radius, and significant content out to about 3 ~tm. The Cloud C.l model has a relatively broad distribution, extending from 1 to 14 im and typifying the droplet spectra in cumulus clouds of moderate thickness. Values of the four parameters in (2.30) corresponding thesecases models 3. Intoboth the are shownindex on thernfigure. particle concentrations are 100 refractive = 1.33 The (no absorption) so that w~=for 1.0.both Themodels haze model thuscmsimulates a wet haze, in which the particles have the refractive index of water. The curves are for a wavelength of 0.7 tim. Both phase functions show a strong forward-scattering peak and a minimum at approximately 0 1100. =

83

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

I

I

I

I

I

100-

-

a

I

Hoze

M

Cloud CI

b 4

5.3333 a 10

I

6~9443

2~373O

6

l~5

O~5

-

10.

-

~

i~0

o1~

=

‘6

~,

I

~

—

Cloud C~l b 3e l673 km~

/

rr~i

0

\

Haze M ‘...a”/3e 0l055 km~

—

20

40

60

80

100

120

,~ •.-

140

160

180

Scattering Angle, 9 Fig. 2.8. Phase functions for two of Deirmendjian’s [90]models. The distributions are shown inset. Note that their scales are different.

The peak at 0°is sharper, and the behavior generally more complex, for the cloud model because of

-

the larger particle sizes. The local maximum at 142°gives rise to the cloudbow, and that at 177°to the

glory, two phenomena of classical meteorological optics. The region of strong scattering gradient within approximately 20°of the forward-scattering direction corresponds to the aureole, the strongest departure from the Rayleigh phase function. For a given aerosol population the extinction coefficient generally depends on wavelength and from

84

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

(2.20) may be written as an integral over the Mie size parameter a as \ /aA\ /A \3 f /aA\ ( faA\2 Qe(a) /A /3e = ~T1~) I,~—)n da = ~ Qe(a)n da.

J

J

.

For the Junge distribution (eq. (2.29)), which is often assumed to approximate the optically important size range of aerosols, this expression reduces to 13e

~c(~)5

Ja2~rQe(a)da ~ A~’.

(2.31)

If v = 4, a typical value, then /3e is seen to decrease as 1/A. Conversely, if the dependence of /3e Ofl wavelength is determined by measurements, then eq. (2.31) may be used to infer the effective slope factor v of a power-law aerosol distribution (e.g. [359]). The optical thickness TM due to aerosol or cloud droplets is defined by TM =

f

/3e

ds,

(2.32)

where the subscript M denotes Mie scattering and the integration extends over the path of interest. One of the standard procedures in actinometry is the estimation, from surface observations of direct-beam solar radiance in narrow spectral intervals, of the total normal optical depth of the atmosphere, TT. When TT is small compared to unity, multiple scattering is unimportant and interactions between molecular and particulate scattering may be disregarded. Then ~1- is the sum of the Rayleigh optical depth r~and the Mie optical depth TM. The contribution of aerosols to total extinction is often described by the turbidity factor D, defined by D = TM/TO

= (TT

—

To)/To.

(2.33)

For a clear sky TT is approximately 0.3, and it is usually less in the red portion of the visible spectrum than in the violet. Employing phase functions and extinction coefficients derived for Junge-type aerosol distributions, Bullrich [38] carried out extensive calculations of the radiance and degree of linear polarization of sunlight scattered by a model atmosphere consisting of molecules and aerosols. Important variables were the solar zenith angle, the normal optical depths for extinction by molecules and by aerosols, and the phase function of the particular aerosol population. Results were presented in the form of radiance and polarization distributions over the entire hemisphere of the sky. All orders of molecular (Rayleigh) scattering were included, but only primary scattering by the aerosols. For realistic aerosol amounts the molecular contribution to total scattering was found to decrease with increasing wavelength and to be a maximum at a scattering angle of about 1200. Discrepancies between measurements of diffuse sky radiation and the calculations were attributed primarily to multiple scattering between aerosols and among aerosols and molecules. An empirical method for introducing higher-order scattering by aerosols was explained by de Bary [87]. The radiance of the skylight was obtained, in the same manner as Bullrich, by adding a multiply-scattered molecular component to a primary-scattered aerosol component. To simulate higher-order aerosol scattering, an additional term was added, which equals the contribution of secondary and higher-order scattering from an artificial Rayleigh atmosphere whose optical depth equals the aerosol optical depth. For clear-sky conditions, reasonably good agreement was found between measurements of sky radiance and calculated values, when allowance was made for surface albedos of about 0.25.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

85

More generally, Deirmendjian [90] showed that an approximate analytical solution for the Stokes vectors of the scattered light can be obtained for a model atmosphere throughout which w0 and P(0) are constant and equal to weighted average values for the Mie and the Rayleigh scattering contributions. Assuming primary scattering only, he gave formulas for the radiances transmitted by and emerging from such an optically homogeneous atmosphere. As multiple scattering effects may not be negligible for optical paths much larger than 0.1, and may in fact become dominant for optical paths approaching 0.5 [272], these formulas are only rough approximations for visible wavelengths in the earth’s atmosphere. Results from a theoretical treatment of horizontally homogeneous, turbid atmospheres including multiple scattering were presented by Sobolev [369].An approximate analytical solution was given for the total illumination at the surface (direct plus diffuse light) which involves the solar zenith angle, the normal optical depth of the atmosphere, the surface albedo, the surface albedo, and a parameter similar to (cos 0) which characterizes the forward elongation of the phase function. Aerosol absorption was neglected. In order to obtain approximate analytical solutions to the radiative transfer equation it is necessary to make assumptions about the form of the phase function, often replacing the complicated Mie scattering function by one that is analytically tractable, and about the vertical distribution of aerosols, the most frequent assumption being a uniformly mixed atmosphere. Taking advantage of progress in numerical methods of solving multiple scattering problems [150,188], many recent investigations of aerosol scattering have not been constrained to these limiting assumptions. For example, Herman et al. [155]calculated the intensity and polarization of diffusely transmitted sunlight at wavelengths of 0.429 and 0.50 ~m allowing for all significant orders of scattering by aerosols and molecules. Various distributions of aerosols were assumed, and results compared with measurements of the radiance and degree of polarization of skylight in the vertical plane of the sun. Most calculations were made for a Junge distribution (eq. (2.29)) with v set equal to 3.5. The concentration parameter c in this distribution was assumed to vary with height in proportion to the concentration given by Elterman [102]in a model aerosol profile. In this profile the concentration is a maximum at the earth’s surface and falls off exponentially with altitude up to 10 km, where it is reduced to approximately l0~~ of the surface value. The proportionality constant was determined by requiring that the aerosol optical depth, calculated from the assumed size distribution and height profile, be equal to some specified value. Figure 2.9 compares measured and theoretical radiances in the plane of the sun’s. vertical for a zenith angle of 40.5°. The solidTMline presents the computed radiance for an aerosol optical depth = 0.08. The lower curve is for a pure Rayleigh atmosphere with 0.23, the line offorobservational data were taken at Los Angeles on clear days when the To == 0.145. The dashed three sets total optical depths ‘TT were 0.36, 0.306 and 0.214. There is reasonably good agreement between observation and theory except in the vicinity of the aureole, the most prominent part of the computed curves, where the theoretical radiances are too high, and for elevation angles below approximately 50° (arccos 0.6) in the direction away from the sun. Calculations for other values of the slope parameter v showed that, so long as TM is held constant, the calculated radiances are essentially unchanged except in the aureole region where the peak becomes less sharp as v is increased. In this connection, Twitty [389] demonstrated with a similar atmospheric model that it may be feasible to deduce the aerosol distribution from accurate measurements of the radiance distribution in the aureole. Herman et al. found that the theoretical values of polarization agreed reasonably well with observations but were more sensitive than the radiances to the assumed value of surface albedo.

86

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

02

I

I

I

TRANSMITTED:~,’’2.5 4~Q~5X5O~31

o 9/7/68

‘0-760

T

0I

1 + 9/8/68

08

1r ‘023 Urn 7 °0375

[Tm0

trT~O225 ~°~766

—--—PURE R.AYLEIGH

)Tp’

UCLA

UCLA

r7 ‘306 A 9/9/68 UCLA

0145)

r1. ~

o0

+;

0

0~2

I

I

I

0-4

06

08

I

0

08

06

0-4

0-2

0

~‘I80) Fig. 2.9. Observed and calculated radiances in the plane of the sun’s vertical [155].Solar zenith angle is 40.5°,corresponding to ~s= cos 8

=

0.760.

Because of the arbitrary assumptions embodied in theoretical aerosol models such as those of de Bary [87] and Herman et al. [155],it is rather remarkable that their predictions agree so closely with observations. To some extent this agreement is explained by the number of adjustable parameters in the models. Very little may be known about the actual surface albedo, the aerosol refractive index, or the aerosol size and height distributions; yet all these variables enter the calculations. By specifying appropriate values for the variables rather close agreement with observations may be achieved, though such agreement is to some extent fortuitous in view of the uncertainties. In recent light-scattering studies there have been efforts to reduce the uncertainties by making independent measurements of aerosol and surface properties to support the radiation observations. The most ambitious of these is the project reported by DeLuisi et al. [92].Along with other analyses and comparisons, they found close agreement between aerosol optical depth as determined from standard actinometric measurements at the surface and as calculated by Mie theory from aerosol distributions measured by aircraft as a function of altitude. Measurement errors and uncertainties in the aerosol distributions seem to be smoothed out by the integration over altitude. Although the aerosol samples varied considerably from one another, the net radiative effect of the aerosols could be approximated by a Junge distribution with v = 3. Consistent with this finding, the wavelength dependence of TM over the range from 0.4 to 0.9 ~m was found to be very weak (or nearly “neutral”) in accord with eq. (2.31). The actual values of TM deviated no more than 20% from the value 0.11. Aircraft measurements of the total solar irradiance as a function of altitude were used to deduce

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

87

the vertical profile of absorption of solar radiation by molecules and aerosols. By experimenting with

solutions of the radiative transfer equation, using as input datathe known sizes and height distribution of aerosols, it was possible to deduce the effective value of n,, the part of the aerosol refractive index which is proportional to the absorption coefficient. This value was determined to be n1

0.01, as

mentioned previously. A comparison of the amounts of solar radiation absorbed by aerosols and by atmospheric gases (primarily water vapor) showed that, on the average, atmospheric heating due to aerosol absorption amounts to one-half the heating rate due to gaseous absorption. This result applies, of course, only to the region of the investigation, the southwestern part of the United States, which is generally characterized by clear, dry air. In the radiative transfer calculations of Part II of [92], it was necessary to incorporate the surface albedo. This was measured as a function of wavelength by instruments carried aboard an airplane. Considerable scatter was evident in the data, but there was a clear tendency for an increase of albedo with wavelength from values less than 0.1 at 0.45 ~m to approximately 0.3 at 1.05 ~m. The significance of surface reflectance in determining the radiance and polarization of sky light in a Rayleigh atmosphere was illustrated by Coulson [73]. It is an equally important factor in studies of radiative effects in turbid atmospheres and was the subject of a careful experimental study by Coulson and Reynolds [75].For various natural surface materials such as loam, silt and clay, blacktop,

rice, alfalfa and bluegrass, they measured the hemispherical reflectance as a function of wavelength and the elevation angle of the sun. This quantity is defined as the ratio of energy reflected in the outer hemisphere to the total (direct plus diffuse) energy incident on the horizontal surface, and it is applied to relatively narrow spectral regions. They found that the reflectance of most mineral surfaces, including blacktop and various soils, generally increases with wavelength throughout the region between 0.320 and 0.795 p.m. The reflectance of green vegetation is typically low in the ultraviolet, blue, and red spectral regions, increasing to its highest values in the infrared, and with a secondary maximum in the green. The reflectance of most surfaces appears to reach a weak maximum at sun elevations between 10°and 20°as a result of combined response to the direct and diffuse beams. Paltridge and Platt [300] have presented other data on the reflecting properties of a variety of surfaces. For climatic modeling, in which large-scale averages of albedo are required, they proposed the formula

A(~)=A1+(l—A,)exp(—k4) where A(4) is the surface albedo (reflectance integrated over wavelength) at solar elevation 4, k is a constant in the order of 0.1 deg~,and A1 is a constant depending on average surface properties. With its strong maximum at 4~= 0°(grazing incidence), this formula seems at variance with the measurements of Coulson and Reynolds, but may only indicate the influence of water surfaces, which have

high reflectivity at grazing incidence. Another experiment in which theoretical calculations based on aerosol measurements were compared with observations was reported by Patterson et al. [304].Simultaneous observations were made of the atmospheric visibility and the size distribution of aerosols near the surface during incidents of blowing dust in West Texas. The visibility, quantitatively defined in terms of the visual range V, is the distance from an observer at which the contrast ratio between a black object and a bright background equals 0.02. If the extinction coefficient ~ is constant along the line of sight, the visual range is then related to the extinction by

V = Iln 0.02I/f3~= 3.9l2/p~.

88

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

For visibilities of a few kilometers and less the molecular contribution to I3~is negligible compared to the aerosol extinction. Using the Mie theory for spheres, Patterson et al. calculated 13e for the measured aerosol distributions, converted the result to V. and compared the calculated and observed visibilities. The assumed refractive index was rn = 1.525 0.005 i. Approximate concordance was found, the calculated value usually being within a factor two of the observed value. Calculations for artificial aerosol distributions and for values of n~between 0 and 0.05 showed that the optically important aerosols are those in the size range 0.6< r < 20 p.m, and that the value of n, has little influence on the computed visibility. —

In version methods Illustrations were given earlier of the determination of aerosol properties from radiation measurements: the Junge parameter v was inferred from the spectral dependence of TM [359];the imaginary part of the refractive index was obtained by repeatedly solving the radiative transfer equation for different assumed values of n, until the observed distribution of solar irradiance with height was reasonably approximated [92]. These are two examples of solutions to inverse problems in radiative transfer theory, in which properties of the scattering medium are inferred from radiation observations. It is a characteristic of meteorology that considerable effort and enthusiasm are applied to inverse problems because of the difficulty of measuring properties of the atmosphere and its constituents by direct means. A compelling reason for studying radiative transfer is thus to develop techniques by which natural and artificial radiations may be used for atmospheric “sounding”. Some of the methods recently demonstrated or proposed for deducing information about aerosols from scattered sunlight are mentioned here. Techniques involving laser illumination are included in section 5. Following the eruption of Mt. Agung near Bali, Indonesia, in 1963, the aerosol optical depth above some nearby stations was observed to increase significantly. Over southeast Australia, for instance, there was a 20% reduction in direct solar radiation. As an example of what can be inferred through relatively simple arguments, Neumann [291] used this observation plus independent estimates by Deirmendjian of the total mass of ejected aerosol particles and deduced that these particles were of a few tenths of a micrometer in radius, with a concentration of about 5 cm~3,and with very small absorption coefficient. In a more elaborate attempt to determine aerosol size distribution, de Bary [88]compared observed and theoretical values of the sky radiance in the solar almucantar. Satisfactory agreement was found by assuming a Junge-type aerosol distribution with v = 4. The assumed refractive index was m = 1.5— 0.02 i and the weak absorption implicit in the imaginary part was found to be important. Initially a surface albedo of zero was assumed, but it was found that better agreement was achieved for an assumed value of a few percent. Twitty’s proposal [389]for inferring aerosol sizes from the radiance distribution in the aureole was applied [92] with results that appear satisfactory only to a first approximation. Angstrom [7] has suggested that the circumsolar radiation, which is the radiance integrated over the region of the aureole, may be used to infer the turbidity or, equivalently, the aerosol optical depth. Crucial in all such inversion techniques is the question of how sensitive the radiation measurement is to changes in the aerosol distribution. Two rather specific studies have been aimed at this question. McKellar [274] showed that, as a first approximation, different hazes have the same scattering properties if they have the same value of the parameter r8/r6 where the overbar represents an average over the size distribution. Bergstrom [23]found that even for the same size distribution, however, the relative importance of particles of different sizes within the distribution depends on the imaginary part

KR. Rogers and Gabor Vali, Recent developments in meteorological physics

89

of the refractive index. As n, increases, the importance of small particles (r<0.1 p.m) for the extinction of visible radiation increases considerably. In theoretical studies of this kind it is assumed that n, is independent of particle size. For real aerosols the effects are likely to be complicated by a dependence of n, upon r. That the polarization of skylight has a strong dependence on aerosols was explained by Sekera [356].More recently, Sekera described methods employing satellite measurements for determining the concentration and vertical distribution of aerosols as well as the reflecting properties of the underlying surface [357].Takashima [377]calculated the intensity and degree of polarization of diffusely reflected radiation from a turbid atmosphere at several wavelengths in the visible spectrum. Several model aerosol distributions were considered, with refractive indices of m = 1.34 and 1.50. He found that the

radiance and degree of polarization observed at the antisolar point depended strongly on the size distribution and refractive index of the aerosols. A theoretical study of the determination of aerosol optical parameters from ground-based polarimeter measurements was reported by Kuriyan et al. [235]. Noting that the radiative transfer equation for a turbid atmosphere cannot easily be inverted to solve for aerosol properties, they solved the direct problem for a variety of aerosol populations, each based on the modified gamma distribution. Results were tabulated which can be matched with observations to deduce which is the more likely

aerosol population for a given set of polarimeter measurements. Climatic effects There has long been speculation about possible climatic changes resulting from variations in solar radiation. The best recent observations indicate that the solar constant itself is remarkably steady, although variations in the X-ray and extreme ultraviolet spectral components associated with solar activity may amount to several orders of magnitude [366].These changes in solar energy at the short wavelengths cause significant disturbances in the ionosphere, but there is no evidence that they have any influence on processes in the troposphere. Even with the solar constant fixed, there could of course be climatic changes if the scattering or absorbing properties of the atmosphere or the surface were to change. With the growing perception of man’s influence on the terrestrial environment, concern has arisen during the past decade lest the aerosol content of the atmosphere may be increasing and leading to a change in the radiation balance of the earth-atmosphere system. Citing as evidence 10 years of monthly-average turbidity measurements at Moana Loa Observatory, Hawaii, Peterson and Bryson [308]in 1968 concluded that either the effects of the Mt. Agung eruption of 1963 were still being felt,

or a longer-term trend of increasing turbidity was underway. Soon afterwards, Budyko [37]inferred from an analysis of world-wide climatic records that secular changes in the mean surface temperature are closely related to changes in the direct component of the transmitted solar radiation. He used a simple climate model to show further that surprisingly small variations in atmospheric turbidity would

suffice to account for glaciation. Gordon and Davies [129],using a more elaborate general circulation model, found that a 5% reduction in atmospheric transmittance would lead to complete glaciation of the earth, a result indicating three-times less sensitivity than Budyko’s 1~%.Sellers [358]briefly reviewed the predictions of climate models regarding the onset of an ice age. Depending on the assumptions made, the change in solar constant required to initiate an ice age is —2 to 5% relative to its present value. To produce an ice-covered earth requires a change of —6 to —14%, depending again on the particular assumptions. Moreover and as an indication of the uncertainty associated with predictions of climatic change Sellers noted that the present value of the solar constant is compatible —

—

—

90

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

with at least three equilibrium climatic regimes: the present climate; an ice-covered earth; and a climate intermediate between these. If we assume that an increase in turbidity due to aerosols causes a net reduction of solar energy in the earth-atmosphere system, then Sellers’ generalizations are in accord with the earlier reasoning of Budyko and others. In a comprehensive review of global air pollution, Bach [10] discussed possible climatic implications of waste heat production and the increasing atmospheric content of CO2 as well as aerosol effects. It is probably wise at this point to bear in mind Scorer’s [352] observation, “The history of meteorological thinking has shown that catastrophic effects predicted on the basis of a simple model have always been wrong”. Nevertheless, it is appropriate to ask two fundamental questions:

1) Is the atmospheric turbidity increasing? 2) If so, does this necessarily imply a reduction in net solar energy input to the earth and atmosphere? There is not a clear answer to the first question. As to the second, though the calculations are not unambiguous, the weight of recent evidence seems to point to planetary cooling as the first-order effect of an increasing aerosol content. The local effects of cities, industries, and other pollution sources on atmospheric turbidity are obvious. Paltridge and Platt [300] have stated that there is also evidence for increased background levels of turbidity in remote areas adjacent to highly industrialized countries. Howers et al. reported

[114]that the average turbidity over the eastern part of the United States is higher than that for the West, although it is not clear how much of the difference is a result of man’s activities. On a world-wide basis, it has been estimated that the total anthropogenic contribution to global aerosol production may amount to 15%. With the increase in population and industrialization there would be a tendency for this component to increase, although changes in agricultural and industrial practices can change the pattern. For example, the transition from coal to petroleum in heating and industry and the diminishing use of the slash-and-burn practice in agriculture have both resulted in reduced particulate emissions by mass, if not by number. Various measurements of turbidity, based on the depletion of the direct solar beam, have been recorded at a number of sites for the past 50 years. Some studies of the records have suggested an increasing trend; others have not. Owing partly to uncertainties in the measurements, it is not even clear whether an event as spectacular as the Mt. Agung eruption had more than a transient and weak effect on world-wide total solar radiation. In Australia the direct solar radiation was depleted following the eruption but the diffuse radiation increased, with the result that the total radiation did not change perceptibly [327,99]. In October 1963 an anomalously high value of turbidity was observed at nearly all the stations across the United States which measure turbidity on a regular basis [114]. This might have arisen from the eruption in March of the same year, but appeared in the records as only a transient effect. In a careful review of data from the USSR, Japan, Australia and the USA, Dyer [99]concluded that there was no convincing evidence for a world-wide increase in atmospheric turbidity. The same conclusion was reached by Robinson [327]and Paltridge and Platt [300].The need for extreme measurement accuracy was emphasized by Dyer, for changes of only a few percent in turbidity, though nearly undetectable with present techniques, might be climatically significant.* Next we turn to the second fundamental question, what the consequences of an increase in aerosol content would be on the radiation budget of the earth. A variety of analyses have been brought to bear on this deceptively simple question, with conflicting results. Initially, it was reasoned that any increase ‘Evidence for a short-term cooling in the tropical troposphere, amounting to 0.5°C and associated with the Agung eruption, was reported by Newell and Weare, Science 194 (1976) 1413.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

91

in aerosols would cause more solar energy to be scattered back to space and hence raise the planetary albedo of the earth. This would cool the planet, assuming that no other adjustments occurred in response to the cooling tendency, such as a change in cloudiness. It was soon recognized, however, that the absorption of solar radiation by aerosols, though small, would tend to oppose the cooling tendency [51,103, 277]. The quantities that assumed importance were then the fraction of incident solar energy absorbed by an aerosol layer (absorptance) and the fraction of incident energy scattered back to space (hemispheric backscatter or reflectance). On the basis of simple models, criteria were

derived for planetary heating or cooling depending upon the relative importance of absorptance and reflectance. Schneider [349]helped to clarify the problem by noting that the important question is not simply whether the absorptance is comparable to or greater than the reflectance, but whether the combined aerosol-surface albedo is comparable to the albedo of the surface before the aerosols were present. He developed an expression for the effective albedo of the surface-aerosol system by considering a thin, uniform aerosol layer in an otherwise transparent atmosphere. The aerosol layer has absorptance a and reflectance b and overlies a surface with albedo A. The fractional transmittance is then t = I a b; of this, the fraction A is reflected from the surface; in turn t times that amount escapes through the aerosol layer to space. The effective albedo of the system, AE, taking into account all orders of reflection between the layer and the ground, is then given by (2.34) AE=b+tAt+tAbAt+...=b+t2A/(1—Ab). —

—

The aerosol layer has no effect on the planetary albedo if AE = A, causes cooling if AE> A, and conversely for heating. The criterion for a neutral layer (AE = A) was shown by Russell and Grams [336]to be expressible in terms of the ratio p = a/b of absorptance to reflectance. The critical value of p, corresponding to a neutral layer, is

Pc

(1 A)2 2At + [(1 A2)2 + 4A2t2]”2 —

—

—

2A(1+t)

2

( . )

If p > Pc, the result is a heating tendency. Evaluating eq. (2.34) for an assumed global average surface albedo of 0.1, Schneider concluded that for any reasonable combination of the variables A, a and b, the effect of an aerosol layer is to increase the planetary albedo. The conclusion was challenged by Charlson and Pilat [52],who claimed that the assumed value of surface albedo (0.1) was too low, that a global figure of 0.3 was more reasonable and would result in heating if the ratio p = a/b 1. Because of the interest in the quantity a/b, Chylek et al. [58]presented values of this ratio for single spherical particles over a range of particle size and refractive index. The general tendency is for the ratio to increase steadily with n 1 and to be a maximum in the size range between approximately 0.06 and 1.0 p.m. For n close to 1.5, a/b usually exceeds unity in this size range if n1>’O.Ol. Mitchell [277]independently developed a model similar to Schneider’s for the climatic effect of an

aerosol layer. He considered the layer to be optically thin, so that multiple reflections were negligible, but introduced additional parameters to describe the fraction of sensible to total heating at the surface and the fraction of the aerosol population that is in convective contact with the surface. Through these parameters the height of the aerosol layer and the type of surface become significant variables. Assuming (a/b) 1, Mitchell found that tropospheric aerosols would tend to warm the low atmosphere over moist surfaces such as vegetated areas and oceans and to cause cooling only over deserts and urban areas, where the effect would likely be marginal.

92

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

Russell and Grams [336] summarized several analytical models of radiative effects of aerosol layers, including those of Mitchell, Schneider and others. They showed that all the models led to the same criterion for heating if only single-scattering was considered. The critical ratio is then 2/2A, (2.36) Pc = (1 A) —

a result which can be obtained, for example, from eq. (2.35) for the case of an optically thin layer (t-+ 1). This relation is shown in fig. 2.10. Thus the criterion for planetary heating by an aerosol layer is determined by the value of a/b in relation to the surface albedo. Assuming a lognormal aerosol size distribution, Russell and Grams carried out Mie scattering calculations of a/b as a function of the two parameters of the distribution, the geometric mean and the geometric standard deviation. For the refractive index, they chose m = 1.525 0.005 i. Figure 2.11 displays the results of these calculations, the pattern of a/b as a function of the parameters of the distribution. Except for narrow distributions of small particles, the overall tendency is for a/b to increase with both the mean and standard deviation. Plotted on the figure are points obtained from 14 measurements of aerosol size distributions, with measurement uncertainties indicated. The values of p for these distributions thus vary between 5 and 28, allowing for the uncertainties. These exceed the critical value Pc for all surface albedos A > 0.09. These aerosol samples should therefore all result in increased energy in the earth-atmosphere system. Russell and Grams qualified this conclusion by pointing out that these samples contained larger particles than ordinarily encountered and consequently had relatively large values of p. They also cautioned that the Mie theory for spheres may incorrectly estimate the backscattering properties of irregular particles. Finally, with reference specifically to —

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AEROSOL PARAMETER,

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00

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GEOMETRIC MEAN RADIUS (/im)

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0

Fig. 2.10. Critical ratio p~of aerosol absorptance to reflectance, as a function of surface albedo [3361.

Fig. 2.11.andValues of the parameter p as a function geometricmodel mean radius geometric standard deviation of a oflognormal aerosol distribution, assuming m = 1.525 [336J.

— 0.005

I

and A

=

0.5 ~m

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

93

volcanic emissions, the dominant constituent is sulfate, usually in the form of aqueous sulfuric acid or ammonium sulfate, neither of which is absorptive in the visible, so that a = 0 and p = 0.

The models described so far are analytical in their treatment of radiative transfer, involving numerical techniques only in the evaluation of the scattering functions a and b. Several aerosolclimate models have been reported recently which are substantially more elaborate. Yamamoto and Tanaka [439] specified the vertical distributions of concentrations of molecules and aerosols. The aerosol concentration was a maximum at the surface and decreased exponentially with altitude up to 10km, in accordance with Elterman’s model [102]. The full radiative transfer equation was solved, using the method of Chandrasekhar and a matrix formulation for the solution. Absorption by water vapor and ozone was taken into account over the wavelength intervals where it is significant. The

Deirmendjian “Haze C” model was used for the aerosol size distribution, which is quite similar over much of its range to a Junge distribution with v = 4. Calculations were made for six values of the refractive index. In each n = 1.5. The values of n1 ranged from 0 to 0.1. Three values of surface albedo were used: 0, 0.05 (average for ocean), 0.15 (land). Results were very sensitive to the value of n,. In particular, the diffuse component of the downward solar radiation was found to decrease with increasing n1. As a result, the total downward irradiance decreases with increasing n1. The authors concluded that if n1 is small, the planet tends to cool with increasing turbidity. However, if n~>0.05, heating is expected. Reck [324] experimented with an existing radiative-convective model to determine the effect on surface temperature of a low-lying aerosol layer. This model, developed by Manabe and Strickler [264], goes a step beyond merely indicating the initial temperature tendency due to radiative effects, by invoking a convective adjustment which redistributes the temperature in the vertical until convective equilibrium is achieved. In practice, this means the temperature profile tends toward the pseudoadiabatic. For an aerosol refractive index of 1.5 —0.1 i, Reck found that at all latitudes of the northern hemisphere and for surface albedos of 0.1 and 0.3, the effect of an aerosol is to cool the surface. The model indicated that heating could occur but required rather unrealistic conditions, an albedo of 0.6 and an aerosol layer with an optical depth of 0.65, which is approximately ten times the

present background level. It was estimated further that the present background reduces by about 1% the solar radiation reaching the surface at latitudes south of 35°N. Also adding weight to the argument for cooling is a recent study [429]by Wesely and Lipschutz. Measurements in turbid air near Chicago indicated that the gain in diffuse radiation associated with increased turbidity compensates for the depletion of direct radiation by an amount that varies from 58% to 70%, depending on the solar zenith angle. A single-scattering analysis applied to the data showed that for a surface albedo of 0.17 the effect of aerosols is to increase the effective albedo. If the surface albedo is assumed arbitrarily to be twice as large, then heating due to aerosols results for zenith angles below about 45°.Cooling was thus determined from the observations to be the first-order effect of aerosol scattering and absorption. In addition to these rather specific studies of the possible climatic effects of aerosol layers, there

have been a number of related fundamental investigations of radiative effects of aerosols. Dave and Braslau [82] included clouds and aerosols in a multi-layer model of a turbid atmosphere and determined that the effect of an increase in aerosol content is more pronounced on the energy absorbed within the earth-atmosphere system than on its albedo. Halpern and Coulson [144], also employing a very detailed multiple-scattering model, evaluated the effects of low-level aerosol pollutants on the shortwave absorption. The solar spectrum was divided into 80 wavelength intervals in order to simulate the absorption characteristics of various atmospheric gases. The radiative transfer

94

R.R. Rogers and Gabor VaIl, Recent developments in meteorological physics

equation was solved for each of these components taking into account all orders of scattering. The aerosol concentration and size distribution were specified in a “radiation boundary layer” extending from the surface to 2 km. At the top of this layer the direct solar radiance and the diffuse sky light were computed. To an estimated accuracy of ±2%, the upward and downward irradiances in each of the 80 intervals of wavelength were evaluated for 20 equally spaced pressure levels within the 200-mb thick radiation boundary layer. The atmosphere was assumed cloudless and surface reflection was neglected. Table 2.1 indicates the effect of aerosols on the amount of solar radiation absorbed in the radiation boundary layer. For these cases the Deirmendjian Haze L distribution was assumed with a concentration of i04 cm3 independently of pressure in the 200-mb layer. The extinction optical thickness for this distribution equals approximately 0.8 over the visible range of wavelengths. The absorption thickness depends on the imaginary part of the refractive index, which varies in this table from 0 (no absorption) to 0.10 (strong absorption). The table shows, for example, that at a 0°zenith angle the presence of a moderately absorbing aerosol population (m = 1.5—0.01 i) in the radiation boundary layer increases the amount of energy absorbed to 3.5 times its value for gaseous absorption alone. Increasing the imaginary part by an order of magnitude (from 0.01 to 0.10) increases the total absorption by a factor 3 for both 0°and 80°solar zenith angles. In terms of the rate of temperature increase, an absorbed flux of 1.348 x 102 W/m2 corresponds to approximately 0.24 K/hr. The importance of the assumed aerosol size distribution is illustrated in fig. 2.12. This compares the vertical profiles of shortwave flux divergence for the Haze L model and for a Junge distribution with v = 4, both normalized to the same mass of aerosol material and both for uniform concentration in altitude. With the sun at the zenith the Junge distribution results in a reduction of flux divergence of about 15%. At a zenith angle of 80°the differences between the curves are within the limits of computational error. Results of this study showed that realistic concentrations of absorbing aerosols can cause more shortwave absorption in the lowest 200 mb of the atmosphere than gases. The most sensitive aerosol parameter is the imaginary part of the refractive index, although the flux divergence is also sensitive to the assumed vertical distribution of aerosol concentration. Twomey [396,398] has pointed out that pollution increases the number of nuclei which serve as condensation centers for cloud droplets, and that the clouds which form in polluted air will therefore tend to contain more but smaller droplets than those which form in cleaner air. This change in the size distribution will affect the optical thickness of the cloud, even for fixed values of water content and geometric thickness. In all but the thickest clouds, the effect of pollution would be to increase the cloud albedo. Though the effect is small, the tendency would be for pollution to increase the planetary albedo. Table 2.1. Total flux absorbed (Wm2) in a 2km column in the radiation boundary layer resulting from combined effect of gases and aerosols[l441. Refractive index

Solar zenith angie 0°

l.50—0.OOi 1.50—0.011 1.50—0.101

0.389x 1.348 x 3.714x

80° 102 102 102

O.043x 102 0.315 x 102 0.978x 102

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics 83 823-

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0.10 0.20 0.30 0.40 050 0.602 mb1) 070 080 FLUX DIVERGENCE (watts xI0~ cm Fig. 2.12. Vertical profiles of shortwave flux divergence for two model aerosol distributions and two solar zenith angles [144].

In summary, the evidence suggests that aerosol particles which are no more than moderately absorbing (n 1 <0.01 i) will tend to increase the planetary albedo by their direct scattering effect and by their indirect effect on cloud droplet distributions. At the same time, the absorption of solar radiation

by these aerosols in the lowest layers of the atmosphere can be as significant as gaseous absorption and ought to be accounted for in climatic or general circulation models, where the cumulative effect of

weak heating rates becomes important. 2.3. Light scattering by clouds As a first approximation, clouds may be regarded as black (albedo, zero) for terrestrial radiation and white (albedo, unity) for solar radiation. For clouds sufficiently thick these approximations are valid to within about 10%. As they become thinner, however, clouds become progressively more transparent to both longwave and shortwave radiation. Moreover, radiative transfer through clouds depends not only on the geometric thickness, but also on the cloud water content and the nature of the cloud particles their sizes, shapes, and thermodynamic phase. Just as for aerosols, there have been great advances in understanding radiative effects in clouds during the past decade. —

Cloud particle sizes and optical properties

Information on cloud droplet sizes is given in the standard references [111,267,330]. Quite briefly, the droplets in non-precipitating water clouds range in radius from approximately 1 p.m to 30 p.m. The largest of these are present only in mature or developing clouds that are on the verge of producing rain. The more frequently occurring cloud types, such as fair weather cumulus and altostratus, contain few droplets larger than 15 p.m. Clouds forming in maritime environments tend to have somewhat

96

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

larger droplets than continental clouds. Droplet concentrations range from tens per cm3 in maritime

clouds to hundreds per cm3 in continental clouds. The liquid water content (cloud density) depends largely on temperature and the stage of cloud development, and ranges from less than 0.1 g/m3 in cold, thin clouds, to several grams per m3 in warm cumulus clouds. Droplet-size distributions, though often unimodal and positively skewed, are occasionally observed to be bimodal or irregular. A number of analytical models have been used to characterize the general form of droplet-size distributions, of which the normal, lognormal, and gamma distributions are the most popular. Deirmendjian’s models [90],based on the modified gamma distribution (2.30) given earlier, have had the deepest influence on radiation studies. Of his cloud distributions, the one which most nearly typifies tropospheric conditions is model C.l, which was shown in fig. 2.8. Considerably less is known of the composition of ice crystal clouds than of water droplet clouds. Though the crystal habit is determined mainly by temperature, natural mixing processes can bring together crystals having different growth histories and hence different crystal habits. The tendency is for the crystals to be larger (in terms of their linear dimensions) than water droplets but in more dilute concentration. Estimates of the optical constants for water and ice have been refined considerably over the past few years because of the need for accurate values in radiation calculations. Figure 2.13, based on data compiled by Hale and Querry [139], shows the dependence on wavelength of the real and imaginary parts of the refractive index of pure water. In the visible region n 1.35 and n, is less than i0~.In the near infrared n, increases rapidly with wavelength and n undergoes abrupt changes. The same quantities for ice are plotted in fig. 2.14, based largely on the measurements of Schaaf and Williams [345].

WATER

I~

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ni

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Fig. 2.13. Dependence on wavelength of real and imaginary parts of the refractive index of water. Note different scales for n and n.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics 10 I

I

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Wavelength, p.m Fig. 2.14. Dependence on wavelength of real and imaginary parts of refractive index of ice. Based on data from Schaaf and Williams [345]except for the values of n in the visible (Hobbs [1671),and the values from 35 ~m and beyond, indicated by points (Irvine and Pollack [189]).

Single-scattering parameters for clouds

For the case of liquid clouds, once the droplet spectrum and refractive index are specified it is straightforward to calculate the scattering functions using Mie theory. We shall pause here and review some of the results that were presented earlier. Figure 2.4 illustrated the scattering efficiency factor for a model cloud in which the spread of the droplet spectrum was varied. At wavelengths in the visible range the effective size parameter 2ira/A is large, so that the scattering efficiency is approximately 2. These curves were for nonabsorbing droplets, m = 1.33. If some absorption were allowed, its effect would be to reduce the scattering efficiency, as was illustrated in fig. 2.3 for the case of a single sphere. In this event fig. 2.4 would depict the behavior of the extinction efficiency rather than the scattering efficiency. Following from eq. (2.20), the total volume extinction coefficient of a cloud at visible wavelengths may therefore be approximated as twice the sum of the geometric cross sections of the droplets in a unit volume. For the Deirmendjian model C.1, the extinction coefficient equals 16.73 km’. The asymmetry parameter for clouds was plotted in fig. 2.5. At visible wavelengths (cosO) is

approximately equal to 0.8 for nonabsorbing cloud droplets. Allowing for absorption tends to increase this value. Not surprisingly, the single-scattering albedo of a cloud is quite sensitive to the absorption, as was shown in fig. 2.6. For a nonabsorbing cloud ~ = 1. Because n, for pure water is less than i0~ in the visible, it is clear from this figure that w0 will be close to unity. Gaseous absorption, primarily

due to water vapor, tends to reduce the effective value of w0 for clouds. Also given earlier as an example was the phase function for cloud C.1 (fig. 2.8). This is a crucial parameter and must be known as a function of position within the cloud to carry out multiple scattering calculations. As a simplification, it is often assumed that the drop-size distribution is the same throughout the cloud so that P(O) is independent of position.

98

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

Moving on, fig. 2.15 illustrates another fundamental scattering characteristic of clouds, the dependence of scattering and extinction coefficients on wavelength. Curves are shown for cloud C.!, haze model M, and for Rayleigh scattering by air molecules at sea level. The dominance of cloud over the other kinds of scattering is clear. Variations in the C.! and M curves are associated with the wavelength-dependence of the refractive index of water. The weak minimum at about 11 p.m has been noted for other models [334]. For any reasonable cloud model /3~depends only weakly on wavelength over the range considered here. The difference between Pc and /3. arises from absorption and is greatest at wavelengths where n, takes on appreciable values. Since the time of Deirmendjian’s book, single-scattering parameters such as P(6), Wo, /3~and (cos 0) have been presented for a number of water cloud models [149,148, 438, 334, 378]. Less is known of the scattering properties of ice clouds. Huffman and Thursby [184]measured the phase function of artificial clouds of laboratory-grown crystals. Using unpolarized incident light with a wavelength of 0.55 p.m, they found that neither the crystal habit nor size distribution had an important effect on P(0) for scattering angles less than 130°. Compared to water clouds, the ice clouds were found to scatter considerably more light at angles near 90°at the expense of scattering in the forward direction. The authors suggested that turbulence in the test chamber might have caused random orientation of the crystals, obscuring any stronger angular dependence. Later measurements by Huffman [182] at wavelengths of 0.475, 0.515 and 0.745 p.m showed that the phase functions for the different wavelengths were not discernably different from one another. Differences have been reported in the polarization properties of the light scattered by water and ice clouds [351,3421. In theoretical studies of light scattering by ice clouds it is usually assumed that the particles are spheres, differing from water droplets only in the value of refractive index. Harris [153] showed on this basis that it should be possible to distinguish between water and ice clouds by the angular dependence of scattering at wavelengths where the refractive indices are markedly different. Plass and Kattawar [311]employed the spherical assumption in an ice cloud model, as did Fleming and Cox 2y

I0

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Fig. 2.15. Dependence on wavelength of the scattering and extinction coefficients for haze model M and cloud model C. I. The normal Rayleigh extinction coefficient is shown for comparison. (After [901.)

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

99

[110].The latter authors assumed that the radiative effects of cirrus clouds could be approximated by Mie theory applied to 40-p.m ice spheres. Following from this assumption, the shortwave optical thickness of cirrus clouds is related to the ice concentration M (g/m3) and the geometrical thickness h (km)by r=40.7hM. Radiative transfer in clouds As a result of the relatively large scattering coefficients of clouds, multiple scattering effects become significant except over very short paths. Indeed, for wavelengths in or near the visible the radiation transmitted and reflected by natural clouds consists predominantly of radiation which has been scattered many times between entering and leaving the cloud layer. To determine the dependence of scattered energy on parameters such as the cloud thickness, scattering angle, phase function, and single-scattering albedo requires solution of the radiative transfer equation. Some of the exact methods of solving the equation for application to clouds are the matrix approach [401], the adding method [148,237], and the Monte Carlo method [80,311].The advantages and limitations of the various methods are explained in reviews cited earlier [150,188]. Approximate methods of solution have been described by Liou [255],Joseph et al. [197],and Wiscombe [433]. In the Monte Carlo approach, the behavior of individual photons is recorded by computer simulation and radiances obtained by integrating the results from a large number of simulations. The required input data are the single-scattering albedo and phase function for every scattering event, the mean distance between scatterings, the total geometrical thickness of the cloud, and the ground albedo. It is usually assumed that the cloud is a plane-parallel (horizontally homogeneous) layer and that the ground reflects according to Lambert’s law. In addition, mean values of P(0) and w 0 are employed and assumed the same for every scattering event. Danielson et al. [80] reported on

extensive results using such a model. The actual phase function was approximated by the Henyey— Greenstein analytical relation 2)/(1 + g2 2g cos0)312, (2.37) P(0) = (1 g where g (cos0). Comparing this with the Deirmendjian C.! phase function, Danielson et al. determined that g = 0.875 is a suitable value for wavelengths in the visible range. The approximation —

—

(2.37) was thoroughly investigated by Hansen [147], who found it to be satisfactory for most purposes

when applied to cloud or aerosol layers. The normal optical thickness of the cloud in this model is simply related to geometric thickness by r = /3~h.Because of the relatively large value of the extinction coefficient for shortwave radiation, r can readily take on values of the order 102 for clouds that are several kilometers thick. Figure 2.16 summarizes some of the results of Danielson et al., showing the behavior of R, T and A (the fractional irradiances that are reflected, transmitted and absorbed) as functions of the optical thickness of the cloud layer and for single-scattering albedos such that 1 = 10_2, i0~and ~ A surface albedo Ag = 0.1 is assumed. The quantities are related by R+A+(lAg)T 1. —

As r increases, T —*0, and R and A approach limiting values that depend directly on 1

—

w

0. For a given optical thickness, the effect of decreasing wo (i.e. of increasing the absorption occurring at each scattering event) is to increase A and to decrease R and T. Increasing the surface albedo was found to increase the reflectance and transmittance, especially for optically thin clouds. Noting that the measured albedos of thick clouds are not greater than about 0.8, Danielson et a!. inferred from fig. 2.16

100

R.R. Rogers and Gabor VaIi, Recent developments in meteorological physics

I

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R: IO~

CI.8 -1,

/

1111111

T

102

/ --~ .~.—.i _I_ o

-~-

—

-

—

50

-

00

T

50

200

Fig. 2.16. The fraction of the irradiance vertically incident on a cloud which is transmitted (T). reflected (R), and absorbed (A) optical thickness r. Curves shown for three values of the single scattering albedo [801.

as a function of

that the value of 1 w0 must be no smaller than about i03. Irvine and Pollack [189] had shown, however, that owing to the small value of n~for pure water, the value of I w 0 should be in the order of i0~in the visible range. The latter value implies albedos of nearly unity for thick cloud layers, rather than the observed value of 0.8. To explain the apparent discrepancy, Danielson et al. suggested that the condensation nuclei on which the droplets formed might account for some additional absorption. Twomey [392] later concluded that this explanation cannot be supported by what is known of cloud condensation nuclei: they are too small to account for the required absorption. The discrepancy may instead be explained by errors in observed cloud albedos, which are based on a limited number of difficult measurements. It may be added that gaseous absorption would tend to reduce the cloud albedo an effect not included in the model of Danielson et al.. Lacis and Hansen [2371explained that the Irvine and Pollack data imply a cloud absorptance of approximately 5% in the visible range, and that the same absorption would occur in a cloud of nonabsorbing droplets with —

—

—

0.08 cm of water vapor. As a final note on the apparent discrepency, Wendling has shown [4281that horizontal irregularities (striations) in clouds always tend to reduce the albedo a small amount below that of a plane-parallel cloud with the same optical thickness. McKee and Cox [2731,using a cubical cloud model, explained that the effect of finite horizontal extent is to reduce the cloud albedo by about

25% for optical depths in the range 20—80. Twomey [396] summarized the results of a number of theoretical calculations of cloud albedo in a simple figure (fig. 2.17). For thin clouds the albedo is approximately proportional to cloud thickness.

As the thickness grows the albedo increases more slowly. In the context of pollution effects, the cloud albedo is increased by atmospheric contaminants which serve as cloud condensation nuclei, thereby

increasing the optical thickness even for fixed cloud water content and geometric thickness. Hansen and Pollack [149]reported on extensive calculations of the spectral reflectivity of water

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

101

I 0-

08

06 0 -

C

.13

Q4 -

02-

0 I

I

I

IC

00

OPTICAL

000

THICKNESS

Fig. 2.17. Approximate dependence of cloud albedo on optical thickness. Dotted area indicates variability about the mean curve due to variations in (cos 8) which arise from differences in the drop-size distribution [396].

and ice clouds for wavelengths between 1 and 3.5 p.m. The distribution of particle sizes was taken as (2.38) n(r) r6exp(—6r/rm), where rm is the modal radius. For rm = 4 p.m, this is equivalent to the Deirmendjian cloud C.1. The angular dependence of reflectance was included and the adding method of treating multiple scattering was employed. Results indicated that it is possible to use the spectral variation of reflectance to infer the size of the cloud particles and their thermodynamic phase, as well as the total optical depth. Fundamentally, this arises because of the spectral dependence of w 0 and (cos0). The calculations were compared with observations of cirrus and cumulus clouds, and effective particle sizes deduced. An example of the results is shown in fig. 2.18. The circles and solid bars represent observations of cirrus clouds at 38000 ft. after correction for absorption by water vapor and CO2. The uncorrected observations are indicated by triangles and dotted bars. The curves are theoretical reflectances for two model ice clouds and five values of optical thickness, calculated for the same viewing geometry as the observations. From these and other curves for different values of rm it was deduced that the effective values of modal radius and optical thickness for these observations are rm 16 p.m and r 10. In the theoretical work on cloud scattering described thus far the scalar form of the radiative transfer equation was employed. This has been found to be sufficient when only the radiance is required. To investigate polarization phenomena it is necessary to solve for the Stokes parameters of the scattered radiation. Hansen [148] reported on multiple scattering calculations in clouds including

102

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

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R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

the polarization. He found that the polarization characteristics of the reflected radiation are more sensitive than the radiance to cloud microstructure. The cloud model and method of calculation were essentially the same as those used by Hansen and Pollack [149].Figure 2.19 shows the radiance and percent polarization of sunlight at 1.2 p.m reflected by a plane-parallel cloud having a drop-size distribution given by eq. (2.38) with rm = 4 p.m and for several optical thicknesses. The strong angular dependence of the polarization is not obscured so quickly by multiple scattering as are the features in the radiance. Because the angular dependence is a function of particle size and thermodynamic phase, it follows that the polarization should provide a more sensitive method of remote cloud measurement than the radiance. Plass and Kattawar [3111presented results on the radiance and polarization of light reflected and transmitted by a water cloud model and an ice cloud model for wavelengths ranging from 0.7 to

6.05 p.m. The Monte Carlo technique was applied to the individual Stokes parameters. Results showed that the reflected radiance was less for the ice than for the water cloud at all angles and at most infrared wavelengths, especially at 1.7, 2.1 and 3.5 p.m. The percent polarization tended to be less for ice than for water. Deirmendjian-type particle distributions were assumed. The modal radii were 12 p.m for water and 50 p.m for ice, which values seem rather too large to be representative of average

cloud conditions. With few exceptions the theoretical models are based on the unrealistic assumption of planeparallel (and often vertically homogeneous) cloud structure. The marked irregularities in real clouds

undoubtedly affect radiative transfer, though it is not obvious how serious the effects are. Griggs [136],reporting on aircraft measurements of the albedo and absorption of stratus clouds, called attention to the great variability of cloud albedo for a given geometric thickness. Robinson noted [327] that the practical value of much of the elegant work on multiple scattering is limited because of the

IntensIty

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X

T

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—_~

¼ PoIor,zclon

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128

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60

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Fig. 2.19. Radiance and percent polarization of sunlight reflected by a plane parallel cloud with the sun overhead. Percent polarization is defined in terms of Stokes parameters as —100 QIl [148].

104

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

enormous diversity of natural clouds. Modest steps have been taken to introduce spatial irregularities into cloud models [443, 273, 428]; to be sure, these attempts also embody rather arbitrary assumptions.

The most reasonable approach, from a meteorological point of view, is probably to be satisfied with an approximate treatment of multiple scattering. Information about cloud and aerosol structure is approximate at best; if, moreover, the radiation calculations are to be used in a large-scale atmospheric circulation model in support of dynamical equations, then computer limitations will allow only an approximate treatment. It was with these constraints in mind that Lacis and Hansen developed an approximate method [237]for computing the amount of solar radiation absorbed at the earth’s surface and in the atmosphere as a function of altitude. The atmosphere is divided into as many as 50 layers. The monochromatic scattering characteristics of each homogeneous layer are defined by the optical thickness, single-scattering albedo, and phase function. Polarization is neglected; the surface albedo is specified; the adding method of solution is used. Absorption by water vapor is included and depends on the humidity profile. Within the stratosphere an allowance is made for ozone absorption. An example is shown in fig. 2.20 of heating rates in the troposphere due to clouds at different levels. The clear-sky heating profile is given for reference, based on the standard atmosphere and water vapor distribution of McClatchey et al. [272].The optical thickness of the clouds in layers 5 and 7 is taken to be 8, approximating thin altostratus and cumulus clouds. The cloud in layer 3 has r = 2 corresponding to a thin cirrus cloud. In each case the asymmetry parameter (cos0) is taken to be 0.85. The major effect of the clouds is to increase the heating rate within their layer and to decrease the heating rate at lower layers. Absorption by cloud droplets was not considered in this example, and the heating within the cloud arises primarily from the greater photon path due to multiple scattering and in part from the increased humidity (100% relative humidity) in the cloud. This model was designed for use in a large general circulation model, and appears to represent a reasonable balance of computational precision and meteorological reality. l2—~

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Fig. 2.20. Heating rates computed for water vapor absorption, including effects of clouds. Solar zenith angle is 60°and ground albedo is 0.07 [2371.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

105

3. Atmospheric absorption and emission of radiation 3.1. Introduction Radiation in the atmosphere is absorbed by certain gases and by cloud droplets, aerosols and other particulates. The significant gaseous absorbers for both solar and terrestrial radiation are water vapor, ozone and carbon dioxide — all minor atmospheric constituents. Although relatively transparent to solar radiation, cloud-free air is an effective absorber of terrestrial radiation. The absorbing gases, especially water vapor, have complex vibration-rotation spectra which extend from about 0.6 to 20 p.m in wavelength. Water vapor and ozone also have pure rotational spectra extending from about 10 p.m to 2 cm. Whereas their most important effect on sunlight is scattering, clouds are strong

absorbers across the entire infrared region. The complexity of gaseous absorption is depicted in fig. 3.1, a much-used illustration of typical clear-air absorption, as given by Robinson [327]. The top curves show that there is no significant overlap of the terrestrial and solar emission spectra, the division between them falling at a wavelength of approximately 4.5 p.m. Atmospheric emission is thus negligible at the wavelengths of significant

solar energy absorption, and the direct solar beam is attenuated according to Bouguer’s law. For terrestrial radiation, however, the atmospheric emission introduces a source term into the transfer

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Fig. 3.1. (a) Blackbody emission curves for 6000 K and 245 K, corresponding approximately to solar and terrestrial emission spectra. Curves are drawn to different ordinate scales to facilitate comparison. (b) Atmospheric absorption spectrum for a solar beam reaching the ground. Curve represents typical clear-sky conditions in temperate latitudes for a solar zenith angle of 500. (c) The same for a beam reaching the tropopause. (d) Attenuation of the solar beam by Rayleigh scattering, at the ground and at the temperate tropopause [327].

106

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

equation. Assuming thermodynamic equilibrium, the equation may be written dL/ds

ya(LB)

(3.1)

where B = B(A, T) is the Planck blackbody radiance at wavelength A, L is the monochromatic radiance in the direction of ds, and Va is the volume absorption coefficient. This equation neglects scattering, which is a good approximation for terrestrial radiation in clear air. If clouds or haze are present scattering may be significant, in which case two source terms should be included, one for emission and the other scattering. The solution of eq. (3.1) may be expressed as L(!) = L(0) e~°’° + J0~Byae~°’ ‘~ds,

(3.2)

where L(1) is the (monochromatic) radiance at distance I along a path in the direction of ds. The absorption optical depth is defined by b

r(a, b) =

Ia

Va

ds.

(3.3)

Application of (3.2) is complicated by the fact that Va depends not only on atmospheric composition but also on pressure, temperature and the partial pressure of water vapor. More seriously, r is a rapidly and erratically varying function of A owing to the complexity of the spectra of atmospheric gases. Fortunately, most practical problems require only the transmittance in certain frequency bands, so some of the severe dependence of monochromatic quantities on wavelength is integrated out. The calculation of these relatively wideband transmission functions involves assumptions about the shape

of individual absorption lines and the arrangement of lines within bands; as one of the fundamental problems of terrestrial radiation transfer and measurement, this procedure is treated in detail in the authoritative books of Goody [128],Kondratyev [227], and Houghton and Smith [177].According to expert opinion [8], present knowledge of the transmission functions for water vapor, carbon dioxide,

nitrous oxide, and a few other gases is generally adequate for energy budget calculations, in which only the wideband transmittances are needed. For remote sensing applications, more precise information is needed on line structure and its dependence on pressure. Even for energy budget applications, data are inadequate for ozone, methane and several other trace gases. The uncertainties for gaseous absorption are minor compared to those for clouds and aerosol layers. Thick clouds of water droplets, as a first approximation, may be regarded as black bodies in the infrared. Cirrus clouds, on the other hand, have infrared transmittances that are quite variable but usually greater than 0.5. Uncertainties in the radiative effects of ice-crystal clouds, combined with the fact that these clouds are so widely occurring, sets the limit to the accuracy of computations of

radiative exchange in the atmosphere and between the earth and space. Although the extinction optical depth due to aerosols in the infrared is usually an order of magnitude less than in the visible [300], it can be significant in “polluted” conditions. Developments in atmospheric radiation during the past decade have included refinement of

measurements of line and band structure, modeling of radiation transfer in realistic atmospheres, and new estimates of radiative energy budgets. Satellite measurements have figured prominently in much of the observational work, and have provided methods of remotely sensing the temperature and composition of the atmosphere. This section describes some of these recent developments. A more detailed view of satellite research is included in section 5.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

3.2. Effects of gases

107

-

Except for certain remote sensing applications, meteorological interest usually centers on the relatively wideband transmission characteristics of atmospheric gases. As explained by McClatchey et al. [272],four basic approaches have been employed for obtaining low resolution transmittance values for a given path through the atmosphere: -

1) direct measurements over the path; 2) laboratory measurements in simulated conditions; 3) line-by-line (monochromatic) calculations based on detailed knowledge of spectroscopic line

parameters, followed by integration over the spectral band of interest; 4) calculations based on band models. In radiative transfer calculations the approach is generally to use observational data wherever possible in prescribing the transmission functions, and to supplement this information with theoretical arguments. A complete treatment of this sort for the infrared region was given by Rodgers and Walshaw [329].The spectrum was divided into a number of regions wide enough for the fine structure

of the absorption lines to be smoothed out but narrow enough for the Planck function to be regarded as a constant. The bands included were the rotation and 6.3-p.m bands of H20, the 15-p.m band of C02, and the 9.6-p.m band of O~,which account for most of the absorption in the terrestrial spectrum (cf. fig. 3.1). Results were expressed in terms of vertical profiles of radiative cooling rate in a

horizontally-homogeneous model atmosphere. Rodgers and Waishaw explained that the cooling rates, obtained by integrating over the entire spectrum, are insensitive to the details of the band models. This was reiterated by Stone and Manabe [373],who showed that simpler schemes for computing infrared transfer yield cooling rates in reasonable agreement with those of the Rodgers—Walshaw model. A recent refinement in the understanding of longwave radiation transfer concerns absorption in the water vapor continuum. Though weak, the continuum absorption extends across the atmospheric “window” between 8 and 13 p.m. This spectral interval is of especial significance because it accounts for substantial energy transfer and is the most useful band for remote measurement of surface temperature. To resolve some uncertainties about the absorption in the water vapor continuum, Bignell [25] carried out laboratory measurements of the spectral absorption from 11 to 21 p.m.

Temperature and vapor pressure were varied to simulate atmospheric conditions. A 15.5-rn multiple reflection cell was used, giving total path lengths up to 500 m. A high-resolution grating spectrograph gave a resolution in terms of wave number of 1 to 2 cm’. Results suggested two absorbing mechanisms, one arising from pressure-broadened wings of water vapor bands on each side of the window, the other, somewhat akin to self-broadening, proportional to the vapor pressure e. Hence the

mass absorption coefficient is of the form k7aIpvkiP+k2e,

(3.4)

where p.., is the density of water vapor. k1 and k2 are functions of temperature and wavelength which characterize, respectively, the absorption associated with pressure broadening and the “e-type” absorption. At a vapor pressure of 15 mb, the two terms were found to be approximately equal. The pressure-broadened component increases with temperature at about 0.5% per °C,whereas the e-type component has a strong negative temperature dependence of about 2% per °Cover the range from 21 to 45°C. The existence of the e-type absorption was confirmed observationally in the atmosphere by Lee [244], using a balloon-borne radiometer in the band 10.5—13.5 p.m. Its contribution to total absorption

108

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

becomes important in the tropics, where humidity is high, and can increase tropospheric radiative

cooling rates by 30% [76].Although the e-term is now generally accepted as real and is incorporated in accurate calculations of radiative transfer in the window, its physical explanation remains elusive [244, 135, 422].

Further evidence that absorption by gases in the infrared is reasonably well understood is provided by direct comparisons of measured and computed radiances, as shown in fig. 3.2. The measured spectrum was obtained with the infrared interferomerer spectrometer (IRIS) carried on the Nimbus 4 meteorological satellite.* The computed spectrum was based on a spectral transmittance model [2331 which requires data on the surface pressure, temperature and spectral emissivity, along with the atmospheric composition and temperatures. Water vapor absorption was modeled according to eq. (3.4). The environmental data were obtained by radiosonde and rocketsonde measurements taken close in time to the Nimbus overflight. A comparison of measured and theoretical radiances indicated a difference of less than 5% in the water vapor continuum between 18.2 and 23.5 p.m (wave numbers between 425 and 550 cm’) and in the 8.3—13.3 p.m window (750—1200 cm~).In the 15-p.m (667 cm~) CO2 band the difference was 5—10%. As these errors are in the same order as the measurement

uncertainties, it is not possible to specify the error attributable specifically to the calculations. Even closer agreement was found between calculated and laboratory-measured transmittances [233], supporting the accuracy of the computational scheme.

For wavelengths in the solar spectrum, laboratory data on the absorption by atmospheric gases are rather inadequate [299, 263]. Yamamoto [437] presented a synthesis of the available data on the significant bands of H20, CO2 and 02. Paltridge [299]compared the water vapor absorption predicted by Yamamoto’s relation with measurements by airplane of the vertical profile of solar irradiance, finding reasonable agreement. A comprehensive numerical model for calculating the atmospheric extinction of solar radiation, allowing for absorption by C02, H20 and 03, scattering by molecules, the ground reflectivity, and scattering and absorption by aerosols, has recently been described by Kerschgens et al. [207]. Owing to the importance of multiple scattering in the transfer of solar radiation, some aspects of this topic were discussed in section 2. 3.3. Clouds and aerosol layers The extinction of visible light by clouds is determined almost entirely by scattering. In the infrared, however, absorption by cloud particles and by water vapor become the most important causes of attenuation, with scattering playing a secondary role. Yamamoto et al. [438] computed the diffuse spectral transmission, reflectance, and aemission water clouds forcm’. wavelengths 50 p.m t) with spectralofresolution of 50 Chosen between to typify5 and altostratus (wave numbers 200—2000 cm clouds, the drop-size distribution was very similar to Deirmendjian’s model C.1 (see fig. 2.8). A number density of 450 cm3 was assumed, corresponding to a water content of 0.28 g/m3. The cloud temperature was —10°Cand that of the earth’s surface 15 °C.The density of water vapor in the cloud was 1.44 g/m3, the saturation amount, corresponding to a vapor pressure of about 1.7 mb. The cloud was assumed infinite in horizontal extent with uniform composition throughout. Scattering phase functions for the cloud were computed for various wavelengths using values for the real and

imaginary refractive indices of water approximating those given in section 2.3. Including the effect of 4This satellite was launched in April 1970 and is still operational. An overview of meteorogical satellite research is given in section 5.

109

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

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110

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

water vapor absorption, the single scattering albedo of the cloud (for a given wavelength) is defined as = /3s/(/3e

+

Va),

where f3.. and f3~are the volume scattering and extinction coefficients of the cloud and

Va is the volume absorption coefficient of water vapor. The full radiative transfer equation, including scattering and emission source terms, was solved for different cloud thicknesses using the formulation of Chandrasekhar [49]. All significant orders of multiple scattering were taken into account. Some of the results are shown in fig. 3.3. Figure 3.3a indicates the spectral transmittance for three values of cloud thickness, assuming diffuse incident radiation. Transmittance is considerably reduced by a cloud only 10 m thick. Minima occur in the 6.3-p.m water vapor band and in the far infrared. The broad maximum in transmittance centered at about 10 p.m is explained by the water vapor window and a minimum in cloud droplet

extinction at about 12 p.m. The spectral emissivity is shown in fig. 3.3b. This quantity increases with cloud thickness but is smaller than unity even for an infinitely thick cloud. Maxima occur in the 6.3-p.m region, in the region around 12 p.m, and in the far infrared. The cloud spectral reflectance, fig. 3.3c, is greatest at short wavelengths, increases with cloud thickness, and has minima associated with water vapor and droplet extinction. At any wavelength the sum of reflectance, transmittance, and emissivity equals unity. Yamamoto et al. calculated mean values of these quantities by integrating the spectral curves over wave number, weighted by the appropriate Planck function. For thin clouds, the mean transmittance and emissivity change rapidly with increasing thickness, the transmittance approaching zero and the emissivity approaching an asymptotic value of 0.97. The mean reflectivity, always small, rises to an asymptotic value of 0.03. The limiting values are reached, to good approximation, at a thickness of about lOOm. The rate at which the asymptotic values are approached, and to a minor extent the values themselves, depend on the cloud optical thickness, which in turn depends on the drop-size distribution and water content. Hunt [185]reported on theoretical calculations of radiative characteristics of water and ice clouds at wavelengths of 2.3, 3.5, 3.8, 8.5 and 11 p.m. Two cloud droplet spectra of the Deirmendjian type (eq. (2.30)) were employed, having modal radii of 4 and 10 p.m. For cirrus clouds, three size distributions of ice particles (assumed spheres) were used, with modal radii of 16, 32 and 50 p.m. Water vapor absorption was neglected in the scattering calculations, which is probably a reasonable assumption for the two longest wavelengths. Hunt concluded that the infrared radiative properties of clouds are sensitive to the size distribution of the cloud particles. With increasing particle size the asymmetry parameter (cos 9) increases, the effect of which is to increase the

transmittance and decrease the reflectance (for a fixed water content). Emissivities of ice clouds are less than those of water clouds of the same thickness as a result of their lower water contents. Measured emissivities of cirrus clouds, though widely variable, are consistent with theoretical predictions in that they are usually 0.5 or less for a 1-km thickness [300]. Davis [84], for example, compared airborne-radiometer measurements of infrared cirrus transmittance t with laser-measured cloud thickness H, finding that t 0.5 typically for a thickness of about 4 km. Moreover, transmittance decreases with thickness approximately according to t

=

exp{— aH e”},

where a and b are constants. Without regard for their thickness, Allen [4] reported on the emissivities of a large sample of cirrus clouds. The values ranged from 0 to 1.0 with a mean of 0.35 ±0.05. Weak

R.R. Rogers and (abor Vali, Recent developments in meteorological physics

111

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112

R.R. Rogers and Gabor Vali. Recent developments in meteorological physics

emission was noted at times when no visible clouds were present. Thin cirrus clouds, which are probably the most frequently occurring cloud type, thus contribute to radiative transfer by an amount difficult to estimate and also introduce uncertainties into remote sensing applications which use the 8—13 p.m window. Fleming and Cox [110] commented on the significance of cirrus clouds in the radiative energy budget of the atmosphere and suggested approximate methods for including their effects in radiative transfer calculations. The influence of aerosol absorption on infrared transfer is negligible when clouds are present but may be significant in cloud-free air if the aerosol count is high, especially if the relative humidity exceeds about 80%. Grassl [134] carried out calculations of cooling rates due to aerosol and water vapor emission at 23 spectral intervals between 5 and 100 p.m. Results showed a significant contribution by aerosol particles to the longwave flux divergence, strongly dependent on the aerosol size distribution, the relative humidity, and the imaginary part of the aerosol refractive index. In highly “polluted” air, the aerosol contribution to cooling rate can equal that of water vapor; more typically, the aerosol contribution amounts to 5—30% of the total. Information is sparse on the infrared refractive index of aerosol particles, though Volz [418] presented data on Sahara dust, fly ash and volcanic pumice aerosols. He found that Sahara dust, with n, = 1.0 at 10-p.m wavelength, was the strongest absorber in the atmospheric window. In the same wavelength interval coal-fire dust and volcanic ash absorb more strongly than normal atmospheric aerosols. Fischer and Grassl [1071have reported that in the 8—13 p.m range the main absorption feature is a strong peak near 9 p.m caused by sulfate or quartz particles present in all continental aerosol types. In terms of their effects on the clear-sky cooling rate, these aerosols in typical concentrations may contribute from about 2% to the atmospheric cooling in the tropics to 20% in arctic regions. 3.4. Radiative energy budget Satellite observations of the earth and, to some extent, computer models have recently provided a better understanding of the planetary radiation budget. In an early general survey of satellite results, Winston [432] reported on the global distributions of cloudiness and terrestrial radiation. Figure 3.4 illustrates the seasonally averaged albedo and outgoing infrared radiation obtained from radiometer data of Tiros IV and VII. The maxima of emitted radiation occur in the tropics or subtropics of both hemispheres, where the cloudiness (and hence albedo) is a minimum. Toward temperate latitudes in both hemispheres the emitted radiation decreases, with strongest gradients between about latitudes 20° and 50°.Variations in albedo and outgoing radiation are negatively correlated and largely controlled by cloudiness, an exception being over desert regions, where high albedos coincide with high values of outgoing radiation. An extensive analysis of the earth’s radiative energy budget, based on more recent satellite data, was given by Raschke et al. [323]. Measurements by Nimbus 3 of reflected solar radiation and emitted terrestrial radiation were employed for estimates of the planetary albedo, the amount of absorbed solar radiation, the infrared radiation lost to space, and the radiation balance of the earth-atmosphere system. High-resolution radiometric data made it possible to estimate these quantities over relatively small areas and on a daily basis, as well as to obtain global averages. The overall experimental accuracy was estimated to be ±5%. The global planetary albedo was found to be approximately 29%, corresponding to a planetary radiative equilibrium temperature of 255 K. Prior to satellite measurements, the earth’s albedo had

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Fig. 3.4. Latitudinal distributions of albedo (above) and emitted infrared radiation (below) for four seasons [4321.

been estimated as about 36%. The new figure implies that the earth is darker, and consequently warmer, than had previously been thought. Figure 3.5 shows the global distribution of yearly-average emitted radiation. Maximum values are generally located over the oceanic subtropics and minima near the poles, in agreement with Winston’s broader-scale findings. As expected, the pattern of albedo (not shown) exhibits an approximate inverse relation with that of the irradiance. Table 3.1 gives a breakdown by hemisphere of the radiation budget. These result show -that-the net

114

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

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absorbed solar irradiance of 244 W/m2 is essentially balanced by the global thermal radiation, the difference of 3 W/m2 being within the limit of error. The authors advised that these figures are preliminary and that several years of observations should be used for climatological averages; nevertheless, they stressed that the new satellite-derived values are more accurate than those from earlier studies of the radiation budget. Table 3.1 Components of the annual hemispheric and global radiation budgets (after ref. [323~) Solar radiation (W/m2)

Northern hemisphere Southern hemisphere Globe

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337 343 341

240 247 244

28.7 28.0 28.4

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—1 7

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A more recent climatological study of satellite data [263]gave estimates of the components of the solar radiation balance, including atmospheric absorption. Depending on the season and location, the fractional atmospheric absorption of incident irradiance was found to vary from 23% to 43%. The global average was 29%, a figure much higher than any previous estimate and in need of confirmation.

The average albedo was found to be 29%, which leaves a remainder of 42% to be absorbed at the surface.

Possible changes in the earth’s energy budget as a consequence of increasing turbidity were discussed in section 2.2. Whereas it is not certain that the global turbidity is in fact changing, there is

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

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unequivocal evidence that the atmosphere’s CO2 content is increasing [e.g.420]. Concern has recently arisen that another radiatively important gas, 03, may decrease in the future as a result of anthropogenic factors. In particular, certain oxides of nitrogen and atomic chlorine act as catalysts in the destruction of ozone by ultraviolet radiation; these trace gases are thought to be increasing in concentration because of growing use of nitrogen fertilizers and of man-made chlorofluoromethanes, which serve as spray propellants and as refrigeration coolants. Implications of these increases in gas concentration — not at all clear at this time — fall in the realm of atmospheric chemistry and outside the scope of this review. Two examples will be mentioned, however, of the use of radiative transfer models in assessing the effects of changes in gas concentration. Dave and Halpern [83]varied the amount of stratospheric ozone in a plane-parallel, horizontally homogeneous model atmosphere and calculated the effects of these changes on the direct ‘and diffuse solar radiation reaching the surface. They found that decreasing the ozone amount results in a strong increase in the direct component but only a moderate increase in the diffuse component. The dependence of these changes on wavelength, solar zenith angle, and surface reflectivity was described. The effect of changes in carbon dioxide on infrared radiative transfer was considered by Zdunkowski et al. [444].Their calculations assumed that water vapor and CO2 are the only absorbing gases. It was found that doubling the CO2 concentration increases the temperature near the ground by approximately ~°Cif clouds are absent. A seven-fold increase is required to increase the temperature by 1 °C.For 50% cloudiness, doubling the CO2 again leads to an increase in temperature of about ~°C. The effect of CO2 variations is weak because of the overlap at 15 p.m of the H2O and CO2 bands. Earlier estimates of the influence of CO2 on equilibrium temperature had not properly allowed for the overlap and had therefore indicated greater sensitivity. The model employed by Zdunkowski et al. did not provide for any dynamic adjustment in response to the temperature changes and hence describes

short-term, rather than climatic, effects. 4. Physics of clouds and precipitation 4.1. Introduction

Cloud physics encompasses studies of the formation and internal development of clouds, of the products of clouds — such as precipitation, electric effects and optical phenomena — and of the interactions of clouds with electromagnetic radiation, acoustic waves, and with gases and aerosols of

the atmosphere. The importance of cloud physics derives from (1) the role of clouds and precipitation in the atmospheric water cycle, (2) the direct dynamic influences of clouds on atmospheric motions through the energy exchange associated with phase transformations of water, (3) the effects of clouds on the radiation balance of the earth-atmosphere system, (4) the interactions of clouds and precipitation with other gaseous and particulate — components of the atmosphere, and (5) a large number of lesser factors. In a somewhat different vein, cloud physics takes on importance as the key ingredient in artificial weather modification. Cloud physics is not an old science. Visual observations of clouds and the inspection of precipitation elements reaching the ground led, over the centuries, only to crude ideas about cloud processes [276].Controlled laboratory experiments for illuminating cloud mechanisms had to await a —

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R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

reasonable degree of clarification of ideas, and even now they remain of limited scope and utility. So, essentially all major tenets in cloud physics originated in the twentieth century; detailed investigations were made possible only by the use of aircraft for flights through the clouds, and by the application of microwave radar for remote sensing of precipitation. It is perhaps not surprising, then, that cloud physics is still in a state of rapid evolution with most of the fundamental theories undergoing continuous re-examination and with only qualitative ideas existing on many crucial issues. Partly by the nature of the phenomena being investigated, partly by the basic scientific disciplines which are the backbone of the studies, cloud physics is often subdivided into specialties like cloud dynamics, cloud microphysics, radiative properties, cloud electrification, etc. This is a meaningful classification to cloud physicists for identifying their areas of interest. This review could well have been organized along the same lines of division. But, for the benefit of non-specialists, it seemed better to organize the material around the major “issues” of cloud physics, that is, to examine the extent to which explanations can be given for basic cloud phenomena. If issues were to be identified in a broad sense, then a very large number of them could be raised due to the wide-ranging impacts of clouds on our environment. Hence, to limit the scope of this review, and to yield to the authors’ prejudice, only four topics will be taken up: 1. The formation of clouds 2. The development of rain in clouds not containing ice 3. The origins of ice in clouds 4. Precipitation processes involving ice. These topics cover the role of clouds in the hydrological cycle and are fundamental in determining most other characteristics of clouds; they can be viewed as the central topics in cloud physics. In the following discussion, emphasis is on microphysical aspects, e.g., on considerations of the elements of clouds cloud droplets, ice crystals, etc. with the fluid-dynamical behavior of the cloud as a whole not covered in detail. Other reviews [70,320, 3611 contain full treatments of cloud dynamics. In this review article, attention is further restricted to the earth’s atmosphere, although there are many fascinating parallels and contrasts with other planetary atmospheres which would be useful to discuss. Furthermore, since clouds are almost exclusively phenomena of the troposphere, the focus will be on the lowest 15 km of the atmosphere. The fundamentals of the reviewed topics will be touched upon only to the extent necessary to introduce the discussion; background for the material covered here can be found in textbooks on cloud physics [111,40,267,330], or in the excellent review article of Latham [241]. Recent summaries of advances and of outstanding problems in cloud physics are given in [71] and [33]. —

—

4.2. The formation of clouds The basic mechanism of cloud formation is the cooling of moist air below its dewpoint, with the additional requirement for atmospheric clouds that the air also contain aerosol particles which can serve as centers or nuclei of condensation. The wide variety in atmospheric cloud forms arises from the multitude of ways in which these elements combine. By far the greatest proportion of clouds form as water clouds (liquid droplets, not ice); in fact there is no known evidence for ice clouds to form directly from the vapor, i.e., without at least the transient formation of a liquid cloud. This fact greatly reduces the complexity of the problem. The factors controlling cloud condensation may be differentiated into dynamical and compositional —

—

ones. Dynamical variables include the manner, especially the rate, of cooling of the air, and the

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

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compositional variable is the aerosol content. The manner, or cause, of cooling, can be radiative heat loss, advection over cold surfaces, mixing of air masses of different temperature, or expansion due to vertical lifting; the associated rates of cooling can range from infinitesimal to an extreme of about 10°Cper minute. The aerosol content, considering concentration, size, and chemical variables, can be

nearly of infinite variety. As a result, cloud droplet concentrations range from about 10 to 1000 cm3 and cloud liquid water contents from less than 0.01 g/m3 to 10 g/m3. There is an important feedback mechanism between cloud formation and the dynamics of their movement: the release of latent heat of condensation. There also exist more subtle, but globaly very

important, feedback mechanisms between cloud formation and aerosol content, through the condensation process itself, through the scavenging of aerosols by clouds and precipitation, and by the creation of new aerosol particles as the residue of evaporated cloud droplets. In the context of the formation of a particular cloud, only the dynamical feedback has to be considered. Cloud formation is visualized most conveniently in terms of a parcel of air undergoing expansion while being lifted vertically, which is descriptive of cumulus cloud development. The temporal and spatial scales are readily comprehended and the magnitudes of the vertical velocities are appreciable (1—10 mIs); as a result these clouds are well suited to experimental investigations. It is more difficult to discuss stratus clouds, as the point of origin and the subsequent path of an air parcel are less distinct; but conceptually at least, stratus can be treated the same way as cumulus. The fundamentals of cloud formation are also the same for clouds formed by radiation or by advective cooling, and also for cloud formation by mixing provided that the spatial inhomogeneities in thermodynamical variables and in aerosol content are taken into consideration. General references on cloud physics [for example, 267, 111,40,203] contain quantitative treatments of the processes of cloud formation and give references to the original papers. Here the physical processes are described without mathematical treatment, the main purpose being to provide a basis for subsequently discussing still outstanding problems and recent research on these problems.

Cloud condensation In a rising and cooling parcel of subsaturated air, the saturation ratio S, defined as the ambient vapor pressure divided by the equilibrium vapor pressure over a plane water surface at the same temperature, increases steadily toward S = 1 with very little uptake of water by the aerosol particles. Particles which are soluble in their entirety, or in part, deliquesce at some S < 1 (mostly between 0.5 and 0.8) and subsequently remain in equilibrium with the vapor by growing and becoming dilute.

Equilibrium breaks down as S exceeds unity by some quite small amount and the growth of droplets beyond that point is controlled only by the supply of vapor. Droplets are then around 0.1 to 1 p.m in diameter. With insoluble aerosols, a similar situation arises, as nucleation takes place on most kinds of

surfaces at relatively small supersaturations. After the barrier against growth is removed, a balance is very quickly established between vapor supply and the uptake of water by the growing droplets. Reaching this balance marks the maximum supersaturation for the air parcel and also stops further aerosol particles from growing beyond their equilibrium sizes. The abundance of tropospheric aerosols is sufficient under all circumstances to have the highest supersaturations occur within the first

minute (the first few hundred meters) of rise after reaching the dewpoint. The first problem in treating cloud condensation within this framework is how to specify the relevant characteristics of the aerosol. Equilibrium conditions for soluble particles are predictable

from their sizes and chemical composition by standard methods. The natural procedure for describing

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R.R. Rogers and Gabor Va/i, Recent developments in meteorological physics

cloud formation would therefore be to start from a specification of particle sizes and composition. This is not practical, however, as the data matrix (size versus composition) is virtually impossible to secure observationally with the required precision. Theoretical examinations of the problem [200,109] have indicated that compositional changes have negligible influence compared to the size distribution of the aerosol. Therefore, the condensational behavior of an aerosol can be approximated by specifying the size distribution and using some average value for the solubiity [161,376], though this can lead to errors in the computed concentration of cloud droplets by several tens of a percent. The procedure that has been found most practical is to determine by direct measurement the propensity of a given aerosol population for promoting condensation: the number of particles which become activated (overcome the growth barrier) is determined as a function of supersaturation. Such measurements of what is termed the cloud condensation nucleus (CCN) spectrum are now possible with accuracies certainly better than a factor of 2, and gradually approaching 10% [181].The range of such measurements is usually from about 0.3% to about 1% in supersaturation, which covers the majority of meteorological situations. Efforts are continuing toward the development of techniques for measuring the concentration of CCN active at supersaturations less than 0.3%. These “very active” nuclei may be quite important, despite their low concentrations, in shaping the large-size tail of the cloud droplet size distributions. CCN spectra almost always can be described by a power-law relationship of the form N=cs”

(4.1) 3, and c and k are where s = (S 1) x 100% denotes the supersaturation, N = concentration in cm constants. The values of the two parameters depend on air mass type; typical ranges of values are: for maritime air masses c = 30—300 cm3, k = 0.3—1.0 and for continental air masses c = 300—3000 cm3, k=0.2—2.0. With the CCN spectrum thus empirically defined, the problem of cloud formation is reduced to determining, for given dynamical conditions, just what the supersaturation is going to be as a function of time, and what the time rate of growth of the droplets will be. The observable against which the theory has to be gauged is the cloud droplet size distribution as a function of time, or distance above cloud base. The most crucial quantity, the supersaturation, cannot be measured with sufficient accuracy inside clouds at the present (vapor pressure would have to be measured with 0.1% accuracy, or dewpoint to about ±0.1°C,with instrument response times in the order of fractions of a second). The most straightforward procedure for the calculations is to obtain simultaneous solutions at small time increments of the equations describing droplet growth and the thermodynamics of the cloud parcel. Droplet growth is initiated in the calculations for numbers of drops indicated by the CCN spectrum at each supersaturation up to the maximum, and each droplet group is followed separately in the size calculations. An example of the type of information which results from such calculations is shown in fig. 4.1, where the most important cloud parameters are shown as functions of altitude above the level where S = 1 is reached. Vertical air velocities of 0.5 and 2 rn/s are assumed, and are taken to be constant for the purposes of these calculations. A further assumption is that all droplets are of 2 p.m radius at the moment they become activated. By varying the updraft profiles and initial droplet sizes it can be shown that the assumptions of constant updraft and the same initial radius are not restrictive for demonstratingthe main —

features of cloud development over the first few tens of meters. The most important points are the sharp rise and settling down of supersaturation, the rapid leveling offof the cloud droplet concentration and the relative narrowness of the droplet size distribution. This latter aspect of the problem will be further discussed. Similar numerical integrations of the droplet growth equations for an ensemble of drops have

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Fig. 4.1. Time evolution of cloud characteristics for air rising with either 0.5 or 2 m/s velocity beyond the condensation level, which was taken to be at 850mb and + 10 6C. The condensation nucleus spectrum was taken to have the form (4.1) with c = 800 cm3, k = 0.7 (data derived by Mr. R. Kelly using the model of Young [442]).

been described by many authors since the first such calculation by Howell [180].Apart from minor differences the various calculations yield essentially the same conclusions. For CCN spectra of the form (4.1), Twomey [391]gave an analytical solution for the peak supersaturation and hence for the total droplet concentration as a function of updraft velocity. Twomey’s equations yield values very similar to the results of numerical integrations typified by fig. 4.1 but are not as complete because the droplet size distribution is not obtainable from them. Some

interesting analyses of Twomey’s equations are given in [31]and [195]. Sedunov [354,355] derived an approximate analytical solution similar to Twomey’s but more elaborate, and also produced an expression for the size distribution of the droplets. This appears to be a fairly comprehensive treatment of the problem but it has not been carried much beyond the formal

derivations. The calculations discussed above treat droplet growth as a quasi-steady state process with each

point in space (i.e. each CCN or droplet) experiencing the average properties of the system. Even with these simplifying assumptions the calculations demonstrate well the major features of initial cloud formation. Field studies of individual, non-precipitating cumulus clouds [402,423, 108] involving the measurement of CCN spectra below cloud base and of updraft velocities and cloud droplet size distributions at short distances above cloud base, and laboratory tests [417,239], revealed that cloud droplet concentrations reach their maximum value well within the first 100 m of cloud ascent, that

these concentrations increase with increasing updraft velocity and are within about ±20%of calculated values for particular updraft/CCN spectra combinations, and that the cloud droplet size

spectra are highly peaked. Agreement on these points is an indication of the basic validity of the theoretical formulations. Nonetheless, three aspects of the problem keep it very much in the forefront of interest. First, the

assumptions usually made about droplet growth in a cloud are known to be unrealistic. Second, it is

120

R.R. Rogers and Gabor Va/i, Recent developments in meteorological physics

difficult to secure good sets of field observations because of the need for in-situ sampling with highly sophisticated instrumentation and because of the variabilities of natural clouds; laboratory tests on realistic scales are perhaps even more difficult. The third and most important motivation for continuing research is that observed cloud droplet size distributions are significantly broader than calculations predict. This is of very great importance because of the subsequent evolution of the droplet spectrum and the generation of precipitation. The basic tendency in a population of cloud droplets formed by condensation is for the droplet distribution to become narrower as the process continues. The predicted breadth of the distributions at about 100 m above the condensation level corresponds to values of the dispersion (standard deviation divided by mean droplet size) of about 0.10 to 0.15 depending on the slope of the CCN spectrum and on the updraft velocity. Observations, on the other hand, indicate that close to cloud base the dispersion is usually in the range 0.1 to 0.4, with an average value near 0.2, and that the value increases with altitude. Clouds formed by slow expansion in a 3200 m3 chamber were found to have

even broader droplet spectra [417,239], but experimental difficulties lessen confidence in these results. As mentioned before, the usual formulations of cloud condensation take droplet growth to be a

steady-state process, and treat the diffusion problem as if a single drop grew in an infinite vapor medium, but with the condensed mass divided among a large number of drops as determined by the CCN spectrum. Some attempts have been made to eliminate these simplifying assumptions. A kinetic

equation for the diffusional growth of an ensemble of droplets has been formulated by Stepanov [372] based on nonequilibrium statistical mechanics. This an important development but the results appear quite intractable. An interesting attempt was made by Srivastava [370]to consider local rather than average properties of the vapor field for determining growth rates of droplets, but difficulties were encountered in the mathematical development [202,371]. Sedunov’s development of the theory of condensation [355] also incorporates nonstationary effects; further exploitation of this theory will probably prove to be very useful. Many aspects of the conventional theory and of the observations have received close scrutiny in the search for explanations of the broadening of cloud droplet spectra. On the theoretical side, the factors whose possible spectrum-broadening influence has been examined are: (1) the condensation coefficient, (2) temporal variations in updraft velocity, (3) spatial inhomogeneities, and (4) mixing of different air parcels. With respect to the empirical findings, there are two further areas of difficulty: (5) to assure the representativeness of the cloud droplet and updraft measurements, and (6) to obtain CCN measurements at small (<0.3%) supersaturations. A brief discussion of each of these points will illuminate the status of the problem. Droplet growth (or evaporation) is receiving increasingly detailed treatments [119,160, 364, 205, 50]. The growth problem is especially difficult during initial cloud formation because of non-continuum transport near the drop surface [119,44, 124, 160]. At the drop surface, the energy and mass accommodation coefficients have to be considered. The latter, conventionally called the condensation or sticking coefficient, has an especially great effect on drop growth through the creation of a discontinuity in vapor density at the surface, so that droplet growth rate is reduced. The magnitude of the effect depends on the value of the condensation coefficient and decreases with increasing droplet size. Uncertainty about the value of the condensation coefficient effectively precludes critical tests of droplet growth theories [415].Measurements with bulk water [summarized in 108] show large scatter, with the majority of values in the range 0.02 to 0.04. Direct observations of micron-size droplets

formed on natural CCN [56,362] point to a value near 0.03, a value consistent with measurements performed at higher supersaturations [414]. For spray droplets a value near unity was obtained [251]

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

121

but the large scatter of the measurements casts doubt on this value. Warner showed [423] that the

spread of cloud droplet spectra increases if lower values of the condensation coefficient are used but quantitative agreement with observations could not be reached with realistic values of the coefficient. In spite of considerable work on this question, the full complexity of the problem has not yet been dealt with. Because droplets formed on natural CCN contain dissolved substances, and may contain surface-active materials as well [314,203],the condensation coefficients may vary over quite wide ranges for different situations, and will most likely change in value as the droplets grow from the size of the original nucleus (0.01 to 0.1 p.m) to several tens of microns in diameter. The use of experimentally determined CCN spectra gets around this problem in principle, but in practice the

measuring instruments need to have longer sensitive times than is available in current devices, and measurements are needed at lower supersaturations. On the other hand, for nuclei activated in current diffusion-chamber type CCN counters, the value of 0.03 for the condensation coefficient can be applied with fair confidence as that is the value derived from measurements with such diffusion chambers [56]. When attempting to correlate cloud droplet spectra with the vertical velocity of the air at the point of observation, it is implicitly assumed that the local updraft velocity adequately characterizes the movement of the air from the condensation level up to the observation level. In the model calculations some simple updraft profiles are assumed, or calculated, for a closed, buoyant parcel. For the short vertical distances involved (few hundred meters maximum) these assumptions seem reasonable since the clouds of interest are typically a kilometer or so in their horizontal dimensions. Yet, it must be recognized that actual cloud updrafts are more complex. To test whether the droplet spectrum might be broadened by variations in the updraft velocity, a number of calculations have been carried out. Warner [423]and Bartlett and Jonas [14]added randomly varying velocity increments to the mean velocity and reached the conclusion that such velocity fluctuations do not produce significant changes in the droplet spectra. The same conclusion was reached by Kornfeld [229], who compared the

evolution of droplet spectra in uniform and in sinusoidally varying updrafts. Clark [60]argued that the Lagrangian framework used by Warner and Kornfeld does not adequately represent the irreversible effects that turbulence might induce, and constructed an Eulerian model to deal with the problem. Preliminary results showed a broadening of the droplet spectrum due to turbulence, but not a large enough effect to account for some of the apparent discrepancies. Using the theory of fluctuations developed by Tatarski [380],Sedunov [249,355] developed a theory for cloud formation in a turbulent atmosphere. This theory considers fluctuations in supersaturation due to fluctuations in the vertical velocity of the air and also accounts for inhomogeneities in droplet concentration or size. The resultant expressions are quite complex and have been applied to simplified cases only. In the form

presented, although the growth rates of drops are influenced by the stochastic variations, the drops attain the same size when reaching a given altitude in the cloud. The author considered his results to be only the beginnings of a complete treatment, and postulated that a spectrum-broadening effect would be evident in the final result. Further elaborations of the formal theory were given by Stepanov [372] but without numerical results. In all, it appears still questionable whether irregularities of air

motion can lead to the observed broadening of droplet distributions in the newly condensed portions of convective clouds. A pair of interesting papers examined the consequences of spatial inhomogeneities on the size distribution of droplets [365].The point of departure of this work is that the usual measurements of cloud droplet size distributions represent averages taken over some horizontal distance. When the influences of inhomogeneities in CCN and in vertical velocities are considered it is shown that with

122

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

averaging distances of the order of 100 m droplet spectra with realistic dispersions result. Also, this theory correctly predicts a number of other features of newly-formed clouds. While the importance of spatial inhomogeneities in cloud parameters cannot be doubted, it is also clear that such influences are not necessary for the development of broad spectra since some of the measurements of droplet sizes are in fact taken over distances less than 1 m.

Another aspect of the problem which received attention is the assumption in simple model calculations that the cloud can be treated as a closed parcel. In reality, air as it rises will mix with the surroundings to greater or lesser degrees. This type of mixing, usually referred to as entrainment, constitutes a major problem in adequately describing the microphysics and dynamics of convective clouds; here the question is examined only insofar as it influences initial cloud formation. The possible droplet-broadening effect of mixing was investigated by Warner [424] both theoretically and by examining empirical data. His calculations showed that while mixing would in fact broaden the size distribution, the shape of the predicted distributions was highly unrealistic. The empirical data

revealed no simple relationship between the degree of dilution of cloud water content from its adiabatic value and the spread of the droplet spectra. Hence, Warner concluded that mixing is not a controlling factor in determining the size distributions of cloud droplets just after condensation. Mason and Jonas [269] examined the situation in which a rising air parcel mixes with its surroundings into which a previously risen cloud parcel has partially evaporated. With this model the evolution of the droplet spectra exhibited quite realistic trends. The recent laboratory experiments of Latham and Reed [242]shed some light on the consequences of mixing cloudy air with subsaturated air. Their results show the effect of mixing to be a reduction in droplet concentrations without changes in the spread of the size spectra. In light of the many remaining uncertainties, there appears to be considerable scope in this area for further theoretical and experimental work. The two problems discussed above, updraft fluctuations and mixing, are closely related to the observational difficulty of ascertaining the representativeness of data collected by flights through clouds. While it is possible to find cloud portions which are fairly homogeneous in their average properties, the variability of the individual parameters such as droplet concentration and liquid water content usually exceed the variability which would result from truly random dispersion of the droplets, and there is always an appreciable degree of fluctuation in such parameters as temperature and updraft velocity as well. The paucity of high-resolution, continuous data, combined with inherent uncertainties of the measurements, make examination of this problem quite difficult, but the situation

should rapidly improve in the near future. Mention has already been made of the difficulty of obtaining measurements of CCN at super-

saturations below about 0.3%. From the point of view of explaining the shapes of cloud droplet spectra, this question is important because the possibility exists that large particles which would

become CCN at very small supersaturations remain undetected. There are two reasons for the problem: the difficulties of creating and accurately controlling supersaturations of small magnitude, and of making measurements on large particles (>1 p.m) which have appreciable fall velocity and which, therefore, present special problems in sampling and handling. A possible avenue for circum-

venting this shortcoming of CCN counters is to sample the large-particle content of the air directly and subsequently to determine the chemical composition, or at least the hygroscopicity of the

collected aerosol. The results of such measurements were reported, for example, by Hindmann et al. [1611.One unsatisfactory aspect of this approach, apart from its inherent tediousness, is the possibility that the particle composition may change due to chemical reactions between~the time of sampling and the time of testing. The measurement concept of Laktinov [238,6] may prove valuable in solving the

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

123

problem. If the chemical composition of the aerosol is assumed known (within fairly wide limits at least) then the sizes of haze particles at saturation (S = 1) can be used to infer their critical supersaturation and hence CCN activity. Very limited field data are available from instruments using this principle, and no use has been made so far of such measurements to compare against initial cloud droplet spectra. It is difficult to decide which of the various reasons just examined is most likely to hold the key to satisfactory explanations of initial cloud formation. The challenge remains to pursue further each of the problem areas mentioned. The importance of the problem certainly justifies such efforts. Origins of CCN A problem of great importance which is closely related to the formation of clouds, and which goes even beyond that, is the origin, chemical composition and residence time of cloud-condensation nuclei

(CCN). Ever since the original suggestion of Melander in 1897 [275]it was widely held that particles of sea-salt generated at ocean surfaces make up the great majority of hygroscopic nuclei needed for cloud condensation at small supersaturations. Recent evidence however points in a different direction.

Direct chemical analysis of CCN is not possible because they cannot be isolated from the more numerous total atmospheric aerosol. Chemical analysis of cloud and of rain-water are difficult to interpret because clouds also scavenge very efficiently aerosols other than the CCN. Perhaps the best clues about the identity of CCN come from thermal decomposition studies [394,95]: the volatility of CCN is more like that of ammonium sulfate than that of sodium chloride. This suggests a predominantly continental origin for the CCN, and there is more evidence for this conclusion from studies of the sizes and global distribution of CCN. Laboratory measurements by Gerber, Hoppel and Wojciechowski [126]of the relationship between particle size and activation supersaturation for NaC1 and for (NH4)2 SO4 have given excellent agreement with theory (as given in [267]for example), dispelling doubts on this point which were raised by Katz and Kocmond [204]. Measurements of the size-supersaturation relationship by Twomey [399] for aerosol particles generated from rural air by ultraviolet radiation, after removal of the original aerosol content, showed that the observations fit well to the theoretical relationship for (NH4)2SO4 or H25O4 the particles produced in this manner definitely could not be NaCI. Hindmann et al. [161]deduced, from simultaneous measurements of CCN spectra and aerosol size distributions, that the size-supersaturation relationship for aerosols in the lower troposphere of Washington State, U.S.A., is very close to those arrived at by Gerber et al. and by Twomey. The combination of these three studies constitutes strong support for identifying natural CCN as soluble sulfur compounds. The sizes of the CCN in the Twomey study [399] were indicated to be between 1 x 10_6 and 3 x 10~cm radius for supersaturations in the range 1% to 0.3%. In an earlier paper [395],Twomey reported that the sizes of natural CCN for 0.75% supersaturation are close to 10~cm radius. Comparison of the two results suggests a similarity between natural CCN and the photolytic gas-to-particle conversion products. Measurements of CCN at various geographical locations and at altitudes of up to 12km [403, 174] revealed greater concentrations over continental areas and in air above low-level maritime airmasses than in maritime air. The picture suggested by these observations is strikingly similar to that deduced by Junge [198]for global aerosol distributions. While the explanation for these patterns is still far from complete, there is a strong implication for the importance of photolytic particle generation from gases

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of continental origin. A similar deduction can be made from the observations of diurnal variations in CCN concentrations [400, 246] which point toward a formation of CCN by coagulation of photolytically-generated gas conversion products, or of other primary aerosols. The mounting evidence for the role of sulfur compounds as CCN is important in pointing to the possible modification of clouds that could result from increases in the sulfur content of the atmosphere (cf. section 2.2 of this article). The results discussed above do not rule out the possibility that large sea-salt particles play an important role in generating the large-diameter ends of cloud droplet size distributions by becoming activated at supersaturations <0.3%. Important evidence to this effect is found, for example, in the observations of Woodcock and Duce [435], who traced the origins of raindrops in Hawaii to salt particles of 0.2 to 1.0 p.m radii. The transport of sea-salt particles inland from the coast was recently studied by King and Maher [212], showing that appreciable salt aerosol concentrations can be expected even 1000 km away from sea coasts. Ubdoubtedly, further research will reveal more precise information about the origins, transformations and transport of natural CCN. The complexities of the chemistry of the troposphere are bound to invalidate any simple views about the nature of CCN, yet the evidence now in hand appears to present a picture consistent enough to justify a fair degree of confidence in its basic validity. 4.3. The “warm-rain” process Fundamental questions about the formation of rain in all-liquid clouds, beyond those of initial cloud formation, relate to the evolution of droplet spectra by continued condensation, to the coalescense of droplets to form larger drops, and to possible subsequent modifications of the rain by evaporation,

drop breakup, etc. A major question is the manner of interaction of the microphysical processes with the dynamics of the cloud and the cloud environment. The microphysical problems involve such details as the collision and coalescence efficiencies of water drops, the shapes and terminal velocities of freely-falling drops, the stability of drops, and the methods of solution of the coalescence equations. Each of these problems, except the last one, is amenable to theoretical and empirical solutions and there has been a steady improvement in the results over the years. Theoretical studies outnumber empirical ones by quite a wide margin, and it has not yet been possible to observe directly and under controlled conditions the rates of coalescence in populations of drops, although this is what is ultimately desired. Field observations are of limited value only, because the complexities of clouds preclude suitable definitions of initial conditions and of entities whose evolution can be followed. The upshot of the situation is that the results of calculations on the formation of rain by coalescence have to be compared against relatively indirect observations such as the rate of development of radar echo intensities, drop-size distributions of rain at the ground, rain intensity as a function of time, etc. Even such comparisons are usually possible only against “typical” patterns rather than against comprehensive data sets for given cases. Against this realm of indefiniteness stands the relative conceptual clarity of the physical processes involved; the hindrances to progress are the limitations of analytical descriptions and the unattainability of comprehensive observations. Partly because the details of the various topics lead to a number of diverging fields of discussion and partly to keep the volume of this review within closer limits, the following material is much less detailed than that of the preceding section and that of 4.4— the two sections which deal with the initial

formation of cloud elements. Also, current developments related to the warm-rain process are by

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nature technical, rather than conceptual, in comparison with the areas of initial cloud formation. Examples will be given in the paragraphs that follow of the types of studies which appeared in the literature over the past several years. These will illustrate the nature of the problems, without fully elaborating on them, and can serve to lead the reader to the relevant literature. Condensation growth of cloud droplets beyond the initial stages was examined analytically by

Kabonov [201]who took up the problem of size spectrum narrowing in a homogeneous, adiabiabatically cooled air parcel. Paluch [301]calculated the droplet size distribution for a cloud parcel which initially contained regions of different humidities and different droplet concentrations, and where droplet sedimentation mixed the contents of such regions. A considerably broader spectrum was found to evolve after 0.5 to 2 km of vertical lifting than would be expected for a homogeneous cloud. The evolution of a population of droplets by coalescence due to gravitational settling depends on the size distribution of the droplets and on the collision and coalescence efficiences. Changes in the concentration of droplets of a given size are due to some combination of smaller drops merging and becoming of the size in question, and due to droplets of that size merging with some other drop and becoming larger. The driving force for the processes is the difference in settling velocities of drops of different sizes. Settling velocities are known with reasonable accuracy [18,19]. The shapes of drops differ from spheres to a sufficiently small extent, except for drops in the millimeter range, to allow the ideal, or geometric, collision rates to be calculated without much error. The actual collision efficiencies are determined by the airflow around the falling drops and by the drag force on the drops. Collision efficiencies are very small for drops of highly disparate sizes (radius ratio >10), but become greater than 0.5 for drop pairs of about 1:5 proportions. The more nearly equal the drops are the more sensitive the calculations become in general to the assumptions used in the flow description. The most recent, and most complete calculations of collision efficiencies are those of Klett and Davis [214], and of Lin and Lee [252].These produced quite good agreement although some important differences remain [213,253]. These newer calculations predict higher collision efficiencies than those in earlier use, especially for droplets of nearly equal sizes where the efficiencies significantly exceed unity. Ryan [338] and Leighton [247] presented calculations which demonstrate the magnitude of the consequences of using different theoretical collision efficiencies. de Almeida [86]gave an analysis of the reasons why the different methods of calculation might disagree. Measurements of collision efficiencies were reported by Levin et al. [250]and by Abbott [1] for different sets of parameters. While Levin et al. found collection efficiencies to be lower than the theoretical values by factors of 2 to 10, Abbott’s data agree with calculations. Levin et al. pointed out that the observed efficiencies are actually a combination of the collision and coalescence efficiencies and interpreted their results to indicate that the coalescence efficiencies are significantly less than unity. Sartor [341] argued that microturbulence in the Abbott measurements led to unrealistically high collision efficiencies, a point which could not be clearly resolved [3]. The influence of electric forces on collision efficiencies was investigated theoretically by Grover and Beard [137], among others, and empirical data were presented by Abbott [2]. Depending on the particle charges and the external electric fields, the collision efficiencies can be greatly enhanced. An important new aspect of the problem has been taken up by de Almeida [85]. In addition to the deterministic forces such as gravity and electric forces, he considered the influence of stochastic elements — primarily turbulence. Only the method of solution has been published so far; a significant

effect can be reasonably anticipated. Mention has already been made of the possibility that the coalescence efficiency of drops might be less than unity, i.e., that drops whose undistorted surfaces are predicted to make contact will actually

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not merge. Foote’s analysis [115] describes this problem. The observations of Whelpdale and List [431] indicated coalescence efficiencies of 0.8 or higher for drops of very dissimilar sizes. Levin et al. [250] found values near 0.2 for drops whose radii were in ratio of 1:5. This question is far from settled and is not going to be easily resolved, as chemical impurities, electric charges and drop oscillations can exert large influences, so that simulation of atmospheric conditions is extremely difficult. This is another. area where subtle atmospheric variabilities may have rather profound influence on cloud processes. We turn now to the various solutions of the coalescence equation. This problem is closely analogous to the coagulation of aerosols or colloids. Since the point was originally raised by Telford [381],modern treatments all recognize the stochastic nature of the coalescence process in populations

of droplets. Analytical solutions are only possible for highly simplified forms of the collection kernel [e.g. 412] and most work has been on numerical solutions by various techniques. Berry and Reinhardt [24] gave a thorough examination of the problem and presented examples of calculated results. Monte Carlo techniques of solution were applied by several other authors [240,326, 339]. There are large numbers of analyses of the accuracy and efficiency of the numerical procedures. The calculations clearly demonstrate how the droplet spectrum begins to broaden and develops a second peak at larger sizes as the process continues. Most important, the sensitivity of the rate at which large drops form to the spread of sizes in the initial spectrum is clearly evident from the results. This is the reason why the shape of the droplet spectrum produced by condensational growth is so critical. The evolution of clouds with both condensation and coalescence growth included was treated by Mason and Jonas [270] and Leighton and Rogers [248]. These calculations showed the importance of continued condensation for speeding up coalescence growth. The next phase of complexity of the warm-rain process is tackled in a number of model calculations on the evolution of whole clouds. In such models the evolution of precipitation is followed from the initial uplifting of a parcel of air, within some interactive dynamical framework. The environmental soundings of temperature, dewpoint, and in some cases of the wind profile, and some initiation of vertical motion are specified. In addition some assumptions are made about microphysical factors such as CCN or cloud droplet spectra, coalescence kernels, and so forth, and about dynamical aspects such as entrainment (mixing with the environment) and turbulence intensities. The most significant difference between these models and the calculations discussed earlier is that interactions among different parcels of the cloud are included. Attention is not restricted to a “closed” parcel; drops from one region are allowed to sediment into another. The models range widely in sophistication, and in placing emphasis on one or another facet of the problem. Convective clouds have received more attention than stratus clouds for reasons which have already been discussed in section 4.1. Some recently published models of the warm-rain process are references [68], [79], [293], [348], [208],[413] and [376]. The logic of the topic would dictate that the observational evidence about the warm-rain process be reviewed next. However, such a discussion would have to delve into the full complexities of the meteorological situations which were studied the volume of material would task writer and reader alike. As a simple generalization it may be said that coalescence is the only process for the development of precipitation in clouds which do not contain ice, so certainly those which are entirely at temperatures warmer than 0°C.The principal domain of the warm-rain process is therefore the tropics, where warm temperatures and high specific humidities combine to have clouds precipitate efficiently without reaching vertical depths that would generate ice in the clouds. This is not to say that the coalescence process can not be effective in clouds which contain some ice particles as well. In some situations both processes of precipitation formation — coalescence and ice growth may be —

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active side-by-side. Synergism may also occur, such as a large drop formed by coalescence freezing

and becoming the center of a hailstone. In this regard as in all other ways, clouds provide an endless variety. 4.4. The formation of ice in clouds

In most clouds ice is a minority constituent even at temperatures considerably below the threshold of thermodynamic stability for ice. Clouds consisting entirely or predominantly of ice, such as cirrus clouds and some cold-temperature clouds at the end of their life cycles, are not significant precipitation producers and are of interest primarily because of their influence on radiation transfer through the atmosphere. In combination with the general scarcity of ice in clouds, the great variability

of ice elements, both in quantity and in type, make the formation and development of ice in clouds a much more complex topic than the analogous problem involving liquid water only. Of greatest importance for precipitation development are clouds in which ice particles coexist with supercooled water droplets. Because ice has a lower vapor pressure than water at the same temperature, in a mixture of ice and supercooled water elements ice particles grow rapidly by

diffusion while water droplets evaporate. The growth is accelerated when ice crystals become large enough to fall relative to water drops and begin to collect rime. The primary reason for the frequent occurrence of mixed-phase clouds is the relative scarcity of nuclei of ice formation. Even with 15 °C of supercooling, the concentration of atmospheric ice nuclei is only of the order of 102 to i03 m3, which is about io~times fewer than the concentration of cloud droplets. (“Seeding” of clouds with artificial ice nuclei aims to take advantage of this scarcity of natural nuclei to increase the efficiency of precipitation from clouds.) Although there are as yet no quantitative estimates of the fraction of precipitation which, in given

regions, or globally, is initiated by the growth of ice elements in clouds, it appears that ice processes dominate over the coalescence-of-liquid-drops process in many clouds of diverse types, including situations where the precipitation reaching the ground is rain. Some of the classical and recent papers on the role of ice in precipitation production are references [175,22, 105, 29, 223, 258, 125, 219, 98, 43]. The most extreme manifestation of ice forms in the atmosphere is hail, which has been observed in sizes up to about 15 cm diameter and which has great economic impact through the damage it causes;

explanation of hail growth and the prospect of mitigation hail damage by cloud seeding are of great concern to cloud physicists. In all areas of ice research there has been intense activity over the past 20 years, and in spite of the major gaps in knowledge which still exist, important advances have been made. Much of the new knowledge is still only qualitative so that the degree of refinement and coherence in definitions of the ice processes is considerably behind that of the condensation-coalescence process. Origins of ice in clouds

The most fundamental question in relation to ice-phase recipitation is: what are the concentrations of ice elements in various cloud types? The simplicity of this question is, of course, deceptive since meaningful answers have also to include descriptions of the spatial variation and temporal evolution

of ice concentrations. Whereas liquid clouds in general form and develop within the very narrow regime of 100 to 101% relative humidity, ice formation can take place and ice growth continue anywhere between 100 and about 150% relative humidity with respect to ice, or between 0°and 40°Cof supercooling, depending

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on whether the ice is nucleated from the vapor or the liquid. From this fact derives the great complexity and variability of occurrence of ice in the atmosphere. Early estimates of ice crystal concentrations were derived from observations of snowfall at the ground, but interpretation of such data in terms of concentrations at the regions of origin of the ice is

not straightforward. More certain data can be acquired by in situ sampling from aircraft, and to a limited extent, at mountain tops. The difficulties and expense of such measurements have been serious limitations to assembling comprehensive data sets.

Perhaps the best way to summarize current knowledge about the origins and concentrations of ice elements in clouds is to start from the best documented cloud situations and physical process, and to proceed toward areas of increasing uncertainties. Thus, ice nucleation will be discussed after a summary of secondary processes of ice generation. Recognition and evaluation of the secondary processes have provided important recent clarifications of the boundaries of the phenomena which are to be explained as nucleation processes. Rime-splintering. A major discrepancy between observed ice crystal concentrations and the best available estimates of atmospheric ice nucleus concentrations was noted by Dobson in 1949 [96]. In some clouds ice crystal concentrations exceed nucleus concentrations by several orders of magnitude. The discrepancy was recognized by Koenig [223] and Braham [29] as an important problem in explaining precipitation development and the topic has received much attention over the past 15 years.

The most thorough documentation of nucleus/ice crystal disparities is for maritime cumulus clouds, through the work of Mossop, Ono and co-workers [280,281, 288, 285, 295, 287, 289]. With tops warmer than 10°C(the temperature of cloud top defines the maximum supercooling in the cloud) these clouds have been observed about 1 hour after their formation to contain as many as l0~m3 ice crystals. Measurements of ice nucleus concentrations typically yielded values 4 orders of magnitude smaller for the same temperatures. Circumstantial evidence from the characteristics of the clouds, and later laboratory demonstration of a process of ice splintering [143,286,283, 284], led to the recognition that the high crystal concentrations were not produced by nucleation. The process involved is the ejection of splinters of submicron sizes when cloud droplets freeze onto previously existing ice crystals (riming). Such splinters in a cloud rapidly grow into pristine crystals which bear no recognizable trace —

of their origin. Upon continued growth the crystals begin to rime, giving rise to a chain-reaction of ice crystal generation. The dependence of the rate of splinter production on temperature and on droplet sizes has been delineated by the laboratory experiments of Hallett and Mossop [143,286, 284]. They found that the rate of splinter production was greatest near —5 °Cand diminished to negligible levels within about 3°Con either side of that temperature. Also, splinter production was found to require the presence of cloud droplets >24 p.m in diameter and the rate of production correlated positively with the concentration of drops of those sizes. The most recent results of Mossop [284] show that the rate of splinter production near —5 °C is roughly proportional to the product of the concentration of droplets >24 p.m and that of droplets <12 p.m. Dependence on the cencentrations of both large and small droplets is indicative of how delicately the efficiency of the process depends on prevailing conditions. The necessary conditions for the Hallett—Mossop mechanism prevail most commonly in

maritime cumulus clouds in accord with the original field evidence for ice multiplication. The physical mechanism involved in the production of splinters has not yet been identified; this is the only

discomforting aspect of an otherwise very important clarification of one way in which ice can form in the atmosphere. Drop shattering. Clouds other than maritime cumulus are also known occasionally to contain

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unexpectedly high ice crystal concentrations [223,224,9, 166, 168, 296, 169, 261]. The criteria for the rime-splintering mechanism are not satisfied by many of these clouds and for most of these situations there are as yet no viable explanations. Prior to the emergence of the rime-splintering mechanism as a very effective process, and because of the need to find mechanisms of secondary ice production which operate at conditions different from the Hallett—Mossop mechanism, a large number of potential ice multiplication processes have been explored. Many experiments have been conducted to observe the behavior of water drops as they freeze and in particular to examine the probability that the drops break into fragments or eject splinters upon freezing. Earlier evidence is reviewed and some new results of this type are given in [321],[2261and [375].With minor variations the results of the different studies are in agreement on the following points. Drops from about 50 p.m up to several mm in diameter can be expected upon freezing to produce two pieces of ice on the average. The multiplication ratio of 2 (ratio of resultant ice elements

to original freezing events) is an average value about which there is considerable variability; many freezings leave the drop intact, some result in the splitting of the original drop into two, and some rare events result in large numbers of fragments. Fragmentation of freezing drops appears to be occurring with about the same probability over the temperature range of the experiments, —5 °C to —25 °C, although in [375]some temperature-dependence was noted. A dropsize-dependence was also reported in [375].

The primary cause of drop shattering on freezing is the build-up of pressure inside the drop as the ice envelope around the drop thickens. Factors which influence the process, and which are not altogether

readily reproduced in the laboratory experiments, are the dissolved gas content of the drops and the airflow, including microturbulence, around the freely-falling drop. There is also some influence of the location where freezing is initiated — within the drop orat its surface. A theoretical analysis of stresses in freezing drops and of their fragmentation has been given [446]. While drop shattering can be expected to contribute to ice development in atmospheric clouds, the large drops which are likely to shatter are present only under limited circumstances. The efficiency of the process is small in comparison with rime-splintering, but it is not restricted to a narrow regime of temperatures. Chisnell and Latham [54] constructed a model to asses the rate of ice development in clouds in

which the rime-splintering and the drop-shattering processes are at work. This model takes into account the stochastic nature of collisions between cloud elements and provides a useful basis for estimating the effects of multiplication mechanisms in clouds of differing compositions. Field observations may be compared against these calculations to deduce which processes might be operative in given circumstances. Mason [268] invoked a kinematic model of successive buoyant parcels rising through the same

cloud mass to estimate the rate of cloud glaciation which would result from the Hallett—Mossop rime-splintering mechanism. Aspects of the model, and of the assumptions used, were shown by subsequent discussions [55]to result in underestimates of the speed with which the ice content of a cloud would increase.

Based on laboratory evidence and on -model calculations, the conclusion seems warranted that for maritime clouds near —5 °Cthe rime-splintering mechanism is capable of producing the observed concentrations of ice crystals in times which are consistent with the observed ages of the clouds.

There are as yet no documented cases of the drop-shattering mechanism alone significantly influencing the development of cloud glaciation.

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Crystal fragmentation. The breaking of snow crystals within clouds is a direct means of ice multiplication. Some field and laboratory evidence for such fragmentation exists. After several earlier suggestions of the likelihood of snow crystals breaking apart in clouds, Vardiman [410], Hobbs and

Farber [172]and Jiusto and Weickman [194]reported observations that snowfall at the surface often contains crystal fragments, especially when the predominant crystal types are of the branched variety. The appearance of these fragments indicates that the original crystals broke into a small number (two to four) of pieces, and further raise the possibility that tiny fragments may be produced by fracture. Large numbers of such fragments, which could grow into new crystals and subsequently undergo further fragmentation, could lead to high multiplication ratios. Two types of processes can lead to crystal fragmentation: collisions between crystals, and thermal shock associated with riming. The mechanical strength of ice being quite high (Young’s modulus, E lOb Pa) for bulk ice and presumably even higher for vapor-grown single cystals, collisions have to be fairly energetic to result in fracture. This view is supported by the common existence in the atmosphere of very delicate unbroken snow crystals [21]. Vardiman and Grant [411] examined this question quantitatively and concluded that significant multiplication will only occur if lightly rimed branched or spatial crystals coexist in the cloud with graupel. In such situations the collisions are energetic enough and the crystals most prone to fragmentation. From laboratory experiments they

estimated the number of fragments per collision to be as high as 30. The possibility that thermal shock, caused by the heat released by a drop freezing onto a crystal, could lead to sufficient stress to break the crystal was studied theoretically and experimentally by King

and Fletcher [210,2111.Although their results indicate that at temperatures below about —10°Cstresses in thin crystals can approach the strength of bulk ice, they concluded that thermal shock is probably not an important factor in causing crystal fragmentation. It has also been proposed that whiskers from growing crystals are easily broken off, or are shed when crystals undergo some evaporation in slightly subsaturated regions of clouds [292,335]. This type of fragmentation could be an efficient process of glaciation. However, the existence of whiskers on crystals in natural clouds is unproven and appears doubtful in light of the fact that whiskers have not been detected in any studies of natural snow crystals. In summary, evidence for crystal fragmentation in the atmosphere is the finding of broken pieces of

crystals. It therefore seems reasonable that some multiplication, by ratios of at least 2 to 4, is occurring under conditions when relatively fragile crystals coexist with graupel, or perhaps when narrow-branched crystals begin to collect rime droplets. There is some possibility, though no conclusive evidence, that higher multiplication ratios can also occur. Fragmentation can be expected to be most important in deep wintertime clouds and in multicellular convective clouds of substantial vertical extent. Considerably more work, and especially field observations, will be necessary to establish more definitely the role and manner of crystal fragmentation in clouds. Shock waves. It is well known that tapping the wall of a container of supercooled water will often result in freezing. It is possible that this happens because nuclei become dislodged from the wall or because of the relative motion of water and the solid wall surface [101, 319]. Observations of such “dynamic nucleation” in the laboratory led to the examination of possible similar occurrences in the atmosphere. Shock waves from lightning discharges are the strongest dynamic disturbances in the atmosphere and the possible nucleating effect of such shock waves has been tested in the laboratory [101,131, 307,312]. The consensus of the results is that even at small distances from the lightning channel shock intensities are too weak to lead to any nucleation.

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Cirrus seeding. Situations exist in the atmosphere in which the ice content of a cloud can be increased and precipitation initiated by the fall of ice crystals into the cloud from higher clouds, most notably from cirrus. (Cirrus clouds are composed entirely of ice crystals because of the low temperatures, 30°to —50°C,at which they form.) Evidence for such initiation of snowfall was first deduced from radar patterns [93,164]. Braham [30]called attention to cirrus-seeding as a possible —

cause of unexpectedly high ice crystal concentrations in clouds of relatively warm temperatures, and

as initiators of summer rain showers. Calculations [32,141, 57] and observations [32,158, 194] show that cirrus crystals can survive long distances of fall through cloud-free air if the air is not too dry.

Survival distances of several kilometers are not unusual. Hence there can be little doubt that in some clouds ice crystals not of local origin make a significant or even dominant contribution to the ice content of the clouds. In contrast with the processes discussed in earlier paragraphs, cirrus-seeding is not truly a secondary ice generation process; rather, it is an example of cloud interactions. The importance of cirrus seeding is most pronounced in situations when several layers of clouds exist at

different altitudes and where the lowest layer, which has the greatest potential for precipitation production because of its highest water content, contains very little ice without the seed crystals. While the phenomenon itself is now clearly documented, good measures of the importance of the process are not yet available. Homogeneous freezing-nucleation. The rate of crystallization in water is so rapid that in any sample of water only the first nucleation event can be detected. Hence, homogeneous nucleation can be observed only in very small amounts of water, that is in droplets of a few p.m diameter, for which the probability of containing heteronuclei is small. This factor brought the study of homogeneous freezing-nucleation into the realm of cloud physics at an early date. The experiments of Schaefer [346], Cwiong [78],Fournier d’Albe [117],and others revealed that ice is nucleated homogeneously at temperatures near —40°C. There is an extensive literature on studies of homogeneous freezing. Major questions relate to the kinetics of the process and to uncertainties about the ice-water interfacial energy. Since there are no reliable independent determinations of the ice-water interfacial energy, the problem cannot be brought to a unique resolution. Wood and Walton [434] found that nucleation rates determined by a droplet-dispersion technique could be satisfactorily explained using the Volmer—Becker—Doring type theory if the ice-water interfacial energy is assumed to be temperature dependent with the value of about 3.0 J m2 at 0°Cand about 2.2 J m2 at —36°C.They also pointed out a number of other avenues for bringing theory and observation into agreement. Knight [217]examined available data on the pressure-dependence of homogeneous freezing temperatures and showed that this dependence is correctly accounted for by classical thermodynamic theory if some reasonable assumptions are made, but, once again, the agreement cannot be construed to be a definite proof of the theory. Eadie [100] constructed a statistical-thermodynamical theory starting from the Némethy—Scheraga model of liquid

water and produced correct predictions of the nucleation temperatures, but the theory was not extended to considerations of the kinetics of nucleation. Some progress has also been made in formulating molecular-cluster theories of nucleation [138and 34] but these theories are not yet sufficiently advanced to produce checkable predictions. The empirical evidence on homogeneous freezing-nucleation is fairly consistent: the nucleation rate becomes appreciable for micron—size droplets at temperatures between —35 °C and —40 °C [434,231,and many previous reports}. This result in itself is nearly sufficient to deal with the contribution of homogeneous freezing nucleation to the formation of ice in the atmosphere. Tempera-

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tures of —35 to —40°Ccorrespond to altitudes around 8 km in the atmosphere where clouds have low enough water contents to be unimportant as direct precipitation producers. Also, clouds which rise to these altitudes from lower levels will have lost most of their water content as precipitation by the time they ascend to —35 °Cand above. Therefore, the greatest importance of these clouds is in influencing the radiation balance of the atmosphere. Reference has been made already to the role of crystals falling from cirrus clouds and triggering precipitation from clouds at lower altitudes; this indirect role of homogeneous nucleation in the atmosphere is perhaps greater than commonly realized. Clouds which occur at temperatures colder than —35 °Care layer-type clouds, especially cirrus and altostratus. “False cirrus”, that is the spreading anvils of large thunderstorms, also belong to this group. Observations of cirrus clouds [426,158, 194, 220, among others] confirm that they are composed of ice crystals with liquid droplets forming only transiently. This is consistent with the idea that homogeneous nucleation would rapidly transform any condensate into ice. Observations are not extensive enough at various temperatures on either side of —35 °Cto detect a discontinuity in the concentrations of ice crystals and thereby establish the contribution of homogeneous nucleation. In any event, knowledge of the homogeneous nucleation temperatures allows one to predict that no atmospheric clouds will contain liquid water below about —40°C. Homogeneous nucleation also plays a role when clouds are seeded with dry-ice, with propane spray [159], or by rapid expansion of compressed air [427]. With such seeding, local regions are cooled to temperatures well below —40°C so that all affected cloud droplets freeze. The frozen droplets subsequently grow into larger ice crystals by gradually mixing with the non-nucleated regions of the supercooled cloud. This process is of practical importance in the clearing of supercooled fogs at airports. In summary, while fundamental questions remain concerning the physics of homogeneous ice nucleation, laboratory and field observations are in good agreement on the occurrence of homogeneous freezing, and these observations are well supported by existing theories. Heterogeneous ice nucleation. To an even greater extent than for the other origins of ice already discussed, it is a formidable task to predict the contribution of heterogeneous nucleation processes to the number and rate of development of ice elements in clouds as functions of environmental and other conditions. Considering the complex and dynamic character of aerosol populations, the interactions of

aerosols with changing arrays of hydrometeors, and the diversity of possible modes of heterogeneous ice nucleation, it is understandable why current knowledge falls short of providing workable explanations. Ice nucleation studies have always occupied a focal position in cloud physics because of the importance of ice in the atmosphere and because of the potential for weather modification through ice-nuclei seeding. In spite of this attention, progress in understanding has been slow. The main questions which are addressed in ice nucleation studies are: What are the elemental nucleation processes contributing to atmospheric ice formation, and what are the mechanisms of nucleation involved? How can measurements of ice nucleus concentrations be made? What are the origins of atmospheric ice nuclei? How can clouds be seeded artifically with ice nuclei? The very simplest generalization that can be made about atmospheric ice nuclei is that their concentration is typically very low in comparison to the total aerosol concentration and that the concentration of active nuclei increases with decreasing temperature. Figure 4.2 illustrates this comparison. The concentration range of ice nuclei shown in the figure is schematic and does not

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108

—

0 ~

~ ~/

I .,,

\ 10° -30

I

-20 TEMPERATURE

I

-10

\ I

0

(C)

Fig. 4.2. The normal range of concentrations for ice nuclei at different temperatures, as compared to CCN and aerosol concentrations.

encompass some of the extremes that have been observed; moreover, as indicated in the graph, the two ends of the temperature range are essentially extrapolations only. Elemental nucleation processes. The possible heterogeneous processes of ice formation are deposition from the vapor and the freezing of liquid water. For deposition, the rate of nucleation is most strongly dependent on supersaturation with respectto ice, and for freezing it is a strong function of temperature. The variations in nucleation rate are so rapid that threshold values can be effectively defined. Such thresholds have been observed for various substances, Fukuta and Shaller’s data being perhaps the most comprehensive [123].Actually, precise differentiation between freezing and deposition is not always possible. There are two main reasons for this: the formation of intermediate molecular, not bulk, phases, and the frequent presence of chemical species other than water and the nucleating substrate. Absorbed water molecules on a nucleus may well form “liquid” or “semi-liquid” layers or patches, and ice may then nucleate in this liquid, rather than by the classical picture of molecules from the vapor transferring to and from the ice embryo. The findings of Klier et al. [215], Barchet and Corrmn [121and of Fukuta and Paik [122]show that absorbed water molecules on substrates form an important link in the nucleation of ice from the vapor. This makes theoretical treatment of deposition kinetics extremely difficult; indeed, there is as yet no formal treatment of nucleation within absorbed water. The necessity to deal with 3-component (or multi-component) systems is present in almost all situations of atmospheric ice nucleation. The most obvious such situation is freezing nucleation in water containing dissolved materials. The diversity of ways in which nucleating ability is affected demonstrates the complexity of the interactions (e.g. Reischel and Vali [325]).Practical manifestations of this situation are the freezing of cloud droplets soon after condensation (while solute concentrations are high), and the nucleation of ice by partly soluble particles which form brines of high concentration at relative humidities less than 100%. Evidence now available points to a lesser role for deposition nuclei than for freezing nuclei. Typically the concentration of deposition nuclei is only about 1 to 10% of the total. However, since

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R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

deposition nuclei can become activated below water-saturated conditions [183],and since there are many instances of ice-supersaturated but water-subsaturated regions in the atmosphere, deposition nuclei can play a significant role. A special case of ice nucleation which has received considerable attention is the so-called

“contact-nucleation”. Gokhale and Goold [127],and others, found that more nucleation events result from allowing laboratory dispersions of various substances to settle onto the surfaces of supercooled water drops than from an equal concentration of particles suspended in the drops. Vali [406] confirmed that natural aerosols show the same property. In fact, the possibility exists that contact nuclei are the most numerous, and hence the most important, class of atmospheric ice nuclei. Yet, the details of contact-nucleation remain obscure. Fletcher [113],Cooper [66]and Fukuta [1211offered possible explanations for the phenomenon and a whole body of literature exists on the inducement of nucleation by dynamical processes such as mechanical and electrical disturbances, but it still appears impossible to arrive at any conclusion regarding the mechanisms involved in contact-nucleation. Observations [406] indicate that nucleation does not result instantaneously upon contact between aerosol particle and water drop, but that several minutes may be needed, during which time the particle remains on the surface of the drop. Such slowness may be indicative of processes such as surface etching, partial dissolution, adsorption of dissolved ions or molecules, etc., as the preliminary

steps leading to nucleation. Alternatively, the explanation for contact-nucleation may lie in special properties of the surface layer of water, in gradual wetting of the particle surface, or in the random reorientations of particles which eventually bring a nucleation site into favorable position. Young’s calculations [441] of the rate of contact-nucleation in the atmosphere revealed some interesting features of the process, such as dependence on nucleus size and humidity, but his estimate of the

concentration of atmospheric contact-nuclei is almost certainly too high by many orders of magnitude. Because of the importance of contact-nucleation in the atmosphere, there is a definite need for further studies of the phenomenon so that the fundamental nature of the process can be illuminated and the atmospheric concentrations of contact nuclei determined. Yet another possible mode of ice nucleation in the atmosphere is “condensation-freezing”, the nucleation of ice by a particle which previously served as a cloud-condensation nucleus. By virtue

of such a dual role for the particle the process is different from freezing of a drop initiated by a particle the drop collected. Also, conditions may be exceptionally favorable for nucleation to occur at the early stages of condensation. Evidence for this process in the atmosphere is circumstantial [67]and there is no information as yet on the mechanism involved or on the factors of importance. The foregoing discussion considered the macroscopic conditions for ice nucleation. Numerous

studies have addressed the question of the molecular mechanisms involved in ice nucleation. Knight [217]recently presented a comprehensive review of concepts relating to this question; the topic will not be taken up here. Ice nucleus measurements. The lack of basic understanding of the processes of atmospheric ice nucleation almost automatically precludes the possibility of performing critical measurements, unless a complete simulation of atmospheric conditions can be achieved. The creation of clouds in controlled chambers, coupled with some means of detecting the ice particles which form, has been the approach taken in early attempts to measure ice nuclei. This still is a possible avenue, but as more and more is learned of the complexities of the phenomena so the hope of satisfactory simulation recedes. Measurements of ice nuclei with instruments which detect nuclei of given modes of activity (deposition, contact-freezing, condensation-freezing, etc.) have been recently attempted ([183], [406])

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

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with promising but still incomplete results. Further material on this topic can be found in [133,405 and 407]. Origins of atmospheric ice nuclei. The rarity and heterogeneity of ice nuclei make it difficult to establish their chemical and physical identity, their origins, and their behavior in the atmosphere. Accordingly, the information on ice nuclei is both tenuous and meager. Examinations of particles at the centers of snow crystals by microscopic and microanalytic techniques indicate that the most common nuclei are clay minerals of a few tenths of several micrometers in diameter [232,190, 303]. Unfortunately, such observations are ambiguous, because of the arbitrariness of ascribing the crystal’s origin to one of the readily identified large particles over the much more numerous smaller particles which are also present in the crystals. Laboratory studies have identified a large variety of materials which possess greater or lesser degrees of ice-nucleating ability, and many of these materials are also known to be prevalent in the atmosphere [267].Unrealistic aspects of the laboratory tests, both in terms of measurement procedures and in reproducing the possible natural states of the materials, limit the value of the information derived. The most recent additions to the list of potential atmospheric ice nuclei are organic products of treeleaf decay, oceanic plankton and certain types of bacteria [350,408]. Other clues to the origins of ice nuclei are the spatial distribution, especially the altitudedependence of ice nucleus concentrations, and correlations between changes in nucleus concentrations and other aerosol or weather characteristics. A great deal of effort in these directions led to very few conclusions, because the variability of ice nucleus concentrations is comparable in magnitude

to the uncertainties of the measurements. Recent work of this type is best summarized in [3181. The fact that no clear answer has yet emerged is perhaps an indication that natural ice nuclei originate from various sources and do not form a chemically homogeneous population. Indications are that nuclei originate at the earth’s surface and that they can be transported through long distances in the atmosphere. As measurement techniques improve it can be expected that more will also be learned about the nature and origin of the nuclei. Artificial ice nuclei. Silver iodide was discovered as a potent ice nucleus three decades ago [419]and has been studied extensively ever since. Other materials of similarly great activity are now known (Pb2I, CuS, metaldehyde, among others). An enormous volume of literature exists on both the fundamental nature of the activity of artificial nuclei and their practical application in cloud seeding. Most of this literature is widely scattered and no comprehensive reviews are available. References can be made to books on cloud physics [267,111], to a brief discussion of the topic by Boucher [281,to the proceedings of ice nucleation

conferences [317],or to journals. The essence of the matter is that the activities of the various materials are due to a number of different factors [217] and are easily influenced in practice by the manner of generation, by environmental conditions (such as UV radiation), and by the way the nuclei are introduced into the clouds. As a result, cloud seeding represents a challenging scientific and technological problem. General aspects of cloud seeding are covered in [156,302]. Heterogenous ice nucleation in clouds. Early parts of this section dealt with the various secondary processes of ice crystal generation and the preceding paragraphs summarized what is known about heterogeneous ice nucleation in the atmosphere. The real extent of the contribution of heterogeneous nucleation to ice formation in the atmosphere is difficult to delineate wherever secondary processes come into play, though it can be argued with some certainty that heterogeneous nucleation is the process which precedes and initiates the secondary processes. Proving this assumption is one of the

136

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

important tasks for future research. It is significant that there are types of clouds for which the ice crystal concentrations are of comparable magnitudes to the ice nucleus concentrations, and where no known secondary processes act. These are: continental cumulus studied in Israel by Gagin [125], stratus studied in Australia by Mossop et al. [285] and in Alaska by Jayaweera and Obtake [1921,and orographic clouds studied in Wyoming by Cooper and Vali [671.In each the ice crystal concentrations exhibit a monotonic increase with decreasing temperature and the concentrations are within about an order of magnitude of the average values of ice nucleus concentrations. In no case has it been possible to demonstrate a one-to-one agreement between nucleus and crystal concentrations for individual clouds, or air masses; such complete proofs will have to await the improvement of nucleus measurement techniques. The evidence is quite suggestive, nonetheless, that heterogeneous nucleation is the sole origin of ice in a wide variety of clouds. Further work will be needed to prove the applicability of this concept, which incidently was also the working hypothesis of early cloud physics research. Summary of origins of ice in clouds. Most of the material discussed in this section is summarized in fig. 4.3 and table 4.1 through examples of cloud types for which the origin of ice has been established relatively well. As illustrated, the temperature-concentration regimes differ for clouds in which the rime-splintering process is at work, for clouds where homogeneous freezing can act, and for the other broad range where heterogeneous nucleation is the most probable origin of ice. Not included in the figure are groups of observations which yielded data over such wide ranges that depiction would have been difficult, or where the origin of ice could not be ascertained. Notable among the studies omitted are the cumulus investigations of Braham in Missouri, U.S.A. [29,2231, the examinations of orographic clouds and cyclonic storms by Hobbs and Atkinson [169], and the observations of

stratiform clouds by Heymsfield [157].Mention should also be made of the results of Magono and Lee [261], who showed that the concentrations of ice crystals in snow-clouds increased with the age of the clouds and with decreasing temperatures. Important evidence on the development of ice in convective

clouds in Colorado, U.S.A., was presented by Dye et al. [98],who showed that glaciation in the lower regions of clouds (below —20 °C)is initiated by ice particles falling or transported down from higher, older regions of clouds.

:::i w-30-

3 4

~

_—‘~~,-

4V U.

~

_—..,J -.--,-, 6

I-

~

fi, IJ’.___

...~ ~

/

/

6~ 5

I

_..

0

4~

III~

0°

tO’

02

10°

CONCENTRATION

IO~

0’

06

1m°)

Fig. 4.3. Observed ice crystal concentrations in different cloud types. Numbers refer to entries in table 4.1.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

137

Table 4.1. List of cloud types included in fig. 4.3 Cloud type

Origin of ice elements

Reference

I. 2. 3. 4. 5.

Rime-splintering Homogeneous freezing Heterogeneous nucleation Heterogeneous nucleation Heterogeneous nucleation

282 32, 158 125 192 67

Heterogeneous nucleation

282

Maritime cumulus (Old) Cirrus Continental cumulus Arctic stratus Orographic clouds

6 1.~Cumulus (New) Layer clouds

In addition to empirical studies, some aspects of the problem have been usefully examined by modeling calculations. A great deal of progress has been made in the construction of numerical models which can follow in detail the evolution of populations of droplets and ice crystals, under various assumptions about the dynamical behavior of the clouds [442,171, 313, 59,69, 353]. Such

models are useful, for example, for examining implications of various assumptions about the modes and rates of heteorogeneous nucleation although, because of the large numbers of variables involved, it is difficult to isolate specific effects. In all, progress in understanding the origins of ice in clouds has been significant over the past ten years or so. Yet a great many open questions remain. In broad terms, the most important problem areas can be said to be these: assumptions about heterogeneous nuclei and nucleation must be

replaced by established cause-effect relationships, and the processes need to be identified which produce copious concentrations of ice crystals in situations where the presently-known processes are ineffective. 4.5. Growth and precipitation of ice In prefacing section 4.4, the mechanism was mentioned through which precipitation can develop in mixed-phase clouds owing to the colloidal instability of an ice-water cloud system. This mechanism is usually referred to as the Bergeron—Findeisen process, in reference to the scientists who first recognized and analyzed the phenomenon. The growth of ice elements by the diffusional transfer of water from supercooled liquid drops is only the first step in the evolution of ice-phase precipitation. Whereas pristine, diffusion-grown ice crystals can become large enough to develop appreciable fall velocities and reach the ground, more significant precipitation in terms of rate and amount is produced by ice forms resulting from rime growth or by aggregation of ice crystals. In an abstract way riming and aggregation are the ice processes corresponding to the coalescence of liquid drops — riming is the combination of ice crystals with supercooled drops, and aggregation is the combination of several ice crystals into larger entities. Vapor growth The variety of snow crystal shapes has long held fascination for natural philosophers, but serious physical interpretations were not attempted until the second third of this century. The towering accomplishment in this regard is that of Nakaya — by growing snow crystals of all the major naturally occurring shapes under controlled laboratory conditions, he proved that the dominant factor

138

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

controlling crystal form, or habit, is the temperature of the cloud in which the crystal grows [290]. Some recent general discussions of snow crystal habits are found in the monograph of Hobbs [1671 and in articles by Knight [2161and by Jiusto and Weickmann [1941.It is now known that, in addition to temperature, crystal habit is influenced by the rate of vapor supply as well. Macroscopically this factor is manifested by supersaturation (principally between water and ice saturation), ventilation, and the proximity of water drops. The dependence of crystal habit on temperature and supersaturation is known quite clearly from laboratory experiments [142,222, 332] and from observations of natural crystals [290,260]. There is excellent agreement between the two sources of data. Crystal shapes change from plate-like to needle, to prismatic, to plate-like, to branched, and back again to platelike and prismatic within well-defined ranges of temperature as the temperature decreases from 0°Cto —25 °C.There is enough confidence in this knowledge that crystal habit is often used to deduce the temperature or altitude of origin of crystals. The effects of ventilation and other factors are less well known. Because the factors controlling crystal habit on the molecular scale are not fully understood, it cannot be said that the reasons for the variations in habit are clear. Processes which contribute to the determination of habit, to greater or lesser degrees, include: the mobility of molecules on the ice surface, the presence and thickness of a liquid-like layer on the ice, dislocations, micro-scale vapor gradients near the crystal surface, mono or polycrystallinity of the frozen cloud droplet from which the crystal grows (if that is the case). There are many interesting details on this topic, but, once again, to enter into those details would lead too far afield. Also, the lack of accurate knowledge in this area does not constitute a serious hindrance to progress in other areas of cloud physics. Hence, the reader is referred to other works on the subject [112,167, among others]. The rate of growth of ice crystals is usually calculated from the equations of Maxwellian diffusion with the shapes of the crystals accounted for on the basis of electrostatic analogy through the incorporation of a capacitance factor. Crystal shapes are taken, for the appropriate temperature regimes, from empirical data, in the form of diameter-thickness or length-diameter relationships. The capacitance factor is calculated for some closely-fitting disk, or oblate or prolate spheroid shapes. Ventilation effects are considered by applying correction factors to the growth rates derived for stationary conditions; empirical fall-velocity data are combined with analyses of the flow-fields around the crystals. Examples of growth-rate calculations are given in [225], [191], [162] and [141]. Empirical studies, [120,20, 340], have been restricted in the laboratory to short time scales. Over the first few minutes of

growth agreement with calculated rates is reasonably good. In a field study of crystal growth rates [173] some deviations from calculated rates were noted, but more precision will be needed in observations before the comparisons can be made conclusive. Enhanced crystal growth rates in electric fields were reported in one study [77]. Growth by riming

Beyond a certain threshold size, falling ice crystals collect cloud droplets by aerodynamic impaction and wake capture, and probably by electrostatic attraction. The rate of collection rapidly increases after that, due to increased fall velocity and greater collection efficiency, so that riming is a

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

139

very important process in precipitation formation. The efficiency of precipitation — the ratio of precipitation produced to cloud water available — is greatly enhanced by riming. Observational evidence on crystal riming is typified by the data presented in [294],[194],[209]and [151].Crystals usually begin to rime along their edges, although droplets are sometimes uniformly distributed on the crystal surface. As riming proceeds, the droplets tend to form clusters which build out from the crystal into a low-density structure. Eventually the crystal becomes completely covered, by which time its mass is many times that of the original vapor-grown crystal. If the cloud is thin, so that the accretion rate is slow, or the crystal falls from the cloud into an ice-supersaturated zone, the rime droplets begin to develop crystal facets (of the same orientation as the original crystal), but this is a relatively rare occurrence. An important feature of riming is that the structure of heavy rime is often so loose that some rime can break off relatively easily leading to accelerated cloud glaciation. Recent laboratory studies of rime growth have been reported by Sasyo [343,344], Pedori et al. [306] and by List and Schemenauer [256].Theoretical examinations of droplet collection were given by Pitter and Pruppacher [310,309]and by Schlamp et al. [347].Jiusto [193]constructed a simple model of crystal development in clouds, including growth by riming. In general, it appears that riming growth of crystals has not received attention commensurate with the importance of the phenomenon. While it is true that the great diversity of ways in which riming appears to occur in nature is quite forbidding, the problem is not intractable. Perhaps the availability of better observations in clouds will provide some new impetus for studying the process. Graupel and hail Graupel particles are small (0.1 to 1 cm), low density pellets often called “soft hail” or “snow pellets”. They consist of rime, i.e. frozen-together cloud droplets. As such they represent the extreme of crystal riming, but deserve to be discussed separately because they are not always simply derived from rimed crystals. Graupel of hexagonal shapes are clearly showing their origins as broadlybranched crystals. Irregularly spherical graupel (lump graupel) may develop by riming of a crystal of no definite symmetry which falls while riming in a random tumbling fashion. Graupel of conical shapes, according to Knight and Knight [218],bear no detectable evidence of their origins and pose an interesting problem. As graupel particles are a frequent form of precipitation from both winter and summer clouds of the temperate zones their growth processes deserve further attention. The ultimate results of accretional growth are hailstones which may reach 15 cm diameter and are found frequently in sizes around 3—6 cm. What differentiates hail from graupel is its density. As size and fall velocity increase, the collection rate of supercooled water becomes so high that the surface of the hailstone warms above the ambient temperature (due to the latent heat of fusion of the accreted drops). As shown by MackIm [259],the density of accreted ice increases with greater impact velocity and higher temperature. If the surface of the hailstone warms to 0°C it grows with a wet skin. As conditions for hailstone growthare high liquid water contentsin the supercooled portions of clouds, and long growth trajectories, hailstones are the products of large, intense summertime storms. The development of hailstorms and of hailstones has been the subject of intensive study for many decades. There is a large body of literature dealing with the internal crystal structure of hailstones, the distribution of air bubbles in the ice, the shapes of haiistones and their aerodynamics, the hydrogen/deuterium isotopic ratios of the ice, and with analyses of the cores of hailstones, all with the aim of deducing the growth conditions of the hailstone throughout its life. A similarly extensive literature deals with studies of hailstorms, and with attempts to modify the storms by seeding with ice

140

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

nuclei to reduce hailstone size through competition. Rather than treating this subject here superfically, the reader is referred to a recent compendium on the subject [116]. Crystal aggregation (snowflakes) Ice crystals in clouds often form clusters of up to many tens of crystals. Such aggregation does not

necessarily lead to higher precipitation rates than would result from the fall of the individual crystals; it can mean a small enhancement through somewhat increased fall velocities, and if falling into air warmer than 0°Cthe aggregates generate larger raindrops which do have higher fall velocities and are less likely to evaporate. The factors governing crystal aggregation are not well known. It is evident from observations that

branched crystals and complex three-dimensional crystals are more likely to form aggregates than the more compact crystal forms. There appears to be a tendency for aggregation to be more prevalent at temperatures near 0°Cand if the crystal concentration in the clouds is high [170].There is also some evidence for the role of electric forces [262]in bringing crystals together. The crystals are held together by mechanical interlocking when their shapes allow it, but this does not appear to be a necessary condition. The adhesion of ice crystals upon contact, by the rapid formation of a neck about the contact point, appears to be well proven by the laboratory experiments of Hobbs [165]. Laboratory experiments on aggregation have been attempted but were restricted to working with small crystals (<100 sm). The motion of aggregates was analyzed by Sasyö [3431 who also presented calculations for the rate of growth of aggregates. Models of ice precipitation

Syntheses of the aspects of ice phase precipitation development into comprehensive models have been attempted with various degrees of elaboration and with a multitude of different aims. In comparison with models of the warm-rain process, models of ice development can have one simplifying feature the dynamical feedback from the growth of ice elements is almost always negligible so that the dynamics of the cloud system can be determined, or assumed, independently of the ice processes. On the other hand, there are many more processes to consider with ice precipitation than in the all-liquid case. Also, initial conditions, such as nucleation modes and rates are very poorly known (cf. section 4.4), and there are more uncertainties about riming and aggregation. Mention has been made of the calculations of Jiusto [193]for the development of precipitation by riming and aggregation for arbitrarily assumed ice crystal concentrations and updraft velocities. Much more detailed calculations were produced by Cotton [69]and by Danielsen et al. [79]for cumulus clouds. These studies were followed in rapid order by, among others, the models of Ryan [337]and of Young [442], the latter being perhaps the most detailed available. The specific case considered in Young’s model was the development of precipitation in wintertime orographic clouds, for which a two-dimensional, steady-state flow field was assumed. A similar situation was modeled by Hobbs et al. [171].An example of calculations for stratiform clouds is given in [39].Hailstorm models which include microphysics were given by Danielsen et al. [79],Chisholm and English [53],and Orville and Kopf [297],among others. All the models represent significant efforts both in the selection of appropriate physical conditions and in numerical techniques. The models exhibit many realistic features and have been found useful in sorting out essential features from inconsequential ones. But that is about as far as the utility of the models goes at present. Proper verification of models, or the construction of valid descriptions of the natural processes (depending on which way the problem is viewed) is going to have to await the —

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

141

development of more detailed models, but probably more important, the clarification of many outstanding questions about the physics of the processes involved. 4.6. Precipitation modification

Throughout this review of cloud and precipitation physics the emphasis has been on the description and explanation of natural processes with only occasional references to the possibility of intentional or inadvertent modifications of these processes. Ever since the original discovery of cloud seeding with ice nuclei in 1949, cloud physics and weather modification have been closely aligned disciplines. To attempt to summarize the science and technology of precipitation modification is beyond the scope of this review. The point of mentioning the subject here is merely to call attention to the close connection between the topics of the preceding sections and the attempts at rain enhancement, snow augmentation, snow redistribution, and hail supression, and to direct the interested reader to some starting points in the literature on precipitation modification [156,116, 316].

5. New methods of observing the atmosphere 5.1. Introduction With the exception of new equipment for in-situ cloud measurements, most observational advances in meteorology during the past decade have been by methods generally classed as remote sensing. Keeping apace with new developments in atmospheric observation by microwave radar, there has been increasing use of laser radars and acoustic radars, especially in studies of the structure of the lower atmosphere. The satellite as an instrument for atmospheric sounding has been proved during this period, and the closely related concept of ground-based sounding by passive radiometry shown to be feasible. To document the rapid developments in precision radiometry is beyond the scope of this review; for this the reader is referred to recent publications, for which the review of Houghton and Taylor [179]is a good introduction. The current state of instruments used for routine measurements of solar and terrestrial radiation at the earth’s surface has been reviewed by Coulson [74]. 5.2. Microwave radar

Since the discovery in the 1940’s that radar is able to detect precipitation, radars have become an essential part of operational meteorology. Recent progress in precipitation physics has been closely tied to radar observations, and radar data on the structure and lifetimes of rainclouds are being increasingly employed in dynamical models of clouds and large scale circulations. Developments in the last decade have been prompted largely by technological improvements: coherent signal-processing techniques that enable the measurement of velocities of the precipitation particles through the Doppler frequencies which their motion induces in the echo, and highly sensitive receivers which make it possible to observe the structure and motion of optically clear air. Doppler radar

The relation betwen the received signal power and properties of the precipitation target is

142

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

expressed by the radar-range equation: PrPt G2A2O2cT I ~ 1024212~2jOB

(5.1)

where P. is the short-term time average of the received power, P~is the peak transmitted power, G is the antenna gain, A the wavelength, T the pulse duration, c the velocity of propagation, and R the

range to the target. This form of the equation assumes that the beam is of circular cross section and with a Gaussian pattern; 6 denotes the half-power beamwidth. The factor ~ (often denoted by t~)is called the radar reflectivity of the precipitation and is the sum of the backscatter cross sections of the precipitation particles in a unit volume of space. The radar backscatter cross section is related to the angular scattering cross section o~(O),defined earlier in (2.15), by ‘YB =

(5.2)

4ir~~(ir).

Hence Mie scattering theory may be used for calculating the radar reflectivity of precipitation.

For radar wavelengths of 10 cm and longer, the Rayleigh scattering approximation applies for all precipitation forms with the possible exception of hail. For a small dielectric sphere, the amplitude a of the backscattered signal is related to the cross section ‘YB and diameter D by

2= (ir5fKI2/A4)D6,

‘YB = =

where K

(5.3)

4~a

(m2 1)/(m2+ 2) depends upon the refractive index m of the material. For water and ice, respectively, 1K12 equals approximately 0.93 and 0.21. The radar equation may be written employing —

(5.3) as

—

i~3c

rG~O2 2

IKI

2Z ~,

(5.4)

A wherePrZ is1024l2P~ the radar reflectivity factor, defined by

Z=~D6=JN(D)D6dD.

(5.5)

The reflectivity factor is thus the sum of the sixth-powers of the diameters of the precipitation particles (assumed spheres) in a unit volume of space. It is usefully expressed as an integral over the particle size distribution N(D), with N(D) d D denoting the number of particles per unit volume with diameters in dD. Equations (5.4) and (5.5) are the basis of radar storm detection. They have also made it possible to study the development of precipitation and growth and motion of precipitation systems over large areas. A new dimension to radar studies was added in the early 1960’s, when Doppler radar equipment became available for meteorological research. In such equipment the frequency content of the reflected signals is compared to that in the transmitted wave and frequency shifts are interpreted as arising from the Doppler effect. Thus a frequency shift ~v corresponds to a velocity V according to ii = (2/A) V where ~ denotes a unit vector in the radar-pointing direction. ~,

Meteorological targets induce a spectrum of Doppler shifts because their scattering elements generally do not all move with the same velocity. When the radar beam is pointed vertically, the Doppler spectrum contains information about vertical air motions and precipitation fall speeds. For horizontal viewing, Doppler velocities indicate horizontal motions of the air through which the precipitation is falling. The particles move with the wind to a close approximation, though the echo

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

143

patterns frequently do not move exactly with the wind because of precipitation development or dissipation in preferred regions of the echo. A general formula for the Doppler spectrum is S(v) = (41TPrIö~B)

a~W(a,v) da,

(5.6)

where W(a, v) dadv is the joint probability that a particle picked at random from the scattering volume has radial velocity v = V ~ in the interval dv and returns a signal with amplitude in da. The

mean cross section of the particles is ‘YB~As defined by (5.6) the Doppler spectrum is normalized such that its area, or total spectral content, equals Pr. If the antenna is pointed vertically, (5.6) may be expressed in terms of the particle size distribution as (5.7) S(v + w) = S(— V~) = (Pr/Z) N(D) D6 dD/d Vt, where w denotes the vertical air velocity (positive upwards), V~the terminal fall velocity of a particle of size D, and N[D(V~)] is the size distribution function. This result is based on the assumption of

Rayleigh scattering, and also assumes a monotonic dependence of fall velocity on particle size. Applied to measurements in rain, it has been used for inferring the vertical air velocity (by specifying N(D)) or conversely for inferring the drop-size distribution (by specifying w). Examples of these applications are given in a comprehensive book [15]. Except for the estimation of the drop-size distribution, it is usually not essential to measure the entire form of the Doppler spectrum. Instead, it has been found sufficient to measure the mean Doppler velocity (v), defined as the first moment of the distribution S(v), and the Doppler variance o~, which is the second central moment of the distribution. For near-horizontal viewing, (v) indicates the mean radial component of air motion in the scattering volume and o~reflects the effects of turbulence and wind shear, which cause spectral spreading. In an early application [331],these spectral moments were used to estimate the intensity of turbulence in snow and the partitioning of turbulent energy between scales larger and smaller than the scattering volume. A limitation in the measurement of air motion is that only the radial velocity component can be sensed. The airflow patterns associated with convective rainclouds are complex and markedly three-dimensional; the restriction to measuring one velocity component is a serious shortcoming when investigating these circulations. One means of overcoming the limitation is to scan a region in space with two or more Doppler radars which are separated by a sufficient distance to give each a different view of the same precipitation area. The development of such multiple systems of Doppler radars has provided much new information on the fine-scale circulation patterns of precipitation systems, and is one of the major observational advances in radar meteorology during the last decade. An example of the wind pattern in the lower atmosphere as measured by a dual-Doppler radar system is shown in fig. 5.1. The measurements were made in a light, steady snow, the blowing snowflakes serving as tracers of the air motion. The radars were separated by a 14-km baseline oriented in an approximate north-south direction. They were coordinated to scan simultaneously a common volume of approximate extent 5 x 5 km2 in the horizontal and 1 km in the vertical. The data were smoothed, interpolated, and combined by computer to produce maps as shown in the figure. The mean velocity within each plane was subtracted out, so that the pattern remaining indicates the deviations of wind velocity at each position from the mean wind. The x-axis is oriented towards the NNE, which was the mean direction of the wind over the entire volume sampled.

144

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

Y

7

=

200 m

=

6

6

~

3

.-~-‘.

~ 3

—.~4~.—--- ~

‘‘__/

—-

——

0-

~.

I

I

0

3 km

6

z

X

0

3 km

300

=

Z

km

7

S

400 ni

kfl

6

_______ ______

-

-

-

~ -_- -

;-~k

~

~,

_________ _______-

,

I ~p

~-r.

_ ...

~

-.. -.~. •.

/

s.i. wind

S

700

m

-

km

~ ~

=~•=~~-~---,

._—~.~--

0

? ‘~

\\

1\

~‘~~‘~\\\

N N~.

Fig.

7

—

~

0

6 600 m

6 __________

3

=

-

0-

.~

m

“~

—....

.~

500

—

‘.‘~

\‘..

— /

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/~m/s~”~~

velocity vectors in horizontal planes ranging from 200 to 700 m above the surface. The mean wind vector at each altitude has been subtracted out; what remains are deviations from the mean. Scale indicated by reference vector at 700 m [118].

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

145

Large eddies or vortices are evident at the lowest altitudes, and the patterns tend to become smoother with increasing height. Frisch et al. [118]analyzed these and other patterns in an attempt to separate organized small-scale flow from random turbulent effects. They found some agreement between the observations and theoretical predictions of the structure of secondary flows. Whereas observations such as in fig. 5.1 have made it possible to study the fine-scale irregularity of wind in the free atmosphere, an equally useful application of multiple-Doppler radar systems is the measurement of circulation patterns in severe storms. Figure 5.2 shows the flow pattern in vertical cross section of a mature thunderstorm, obtained again with a dual-Doppler system. The raw data consisted of horizontal velocity components measured at low elevation angles. Vertical velocities were inferred from a set of horizontal velocities at different altitudes by integrating over altitude the mass continuity equation, V~pV =0. This procedure is subject to computational errors, but these are smoothed out to some extent by the computer analysis which produces the flow diagram. The example shown here indicates the complex circulation pattern of the storm — several large vortices, updrafts predominating above 6 km and downdrafts below — and tends to support earlier speculation [36]about the airflow in large thunderstorms. -

Radar reflections from clear air Clear-air echoes, or “radar angels”, have been observed since the early years of microwave radar. They arise from unnoticed birds or insects or, under certain conditions, from fluctuations in the microwave refractive index of optically clear air. Using radars with high sensitivity and fine spatial resolution, it has been possible during recent years to exploit the clear air echoes as a way of probing the atmosphere for information on wind, stability, and turbulence. The theory which most adequately explains clear-air radar reflections is the mixing-in-gradient theory developed by Russian turbulence theorists, e.g. Tatarski [379].The theory and applications to the sounding of stable layers were comprehensively reviewed by Ottersten et al. [298]. The radio refractive index n of air, expressed in terms of “N-units”, is given approximately in terms of the temperature T and vapor pressure e as N

(n

—

1) x

106

= (A/T)(.p

(5.8)

+ BelT),

where a and B are constants. If T is in K and p and e in mb, then this equation is accurate to within I

I

I

I

I

x~4O2km -

~3

Distance North of Basehne Midpoint (km) Fig. 5.2. Vertical cross section of a thunderstorm, indicating air motion, in a plane oriented normal to direction of storm motion. Solid lines reflectivity contours at 20, 28 and 36 dBz levels [230].

are

R.R. Rogers and Gabor VaIl, Recent developments in meteorological physics

0.02% by choosing A = 77.6 and B = 4807 [16].In a layer of air through which the temperature or vapor pressure varies, turbulent mixing distorts the mean structure of these quantities, resulting in small-scale spatial contrasts in temperature or humidity. The consequent variability in refractive index may create detectable radar echoes. The radar reflectivity ~ of a turbulent volume of atmosphere depends upon the intensity of the spatial fluctuations in N which have a scale size equal to half the wavelength. These depend in turn on the mean gradient of N, the strength of the turbulence, and the

distribution of turbulent energy over different scales. For homogeneous turbulence and scales in the inertial subrange [379],it is expected that the turbulent energy spectrum will satisfy the Kolmogorov power law. Then the reflectivity is related to turbulent intensity by ,~=0.38C~A”~,

(5.9)

where A is the radar wavelength and C~is a quantity that measures the strength of the refractive index variations, expressible in terms of the mean index gradient and the eddy diffusion coefficient. Large values of C~require primarily strong gradients of the refractive index. Detectable clear-air radar echoes are therefore usually associated with layers of discontinuity across which the temperature or the humidity undergoes a sharp variation. Stable atmospheric layers separating relatively moist surface-layer air from drier air aloft are often responsible for clear-air echoes and have been the subject of considerable study. When there is wind shear across these layers a kind of shear-flow instability may develop. Radar then provides an ideal means of studying the mechanism of the instability. Figure 5.3 is an example of the wave structure of an inversion layer observed by a high resolution vertically pointing radar. A number of studies based on these and similar observations have suggested that the wave pattern indicates gravity waves in the atmosphere and that the appearance of breaking waves often implies a shear-induced instability of the Kelvin—Helmoltz type. As a precursor to the formation of cumulus clouds, convective elements or thermals begin to ascend in low-level air. Refractive index variations in some conditions make these convective A

HEIGHT

24 AuG

970

(rn)

L.a 2G0

——-‘--•--——----—-——r—~--~--~~---

~ t43&

1442

1446

1450

TIME (Psr)

Fig. 5.3. Wave structure observed by radar in optically clear air, indicating shear instability [130].

147

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

~3T~

__

L~ ~

_ —~

L

Fig. 5.4. Clear-air convective elements observed by high.power radar. This is a sector of a low.elevation plan view, with range rings shown at 10 and 15 nautical miles. (Courtesy T.G. Konrad, Applied Physics Laboratory, Johns Hopkins University.)

elements visible to radars with sufficient sensitivity, providing an effective means of observing the structure of convection prior to cloud formation. Figure 5.4 shows the echo pattern observed by a sensitive radar in clear air near the ground at a time when surface heating was causing convective overturning. The ring-shaped structures are individual convective cells made visible by the refractive index gradients at their edges. Measurements by airplane of the temperature and humidity of the cells showed that the air within the cells was more humid and cooler than the ambient air. At the altitudes where patterns of this kind are observed, the upward-moving air has evidently extended above the level of neutral buoyancy. By this process of cellular convection the low-level mixed layer deepens. Clouds form if the convection reaches the condensation level. Konrad [2281reported that the height of the convective field increases at a rate of 4.5 to 6 m/min. At the onset, the cells are about 100—200 m in diameter. They increase in size with time and altitude above the ground, attaining diameters of several kilometers. A link between convective currents or bubbles in the clear air and the thermals visible as turrets in growing cumulus clouds seems thus to be established. 5.3. Laser applications

A number of experiments have been conducted to demonstrate the feasibility of using laser light to

148

R.R. Rogers and Gabor Vali. Recent developments in meteorological physics

measure remotely certain properties of the atmosphere or atmospheric scatterers. The meteorological information must be inferred from the light which reaches a receiver after being scattered or transmitted by the atmosphere. Laser studies are thus intimately related to atmospheric optics. Laser experiments may for convenience be classified as transmission experiments or scattering

experiments, depending upon whether the meteorological effects are deduced from transmitted or scattered energy. Transmission experiments are employed for measuring path-integrated quantities such as wind speed, turbulence intensity, and rain rate along the path of the laser beam. Scattering is used for sensing atmospheric particles and inferring their properties, and for measuring the concentration and distribution of certain atmospheric gases. When employed for backscattering measurements, a pulsed laser operates in a manner analogous to pulsed radar and is sometimes called lidar. A recent review of lidar applications was given by Collis and Russell [64].The related subject of laser-light transmission through the atmosphere was reviewed by Zuev [448]. Experiments in light transmission

With the discovery that lasers which nominally operate at a fixed frequency are tunable, within a narrow frequency range, it has become possible to measure absorption spectra of atmospheric gases with extremely high resolution. Figure 5.5, for example, is the absorption spectrum of atmospheric water vapor as measured with a ruby laser spectrometer. The strong dependence of absorption on frequency provides a method of measuring the path-integrated concentration of water vapor or of other absorbing gases. By transmitting sequentially over the same path two laser beams with slightly different frequencies, one located between absorption lines and the other on a line center, it is possible to deduce from Bouguer’s law the amount of absorbing gas along the path. In practice this measurement is subject to errors because of uncertainties in the precise line structure of the gas. Only tunable lasers of ultra-high resolution, such as the one which produced the data in fig. 5.5, are suitable for this application. The technique, referred to as differential absorption, has been attempted for several trace gases of the atmosphere as well as water vapor. The principles of the technique and estimated accuracies are described in the reviews cited earlier [448,64]. Applications to pollution monitoring are discussed by Hinkley et al. [163]. Laser pulses propagating through the atmosphere not only suffer absorption and scattering, but are influenced as well by turbulent fluctuations of the refractive index of the air. These fluctuations cause random scintillation in the received energy. Characterized by the variance a, of the logarithm of intensity, weak scintillations have been adequately explained by the theory of turbulent mixing [379]

~

__________________

694.35

694.36

69437

694.38

WAVELENGTH,

694.39

694.40

X(nm)

Fig. 5.5. Fine structure of water vapor absorption spectrum over a region 0.7 cm~wide, obtained with high-resolution laser spectrometer [448].

R.R. Rogers and Gabor Vali~Recent developments in meteorological physics

149

and are related to the path length l and the refractive index structural characteristic C~by °C C~l’~’~.

(5.10)

Measurements of scintillations may therefore be used to determine the integrated value of the turbulence parameter C~along the propagation path. A recent finding is that a saturation phenomenon occurs when a•~ exceeds approximately 0.6. Then the variance no longer increases with C~or I at the rate predicted by (5.10). An approximate formula is given by Zuev [448] for the saturation condition. Theories to explain the phenomenon have been offered [61]and different theories reconciled [63]. Of more interest to conventional meteorology is the use of scintillation measurements to determine the path-integrated wind blowing across the propagation path. In the simplest configuration, two detectors are used, displaced horizontally from one another by a small distance. A single laser beam is transmitted from a location some distance away and received simultaneously at the two detectors. The received intensities are correlated with one another for different time lags. The cross-beam velocity is inferred from the time delay which maximizes the correlation. This technique was confirmed experimentally [243]and has been extended to give information about the wind as a function of position along the beam [245]and to employ natural illumination instead of laser light [62].It has been proposed to use rain-induced scintillations, which usually are of higher frequencies than those caused by turbulence, to deduce properties of the rain along the propagation path [436]. Scattering experiments

When pulsed lasers are used for atmospheric scattering studies, the transmitter and receiver are usually located together for convenience. The system thus constitutes a laser radar, concerning which a considerable body of literature has developed. Such systems are of two types: those designed to measure elastic scattering, in which the scattered energy has the same wavelength as the incident energy; and those designed for studying Raman scattering or resonant fluorescence, in which the scattered energy is shifted in wavelength by some characteristic amount from the incident energy. In the case of elastic scattering, the basis for interpreting observations is the lidar equation, analogous to (5.1) for microwave radar. Neglecting multiple scattering, this equation may be written [64] Pr(R) = PtAr(cTl2)131T(R)R_2exp[2f

y(r) dr],

(5.11)

where Pr is the instantaneous received power at time 2R/c, c is the velocity of light, P~is the transmitted time 0, r is the pulse duration, y is the volume coefficient at range R (units, 2lm3), and power ~,(R) at = p (IT) at range R is the volume scattering coefficient for back-scattering (units, m m2m3sr5. Ar is the effective receiver area, so that ArR2 is the solid angle subtended by the receiver at range R. For pure Rayleigh scattering, ‘y and ~ can be evaluated from the relations given in section 2.1. The Rayleigh backscattering coefficient for sea level conditions is 1.3 x l0~A4 m~sr~ where A is in ~m. The extinction and backscattering coefficients are related by =

(318ir)y. According to Collis and Russell [64],the Rayleigh backscattering coefficient is large enough to be =

150

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

detectable at close range by lidars of modest capability. Mie scattering usually dominates, however, and because of the complexity of the scattering functions simple relations such as these are not applicable. In theoretical studies the starting point is to assume that the aerosols are homogeneous spheres for which the Mie theory applies. Because of the predominance of aerosol over molecular scattering, lidar has been shown to be effective in such applications as observing haze layers and monitoring particulate emissions from pollution sources. Even without knowing the scattering properties of the aerosol, data of the kind shown in fig. 5.6 provide unique information on the vertical extent of haze layers and, to some extent, the aerosol concentration in the layers. This example was obtained with a vertically-pointing lidar during the course of a typical summer day over downtown St. Louis, Missouri. Logarithmic reception was employed, and before being displayed the receiver output was corrected by the R~2falloff of power with range. In this form of presentation, the displayed quantity is the so-called S-function defined from (5.11) by S(R) = 10 log[f3~(R)t2(R)/f3.,~(R 2(R 0)t

0)],

(5.12)

2(y) = exp(—2f y(r) dr) is the two-way transmittance. If the relation between f3,. and y is where known, tthen both may be determined as functions of R from the derivative of S with respect to R [64]. At 0730 local time, fig. 5.6 shows a haze layer from the ground to 2.3 km, capped by a thin layer of more strongly reflecting stratus cloud that had formed during the night. By 1030 the cloud layer had dissipated and a relatively dense layer of aerosols near the surface had begun to grow thicker. Shortly afterwards the irregular appearance of the top of this layer indicates the onset of convection, with convective bubbles penetrating increasingly higher into the stable layers. By mid-afternoon the mixed layer reached 2 km and some condensation occurred indicated especially by the intense reflections at about 1500. Observations of this kind are useful in delineating the volume of the mixed layer of the lower atmosphere, which bears on the possible concentrations of low-level pollutants. Moreover, in a manner similar to that of the ultrasensitive radars described earlier, these observations indicate the connection between clear-air convective elements and cloud formation. Another lidar application is the observation of depolarization by clouds to distinguish between water droplets and ice crystals. If linearly-polarized light is incident on a perfectly symmetrical and homogeneous particle, the backscattered light will be polarized in the same sense as the incident light. If multiple scattering is negligible, it follows that the light scattered from cloud droplets should have essentially the same polarization as the incident light. Laboratory observations of water clouds have shown this to be the case, with linear depolarization ratios typically less than 0.03 [351].Because of their irregular shapes, ice crystals induce a cross-polarized scattered component, which may be nearly as strong as the parallel-polarized component. Schotland et al. [351]reported values of the linear depolarization ratio in natural ice clouds as large as 0.8. More recently, Sassen [342]cited a value of 0.5 as typical for ice, though in some conditions, such as for melting snowflakes, the values can be much higher. The use of lasers in Raman-scattering studies of the atmosphere has been reviewed by Inaba [186]. This application enables the measurement of water vapor and certain other atmospheric constituents by monitoring the backscattered energy which is shifted in frequency from that of the transmitted energy by an amount specific to the constituent of interest. Because the Raman cross sections are about three orders of magnitude less than the Rayleigh cross section at the same frequency, experiments are made difficult by the weak received signals. Nevertheless, Strauch et al. [374] —

R.R. Rogers and Gabor Vail. Recent developments in meteorological physics

151

U U

0

E

a

a 0 U a

a

a

...

•

.,

C

a ~f ,..

.-

C

~ ‘Isuol,, — 3a~i,L,v

a

,,, s,-ar.a*.

0

•fl.,e,.

‘‘‘

a

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0

—-

C

e

152

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

demonstrated the successful measurement of the water vapor content in a sampled volume of 5-m length and 20-cm diameter using a pulsed nitrogen laser. Although atmospheric studies by light reflections actually predated microwave radar [64],lidar meteorology is still in an early stage of development. Several attractive measurement concepts have been proven feasible but are far from being routinely applicable. Yet because of its contribution to pollution studies and its potential for water vapor measurements, lidar will continue to be a useful instrument in meteorological research. 5.4. Acoustic radar

Sound waves are scattered by spatial variations in temperature and wind velocity, as well as by particulate material in the atmosphere. Pulses of acoustic energy therefore provide a method of atmospheric sounding. Like lidar, acoustic radar has proven to be useful in studying stability and

mixing processes in the lower atmosphere. Moreover, the low level wind can be determined by measuring the Doppler frequency shift in the scattered sound. Sound propagation and scattering

Following from the adiabatic formula for sound velocity, the speed of sound c in the atmosphere is given by c =20.05(l+0.19e/p)~T,

(5.13)

where c is in m/s, T in K, and e and p denote the vapor pressure and the total atmospheric pressure, respectively [279]. The sound intensity, measured for example in W/m2, is attenuated with distance according to Bouguer’s law (eq.(2. 1)), where in this case the extinction coefficient depends primarily on scattering and molecular absorption. Because of interactions between molecules of oxygen and water vapor there is a strong dependence of the molecular absorption coefficient upon humidity, as shown in fig. 5.7. For frequencies between 2 and 12.5 kHz, the absorption peak occurs between 6% and 20% relative humidity, increasing with frequency. The absorption also increases weakly with temperature. Scattering of sound waves in the clear air is caused by spatial fluctuations in the velocity of propagation, which in turn arise from variations in wind speed, temperature, or humidity. Monin’s [278]solution to the scattering problem, as given by Hall [140],is ~(O) = 0.055A

“s

cos2o[(C2jc2) cos2~O+ 0.l3C~T2](sin~OY”3,

(5.14)

where ~(O)(in m’sr’) is the angular scattering coefficient, T is the average temperature and c the speed of sound in an undisturbed atmosphere at temperature T. C~ and ~ are structure parameters characterizing the fluctuations in velocity and temperature, analogous to C~for the refractive index. This form of the solution neglects humidity fluctuations, assumes that the temperature and velocity fluctuations are uncorrelated, and assumes a Kolmogorov turbulence spectrum. The separate contributions of temperature and velocity fluctuations have been accurately confirmed by laboratory measurements [11].According to Hall [140],the contribution of scattering to total extinction is highly variable, depending upon the values of C~-and C~,but can be of the same order of magnitude as that of molecular absorption.

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

0.09

153

____

0.08———--

~O.06

—

_/

-

—

-

~—

~O.O5~-~

—

~

~

—

~

_~__~

-.

h~_~~

-

-

_~__

-

-:~-25

~002

0

~

1~z::~ 20

~E

5060 RELATIVE

Fig.

5.7.

volume absorption

coefficient of sound in

70

HtJMtOITY , PERCENT

!;~ ~11

air at 20°C and standard pressure, as a function of relative humidity and for various frequencies [152].

Acoustic backscattering The effective “radar equation” for an acoustic sounder may be written as (5.11), where f3,. now refers to the acoustic backscatter coefficient and ‘y to the acoustic extinction coefficient. It follows from (5.14) that only the temperature fluctuations contribute to backscattering, because of the cos2(O/2) weighting of the velocity term. Values of C~in the atmosphere, especially in layers with strong vertical gradients of temperature, are often detectable at ranges out to several kilometers by typical acoustic radars. Sensitivity is limited by the noise of wind across the receiver, insects, birds, and especially human transportation and industrial activities. Consequently acoustic radars serve best for atmospheric probing in quiet, rural locations. Figure 5.8 is an example of data collected in a study of turbulence in the lowest layer of the atmosphere, including a 1-hour record obtained with a vertically-pointing acoustic radar. The lower three sets of curves are records of temperature and wind speed and direction, measured at three levels on an instrumented tower located 250 m from the acoustic radar. The sounder itself operated at a frequency of950 Hz, with a pulse of 20-ms duration and a power of 8 acoustic watts being transmitted at 2-s intervals. On the facsimile record shown, dark regions indicate strong acoustic backscattering, and hence large values of C2 7~.Although such records cannot be employed for quantitative measurements of ~ in part because of the display limitations and in part because y(r) in (5.11) is generally unknown, they clearly indicate the altitude intervals in which there are strong temperature contrasts. The tower measurements confirmed this interpretation, the elevated layer of strong acoustic reflectivity being associated with a temperature inversion, and its undulations indicating the passage of wave-like instabilities. Bean et al. [17]concluded from this investigation that acoustic sounding is a useful adjunct to boundary-layer turbulence studies. Rayleigh’s theory applies to the scattering of sound by obstacles such as raindrops and snowflakes.

154

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

HASWELL, COLORADO OCTOBER

__

I

~

0~A

9,

969

~ ~B~~C-~-

-

-~

H

E

2

1

-64-

—

—

I

\

,‘

~

—

~

...

.

:.:.::.: —s

:1:

Z3 21-

_••‘~.I._,,___

~

~

~‘,w___•_

39M~r~ 49

0800

~30 LOCAL

Melers

0900 TIME

Fig. 5.8. Acoustic echo return (top) with simultaneous records of wind and temperature at three levels on an instrumented tower. Strong echoes appear dark on the display. Period A indicates a diffuse layer; period B has a region of no echoes below a few wave-like returns; period C indicates undulating surfaces with superimposed small-scale instabilities [17].

Little [257]applied the theory to water droplets and derived the following expression for the acoustic

reflectivity (analogous to radar reflectivity) of a population of drops: =

4ir~,~ =

(25ir5/36A4) ~D6.

(5.15)

Considering the signal-to-noise requirements in acoustic radars, he concluded that fog and cloud

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

155

should be detectable to distances of about 300 m, and rain and snow to considerably further ranges. There are, however, no apparent advantages over microwave radar; moreover, the acoustic reflections from hydrometeors could be masked by the signals arising from temperature fluctuations. 5.5. In-situ cloud measurements Advances in the study of cloud microphysical processes are closely tied to progress in the sophistication of observations taken from aircraft. Prior to the use of aircraft, cloud studies were restricted to fogs and precipitation at ground level, and to occasional obserVations taken at mountain tops enveloped by clouds. After some pioneering balloon flights into clouds, systematic observations by aircraft were undertaken just before World War II in Germany, but progress was very slow until after 1945. Diem in 1942 [94]and Weickmann in 1949 [426]were probably the first to report on extensive observations taken from aircraft of cloud droplet spectra and of ice crystal forms, respectively. Rapid progress in the following three decades made aircraft studies practically synonymous with cloud physics research. Recent advances in cloud physics instrumentation resulted primarily from technological progress in electro-optics and in computer systems, especially with devices capable of providing information of hydrometeor size spectra. One of the principal tasks in cloud measurements is to detect, size and count particles of 1 pm to 10 cm in size with large enough sampling rates to resolve the variabilities of clouds. In addition, discrimination between water and ice hydrometeors is needed and the shapes of ice particles need to be recorded. The most important contributions to the measurement technology of clouds are perhaps those of Knollenberg [221],who developed a series of instruments for cloud particle detection and sizing. Knollenberg’s cloud droplet probe relies on the detection of forward scattered light by droplets passing through a laser-illuminated volume. The sample volume is defined by a combination of electronic screening of signals for proper drop transit time and by optical definition of in-focus and out-of-focus events. The size-range of the instrument is from 0.5 to 45 pm droplet diameter in bin widths of 0,5 to 3 pm. The sampled volume at aircraft velocities is about 50—100 cm3s’. With an appropriate system, data can be recorded at rates up to 100 s’, so that about 1 m resolution in cloud structure is possible. An example of data collected with such an instrument is shown in fig. 5.9. These data were taken in a cumulus cloud and the segments shown represent one region of the cloud with variable droplet content and another with quite uniform distribution. Naturally, the spectral information can be readily integrated to yield total droplet concentration and total liquid water contents. The only disadvantage of this instrument is the difficulty of calibration, which is a fairly elaborate bench procedure. The other Knollenberg instrument is the 2-D imaging probe. This device uses a linear array of photodiodes which is illuminated by a CW laser. High rate (up to 4 MHz) sampling of the photodiode outputs produces parallel slices of shadow information which are stored in memory until recorded serially. Image resolution to 25 pm is possible. The sampled volume is 5 x 1O~m3s’ for highresolution probes and 0.15 m3s’ for probes of 100pm resolution. Examples of reconstructed images of ice crystals are shown in fig. 5.10. Two blocks of data are shown taken 3 minutes apart in a winter cloud over Wyoming, U.S.A. The sequence of images runs from left to right in each line with the digits between images giving the length of the time gap (in microseconds) between the passage of the crystals. The top frame has some nearly transparent crystals and both frames contain images of two

156

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

HL

~AHA

H~~A~~EH L

~~AHi ~ H~

~

A A HA A~ too

~A

~A

~A

~A

____

~A

~

~AftA ~ UA ~A~ HA ~ ~ ~ ~AHA

~ L ~Hk HA HA HA

~A

I

30 DIAMETER

(pm)

~

~ ~ L~Hi HA ~AHA HK ~A~A ~A HA ~AHA ~ ~A IA ~Aftk ~ HA1~A ~ HA A ~AHA ~ ~ ft& ~ ~A~A A ~ ~IA HA kHA ~ I~AHA

H~A

~

~A

_

~

I 8 100

~

~

_____

~A

DIAMETER

(tom)

Fig. 5.9. Cloud droplet size spectra obtained with a Knollenberg forward-scattering droplet probe at 0.1 sec intervals in a cumulus cloud on July 21, 1977. Sequences start at lower left and proceed to the right. Block of data on top was taken near the cloud edge, bottom block is from the cloud interior. (Courtesy A. Rodi.)

crystals forming the beginnings of aggregates. The large amount of information which is incorporated in such records in clearly evident from these examples. A photographic imaging system for cloud particles has been devised by Cannon [42]. Using a rotating mirror, synchronized to aircraft speed, and twin flash illumination the Cannon camera is capable of producing high-resolution photographs of particles in the free air-stream. Detection of droplets of about 4 pm diameter has been achieved, and image quality for objects >50 pm is very high. The disadvantage of the system is that the sampled volume is defined only by the depth-of-field of the camera and hence its determination is somewhat subjective.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

157

Fig. 5.10. Shadow images ofice crystals recorded with a Knollenberg 2-D probe. Sequences go from left to right in each line. Heights of each band corresponds to 800 ~sm.Data were taken in Wyoming on February 20, 1975.

Holography has also been applied successfully to cloud studies from aircraft [386].The advantages of holography are the large sampled volume and the preservation of spatial relationships of the imaged particles. Disadvantages are large weight and relatively low resolution (about 20 pm). The devices mentioned so far certainly do not exhaust the list of instruments and techniques which have been applied in the near past to cloud observations. A more comprehensive coverage is given in [414]and [187].In all, the latest generation of instruments has equipped the cloud physicist with tools

158

R.R. Rogers and Gabor VaIl, Recent developments in meteorological physics

that were much needed and which will have a significant influence on future progress of the field. A gap in the capabilities is the reliable and high resolution detection of low concentrations of small (<50 pm) ice particles. The next phase of development may be in the realm of remote sensing devices capable of providing the type of information that can be acquired today only by in-situ probing. 5.6. Remote sensing by satellite The weather satellite has made perhaps the greatest contribution to meteorological observations since the radiosonde balloon. By providing a global view, satellites enable the observation of entire weather systems, thçir development, decay, and interaction with other systems. Wide-ranging scales of organized motion are observable for the first time, giving new perspectives on atmospheric dynamics. This information is obtained from observations at visible wavelengths, the cloud patterns indicating airflow and structure of weather systems. Observations in other parts of the spectrum yield information about atmospheric temperature and composition. An example of a satellite-measured emission spectrum of the atmosphere was given in section 3, where uses of satellite data in studies of the planetary radiation budget were also described. Extended reviews of radiation measurement by satellite were given by Hanel [146]and Houghton and Taylor [1791.More recent and shorter reviews of satellite applications are those of Yates and Bandeen [440] and Allison et al. [5].In an early survey, Wark and Fleming [421]reviewed the historical development of the concept of atmospheric sounding by radiometry. Selected topics in Soviet satellite research were compiled by Vinnichenko and Gorelik [416].

METEOROLOGICAL SATELLITE PROGRAM (1980-1976) CALENDAR YEAR

11N01

163 164 165I 166I 187 168 169 170I 171172 173 I7~ I

61j62

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OPERATIONAL SATEWTES

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EXPERIIENTALSATEWTES

(4/N)

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~

ATS-5 (5/74)

~

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/

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ESSA (2/US)

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Fig. 5.11. Chronology of the U.S. satellite program. (Courtesy D. Atlas. Goddard Space Flight Center, NASA.)

176 I

I

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

159

The American satellite program Figure 5.11 depicts the development of American weather satellites from the first Tiros in 1960, through 1976. The figure is up to date at the time of writing, although several new satellites are planned for 1978 as part of the First GARP Global Experiment [1%]. The first satellites in the Tiros series of the early 60’s were polar-orbiting at altitudes of about 700 km, making 12 to 14 orbits daily. All Nimbus satellites and the last two of the Tiros series (1965) were placed in sun-synchronous orbits, such that a given satellite passes over the equator at the same local time on every pass. Polar orbits of this kind give essentially full coverage of the earth with high spatial resolution but sample any particular region only twice a day. Some later satellites, starting with the ATS and SMS series, are in geostationary orbits at a distance of about 3600 km from the earth. Their period of revolution equals that of the earth’s rotation so that they remain above the same meridian. This enables continuous observation of weather over one section of the earth, but with limited spatial resolution because of the great distance. Satellites transmit data from radiometers, photometers, spectrometers, and interferometers operating

at wavelengths from the ultraviolet to microwave, as well as visible and infrared images of the pattern in their field of view. Data interpretation requires precise solution of radiative transfer problems supported by laboratory measurements of the spectral properties of atmospheric constituents. Infrared emission Measurements of relatively broadband emission in the infrared are used in studies of the earth’s radiation budget, as described in section 3. Narrow-band radiometric measurements in the atmospheric windows around 11 ~&mand 4 ~smare used for determining the temperature of emitting surfaces. The radiance observed at a narrow interval in a window, assuming a clear sky and neglecting the weak atmospheric emission, is L = tB(A, T), (5.16) where is the surface emissivity at the wavelength A of the observation, t is the spectral transmittance of the atmosphere, and B(A, T) is the blackbody function. In order to determine the temperature it is necessary to know and t. The emissivity depends upon surface type and may often be estimated with acceptable accuracy. The transmittance for clear skies depends upon carbon dioxide and water vapor, and also may be estimated. The main uncertainty arises from clouds, which can be too thin to be visible yet of sufficient optical thickness in the infrared to influence the measurement. The effect will usually be a reduction in the temperature estimate because clouds are cooler than the surface below them over most of the globe. Because thick clouds radiate very nearly as blackbodies in the infrared, radiometric measurements in a window over heavy cloud layers may be used to infer the cloud top temperature. Figure 5.12 shows the field of temperature associated with a hurricane, derived from radiometric measurements in the 3.4—4.2 ~iminterval. The colder temperatures indicate cloud, and the highest clouds account for the coldest temperatures. If, indeed, the relation between temperature and altitude is known, then measurements of this kind may be converted to effective cloud-top height. Temperature sounding Radiometric measurements in certain strongly absorbing atmospheric bands are employed for estimating the vertical profile of temperature. What is required is a band in which the dominant

160

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics

z

z

7

z

o

~

~f~y~2

~

___

_____

~ H

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R.R. Rogers and Gabor Vaii, Recent developments in meteorological physics

161

absorption is by a gas that is uniformly mixed in the atmosphere and with known concentration. Appropriate gases for temperature sounding are CO2 in the 4.3-sm or 15- ~m infrared bands and 02 in the 60-GHz microwave band. For the case of infrared radiance emerging from the top of the atmosphere in the zenith direction, the general solution (3.2) of the transfer equation may be written L = L(0)t(0,oo)+f

B7at(Z,°°)dZ,

(5.17)

where L(0) is the radiance emitted at the surface, t(z, on) is the transmittance between altitude z and the top of the atmosphere, B is the Planck function, and ‘y~is the volume absorption coefficient. The first term on the right is equal to tB, as in (5.16). B, ‘y~and t are monochromatic quantities which are functions of altitude. Noting that t(a,b) = exp[—r(a, b)] and that r(a, b) = f ~ dz, we may write Yat(Z, on)

=

at/az.

It is also convenient to introduce y = in (pip0) as the independent variable, where Po is the surface pressure and p is the pressure at altitude z. Then (5.17) becomes L=L(0)t(0,eo)+ B(A,T)(~’Idy. (5.18) —

I

\ayi

Jo

In terms of the mass absorption coefficient k, the optical thickness of the layer between pressure level p and the top of the atmosphere may be written 1 ~ kwdp’, (5.19) g0

where w denotes the mixing ratio of the absorbing gas, i.e. the mass of gas per unit mass of air, and g is the gravitational acceleration. If the absorber is uniformly mixed through the atmosphere then w is a constant and L depends only on the vertical profile of temperature. The factor (at/Oy) in (5.18), known as the weighting function, indicates the relative contribution of different levels of the atmosphere to the total radiance. The form of at/ay depends upon the absorption coefficient and is a strong function of wavelength. For a wavelength in the wing of an absorption line, 2k and is proportional pressure. Hence, for a constant gas mixing ratio, ~ris the at/ay may betoshown to be of the form proportional to p atlay = 2(p/pm)2 {exp (p/p,,,)2}. (5.20) —

Thus the weighting function has a maximum at p = Pm and a half-width in terms of altitude equal to about 10km [179].The pressure Pm where the weighting has its maximum depends upon the absorption coefficient: as k increases Pm moves to lower pressures (higher altitudes). Temperature sounding becomes feasible if for a particular absorbing gas a set of wavelengths can be selected for which Pm varies from low to high altitudes. The object is then to solve for B[A, T(p)], given the measured radiances and the known weighting functions. As a practical matter radiometers have finite bandwidths in which k and at/ay may vary substantially. Nevertheless (5.20) is often a reasonable approximation to the effective bandwidthintegrated weighting functions because most of the energy in a band originates from the collisionbroadened wings of lines. Figure 5.13 shows the weighting functions employed for temperature sounding on Nimbus 5. The

162

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

ScR 2

I~0

30 50 70 100

~X~v’ ~ )‘~\~-.

~

~

~

-

~

ITPR-2

~

~

700

.

looc ~ 00

“ .‘

~

300 500

NEMS-1

~

~

NEM S. 2 -

>~‘~‘~ ITPR. 3

~ ._.—‘—•~•~•~•~

0.1

0.2

0.3

0.4

0.5

Weighting Function, öt /ay

0.7

Fig. 5.13. Weighting functions for the different temperature-sounding instruments on Nimbus 5(440].

infrared temperature profile radiometer (ITPR) employs interference filters to define four channels in the 668 cm’ (15-~sm)CO2 band (668.3, 669.5, 713.8 and 747.0 cm’). The selective chopper radiometer (SCR) [178]incorporates a cell containing a known amount of CO2 which is used to filter the incident radiation and provide, in effect, narrower weighting functions at higher altitudes than otherwise possible. The Nimbus E microwave spectrometer (NEMS) observes at three frequencies (58.8, 54.9 and 53.65 GHz) in the 60-GHz 02 band. The advantage of the microwave channels over those in the infrared is that they are relatively unaffected by non-precipitating clouds. The relevant transfer equation for microwaves is more complicated than (5.18), however, as it requires a third term to

account for energy which is emitted downward by the atmosphere, reflected at the surface, and transmitted upward to the satellite [440].Moreover, the surface emissivity is difficult to estimate for microwaves, limiting the accuracy of temperature measurements at low levels. Given the radiances measured in the different channels, there are essentially two methods of solving the inverse radiative transfer problem and determining B(A, T), and hence the temperature, as a function of pressure. Neither method is capable of reproducing the details of a temperature proffle exactly. A basic limitation is that radiances are measured in only a finite number of channels, and these measurements are not independent because of the overlapping of the weighting functions. Measurement errors, theoretical uncertainties about the precise shape of absorption lines, and imperfect knowledge of the concentrations of minor atmospheric constituents also limit the accuracy of temperature retrieval. In fact, as a consequence of noise in the different channels, it has been found that there is essentially no advantage in increasing the number of channels beyond 6—8, as the

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

163

information content of a set of measurements remains the same [65,305]. With new low-noise, high-resolution spectrometers it may be possible to achieve 10 useful channels. The rather modest information content has been shown to be a fundamental limitation of radiance measurements [393]. The first method of inverting a set of radiances to solve for the temperature proffle is statistical [328].For each channel, eq. (5.18)is linearized by expandingin terms of {B[A, T(p)] B} where B(A, T) is —

based on a climatological mean atmosphere or a forecast of the temperature profile. An iterative method is then used to determine B(A, T) which satisfies the equations in the maximum likelihood sense. The

accuracy of the solution depends upon how closely B approximates the actual sounding. The better the a priori information, the better the final estimate of temperature. The second method [45,46,47] entails use of an iterative algorithm to obtain the temperature proffle, starting from an initial guess. In this procedure, the solution is unique, independent of the starting proffle, but requires fewer iterations if the initial guess is good. Both methods have been applied to observational data and shown to yield clear-sky temperature proffles with an accuracy generally better than 3 K. Figure 5.14, for example, shows two temperature profiles obtained over Berlin by Nimbus 3 and compared with nearly simultaneous radiosonde data. Inversion was accomplished by a statistical technique, using data from the satellite infrared spectrometer system — seven narrow channels in the 15-sm CO2 band plus another in the 1 1-sm window. The overall agreement between satellite and radiosonde proffles is good, though the fine-scale irregularities are not resolvable by radiance measurements. An example of temperature retrieval using Chahine’s iterative method is shown in fig. 5.15. Rather than from satellite radiometers, data were obtained from a high-altitude balloon-borne grating spectrometer in the 4.3-~smCO2 band (2160—2360 cm’). Spectral radiances were measured with an estimated accuracy of 3%. A theoretical error analysis indicated that temperature should be measur10

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Fig. 5.14. Two examples of temperature profiles obtained from infrared spectrometer system on Nimbus 3 (367].

300

164

R.R. Rogers and Gabor Vail, Recent developments in meteorological physics 2

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Teniperature (K) Fig. 5.15. Temperature profile obtained from balloon-borne spectrometer at 3.5-mb level, compared with data from rocketsonde and radiosonde (360].

able to within approximately 2 K. Results, such as those shown in this example, appeared to have an accuracy in agreement with theoretical prediction. Clouds pose the severest complication for infrared temperature sounding. If their radiative properties were known accurately, the radiative transfer formulation could be modified accordingly, but this is not generally the case. If the field of view of the radiometer is sufficiently narrow, there will be a good chance of obtaining some soundings in cloud-free air. Barnett et al. [13]have suggested that the gross effect of clouds may then be removed by applying a running ifiter to the records, which in effect preserves the horizontal structure with scales greater than 500 km. A more elegant way of allowing for clouds in the infrared has been demonstrated by Chahine et al. [48].The iterative algorithm for profile inversion was elaborated to accommodate multiple cloud layers. Observations are required at two sets of frequencies and in two adjacent regions in which the cloud properties are different. No a priori knowledge of cloud spectral properties or the number of cloud layers is needed. Starting with an initial guess, a unique temperature proffle is obtained by iteration. Results derived from measurements with an airplane-borne multidetector grating spectrometer, employing channels in the 4.3-sm and 13.4-sm CO2 bands, indicated that the atmospheric temperature profile was measurable with an rms error of I K relative to radiosonde measurements.

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

165

As mentioned earlier, microwave radiometric measurements are only slightly affected by the presence of clouds. Waters et al. [425]described temperature retrieval based on three microwave channels of the NEMS system on Nimbus 5 (cf. fig. 5.13). The temperatures were found to be accurate to within 2—3 K at altitudes above approximately the 800-mb level, but to be considerably less accurate at lower altitudes. There is a substantial literature on passive radiometric measurement from the ground at microwave frequencies [e.g. 384,89,430]. For temperature sounding, the approach is to build up a set of radiance data by varying the elevation angle of the receiving antenna. For each elevation there is a theoretical weighting function, analogous to atlay for the satellite case, which indicates the relative contribution of different altitude levels to the measured radiance. These weighting functions all peak at the surface, but extend to higher altitudes as the elevation angle is increased. Temperature sounding is feasible up to an altitude of about 3 km, with a vertical resolution that rapidly decreases with altitude, owing to the nature of the weighting functions. Accuracies generally to within 2 K are reported. Sounding of gaseous constituents

Once the temperature profile is known, then B(A, T) in (5.18) is determined and the weighting function (at/ay) may be regarded as unknown. Using numerical methods similar to those described for deriving the temperature, the vertical distributions of trace gases may in principle be determined from measurements in spectral regions where they strongly absorb. This technique has been applied with most success to the measurement of water vapor. Nimbus 4 is equipped with an infrared spectrometer that measures the radiation upwelling from the atmosphere in 14 narrow channels. The first eight (11—15 jam) are used for temperature profiles and the remaining six (18—36 jam), in the rotational H20

band, for calculating water vapor profiles. The effective weighting functions for these channels, for typical midlatitude distributions of temperature and water vapor, are shown in fig. 5.16.

As a consequence of the form of the weighting functions, water vapor estimates are most accurate in the middle troposphere. Smith and Howell [368]reported that errors in the derived relative humidity are generally less than 20% in the 400—600 mb layer. In the lower troposphere errors are larger but still less than 30%. The determination of humidity depends crucially on temperature; a systematic error of 2°Cresults in a 12—25% error in relative humidity. Wark et al. [422]called attention to the importance of the e-type absorption in deriving the water vapor amount. The presence of clouds complicates the problem of humidity determination and introduces substantial error unless properly accounted for. The microwave spectrometer on Nimbus 5 includes a single channel at about 20 GHz for determining the total, path-integrated water vapor. Early experiments [322]utilized a channel in the 6.3-sam H2O band for the same purpose. The vertical distribution of ozone has been inferred from measurements of its emission in the 9.6-jam band by the interferometer spectrometer on Nimbus 3 and 4. Explained by Prabhakara et al. [315],the method entails a statistical fit to a gross climatological average profile of ozone concentration. Though the information content of the radiance measurements is rather seriously limited, comparisons of derived profiles with balloon-soundings show good agreement (see fig. 5.17). The total vertically-integrated ozone amount can be determined with a spatial resolution of 145 km and an accuracy (relative to standard ozone measurements) of ±6%. Nimbus 4 carries an ultraviolet spectrometer which measures the spectrum of ultraviolet radiation reflected by the upper atmosphere to the satellite. The ozone distribution may also be determined from these measurements. A brief explanation of the theory is given by Houghton and Taylor [179]; more details and a summary of two years of data are given by Heath et al. [1541.

R.R. Rogers and Gabor Vaii, Recent developments in meteorological physics

166

106 I

I

200

-

(1)

300

-

(2) 400

(3

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500 —

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700-

I

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Fig. 5.16. Water vapor weighting functions for the six water vapor channels of the infrared spectrometer on Nimbus 4

4C

I

[365].

I

DERIVED PROFILE = 317cm

TOTAL OZONE 30’

I-.

20

I “I I

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BALLOON OZONE SOUNDING TOTAL OZONE

0

0

.005

I

=

.308 cm

I

.01 015 .02 OZONE CONCENTRATION (cm SIP/Kin)

.025

Fig. 5.17. Comparison of balloon-borne ozonesonde measurements with radiance.derived profile over Point Mugu, California [315].

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

167

Impact on weather prediction The ultimate goal of the meteorological satellite program, as indeed for most research in meteorology, is to improve the accuracy of weather forecasts. In the modern context, weather prediction is made by numerically integrating the equations of atmospheric motion and thermodynamics. The main contribution of satellites to these numerical models is the specification of initial conditions. The expectation has been that radiance-derived temperature soundings, made available over regions such as the oceans, where conventional observations are scarce, would improve the accuracy of numerical forecasts. While there is no doubt that satellites have contributed to a better comprehension of the organization of atmospheric circulations and weather systems, it is not at all clear that the new data have affected the accuracy of forecasts. Possible reasons for this lack of influence have been cited by Tracton and McPherson [385]: 1) Although the remote-sounding data are generally of reasonable quality, they are not as good as conventional data; moreover, the errors in satellite data tend systematically to reduce the amplitudes of weather systems. 2) The remote-sounding data do not actually fill a void over the ocean, but compete with accurate but relatively sparse conventional data. 3) The operational numerical prediction models are not designed to take full advantage of the satellite-derived temperature profiles. 4) The forecast accuracy seems usually to be limited more by the prediction model than the quality of data. The challenge thus appears to be to improve both the numerical models and the quality of the radiance data in order to exploit the potential of satellite observations. Appendix 1. List of principal symbols A Albedo

B Planck blackbody function C~ Parameter describing refractive index fluctuations C~ Parameter describing velocity fluctuations

C~. Parameter describing temperature fluctuations c Velocity of wave propagation (cos 0) Asymmetry parameter D Turbidity factor; diameter ds Incremental path length dfl Incremental solid angle E Irradiance g Gravitational acceleration

1(0) Intensity of radiation in direction 0 k Mass absorption coefficient L(0) Radiance in direction 0 I Path length m Complex refractive index N Particle number density (concentration) N(D) Drop-size distribution of rain

R.R. Rogers and Gabor Va!i, Recent developments in meteorological physics

168

n Refractive index; real part of m

fn,la

Refractive index of air

Imaginary part of m n(r) Particle-size distribution in terms of radius

P(0) Phase function for angular scattering P Phase matrix P~ Transmitted power Pr p

Received power Pressure

Absorption efficiency factor Qe Extinction efficiency factor Qa

Q~

Scattering efficiency factor

R Radar range r Radius S(v) Doppler velocity spectrum S1, S2 Mie-scattering complex amplitudes T Temperature t Transmittance tR

Rayleigh transmittance

V Volume; visual range v Parameter in Junge aerosol distribution y Dimensionless pressure coordinate

Z Radar reflectivity factor a Mie size parameter Volume j3, Volume 3R Volume 1 I3~(0) Volume 13R(O) Volume f3e

total extinction coefficient (particles) total total scattering (Rayleigh) coefficient scattering coefficient angular scattering coefficient

angular (Rayleigh) scattering coefficient

f3,. Volume scattering coefficient for backscattering

y Volume extinction coefficient (particles and gases) Volume absorption coefficient (gases)

‘Ya

& Depolarization factor Emissivity i~

Radar reflectivity

0 Scattering angle A Wavelength

p Density; ratio of aerosol absorptance to reflectance Pv Density of water vapor ~ Summation over unit volume ~ Radar (backscatter) cross section o~, Total scattering cross section of a particle 0R Rayleigh total scattering cross section o~(O) Angular scattering cross section of a particle 0R(O) Rayleigh angular scattering cross section

R.R. Rogers and Gabor Vali, Recent developments in meteorological physics

169

Optical thickness Mie optical thickness TR Rayleigh optical thickness TT Total optical thickness r0 Normal optical thickness of Rayleigh atmosphere w0 Albedo for single scattering T

TM

References [1] CE. Abbott, J. Geophys. Res. 79 (1974) 3098. [2] CE. Abbott, J. Appi. Meteor. 14 (1975) 87. [3] C.E. Abbott, J. Geophys. Res. 80 (1975) 1699. [4] JR. Allen, J. AppI. Meteor. 10 (1971) 260. [5] U. Allison, A. Asking, W.R. Bandeen, W.E. Shenk and R. Wexler, Rev. Geophys. Space Phys. 13 (1975) 737. [6] DJ. Alofs and J. Podziniek, J. Appi. Meteor. 13 (1974) 511. [7]A. Angstrom, AppI. Opt. 13 (1974) 474. [8] Atmospheric Radiation Working Group (ARWG), Bull. Amer. Meteor. Soc. 53(1972) 950. [9] A.H. Auer and J.D. Marwitz, J. Atm. Sci. 26 (1969) 1342. [10] W. Bach, Rev. Geophys. Space Phys. 14 (1976) 429. [ill W. Baerg and W.H. Schwartz, J. Acoust. Soc. Amer. 39 (1965) 1125. [12] W.R. Barchet and ML. Corrin, J. Phys. Chem. 76 (1972) 2280. [13] J.J. Barnett, R.S. Harwood, J.T. Houghton, C.G. Morgan, C.D. Rodgers and EJ. Williamson, Quart. J. Roy. Meteor. Soc. 101 (1975) 423. [14] J.T. Bartlett and P.R. Jonas, Quart. J. Roy. Meteor. Soc. 98 (1972) 150. [15] L.J. Battan, Radar observation of the atmosphere (University of Chicago Press, 1973). [16] B.R. Bean and EJ. Dutton, Radio meteorology (National Bureau of Standards, Monograph 92, 1966). [17] B.R. Bean, A.S. Frisch, L.G. McAllister and JR. Pollard, Bound. Layer Meteor. 4 (1973) 449. [18] KY. Beard, J. Atmos. Sci. 33(1976) 851. [19] Ky. Beard, J. Atmos. Sci. 34 (1977)1293. [20] K.V. Beard and HR. Pruppacher, J. Atmos. Sci. 28 (1971)1455. [21] W.A. Bentley and Wi. Humphreys, Snow crystals (McGraw Hill Book Co., Inc., 1931 and Dover Pubi. Inc., 1962). [22] T. Bergeron, Proc. 5th Assembly U.G.G.I., Lisbon (1935) p. 156. [23] R.W. Bergstrom, Beitr. z. Phys. Atmos. 46 (1973) 223. [24] E.X. Berry and RU. Reinhardt, J. Atmos. Sci. 31(1974)1814,1825, 2118 and 2127. [25] Ki. Bignell, Quart. J. Roy. Meteor. Soc. 96 (1970) 390. [26] W.G. Blättner, H.G. Horak, D.G. Collins and M.B. Wells, AppI. Opt. 13(1974) 534. [27] M. Born and E. Wolf, Principles of optics (Pergamon Press, 1964). [28] E.A. Boucher, Nucleation in the atmosphere, in: Nucleation, ed. AC. Zettlemojer (Marcel Dekker, Inc., 1969). [29] R.R. Braham, J. Atmos. Sd. 21 (1964) 640. [301R.R. Braham, J. Atmos. Sic. 24(1967)311. [31] R.R. Braham, J. Atmos. Sci. 33(1976)343. [32] R.R. Braham and P. Spyers-Duran, J. Appl. Meteor. 6 (1976)1053. [33] R.R. Braham and P. Squires, Bull, Amer. Meteor. Soc. 55(1974)543. [34] C.L. Briant and JJ. Burton, I. Chem. Phys. 60 (1974) 2849. [35] J.R. Brock, in: Aerosols and atmospheric chemistry, ed. G.M. Hidy (Academic Press, 1972). [36] K.A. Browning and F.H. Ludlam, Quart. J. Roy. Meteor. Soc. 88 (1962)117. [37] M.I. Budyko, Tellus 21(1969)611. [38] K. Bullrich, in: Advances in geophysics, Vol. 10, eds. H.E. Landsberg and J. Van Mieghem (Academic Press, 1964). [391MV. Buykov and AM. Pirnach, Izv. Akad. Nauk SSSR Fiz. Atm. Okeana 9 (1973) 486 (Original); Izv. Atm. Oceanic Phys. 9 (1973) 270 (English translation). [40]H.R. Byers, Elements of cloud physics (University of Chicago Press, 1965). [41] R.D. Cadle and G.W. Grams, Rev. Geophys. Space Phys. 13 (1975) 475. [42]T.W. Cannon, Rev. Sci. Instr. 45 (1974) 1448. [43]T.W. Cannon, J.E. Dye and V. Toutenhoofd, J. Atmos. Sci. 31(1974) 2148. [44]J.C. Carstens and J.L. Kassner, J. Rech. Atmos. 3(1968)33.

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