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Gaussian Thermal Flux model—I theory
Solar Energy, VoL 32, No. 4, pp. 505 514, 1984 Printed in Great Britain.
0038~)92X/84 $3.00+.00 ,/~ 1984 Pergamon Press Ltd.
GAUSSIAN THERMAL FLUX MODEL--I THEORY R. F. KAMADAand R. G. FLOCCHINI Department of Land, Air and Water Resources, University of California, Davis, CA 95616 U.S.A.
(Received 21 September 1982; accepted 4 April 1983) Abstract--The Gaussian Thermal Flux model is a fast thermal radiation sub-model which operates with minimal input and predicts flux on sloped surfaces for both clear and cloudy days. This paper presents the model's theoretical basis. It is distinguished from empirical models by its detailed two dimensional treatment of clouds, its computation of air layer transmissivity above and below cloud layers and its consideration of cloud height effects. The model improves upon existing flux emissivity and multi-band or line models by being independent of soundings and by its use of gaussian quadrature to speed the numerical integrations. It may be input entirely from standard screen level meteorological measurements.
INTRODUCTION
Thermal radiation is an important term in energy balance calculations for solar homes and other solar collection devices. The instantaneous thermal flux, F~$ on the collector surface of a solar energy device may exceed 500 wm -2 or more than half the peak solar flux, even on a clear day. Daily integrated thermal flux totals often exceed solar. Moreover, the reflective, absorptive, and emissive properties of typical collector surfaces differ between solar and thermal infrared wavelength regimes. Therefore, in order to specify the energy balance equations for a dynamic computer simulation of a solar building or device, a fast, thermal radiation sub-model is needed which can operate with minimal input, handle both clear and cloudy weather, and predict flux on sloped surfaces. The Gaussian Thermal Flux (GTF) model described here attempts to fulfill these objectives. Atmospheric thermal flux models fall into three major categories: simple empirical expressions, gray body flux emissivity models of which the G T F model is a recent example, and multi-band, or line-by-line models. Each has certain strengths and weaknesses in calculating downward flux received at the earth's surface. Empirical models are simple enough for hand calculation and accurate within perhaps 10~ for clear sky conditions. Currently, however, they do not treat clouds, or latitude and elevation extremes nearly as accurately, since they were developed from data acquired at specific sites. Existing flux emissivity models require computer processing and sounding data. Line-by-line and multi-band models are too tedious and bulky for an operational solar simulation, require sounding data, and at best are scarcely more accurate than simpler models. To circumvent these difficulties the atmospheric portion of the G T F model was developed by extending Sasamori's[1] flux emissivity model to include clouds and operate without soundings. The atmospheric model was then used as a basis for incorporating the angularly anisotropic sky radiance and the ground contribution in a general ambient thermal flux model for sloped surfaces. Table 1 lists a number of empirical models which SE Vol. 32, No. 4--E
505
correlate screen level (1 to 2 m above the surface) temperature, T,, and/or vapor pressure e,, and the effective sky emissivity, Esky.Eskyis used to obtain F,$, via the Stefan-Boltzman expression, F ~ = Esky o T ~ 4. The formulae lead from Angstrom's [2] early effort up to recent attempts by Idso[8] and Berdahl[9]. The surface dewpoint temperature, Taps, and e~ are related by the Clausius-Clapeyron equation:
e=eoexpFL,,(1 • LR, kTo
1 )]
T-dp~
(10)
where L,, = latent heat of vaporization, R,. = water vapor gas constant, and To = triple point of water. Among these expressions the coefficients for (1) and (2) are site specific while (3) poses a T 6 dependence difficult to justify theoretically. The empirical coefficients k and l of (8) are distinct for day or night conditions. Equation (7) is based on night data only and claims some theoretical justification by posing a term of the same form as is used to describe chemical reaction rates. In this way it incorporates the 1.5 kcal mole ~activation energy for the splitting of water vapor dimer. Idso claims that dimer formation accounts for absorption in the infrared continuum portion of the water vapor absorption spectrum. The other coefficients for (7) and (8) are best fits to data from several sites each, such that their agreement with data is often within five or ten percent. The very recent complex expression by Centeno[10], where N = integrated sky transparency or nebulosity, y = relative humidity, attempts to account for clouds and surface elevation, z, in addition to temperature and water vapor content. Its accuracy under cloudy conditions is difficult to judge, since overall correlation or error figures were not supplied with its presentation. However, cloud contributions to flux are a strong function of percent cloud cover and cloud height, factors not directly represented in (9). Thus its ability to accurately account for clouds is unclear. In developing physical models for flux under cloudy skies one must account for cloud emission. F,.,
506
R. F. KAMADAand R. G. FLOCCHINI Table 1. Empirical models for effective sky emissivity Equation
Author
Ref.#
Date
Esky = a " ~*10-Yes Esky = a+b*es 2 Esky = 6T s
ER-#
Angstrom
[2]
1915
(i)
Brunt
[3]
1932
(2)
Swinbank
[4]
1963
(3)
Esky = l-c*exp[d(273-Ts)2]
Idso
[5]
1969
(4)
Esky = f'el/7 s
Brutsaert [6]
1975
(5)
Esky = g+h*in((Tdps+273)/273)
Clark
[7]
1978
(6)
Esky = i+3*es*eXp(1500/Tdp s)
Idso
[8]
1940
(7)
Esky = k+l*Tdp s
Berdahl
[9]
1981
(8)
Centeno
[I0] 1981
(9)
Esky = (I-N)*[57723+,9555(.6017)z] *I0-4*T s 1.1893 ~^ ¥0.0665 + N[l-(3000+1751*z0"652)* ¥-3/2 . Ts-l]4
as well as the transmissivity of the air layer below cloud level, z:. Although F c may be estimated from ground based observation, z: is less tractable. Attempts to empirically describe T: as per Exell[ll] or Geiger[12] have been site specific and applicable only to low clouds, z/may be calculated, given sufficient sounding data and some treatment of the radiative transfer equation. However, a full scale treatment is tedious and generally avoided. Moreover, the global grid for twice a day soundings is quite sparse with few readings in the planetary boundary layer where F,J, is most affected. The GTF model was developed to respond to these difficulties. THEORY
The GTF model is a "flux-emissivity-gray body" model which implies that the atmosphere is assumed to absorb and emit equally well at all wavelengths as a spectrally gray body and that for a plane parallel atmosphere the flux emissivity for optical depth or path length, u, is approximated by the beam emissivity for optical depth 1.66 u. Not only does this avoid spectral integration, but also the zenith and azimuth integrations performed in the multi-band and line-by-line beam models exemplified by Kneizys et aL[13], Rogers and Walshaw[14], or Goody[15]. The Curtis-Godson approximation allows the path length integrals to be normalized to standard temperature and pressure in calculating emissivity increments and, since both Rayleigh and Mie scattering are negligible in the thermal infrared, the scattering phase functions normally imbedded in the radiative transfer equation may be ignored. The result is that of the minimum five nested numerical integrations needed for full scale treatment, the only ones remain-
ing are for the absrrber path length and layer emissivity. Examples of this approach are given by Atwater and Ball[16], Bliss[17], Sasamori[1], Staley and Jurica[18] or Stone and Manabe[19]. The Staley-Jurica model is quite popular. However, it assumes that the emission losses due to the carbon dioxide-water and ozone-water vapor band overlaps cancel the ozone emission, allowing both to be ignored. Sasamori's model instead calculates the ozone flux, then also subtracts band overlaps from the unalloyed water vapor, carbon dioxide and ozone absorptions using empirical curve fit expressions. These expressions are more convenient and presumably more accurate than the tables of Staley and Jurica[20] or Elsasser and Culbertson [21]. Sasamori has shown that the latter contains an error causing an underestimate in emissivity. Sasamori's model appeared to have the best combination of facility and accuracy of those apparent in the literature and thus was selected as a basis for the atmospheric model. The remaining assumptions used in the GTF model are included in Table 2.
1. Emissivity and flux calculations A normalized defined as
emissivity-absorptivity can
dP~ E(u, T) _
IE(e, T) -
4aT3
(1 - *:)-d-f
l °~dPv dv Jo dT
be
dv (11)
where Pv is the Planck source function over frequency, and E is the absorptivity-emissivity, such
507
Gaussian thermal flux model--I. Theory Table 2. Basic assumptions for the GTF model I.
Gray body absorption-emission
2.
Flux
3.
Plane
emissivity parallel
of
path
in thermal infrared
length u = beam
atmosphere
except
for
emissivity
total
for 1.66 u
thermal
emission
calculation 4. .Curtis-Godson approximation 5.
Water vapor path length a function of surface dewpoint temperature
6.
No scattering
7.
Absorptivity of tropospheric absorbers independent of temperature
8.
Adjusted standard T, p, and absorber profiles for clear skies and >1 I(m above clouds
9.
Bulk emissivities E
for clouds, no microphysics C
I0.
No multiple cloud layers considered
ii.
2-D atmosphere, no cloud sides or advection
12.
Not for fronts, cumulus towers, cumulus congestus
13.
Negligible dust effects
14.
No sounding data
that the downward flux is written f. T~
Fs~ = 4 ° | E0{u(r') - u(L), ~r'}r'3d~r '. (12) dO
This integral may be parsed into tropospheric and supertropospheric portions in order to account for differences in the behavior of the emissivity functions in these two regions such that f. r~ Fs~ =4tr | ff~{u(T')--u(Ts),
r'}r'3dr'
J Tr
t"TT +4~ [ g { u ( r T ) - u ( L ) , r ' } r ' ~ d r ' (13) dO
where Tr is the tropopause temperature. Since the absorption lines and the Planck function's spectral intensity distribution are temperature dependent, one would expect E to vary with temperature. However, for troposphere temperature ranges, emissivity variations in the rotational bands nearly cancel those in the 6.3/~ water vapoi" band, resulting in a total water vapor absorptivity quite insensitive to temperature. Since the combined effect of the other absorbers is small compared to water vapor, they too are also assumed to be temperature independent. Sasamori's curve fit expressions are presented in Appendix 1. According to (12) the emissivity-absorptivities must be vertically integrated throughout the atmo-
sphere. The "adding method" (22) is the standard approach for evaluating the emissivity integral. Since it simply adds the emissivity increment at the top of each layer to the preceding layers, it amounts to a less than linear, zeroth order Riemann sum approximation to the integral and requires many layers for accuracy. In contrast gaussian quadrature in effect integrates an orthonormal polynomial curve of order 2 m - 1 fitted to the m points used for evaluation. Since the profiles extrapolated from surface data are modifications of smooth, standard profiles, m can be a small number, say 2-5. Use of gaussian quadrature in evaluating the absorber path length and emissivity integrals greatly enhances computational speed and accuracy (see Appendix 2). For simplicity stratospheric flux is assumed to all be emitted from an imaginary plane at the tropopause. It's effective emissivity is obtained by summing contributions from the total absorber path lengths up to 45 km for CO: and 03 and up to 10 km for H20. The seasonally and latitudinally adjusted temperature at 12 km is also used. The above is sufficient to compute Fs~ for clear skies. However, for cloudy skies adjusted standard profiles cannot account for water vapor enrichment below the cloud base. One must parse tropospheric air into layers above and below the cloud levels. Then Fs~ consists of flux from a clear pseudo-sky, calculated for an atmosphere lacking the cloud layer entirely, plus rj times the cloud's emission, minus the
508
R.F. KAMADAand R. G. FLOCCHINI ENVIRONMENTAL REGION CLOUDREGION 45 Km
f
":g
STRATOSPHERE
4 pt. GAUSS
l-
UPPER TROPOSPHERE
I0 Km
4pt. GAUSS
4 pt. GAUSS I Km SUPRA-
~ I Krn CLOUD LAYER
300m TRANSITION ZONE 4pt. GAUSS
l
BOUNDARY LAYER
2pt; GAUSS
A general expression for the effective absorber path length is
- - - 30m
SURFACELAYER
Fig. 1. Layers and regions of atmosphere for general flux calculation.
cloud's absorption of flux which would otherwise" have reached the surface. As per Fig. 1 care must b e taken to distinguish pseudo-sky from a clear, real sky, since their water vapor profiles may differ substantially. Thus, for fractional cloud cover, CC, the gaussian quadrature form of (13) becomes
1 ;f2 fp~,,fTo~,,,2
, qt
)tT) d.
(l,)
where g = mean sea level gravitational acceleration, p = pressure, P0 = 1 atmosphere, and q = specific humidity. Compromises between strong and weak line pressure and temperature corrections are given as n = 0.7 for H20, 0.86 for CO2, and 0.3 for 03[23 ]. The weighting is a form of the Curtis-Godson approximation which accounts for emission line broadening with temperature and pressure. It normalizes path lengths for non-isothermal atmospheres to an effective path length (based on a reference temperature and pressure) which matches both the square root variation of emission with path length in the strong line limit and the linear variation in the weak line limit. For CO2 the evaluation is trivial, since q can be taken to be 0.00034. Assuming negligible temperature correction, we obtain
Uco2
q - 186 1.86). 1.86gp 0.86~° f --P
(16)
Ozone and water vapor profiles, however, vary greatly with altitude, latitude, season and local conditions. To account for this, latitude-season tables by Robinson[24] and Smith[25] for ozone and water vapor are normalized to the standard vertical profiles for absorber densities provided in the Air Force Geophysics Lab Lowtran IR flux model[13]. Hydrostatic equilibrium allows us to write the ozone path length in gaussian quadrature form as
m F~.L= a(1 -
CC)2(z~ - z¢) ~ WkE,.~Tk- .3 dT'd~
u03
k=l
+ 2(z~ -
,,
_
3dT '
zr) Z WkE,,~T" ~ k=l
-
~o3,~J"
'3dT'
Tt -.Fz~
•
k=l
+
E'c,drr 4
03]
2pO.3b=, kpo3,p ~ J u03, ROBINSON]
+ E¢,~Tr4
+ aCC2(z~ - z~)k~=, WkEcMTk - '3dT' -~Z + 2(Z<-- ZT) ~
-':"Z 2 - ZIF ~ W
(14)
Multiple cloud layers are not considered, since the quality of its ground based estimates is dubious. Moreover, to obtain net CC a statistical formula for the probability of overlap must be invoked, rendering any short term flux calculations meaningless.
2. Clear sky absorber path lengths Before determining emissivities and fluxes, one must determine the absorber path lengths themselves.
(17)
For water vapor density, deviations from the mean tend to decrease with height. Thus, when lacking sounding data, it seems reasonable to assume a normalized standard profile some distance above the cloud layer along with standard temperatures and pressures extrapolated from surface measurements. A simple way to account for local distinctions and diurnal effects in the temperature lapse rate is to assume that the defect between local temperature and the mean at that latitude, altitude, and time of year decreases exponentially with height. A formulation such as T = L + (T~.- 7~) exp(0.5z). where z is in km seems to track diurnal effects reasonably well. For further refinement one would need to include a boundary layer model to estimate
509
Gaussian thermal flux model--I. Theory the diurnal changes in the height of the mixed layer and strength of the night time inversion as discussed in more detail in part II. The above formulation is used for clear sky conditions and for heights greater than 1 km above a cloud layer. Thus above clouds or for clear skies one has in gaussian form .2" 2 -- ZI[ ~,,
u.,o=
k'n
,L k=
I
uH2o, SMITH UH~O,L O ~ N J
] (18)
Below clouds, however, standard profiles underestimate H20 path lengths. Thus we wish to express the path length integral in a form such that profiles for humidity, pressure, and temperature may be varied according to given conditions. To do so note that where E = 0.622
dp = - pg dz p
=
P
(19) (20)
Specific humidity does tend to decrease within the surface layer. However, buoyant plumes from an extended super adiabatic surface layer tend to form cumulus clouds. Such a convective cell situation would tend to dissipate stratus before any cumulus would form. Thus the presence of stratus suggests that the surface layer contribution is minor. Moreover, once formed, a substantial cloud layer tends to inhibit surface layer growth by limiting surface heating. For these reasons we assume for stratus clouds that the surface layer lies below the screen level (1 to 2 m) at which the standard surface meteorological measurements are taken. Then, whether the air between screen level z s and zc is well-mixed or consists of lifted air from a well-mixed region, the temperature lapse rate tends to be moist adiabatic and can be obtained from
(21)
RjT~
T=
give the expressions for the specific humidity, hydrostatic equilibrium and the gas law, where Rd = dry air gas constant, and T~. = T(1 + (1 - Qe/p), the virtual temperature. Therefore, -q dp = - Ee dz g RjT~,
(22)
and
UH2O- R~n0.7 /
- - - - - dz ae0 .)z, I + ( I _ E ) e P
g
dz
RAT,,"
[g r d& \ R d J , , T,/
In practice R/Cp is nearly equal t o Rd/CM, the dry air ratio. From (26) alone the temperature Tc at zc can be inferred from a pressure profile giving Pc- Thus for stratus clouds (23) becomes
T-3/2 dz
(27)
Therefore, the general expression for pressure is p = p, e x p ) - - i - - ) .
(26)
(23)
gives a general expression for the water vapor path length. From (20) and (21) dlnp
Ts(P) R/cp.
_EeoT~/2 ~2 0.7 expt--/---LRv \ To
ETIo,'2 (-'2 epO.7T - 3/2
'
(25)
J
3. Cloudy sky absorber path lengths
Ee q =-P
Stratus clouds are perhaps simplest to deal with, since they tend to form in the mixed layer or by adiabatic ascent from the mixed layer. In either case the specific humidity should remain approximately constant up to the cloud base height, z~. That is,
e = e,p/p,.
-07 fZ°~l/2]
t, J
4. Effect of a stratiform cloud layer
(24)
Given sounding data (23) may be evaluated. However, if unavailable, it may be assumed that among the vertical profiles for meteorological quantities, pressure varies least in form from an adjusted standard. Thus, it is used as the basis from which the other variables in (23) are estimated. To do so the effect of different cloud types on temperature profiles as well as absorber path lengths must also be treated.
(27) cannot be used for middle or upper level stratiform clouds, since they may form by lifting starting from above the mixed layer. In such cases T and Tc may be inferred from adjusted standard profiles. Assuming that the relative humidity 7 increases linearly with height, the vapor pressure may be given by
e = 7esat = 7 e 0 e x p / - - - q
--
LR~\ To
(28)
--
where e0 = 611 paScals and e~at = saturation vapor pressure. Such that for higher stratiform clouds
_ T~/2 f z2 7e~tp°7 T - 3/2 tl+tp)
Te•t)
(29) is perhaps less accurate than (27). However, the
R. F. KAMADAand R. G. FLOCCHINI
510
importance of T, UH20, and F< decreases with height, since the transmittance of the intervening air, rf, also decreases with height.
5. Effect o)c a cumuliform cloud layer Cumulus presence complicates flux models because a large number of assumptions concerning profiles must be made to render such models tractable. Not only does the real cumulus conl~aining atmosphere tend to consist of several distinct layers in the vertical, but the horizontal region of environmental air surrounding the cloud region also has different properties and vertical segmentation. Referring to Fig. 1 again in detail, a superadiabatic surface layer with constant 30 m depth is assumed. In the cloud region an adiabatic layer extends to z: and in the environmental region to z ~ - 300 meters. Above this is a 300 m thick transition zone in the environmental region, followed by the cloud layer, a kilometer thick supra-cloud layer and the remainder of the atmosphere. Sans sounding data T and q are assumed to decrease by 2°C and 20 per cent. In the adiabatic layer specific humidity is held constant as with stratus cloud cover. Zc is inferred from the adiabatic lifting condensation level, LCL, of a parcel released from the top of the cloud plume generating surface layer. For simplicity this convective condensation level, CCL, is taken to be the LCL of a parcel from screen level plus 300 m [26]. It is assumed at the LCL that
T = Tdp or (30)
T, + LCL d z = Tdp~+ LCL
Using the differential form of the Clausius-Clapeyron relation and the Poisson equation we can obtain for the LCL
1--3 g K g - L A change of - 2 g Kg -I for q in this zone is assumed. Since a fixed decrease in T in the constant depth surface layer is assumed with a moist adiabatic lapse rate thereafter, one easily obtains a temperature T at the cloud base height of about T: = T, - 2°C -- 9.7 z< °C/km from which a vapor pressure at that height
VLof 1
ec = eeo exp . . . .
L, t o
LCL
dz
EL~T
t
T,
/I
If the Aq in the superadiabatic and transition layers are assumed to be roughly equal, then the near constant q in the adiabatic layer of environmental air implies
ee,,,
p fece + e~'~.
2\p<
=
--9.76 1 + 5.42 x 103e~,t/Tp
dT
i£ ~
-
x 106e~t/T2p °C/km
so that the temperature of both the environmental and cloud air is reckoned to be
and x
Note that, if z~ is accurately measured by theodolite or lidar as at major airports, (30) may be solved for Tap = T~ from which a revised water vapor path length may be computed in both cumuliform and stratiform cases. From the edge of the cloud region the adiabatic layer gradually drops two to three hundred meters as one moves into the environmental air region between the clouds. Thus, in the environmental region the adiabatic layer is assumed to extend to about the LCL. In the transition zone the potential temperatures 0 and 0~ increase by I°C and q rapidly decreases, by
(35)
p#
Since one assumes a bulk flux emissivity E~ for the cloud layer, q or T within the cumulus layer is of no concern. However, horizontal temperature differences between cloud and non-cloud air are typically less than 1 degree C. Thus one notes that the temperature at the top of the cumulus layer is usually about 2AT/3 higher than that corresponding to dry adiabatic ascent from z<, where AT is the difference in the temperatures produced at the top of the layer by dry and saturated adiabatic ascents. The saturated adiabatic lapse rate is given' approximately by
T = T:--9.76(1--3~1
(32)
(34)
e = edg/pc.
(31)
z< = CCL = LCL + 300 m.
1 k-1
may be calculated. Below it with q constant
dz Tdps-- L (Tdp~-- T,)(1 + 0.87~) d T ( l - cpTf)-//1 6.46 × --/10-4T~p~'~(9.76)
(33)
(z
-
1 + 5.42 x 103e~at/Tp ~ 1 + 8.39 x 106esat/T2pj]
zD.
(36)
In this case e~t is taken to be a constant 0.7e<. With typical dewpoint depressions of 5°C the vapor pressure in the environmental air at cumulus layer heights is assumed to be
,)]
r.1 L,~,\To (r- 5)
ec,v = e0 exp/---q - -
.
(37)
Disregarding extended cumulus towers common in the tropics, representative 1 and 2 km depths were assigned to cumuliform and cumulonimbus clouds in the mid-latitudes. The emissivity of all nimbus clouds is assumed to be unity rendering flux calculations above z< superfluous in those cases.
511
Gaussian thermal flux model--I. Theory Table 3. Vapor pressure and temperature expressions for air layers in the cumuliform region Layer
Height
Upper air
Vapor Pressure
Temperature
corrected standard profile
corrected standard profile ~=30CKm
"
-I
>Cloud
CCL+2,3 Km
Cumulus
CCL+I,2 Km
inapplicable
~sat - ~d (z-zc) TC + - 3
Adiabatic
CCL
ecp/pc
T
Surface
30 m e t e r s
L e°exPlR-~ (~o _ ~)1 1 v _
The peak height of the average cloud plume naturally defines the top of the cumulus layer. Thus buoyancy must be negative above this height. In order to counteract plume momentum, the temperature lapse rate above the cloud must be considerably less than saturated adiabatic. Indeed, an inversion often exists. Thus a conservative lapse rate of 3°C is assigned to the first kilometer above the cloud tops. Above this point vapor pressures and temperatures are assigned adjusted standard profiles. The assignments for e and T may be summarized by Tables 3 and 4. Where fl, flsat, and fld are the general, saturated, and dry adiabatic lapse rates. These values may then be substituted into gaussian quadrature approximations to (23) to yield water vapor path lengths and temperature profiles above and below cumulus cloud levels.
-2°C - 9.7"z
s
°C Km-1
T s - z/.015oC Km -I
a ground based observer. To account for this situation lifting fog is assumed when, together with total overcast, the screen level dry bulb temperature is less than 0.3°C above the dewpoint. Provided the solar disk is visible, this fog is then assumed to contribute 8 per cent to an Fs~ based on clear sky conditions. Thick fogs are assumed to be near black and the total flux given by Fs~ = 0.95trTs 4.
7. High clouds A standard height of 5 k m and E~=0.35 are assigned to cirrus and cirrostratus. Otherwise, standard profiles are assumed since the effect of high -clouds on surface flux is quite small. THERMAL FLUX INCIDENT ON INCLINED SURFACES
Thus far only downward atmospheric thermal flux incident upon a horizontal surface has been considered. For solar energy applications the more general case of ambient thermal flux incident upon sloped surfaces must also be treated. This involves the angularly anisotropic distribution of sky flux and the ground flux contribution. Cole[29] has done empir-
6. Fogs Preliminary evaluation suggested that, if thin radiation or advection fogs lift (presumably as surface heat evaporates near surface condensate), they can easily be mistaken for much thicker low stratus by
Table 4. Vapor pressure and temperature expressions for air layers in the environmental region Layer
Height
Upper air
Vapor Pressure
Temperature
corrected standard profile
corrected standard profile
>Cloud
CCL+2,3 Kin
ADewpoint = 5°C
~ = 3oc Km -I
Cumulus
CCL+I,2 Km
L eoeXe[ ~
rc +
Adiabatic
CCL
~(ece2.Pe+ ~s )
Ts -2°C - 9"7"z°C Km-I
Surface
30 m e t e r s
L 1 - ~)] 1 eoeXp [~-~ (~-v o
T
(I
v ~oo
_ T~5) ]
~sat - ~d
3
(Z-Zc)
es
s
- z/.015°C Km -I
512
R. F. KAMADAand R. G. FLOCCHINI
ical work in this area by expanding the results of Unsworth and Monteith[30]. Here, instead, a fast numerical procedure is derived to render Kondratyev's[31] exact analytic approach amenable to an operational model. If the infrared albedo of the ground is negligible, calculating the ground flux is trivial. Since most ground surfaces have an emissivity near unity, including the remaining reflected sky flux amounts to nearly black body emission such that one can say
Egr
(38)
where ct is the slope angle for the surface. Kondratyev supplies the equation
h dh) (39)
to account for total sky emission where h = angular elevation of the beam relative to the plane and ff is the azimuth angle. At first glance the beam intensity Jh and the double integral appear tedious to evaluate. However, once again some simple approximations imbedded in a gaussian quadrature formulation render (39) quite tractable. That is, using the cosine of the zenith angle, optical path lengths for flux emissivity are multiplied by air masses computed at the interpolating polynomial's nodes, then as previously suggested, converted to give effective beam emissivity path lengths by dividing by 1.66. The beam intensities can then be evaluated from the resulting emissivities using the same lapse rates and temperature profiles previously calculated, dividing by n to render flux to intensity, such that (39) becomes (see Appendix 2)
F~,op~-E,.~
Fambien t = eslop e -I- Fgr
(41)
LIMITATIONS
One drawback to general applicability is the occurrence of cumulus congestus clouds which may spread over large areas when the upward momentum of the cloud plumes is steered to the horizontal by high winds aloft and a capping inversion, such that (30)-(38) lose accuracy, if is a major fraction of the total sky area. Equation (26) also does not apply well for the case of an approaching warm front presaged by apparently descending stratiform layer clouds, since a sizeable portion of local air below the cloud layer belongs to a different, colder air mass. The cloud layer itself would then sit above an inversion with respect to the observer's vertical line of sight. In general after the formation period vertical drift of the cloud layer or sub-cloud layer inversions may occur. This is less of a problem for cumuliform clouds, since they tend to dissipate quickly without continued active formation at heights strongly correlated with the CCL. As the surface warms and cools during the daytime heating cycle, the CCL of course rises and falls along with zc until the late afternoon when the boundary layer becomes stable and the remaining cumulus tufts tend to dissipate. Along with multiple cloud layers other such particular conditions may be ascertained by an experienced ground based observer. However, for modeling purposes they are not considered, since they cannot be gleaned accurately from the set of standard meteorological surface observations.
CC
Fsl°P~=F~"c°sa(2 ,Io t'~/2~ x [sin ct cos ff cos h + cos ct sin h] cos
because, unlike the gauss procedure which evaluates near the endpoints of the curve, the regions near the angular endpoints which change most rapidly are not accounted. Either way, the final expression for ambient thermal flux incident upon an inclined surface becomes:
cos (": \ 4 t Wk3E1,..:e,,
SUMMARY
k=,
X [sin Ct COSCk COShJ + COSa sin hJ cos hi) (40) where O~
and O, = 2 (3.0 + ~*)Alternatively, one may assume that the horizontal beam intensity is simply that of a black body. Having in effect already computed the vertical beam intensity, one or two additional equi-spaced intensities may be computed and Simpson's rule applied to obtain the total atmospheric flux. This is less accurate
To briefly recapitulate, depending upon conditions, the G T F model may parse the atmosphere into clear and cloudy regions each consisting of up to several layers. The effective path lengths of the major absorberemitters: water vapor, carbon dioxide, and ozone are obtained by evaluating gaussian quadrature approximations for eqns (16)-(18), and/or among (27), (29) and (33)-(37), depending upon cloud conditions. This is done for a number of layers above and below a single cloud layer. Curve fits for emissivities from absorber path lengths then give the effective emittance from each layer. The layer thicknesses are themselves determined by the node points of the gaussian quadrature approximations of the integrals for the transmissivities and downward fluxes of the air masses above and below the cloud. The clear sky flux is determined by the clear sky fraction of the total sky area, and standard vertical profiles of
513
Gaussian thermal flux model--I. Theory temperature, pressure, and absorber concentrations adjusted for latitude, season, local surface conditions, diurnal variations, and cloud formations. The cloudy sky flux is determined by the fractional cloud cover including the flux that would have reached the surface from an otherwise clear pseudo-sky, plus the cloud emission attenuated by the intervening air, minus the amount absorbed by the cloud which would otherwise have reached the surface. The water vapor and temperature profiles for clear pseudo~sky are those for air masses containing clouds rather than from adjusted standard profiles for clear skies. Dust effects are assumed negligible. The spectral overlaps between absorbers and the ozone contribution are both included in the flux calculation. The variation of the sky thermal beam intensity with zenith angle is computed and the ambient flux evaluated by gaussian quadrature approximation of Kondratyev's analytic formulation for sky thermal emission upon a sloped surface. This is then added to the ground emission. The atmospheric model is distinguished from Brunt type empirical models by its detailed 2-dimensional treatment of clouds, its methods for estimating the transmissivity of the air directly above and below the cloud layer, and its consideration of cloud heights. It improves upon existing flux emissivity and band or line-by-line models by being independent of soundings and by using gaussian quadrature to speed the otherwise tedious numerical integrations. The analytically based treatment of ambient thermal flux is free from the condition specific limitations of Cole's empirical approach, but still practicable for computation in conjunction with an atmospheric flux emissivity model. The G T F model may be input entirely from standard screen level meteorological measurements. NOMENCLATURE c~ heat capacity at constant pressure CC fraction of sky covered by clouds CCL convective condensation level (cumulus base level) e partial pressure of water vapor E emissivity-absorptivity F downward thermal flux g mean sea level gravitational acceleration LCL lifting condensation level (stratus base level) t~ latent heat of vaporization of water P pressure P Planck's function = 2rchv3/(c2(exp(hv/kT) - 1)) q specific humidity R gas constant Ra dry air gas constant = 287.05 J Kg-I°K -1 Rr water vapor gas constant = 461.51 J Kg-~°K T temperature 11 effective vertical path length for absorber W gaussian quadrature weighting function Z height in atmosphere above sea level slope angle of inclination relative to horizon temperature lapse rate A difference symbol ( Rd/ R ~,= 0.622 7 relative humidity azimuth angle /7 density
tr Stefan-Boltzmann constant °K z transmissivity 0 potential temperature
5.672"10 8JM 2_
Subscripts
amb ambient (includes sky and ground) c cloud base height in cloud region of sky ce cloud base height in environmental region of sky cld cloudy region of sky d dry air dp dewpoint env environmental (cloudless) region of sky f air layer between ground and cloud base height gr ground h beam angular elevation relative to the horizon p pressure s screen level (1-2 m) above surface sat saturated with water T tropopause v virtual, including water v frequency REFERENCES
1. T. Sasamori, The radiative cooling calculation for application to general circulation experiments. J. Appl. Meteor. 5, 721 (1968). 2. A. Angstrom, A Study of the radiation of the atmosphere. Smithsonian Miscellaneous Collections. 65, 3 (1915). 3. D. Brunt, Notes on radiation in the atmosphere. Quart. J. Roy. Met. Soc. 58, 389 (1932). 4. W. C. Swinbank, Long-wave radiation from clear skies. Quart. J. Roy. Met. Soc. 89, 339 (1963). 5. S. B. Idso and R. D. Jackson, Thermal radiation from the atmosphere. J. Geophy. Res. 74, 5397 (1969). 6. W. Brutsaert, On a derivable formula for long-wave radiation from clear skies. Water Resources Res. 11,742 (1975). 7. G. Clark and C. P. Allen, The estimation of atmospheric radiation for clear and cloudy skies. Proc. 2nd Nat. Passive Solar Conf. Vol. 2, 676 (Philadelphia, 1978). 8. S. B. Idso, A set of equations for full spectrum and 8-14/~ and 10.5-12.5/1 thermal radiation from cloudless skies. J. Water Resources Res. 17, 295 (1981). 9. P. Berdahl and R. Fromberg, An empirical model for clear sky thermal radiation. Solar Energy 29, 299 (1982). 10. M. Centeno V, New formulae for the equivalent night sky emissivity. Solar Energy 28, 489 (1982). 11. R. H. B. Exell, The atmospheric radiation climate of Thailand. Solar Energy 21, 73 (1978). 12. R. Geiger, The Climate Near Ground (Edited by M. N. Stewart et al.), 2nd Edn., Chap. 2. Trans. Harvard Press, Cambridge, Massachusetts (1959). 13. F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, Jr., L. W. Abren, J. E. A. Selby, R. W. Fenn and R. A. McClatchey, Atmospheric transmittance radiance: computer code LOWTRAN 5. A. E. Geophys. Lab. AFGL-TR-80-0067 (1980). 14. C. D. Rogers and C. D. Walshaw, The computation of infrared cooling rate in planetary atmospheres. Quart. J. Roy. Met. Soc. 92, 67 (1966). 15. R. M. Goody, Atmospheric Radiation, 1. Theoretical Basis. Clarendon Press, Oxford (1964). 16. M. A. Atwater and J. T. Ball, Computation of IR sky temperature and comparison with surface temperature. Solar Energy 21, 211 (1978). 17. R. W. Bliss, Atmospheric radiation near the surface of the ground: a summary for engineers. Solar Energy 5, 103 (1961). 18. D. O. Staley and G. M. Jurica, Effective atmospheric emissivity under clear skies. J. Appl. Meteor. IL 349 (1972).
514
R . F . KAMADA and R. G. FLOCCHINI
19. H. M. Stone and S. Manabe, Comparison among various numerical models designed for computing infrared cooling. Mon. Weather Rev. 96, 735 (1968). 20. D. O. Staley and G. M. Jurica, Flux emissivity tables for water vb,pour, carbon dioxide and ozone. J. Appl. Meteor. 9, 365 (1970). 21. W. M. Elsasser and M. F. Culbertson, Atmospheric radiation tables. Meteor. Monogr. 4, 23 (1960). 22. A. A. Lacis, J. E. Hansen, A parameterization for the absorption of solar radiation in the earth's atmosphere. J. Atmos. Sci. 31, 118 (1974). 23. G. W. Paltridge and C. M. Platt, Radiative Processes in Meteorology and Climatology. Elsevier, New York (1976). 24. N. Robinson, Solar Radiation. Elsevier, New York (1966). 25. W. L. Smith, Note on the relationship between total precipitable water and surface dew point. J. Appl. Meteor. 5, 726 (1966). 26. F. H. Ludlam and P. M. Saunders, Shower formation in large cumulus. Tellus 8, 424 (1956). 27. J. V. Iribarne and W. L. Godson, Atmospheric Thermodynamics. D. Reidel, Boston (1973). 28. F. H. Ludlam, Clouds and Storms. Penn. St. Univ. Press, (1980). 29. R. J. Cole, The longwave radiation incident upon inclined surfaces. Solar Energy 22, 459 (1979). 30. M. J. Unsworth and J. L. Monteith, Longwave radiation at the ground (1) Angular distribution o f incoming radiation. Quart. J. Roy. Met. Soc. 101, 1 (1975). 31. K. Y. Kondratyev, Radiation in the Atmosphere. Chap. 6, Academic Press, New York (1969). APPENDIX 1
Emissivity f u n c t i o n s
pospheric temperatures and pressures. To account for the 15 t~ band overlap for CO2-H20 it is assumed that the H20 transmissivity in that spectral r e # o n is given by ZH2o = 1.33 - 0.832(u.2 o + 0.0286) 0.26
such that the effective CO 2 emissivity-absorptivity is the product of the nominal absorptivity and the 15/, H20 transmissivity. That is, ~ TH20,15gJ~CO2" ECO 2,eff
Likewise, for the O 3 - H 2 0 overlap at 9.6/t the water vapor transmissivity is given by z,2o = 1.0 - 0.1 u.2o. Eg.6#03,e ff =
(I.0 -- 0.1UH2o)E03.
For temperatures in the lower stratosphere the temperature independence degrades such that for H20 ~H20 = 8 . 3 4 (0.353Iogl0uH2oo3-0.~)*UH20( -- 0.034551ogloUH20 0,705) - -
For CO2 and 03 it is assumed that temperature = - 6 0 ° C and that E is independent, giving, J~CO2 ~
0 456 0.0825 U~o 2 Uco~< 0.05 cm
~co2 = 0.0461 logl0 Uco2 + 0.074
Uco2 > 0.5 cm
and /~o3 = 0.0122 logt0 (Uo3+ 6.5 x 10 -4) + 0.0385 APPENDIX 2
Gaussian quadrature a p p r o x i m a t i o n to integrals For a general integral the gaussian quadrature approximation is given by
L'H2O= 0-846(UH20 + 3.59 X 10-5) 0.243-- 0.069
b
b -- a "
f(x) dx --" 2 ~ W'oc(Xk)
UU2o < 0.01 cm
k=l
E'mo = 0.24 logl0(Umo + 0.01) + 0.622
where W k is a weighting function and
0.01 < UH2o < 5.0 cm
b+a
Eco2 = 0.0676(Uco~ + 0.01022) °'42t -- 0.00982 Uco2 _< 1.0 cm Eco 2 = 0.0546 loglo Uco2+ 0.0581 Uco2 > 1.0 cm
where ~k are the nodes of the orthonormal polynomial which approximates the curve to be integrated. The gaussian quadrature form for double integrals may be generalized as
J~o3= 0-209(Uo3 + 7 x 1 0 - 5 ) 0"436 - 0.00321
J,x)f(x, y) dy d x -
Uo3_< 0.01 cm
0.01 cm
are curve fit expressions derived by Sasamori[1] for the emissivities of the major atmospheric absorbers for tro-
4
Wg(d(x) - c(x)). k=l
n
E w/(x~, ~)
/~o3 = 0.0212 loglo uo3 + 0.0748 U03 >
b-a Z
j=l
where
YJ = d(x) + c(x) d(x) 2 + 2 c(x)~k
0038~)92X/84 $3.00+.00 ,/~ 1984 Pergamon Press Ltd.
GAUSSIAN THERMAL FLUX MODEL--I THEORY R. F. KAMADAand R. G. FLOCCHINI Department of Land, Air and Water Resources, University of California, Davis, CA 95616 U.S.A.
(Received 21 September 1982; accepted 4 April 1983) Abstract--The Gaussian Thermal Flux model is a fast thermal radiation sub-model which operates with minimal input and predicts flux on sloped surfaces for both clear and cloudy days. This paper presents the model's theoretical basis. It is distinguished from empirical models by its detailed two dimensional treatment of clouds, its computation of air layer transmissivity above and below cloud layers and its consideration of cloud height effects. The model improves upon existing flux emissivity and multi-band or line models by being independent of soundings and by its use of gaussian quadrature to speed the numerical integrations. It may be input entirely from standard screen level meteorological measurements.
INTRODUCTION
Thermal radiation is an important term in energy balance calculations for solar homes and other solar collection devices. The instantaneous thermal flux, F~$ on the collector surface of a solar energy device may exceed 500 wm -2 or more than half the peak solar flux, even on a clear day. Daily integrated thermal flux totals often exceed solar. Moreover, the reflective, absorptive, and emissive properties of typical collector surfaces differ between solar and thermal infrared wavelength regimes. Therefore, in order to specify the energy balance equations for a dynamic computer simulation of a solar building or device, a fast, thermal radiation sub-model is needed which can operate with minimal input, handle both clear and cloudy weather, and predict flux on sloped surfaces. The Gaussian Thermal Flux (GTF) model described here attempts to fulfill these objectives. Atmospheric thermal flux models fall into three major categories: simple empirical expressions, gray body flux emissivity models of which the G T F model is a recent example, and multi-band, or line-by-line models. Each has certain strengths and weaknesses in calculating downward flux received at the earth's surface. Empirical models are simple enough for hand calculation and accurate within perhaps 10~ for clear sky conditions. Currently, however, they do not treat clouds, or latitude and elevation extremes nearly as accurately, since they were developed from data acquired at specific sites. Existing flux emissivity models require computer processing and sounding data. Line-by-line and multi-band models are too tedious and bulky for an operational solar simulation, require sounding data, and at best are scarcely more accurate than simpler models. To circumvent these difficulties the atmospheric portion of the G T F model was developed by extending Sasamori's[1] flux emissivity model to include clouds and operate without soundings. The atmospheric model was then used as a basis for incorporating the angularly anisotropic sky radiance and the ground contribution in a general ambient thermal flux model for sloped surfaces. Table 1 lists a number of empirical models which SE Vol. 32, No. 4--E
505
correlate screen level (1 to 2 m above the surface) temperature, T,, and/or vapor pressure e,, and the effective sky emissivity, Esky.Eskyis used to obtain F,$, via the Stefan-Boltzman expression, F ~ = Esky o T ~ 4. The formulae lead from Angstrom's [2] early effort up to recent attempts by Idso[8] and Berdahl[9]. The surface dewpoint temperature, Taps, and e~ are related by the Clausius-Clapeyron equation:
e=eoexpFL,,(1 • LR, kTo
1 )]
T-dp~
(10)
where L,, = latent heat of vaporization, R,. = water vapor gas constant, and To = triple point of water. Among these expressions the coefficients for (1) and (2) are site specific while (3) poses a T 6 dependence difficult to justify theoretically. The empirical coefficients k and l of (8) are distinct for day or night conditions. Equation (7) is based on night data only and claims some theoretical justification by posing a term of the same form as is used to describe chemical reaction rates. In this way it incorporates the 1.5 kcal mole ~activation energy for the splitting of water vapor dimer. Idso claims that dimer formation accounts for absorption in the infrared continuum portion of the water vapor absorption spectrum. The other coefficients for (7) and (8) are best fits to data from several sites each, such that their agreement with data is often within five or ten percent. The very recent complex expression by Centeno[10], where N = integrated sky transparency or nebulosity, y = relative humidity, attempts to account for clouds and surface elevation, z, in addition to temperature and water vapor content. Its accuracy under cloudy conditions is difficult to judge, since overall correlation or error figures were not supplied with its presentation. However, cloud contributions to flux are a strong function of percent cloud cover and cloud height, factors not directly represented in (9). Thus its ability to accurately account for clouds is unclear. In developing physical models for flux under cloudy skies one must account for cloud emission. F,.,
506
R. F. KAMADAand R. G. FLOCCHINI Table 1. Empirical models for effective sky emissivity Equation
Author
Ref.#
Date
Esky = a " ~*10-Yes Esky = a+b*es 2 Esky = 6T s
ER-#
Angstrom
[2]
1915
(i)
Brunt
[3]
1932
(2)
Swinbank
[4]
1963
(3)
Esky = l-c*exp[d(273-Ts)2]
Idso
[5]
1969
(4)
Esky = f'el/7 s
Brutsaert [6]
1975
(5)
Esky = g+h*in((Tdps+273)/273)
Clark
[7]
1978
(6)
Esky = i+3*es*eXp(1500/Tdp s)
Idso
[8]
1940
(7)
Esky = k+l*Tdp s
Berdahl
[9]
1981
(8)
Centeno
[I0] 1981
(9)
Esky = (I-N)*[57723+,9555(.6017)z] *I0-4*T s 1.1893 ~^ ¥0.0665 + N[l-(3000+1751*z0"652)* ¥-3/2 . Ts-l]4
as well as the transmissivity of the air layer below cloud level, z:. Although F c may be estimated from ground based observation, z: is less tractable. Attempts to empirically describe T: as per Exell[ll] or Geiger[12] have been site specific and applicable only to low clouds, z/may be calculated, given sufficient sounding data and some treatment of the radiative transfer equation. However, a full scale treatment is tedious and generally avoided. Moreover, the global grid for twice a day soundings is quite sparse with few readings in the planetary boundary layer where F,J, is most affected. The GTF model was developed to respond to these difficulties. THEORY
The GTF model is a "flux-emissivity-gray body" model which implies that the atmosphere is assumed to absorb and emit equally well at all wavelengths as a spectrally gray body and that for a plane parallel atmosphere the flux emissivity for optical depth or path length, u, is approximated by the beam emissivity for optical depth 1.66 u. Not only does this avoid spectral integration, but also the zenith and azimuth integrations performed in the multi-band and line-by-line beam models exemplified by Kneizys et aL[13], Rogers and Walshaw[14], or Goody[15]. The Curtis-Godson approximation allows the path length integrals to be normalized to standard temperature and pressure in calculating emissivity increments and, since both Rayleigh and Mie scattering are negligible in the thermal infrared, the scattering phase functions normally imbedded in the radiative transfer equation may be ignored. The result is that of the minimum five nested numerical integrations needed for full scale treatment, the only ones remain-
ing are for the absrrber path length and layer emissivity. Examples of this approach are given by Atwater and Ball[16], Bliss[17], Sasamori[1], Staley and Jurica[18] or Stone and Manabe[19]. The Staley-Jurica model is quite popular. However, it assumes that the emission losses due to the carbon dioxide-water and ozone-water vapor band overlaps cancel the ozone emission, allowing both to be ignored. Sasamori's model instead calculates the ozone flux, then also subtracts band overlaps from the unalloyed water vapor, carbon dioxide and ozone absorptions using empirical curve fit expressions. These expressions are more convenient and presumably more accurate than the tables of Staley and Jurica[20] or Elsasser and Culbertson [21]. Sasamori has shown that the latter contains an error causing an underestimate in emissivity. Sasamori's model appeared to have the best combination of facility and accuracy of those apparent in the literature and thus was selected as a basis for the atmospheric model. The remaining assumptions used in the GTF model are included in Table 2.
1. Emissivity and flux calculations A normalized defined as
emissivity-absorptivity can
dP~ E(u, T) _
IE(e, T) -
4aT3
(1 - *:)-d-f
l °~dPv dv Jo dT
be
dv (11)
where Pv is the Planck source function over frequency, and E is the absorptivity-emissivity, such
507
Gaussian thermal flux model--I. Theory Table 2. Basic assumptions for the GTF model I.
Gray body absorption-emission
2.
Flux
3.
Plane
emissivity parallel
of
path
in thermal infrared
length u = beam
atmosphere
except
for
emissivity
total
for 1.66 u
thermal
emission
calculation 4. .Curtis-Godson approximation 5.
Water vapor path length a function of surface dewpoint temperature
6.
No scattering
7.
Absorptivity of tropospheric absorbers independent of temperature
8.
Adjusted standard T, p, and absorber profiles for clear skies and >1 I(m above clouds
9.
Bulk emissivities E
for clouds, no microphysics C
I0.
No multiple cloud layers considered
ii.
2-D atmosphere, no cloud sides or advection
12.
Not for fronts, cumulus towers, cumulus congestus
13.
Negligible dust effects
14.
No sounding data
that the downward flux is written f. T~
Fs~ = 4 ° | E0{u(r') - u(L), ~r'}r'3d~r '. (12) dO
This integral may be parsed into tropospheric and supertropospheric portions in order to account for differences in the behavior of the emissivity functions in these two regions such that f. r~ Fs~ =4tr | ff~{u(T')--u(Ts),
r'}r'3dr'
J Tr
t"TT +4~ [ g { u ( r T ) - u ( L ) , r ' } r ' ~ d r ' (13) dO
where Tr is the tropopause temperature. Since the absorption lines and the Planck function's spectral intensity distribution are temperature dependent, one would expect E to vary with temperature. However, for troposphere temperature ranges, emissivity variations in the rotational bands nearly cancel those in the 6.3/~ water vapoi" band, resulting in a total water vapor absorptivity quite insensitive to temperature. Since the combined effect of the other absorbers is small compared to water vapor, they too are also assumed to be temperature independent. Sasamori's curve fit expressions are presented in Appendix 1. According to (12) the emissivity-absorptivities must be vertically integrated throughout the atmo-
sphere. The "adding method" (22) is the standard approach for evaluating the emissivity integral. Since it simply adds the emissivity increment at the top of each layer to the preceding layers, it amounts to a less than linear, zeroth order Riemann sum approximation to the integral and requires many layers for accuracy. In contrast gaussian quadrature in effect integrates an orthonormal polynomial curve of order 2 m - 1 fitted to the m points used for evaluation. Since the profiles extrapolated from surface data are modifications of smooth, standard profiles, m can be a small number, say 2-5. Use of gaussian quadrature in evaluating the absorber path length and emissivity integrals greatly enhances computational speed and accuracy (see Appendix 2). For simplicity stratospheric flux is assumed to all be emitted from an imaginary plane at the tropopause. It's effective emissivity is obtained by summing contributions from the total absorber path lengths up to 45 km for CO: and 03 and up to 10 km for H20. The seasonally and latitudinally adjusted temperature at 12 km is also used. The above is sufficient to compute Fs~ for clear skies. However, for cloudy skies adjusted standard profiles cannot account for water vapor enrichment below the cloud base. One must parse tropospheric air into layers above and below the cloud levels. Then Fs~ consists of flux from a clear pseudo-sky, calculated for an atmosphere lacking the cloud layer entirely, plus rj times the cloud's emission, minus the
508
R.F. KAMADAand R. G. FLOCCHINI ENVIRONMENTAL REGION CLOUDREGION 45 Km
f
":g
STRATOSPHERE
4 pt. GAUSS
l-
UPPER TROPOSPHERE
I0 Km
4pt. GAUSS
4 pt. GAUSS I Km SUPRA-
~ I Krn CLOUD LAYER
300m TRANSITION ZONE 4pt. GAUSS
l
BOUNDARY LAYER
2pt; GAUSS
A general expression for the effective absorber path length is
- - - 30m
SURFACELAYER
Fig. 1. Layers and regions of atmosphere for general flux calculation.
cloud's absorption of flux which would otherwise" have reached the surface. As per Fig. 1 care must b e taken to distinguish pseudo-sky from a clear, real sky, since their water vapor profiles may differ substantially. Thus, for fractional cloud cover, CC, the gaussian quadrature form of (13) becomes
1 ;f2 fp~,,fTo~,,,2
, qt
)tT) d.
(l,)
where g = mean sea level gravitational acceleration, p = pressure, P0 = 1 atmosphere, and q = specific humidity. Compromises between strong and weak line pressure and temperature corrections are given as n = 0.7 for H20, 0.86 for CO2, and 0.3 for 03[23 ]. The weighting is a form of the Curtis-Godson approximation which accounts for emission line broadening with temperature and pressure. It normalizes path lengths for non-isothermal atmospheres to an effective path length (based on a reference temperature and pressure) which matches both the square root variation of emission with path length in the strong line limit and the linear variation in the weak line limit. For CO2 the evaluation is trivial, since q can be taken to be 0.00034. Assuming negligible temperature correction, we obtain
Uco2
q - 186 1.86). 1.86gp 0.86~° f --P
(16)
Ozone and water vapor profiles, however, vary greatly with altitude, latitude, season and local conditions. To account for this, latitude-season tables by Robinson[24] and Smith[25] for ozone and water vapor are normalized to the standard vertical profiles for absorber densities provided in the Air Force Geophysics Lab Lowtran IR flux model[13]. Hydrostatic equilibrium allows us to write the ozone path length in gaussian quadrature form as
m F~.L= a(1 -
CC)2(z~ - z¢) ~ WkE,.~Tk- .3 dT'd~
u03
k=l
+ 2(z~ -
,,
_
3dT '
zr) Z WkE,,~T" ~ k=l
-
~o3,~J"
'3dT'
Tt -.Fz~
•
k=l
+
E'c,drr 4
03]
2pO.3b=, kpo3,p ~ J u03, ROBINSON]
+ E¢,~Tr4
+ aCC2(z~ - z~)k~=, WkEcMTk - '3dT' -~Z + 2(Z<-- ZT) ~
-':"Z 2 - ZIF ~ W
(14)
Multiple cloud layers are not considered, since the quality of its ground based estimates is dubious. Moreover, to obtain net CC a statistical formula for the probability of overlap must be invoked, rendering any short term flux calculations meaningless.
2. Clear sky absorber path lengths Before determining emissivities and fluxes, one must determine the absorber path lengths themselves.
(17)
For water vapor density, deviations from the mean tend to decrease with height. Thus, when lacking sounding data, it seems reasonable to assume a normalized standard profile some distance above the cloud layer along with standard temperatures and pressures extrapolated from surface measurements. A simple way to account for local distinctions and diurnal effects in the temperature lapse rate is to assume that the defect between local temperature and the mean at that latitude, altitude, and time of year decreases exponentially with height. A formulation such as T = L + (T~.- 7~) exp(0.5z). where z is in km seems to track diurnal effects reasonably well. For further refinement one would need to include a boundary layer model to estimate
509
Gaussian thermal flux model--I. Theory the diurnal changes in the height of the mixed layer and strength of the night time inversion as discussed in more detail in part II. The above formulation is used for clear sky conditions and for heights greater than 1 km above a cloud layer. Thus above clouds or for clear skies one has in gaussian form .2" 2 -- ZI[ ~,,
u.,o=
k'n
,L k=
I
uH2o, SMITH UH~O,L O ~ N J
] (18)
Below clouds, however, standard profiles underestimate H20 path lengths. Thus we wish to express the path length integral in a form such that profiles for humidity, pressure, and temperature may be varied according to given conditions. To do so note that where E = 0.622
dp = - pg dz p
=
P
(19) (20)
Specific humidity does tend to decrease within the surface layer. However, buoyant plumes from an extended super adiabatic surface layer tend to form cumulus clouds. Such a convective cell situation would tend to dissipate stratus before any cumulus would form. Thus the presence of stratus suggests that the surface layer contribution is minor. Moreover, once formed, a substantial cloud layer tends to inhibit surface layer growth by limiting surface heating. For these reasons we assume for stratus clouds that the surface layer lies below the screen level (1 to 2 m) at which the standard surface meteorological measurements are taken. Then, whether the air between screen level z s and zc is well-mixed or consists of lifted air from a well-mixed region, the temperature lapse rate tends to be moist adiabatic and can be obtained from
(21)
RjT~
T=
give the expressions for the specific humidity, hydrostatic equilibrium and the gas law, where Rd = dry air gas constant, and T~. = T(1 + (1 - Qe/p), the virtual temperature. Therefore, -q dp = - Ee dz g RjT~,
(22)
and
UH2O- R~n0.7 /
- - - - - dz ae0 .)z, I + ( I _ E ) e P
g
dz
RAT,,"
[g r d& \ R d J , , T,/
In practice R/Cp is nearly equal t o Rd/CM, the dry air ratio. From (26) alone the temperature Tc at zc can be inferred from a pressure profile giving Pc- Thus for stratus clouds (23) becomes
T-3/2 dz
(27)
Therefore, the general expression for pressure is p = p, e x p ) - - i - - ) .
(26)
(23)
gives a general expression for the water vapor path length. From (20) and (21) dlnp
Ts(P) R/cp.
_EeoT~/2 ~2 0.7 expt--/---LRv \ To
ETIo,'2 (-'2 epO.7T - 3/2
'
(25)
J
3. Cloudy sky absorber path lengths
Ee q =-P
Stratus clouds are perhaps simplest to deal with, since they tend to form in the mixed layer or by adiabatic ascent from the mixed layer. In either case the specific humidity should remain approximately constant up to the cloud base height, z~. That is,
e = e,p/p,.
-07 fZ°~l/2]
t, J
4. Effect of a stratiform cloud layer
(24)
Given sounding data (23) may be evaluated. However, if unavailable, it may be assumed that among the vertical profiles for meteorological quantities, pressure varies least in form from an adjusted standard. Thus, it is used as the basis from which the other variables in (23) are estimated. To do so the effect of different cloud types on temperature profiles as well as absorber path lengths must also be treated.
(27) cannot be used for middle or upper level stratiform clouds, since they may form by lifting starting from above the mixed layer. In such cases T and Tc may be inferred from adjusted standard profiles. Assuming that the relative humidity 7 increases linearly with height, the vapor pressure may be given by
e = 7esat = 7 e 0 e x p / - - - q
--
LR~\ To
(28)
--
where e0 = 611 paScals and e~at = saturation vapor pressure. Such that for higher stratiform clouds
_ T~/2 f z2 7e~tp°7 T - 3/2 tl+tp)
Te•t)
(29) is perhaps less accurate than (27). However, the
R. F. KAMADAand R. G. FLOCCHINI
510
importance of T, UH20, and F< decreases with height, since the transmittance of the intervening air, rf, also decreases with height.
5. Effect o)c a cumuliform cloud layer Cumulus presence complicates flux models because a large number of assumptions concerning profiles must be made to render such models tractable. Not only does the real cumulus conl~aining atmosphere tend to consist of several distinct layers in the vertical, but the horizontal region of environmental air surrounding the cloud region also has different properties and vertical segmentation. Referring to Fig. 1 again in detail, a superadiabatic surface layer with constant 30 m depth is assumed. In the cloud region an adiabatic layer extends to z: and in the environmental region to z ~ - 300 meters. Above this is a 300 m thick transition zone in the environmental region, followed by the cloud layer, a kilometer thick supra-cloud layer and the remainder of the atmosphere. Sans sounding data T and q are assumed to decrease by 2°C and 20 per cent. In the adiabatic layer specific humidity is held constant as with stratus cloud cover. Zc is inferred from the adiabatic lifting condensation level, LCL, of a parcel released from the top of the cloud plume generating surface layer. For simplicity this convective condensation level, CCL, is taken to be the LCL of a parcel from screen level plus 300 m [26]. It is assumed at the LCL that
T = Tdp or (30)
T, + LCL d z = Tdp~+ LCL
Using the differential form of the Clausius-Clapeyron relation and the Poisson equation we can obtain for the LCL
1--3 g K g - L A change of - 2 g Kg -I for q in this zone is assumed. Since a fixed decrease in T in the constant depth surface layer is assumed with a moist adiabatic lapse rate thereafter, one easily obtains a temperature T at the cloud base height of about T: = T, - 2°C -- 9.7 z< °C/km from which a vapor pressure at that height
VLof 1
ec = eeo exp . . . .
L, t o
LCL
dz
EL~T
t
T,
/I
If the Aq in the superadiabatic and transition layers are assumed to be roughly equal, then the near constant q in the adiabatic layer of environmental air implies
ee,,,
p fece + e~'~.
2\p<
=
--9.76 1 + 5.42 x 103e~,t/Tp
dT
i£ ~
-
x 106e~t/T2p °C/km
so that the temperature of both the environmental and cloud air is reckoned to be
and x
Note that, if z~ is accurately measured by theodolite or lidar as at major airports, (30) may be solved for Tap = T~ from which a revised water vapor path length may be computed in both cumuliform and stratiform cases. From the edge of the cloud region the adiabatic layer gradually drops two to three hundred meters as one moves into the environmental air region between the clouds. Thus, in the environmental region the adiabatic layer is assumed to extend to about the LCL. In the transition zone the potential temperatures 0 and 0~ increase by I°C and q rapidly decreases, by
(35)
p#
Since one assumes a bulk flux emissivity E~ for the cloud layer, q or T within the cumulus layer is of no concern. However, horizontal temperature differences between cloud and non-cloud air are typically less than 1 degree C. Thus one notes that the temperature at the top of the cumulus layer is usually about 2AT/3 higher than that corresponding to dry adiabatic ascent from z<, where AT is the difference in the temperatures produced at the top of the layer by dry and saturated adiabatic ascents. The saturated adiabatic lapse rate is given' approximately by
T = T:--9.76(1--3~1
(32)
(34)
e = edg/pc.
(31)
z< = CCL = LCL + 300 m.
1 k-1
may be calculated. Below it with q constant
dz Tdps-- L (Tdp~-- T,)(1 + 0.87~) d T ( l - cpTf)-//1 6.46 × --/10-4T~p~'~(9.76)
(33)
(z
-
1 + 5.42 x 103e~at/Tp ~ 1 + 8.39 x 106esat/T2pj]
zD.
(36)
In this case e~t is taken to be a constant 0.7e<. With typical dewpoint depressions of 5°C the vapor pressure in the environmental air at cumulus layer heights is assumed to be
,)]
r.1 L,~,\To (r- 5)
ec,v = e0 exp/---q - -
.
(37)
Disregarding extended cumulus towers common in the tropics, representative 1 and 2 km depths were assigned to cumuliform and cumulonimbus clouds in the mid-latitudes. The emissivity of all nimbus clouds is assumed to be unity rendering flux calculations above z< superfluous in those cases.
511
Gaussian thermal flux model--I. Theory Table 3. Vapor pressure and temperature expressions for air layers in the cumuliform region Layer
Height
Upper air
Vapor Pressure
Temperature
corrected standard profile
corrected standard profile ~=30CKm
"
-I
>Cloud
CCL+2,3 Km
Cumulus
CCL+I,2 Km
inapplicable
~sat - ~d (z-zc) TC + - 3
Adiabatic
CCL
ecp/pc
T
Surface
30 m e t e r s
L e°exPlR-~ (~o _ ~)1 1 v _
The peak height of the average cloud plume naturally defines the top of the cumulus layer. Thus buoyancy must be negative above this height. In order to counteract plume momentum, the temperature lapse rate above the cloud must be considerably less than saturated adiabatic. Indeed, an inversion often exists. Thus a conservative lapse rate of 3°C is assigned to the first kilometer above the cloud tops. Above this point vapor pressures and temperatures are assigned adjusted standard profiles. The assignments for e and T may be summarized by Tables 3 and 4. Where fl, flsat, and fld are the general, saturated, and dry adiabatic lapse rates. These values may then be substituted into gaussian quadrature approximations to (23) to yield water vapor path lengths and temperature profiles above and below cumulus cloud levels.
-2°C - 9.7"z
s
°C Km-1
T s - z/.015oC Km -I
a ground based observer. To account for this situation lifting fog is assumed when, together with total overcast, the screen level dry bulb temperature is less than 0.3°C above the dewpoint. Provided the solar disk is visible, this fog is then assumed to contribute 8 per cent to an Fs~ based on clear sky conditions. Thick fogs are assumed to be near black and the total flux given by Fs~ = 0.95trTs 4.
7. High clouds A standard height of 5 k m and E~=0.35 are assigned to cirrus and cirrostratus. Otherwise, standard profiles are assumed since the effect of high -clouds on surface flux is quite small. THERMAL FLUX INCIDENT ON INCLINED SURFACES
Thus far only downward atmospheric thermal flux incident upon a horizontal surface has been considered. For solar energy applications the more general case of ambient thermal flux incident upon sloped surfaces must also be treated. This involves the angularly anisotropic distribution of sky flux and the ground flux contribution. Cole[29] has done empir-
6. Fogs Preliminary evaluation suggested that, if thin radiation or advection fogs lift (presumably as surface heat evaporates near surface condensate), they can easily be mistaken for much thicker low stratus by
Table 4. Vapor pressure and temperature expressions for air layers in the environmental region Layer
Height
Upper air
Vapor Pressure
Temperature
corrected standard profile
corrected standard profile
>Cloud
CCL+2,3 Kin
ADewpoint = 5°C
~ = 3oc Km -I
Cumulus
CCL+I,2 Km
L eoeXe[ ~
rc +
Adiabatic
CCL
~(ece2.Pe+ ~s )
Ts -2°C - 9"7"z°C Km-I
Surface
30 m e t e r s
L 1 - ~)] 1 eoeXp [~-~ (~-v o
T
(I
v ~oo
_ T~5) ]
~sat - ~d
3
(Z-Zc)
es
s
- z/.015°C Km -I
512
R. F. KAMADAand R. G. FLOCCHINI
ical work in this area by expanding the results of Unsworth and Monteith[30]. Here, instead, a fast numerical procedure is derived to render Kondratyev's[31] exact analytic approach amenable to an operational model. If the infrared albedo of the ground is negligible, calculating the ground flux is trivial. Since most ground surfaces have an emissivity near unity, including the remaining reflected sky flux amounts to nearly black body emission such that one can say
Egr
(38)
where ct is the slope angle for the surface. Kondratyev supplies the equation
h dh) (39)
to account for total sky emission where h = angular elevation of the beam relative to the plane and ff is the azimuth angle. At first glance the beam intensity Jh and the double integral appear tedious to evaluate. However, once again some simple approximations imbedded in a gaussian quadrature formulation render (39) quite tractable. That is, using the cosine of the zenith angle, optical path lengths for flux emissivity are multiplied by air masses computed at the interpolating polynomial's nodes, then as previously suggested, converted to give effective beam emissivity path lengths by dividing by 1.66. The beam intensities can then be evaluated from the resulting emissivities using the same lapse rates and temperature profiles previously calculated, dividing by n to render flux to intensity, such that (39) becomes (see Appendix 2)
F~,op~-E,.~
Fambien t = eslop e -I- Fgr
(41)
LIMITATIONS
One drawback to general applicability is the occurrence of cumulus congestus clouds which may spread over large areas when the upward momentum of the cloud plumes is steered to the horizontal by high winds aloft and a capping inversion, such that (30)-(38) lose accuracy, if is a major fraction of the total sky area. Equation (26) also does not apply well for the case of an approaching warm front presaged by apparently descending stratiform layer clouds, since a sizeable portion of local air below the cloud layer belongs to a different, colder air mass. The cloud layer itself would then sit above an inversion with respect to the observer's vertical line of sight. In general after the formation period vertical drift of the cloud layer or sub-cloud layer inversions may occur. This is less of a problem for cumuliform clouds, since they tend to dissipate quickly without continued active formation at heights strongly correlated with the CCL. As the surface warms and cools during the daytime heating cycle, the CCL of course rises and falls along with zc until the late afternoon when the boundary layer becomes stable and the remaining cumulus tufts tend to dissipate. Along with multiple cloud layers other such particular conditions may be ascertained by an experienced ground based observer. However, for modeling purposes they are not considered, since they cannot be gleaned accurately from the set of standard meteorological surface observations.
CC
Fsl°P~=F~"c°sa(2 ,Io t'~/2~ x [sin ct cos ff cos h + cos ct sin h] cos
because, unlike the gauss procedure which evaluates near the endpoints of the curve, the regions near the angular endpoints which change most rapidly are not accounted. Either way, the final expression for ambient thermal flux incident upon an inclined surface becomes:
cos (": \ 4 t Wk3E1,..:e,,
SUMMARY
k=,
X [sin Ct COSCk COShJ + COSa sin hJ cos hi) (40) where O~
and O, = 2 (3.0 + ~*)Alternatively, one may assume that the horizontal beam intensity is simply that of a black body. Having in effect already computed the vertical beam intensity, one or two additional equi-spaced intensities may be computed and Simpson's rule applied to obtain the total atmospheric flux. This is less accurate
To briefly recapitulate, depending upon conditions, the G T F model may parse the atmosphere into clear and cloudy regions each consisting of up to several layers. The effective path lengths of the major absorberemitters: water vapor, carbon dioxide, and ozone are obtained by evaluating gaussian quadrature approximations for eqns (16)-(18), and/or among (27), (29) and (33)-(37), depending upon cloud conditions. This is done for a number of layers above and below a single cloud layer. Curve fits for emissivities from absorber path lengths then give the effective emittance from each layer. The layer thicknesses are themselves determined by the node points of the gaussian quadrature approximations of the integrals for the transmissivities and downward fluxes of the air masses above and below the cloud. The clear sky flux is determined by the clear sky fraction of the total sky area, and standard vertical profiles of
513
Gaussian thermal flux model--I. Theory temperature, pressure, and absorber concentrations adjusted for latitude, season, local surface conditions, diurnal variations, and cloud formations. The cloudy sky flux is determined by the fractional cloud cover including the flux that would have reached the surface from an otherwise clear pseudo-sky, plus the cloud emission attenuated by the intervening air, minus the amount absorbed by the cloud which would otherwise have reached the surface. The water vapor and temperature profiles for clear pseudo~sky are those for air masses containing clouds rather than from adjusted standard profiles for clear skies. Dust effects are assumed negligible. The spectral overlaps between absorbers and the ozone contribution are both included in the flux calculation. The variation of the sky thermal beam intensity with zenith angle is computed and the ambient flux evaluated by gaussian quadrature approximation of Kondratyev's analytic formulation for sky thermal emission upon a sloped surface. This is then added to the ground emission. The atmospheric model is distinguished from Brunt type empirical models by its detailed 2-dimensional treatment of clouds, its methods for estimating the transmissivity of the air directly above and below the cloud layer, and its consideration of cloud heights. It improves upon existing flux emissivity and band or line-by-line models by being independent of soundings and by using gaussian quadrature to speed the otherwise tedious numerical integrations. The analytically based treatment of ambient thermal flux is free from the condition specific limitations of Cole's empirical approach, but still practicable for computation in conjunction with an atmospheric flux emissivity model. The G T F model may be input entirely from standard screen level meteorological measurements. NOMENCLATURE c~ heat capacity at constant pressure CC fraction of sky covered by clouds CCL convective condensation level (cumulus base level) e partial pressure of water vapor E emissivity-absorptivity F downward thermal flux g mean sea level gravitational acceleration LCL lifting condensation level (stratus base level) t~ latent heat of vaporization of water P pressure P Planck's function = 2rchv3/(c2(exp(hv/kT) - 1)) q specific humidity R gas constant Ra dry air gas constant = 287.05 J Kg-I°K -1 Rr water vapor gas constant = 461.51 J Kg-~°K T temperature 11 effective vertical path length for absorber W gaussian quadrature weighting function Z height in atmosphere above sea level slope angle of inclination relative to horizon temperature lapse rate A difference symbol ( Rd/ R ~,= 0.622 7 relative humidity azimuth angle /7 density
tr Stefan-Boltzmann constant °K z transmissivity 0 potential temperature
5.672"10 8JM 2_
Subscripts
amb ambient (includes sky and ground) c cloud base height in cloud region of sky ce cloud base height in environmental region of sky cld cloudy region of sky d dry air dp dewpoint env environmental (cloudless) region of sky f air layer between ground and cloud base height gr ground h beam angular elevation relative to the horizon p pressure s screen level (1-2 m) above surface sat saturated with water T tropopause v virtual, including water v frequency REFERENCES
1. T. Sasamori, The radiative cooling calculation for application to general circulation experiments. J. Appl. Meteor. 5, 721 (1968). 2. A. Angstrom, A Study of the radiation of the atmosphere. Smithsonian Miscellaneous Collections. 65, 3 (1915). 3. D. Brunt, Notes on radiation in the atmosphere. Quart. J. Roy. Met. Soc. 58, 389 (1932). 4. W. C. Swinbank, Long-wave radiation from clear skies. Quart. J. Roy. Met. Soc. 89, 339 (1963). 5. S. B. Idso and R. D. Jackson, Thermal radiation from the atmosphere. J. Geophy. Res. 74, 5397 (1969). 6. W. Brutsaert, On a derivable formula for long-wave radiation from clear skies. Water Resources Res. 11,742 (1975). 7. G. Clark and C. P. Allen, The estimation of atmospheric radiation for clear and cloudy skies. Proc. 2nd Nat. Passive Solar Conf. Vol. 2, 676 (Philadelphia, 1978). 8. S. B. Idso, A set of equations for full spectrum and 8-14/~ and 10.5-12.5/1 thermal radiation from cloudless skies. J. Water Resources Res. 17, 295 (1981). 9. P. Berdahl and R. Fromberg, An empirical model for clear sky thermal radiation. Solar Energy 29, 299 (1982). 10. M. Centeno V, New formulae for the equivalent night sky emissivity. Solar Energy 28, 489 (1982). 11. R. H. B. Exell, The atmospheric radiation climate of Thailand. Solar Energy 21, 73 (1978). 12. R. Geiger, The Climate Near Ground (Edited by M. N. Stewart et al.), 2nd Edn., Chap. 2. Trans. Harvard Press, Cambridge, Massachusetts (1959). 13. F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, Jr., L. W. Abren, J. E. A. Selby, R. W. Fenn and R. A. McClatchey, Atmospheric transmittance radiance: computer code LOWTRAN 5. A. E. Geophys. Lab. AFGL-TR-80-0067 (1980). 14. C. D. Rogers and C. D. Walshaw, The computation of infrared cooling rate in planetary atmospheres. Quart. J. Roy. Met. Soc. 92, 67 (1966). 15. R. M. Goody, Atmospheric Radiation, 1. Theoretical Basis. Clarendon Press, Oxford (1964). 16. M. A. Atwater and J. T. Ball, Computation of IR sky temperature and comparison with surface temperature. Solar Energy 21, 211 (1978). 17. R. W. Bliss, Atmospheric radiation near the surface of the ground: a summary for engineers. Solar Energy 5, 103 (1961). 18. D. O. Staley and G. M. Jurica, Effective atmospheric emissivity under clear skies. J. Appl. Meteor. IL 349 (1972).
514
R . F . KAMADA and R. G. FLOCCHINI
19. H. M. Stone and S. Manabe, Comparison among various numerical models designed for computing infrared cooling. Mon. Weather Rev. 96, 735 (1968). 20. D. O. Staley and G. M. Jurica, Flux emissivity tables for water vb,pour, carbon dioxide and ozone. J. Appl. Meteor. 9, 365 (1970). 21. W. M. Elsasser and M. F. Culbertson, Atmospheric radiation tables. Meteor. Monogr. 4, 23 (1960). 22. A. A. Lacis, J. E. Hansen, A parameterization for the absorption of solar radiation in the earth's atmosphere. J. Atmos. Sci. 31, 118 (1974). 23. G. W. Paltridge and C. M. Platt, Radiative Processes in Meteorology and Climatology. Elsevier, New York (1976). 24. N. Robinson, Solar Radiation. Elsevier, New York (1966). 25. W. L. Smith, Note on the relationship between total precipitable water and surface dew point. J. Appl. Meteor. 5, 726 (1966). 26. F. H. Ludlam and P. M. Saunders, Shower formation in large cumulus. Tellus 8, 424 (1956). 27. J. V. Iribarne and W. L. Godson, Atmospheric Thermodynamics. D. Reidel, Boston (1973). 28. F. H. Ludlam, Clouds and Storms. Penn. St. Univ. Press, (1980). 29. R. J. Cole, The longwave radiation incident upon inclined surfaces. Solar Energy 22, 459 (1979). 30. M. J. Unsworth and J. L. Monteith, Longwave radiation at the ground (1) Angular distribution o f incoming radiation. Quart. J. Roy. Met. Soc. 101, 1 (1975). 31. K. Y. Kondratyev, Radiation in the Atmosphere. Chap. 6, Academic Press, New York (1969). APPENDIX 1
Emissivity f u n c t i o n s
pospheric temperatures and pressures. To account for the 15 t~ band overlap for CO2-H20 it is assumed that the H20 transmissivity in that spectral r e # o n is given by ZH2o = 1.33 - 0.832(u.2 o + 0.0286) 0.26
such that the effective CO 2 emissivity-absorptivity is the product of the nominal absorptivity and the 15/, H20 transmissivity. That is, ~ TH20,15gJ~CO2" ECO 2,eff
Likewise, for the O 3 - H 2 0 overlap at 9.6/t the water vapor transmissivity is given by z,2o = 1.0 - 0.1 u.2o. Eg.6#03,e ff =
(I.0 -- 0.1UH2o)E03.
For temperatures in the lower stratosphere the temperature independence degrades such that for H20 ~H20 = 8 . 3 4 (0.353Iogl0uH2oo3-0.~)*UH20( -- 0.034551ogloUH20 0,705) - -
For CO2 and 03 it is assumed that temperature = - 6 0 ° C and that E is independent, giving, J~CO2 ~
0 456 0.0825 U~o 2 Uco~< 0.05 cm
~co2 = 0.0461 logl0 Uco2 + 0.074
Uco2 > 0.5 cm
and /~o3 = 0.0122 logt0 (Uo3+ 6.5 x 10 -4) + 0.0385 APPENDIX 2
Gaussian quadrature a p p r o x i m a t i o n to integrals For a general integral the gaussian quadrature approximation is given by
L'H2O= 0-846(UH20 + 3.59 X 10-5) 0.243-- 0.069
b
b -- a "
f(x) dx --" 2 ~ W'oc(Xk)
UU2o < 0.01 cm
k=l
E'mo = 0.24 logl0(Umo + 0.01) + 0.622
where W k is a weighting function and
0.01 < UH2o < 5.0 cm
b+a
Eco2 = 0.0676(Uco~ + 0.01022) °'42t -- 0.00982 Uco2 _< 1.0 cm Eco 2 = 0.0546 loglo Uco2+ 0.0581 Uco2 > 1.0 cm
where ~k are the nodes of the orthonormal polynomial which approximates the curve to be integrated. The gaussian quadrature form for double integrals may be generalized as
J~o3= 0-209(Uo3 + 7 x 1 0 - 5 ) 0"436 - 0.00321
J,x)f(x, y) dy d x -
Uo3_< 0.01 cm
0.01 cm
are curve fit expressions derived by Sasamori[1] for the emissivities of the major atmospheric absorbers for tro-
4
Wg(d(x) - c(x)). k=l
n
E w/(x~, ~)
/~o3 = 0.0212 loglo uo3 + 0.0748 U03 >
b-a Z
j=l
where
YJ = d(x) + c(x) d(x) 2 + 2 c(x)~k