Cloud Microphysics

Chapter 3

Cloud Microphysics

☆

. . .fleecy piles dissolved in dew drops. . . Goethe, In Honour of Howard1

Cloud physics consists of two branches: cloud microphysics and cloud dynamics. While the topic of this book is the latter, it is impossible to divorce a discussion of cloud dynamics from knowledge of the microphysics. Just as the discussions in Chapters 5–12 assume a certain level of background knowledge of atmospheric dynamics, so do they assume some background in cloud microphysics. To provide this background, the present chapter summarizes the aspects of cloud microphysics that are crucial to the discussions of later chapters. First, we describe some of the basic microphysical processes that are involved in the formation, growth, shrinkage, breakup, and fallout of cloud and precipitation particles.2 In Section 3.1, we describe the microphysics of warm clouds, where the temperature is everywhere above 0
C. Section 3.2 extends the review of microphysical processes to cold clouds, in which the temperature drops below 0
C and both ice and liquid particles may exist. After this review of the individual microphysical processes that may occur in clouds, we consider in Sections 3.3–3.7 how these microphysical processes occur simultaneously in a real cloud, and how they may be linked to the cloud dynamics through a set of water-continuity equations.

3.1

MICROPHYSICS OF WARM CLOUDS

The liquid drops in a cloud typically begin as molecules condense onto a cloud condensation nucleus (CCN), which is on the order of tenths of a micron in dimension. Once they ☆ “To view the full reference list for the book, click here” 1. Goethe realizes here that the clouds are composed of microscopic particles. 2. These microphysical processes are described in more detail in basic texts on cloud microphysics, such as Fletcher (1966), Mason (1971), Pruppacher and Klett (1997), and Rogers and Yau (1989) as well as in Chapter 6 of Wallace and Hobbs (2006). The physics of ice is presented comprehensively by Hobbs (1974). Cloud Dynamics, Vol. 104. http://dx.doi.org/10.1016/B978-0-12-374266-7.00003-2 Copyright © 2014 Elsevier Inc. All rights reserved.

appear liquid drops grow to larger sizes, first to become cloud droplets on the order of  1–100 μm in diameter. Their growth can continue until they form precipitation particles. Marginal precipitation particles that fall slowly are called drizzle drops, which are  50 to a few hundred microns in diameter, and raindrops, which are a few hundred to 1000s of microns in dimension. This range of drop sizes presents the ambient air with a wide range of cross sectional areas and surface curvatures (Figure 3.1), which must be carefully considered in the microphysics of drop formation, growth, and evaporation.

3.1.1

Nucleation of Drops

The particles in a cloud form by a process referred to as nucleation, in which water molecules change from a less ordered to a more ordered state. For example, vapor molecules in the air may come together by chance collisions to form a liquid-phase drop. To see how this process takes place, consider the conditions required for the formation of a drop of pure water from vapor. This case is called homogeneous nucleation, to distinguish it from the case of heterogeneous nucleation, which refers to the collection of molecules onto a foreign substance. If the embryonic drop of pure water has radius R, then the net energy required to accomplish its nucleation is 4 ΔE ¼ 4πR2 σ vl  πR3 nl ðμv  μl Þ 3

(3.1)

The first term on the right is the work required to create a surface of vapor–liquid interface around the drop. The factor σ vl is the work required to create a unit area of the interface. It is called the surface energy or surface tension. The second term on the right of (3.1) is the energy change associated with the vapor molecules going into the liquid phase. It is expressed as the change in the Gibbs free energy of the system. The Gibbs free energy of a single vapor molecule is μv, while that of a liquid molecule is μl, and the factor nl is the number of water molecules per unit volume in the drop. It can be shown3 that 3. See Equation (6.2) of Wallace and Hobbs (2006).

47

48

PART I Fundamentals

FIGURE 3.2 A spherical cap embryo of liquid (L) in contact with its vapor (V) and a nucleating surface (C). R is the radius of curvature of the embryo. From Fletcher (1966). Republished with permission of Cambridge University Press.

FIGURE 3.1 Typical diameters of liquid hydrometeors and aerosol in the atmosphere.

μv  μl ¼ kB T ln

e es

(3.2)

where kB is Boltzmann’s constant, e is the vapor pressure, and es is the saturation vapor pressure over a plane surface of water. This relation implies that the second term on the right of (3.1) depends on the humidity of the air surrounding the nucleating drop. If a nucleated drop of a given radius nucleated in air of a given humidity has a certain ΔE, then the drop has no chance of growing further if ΔE increases as R increases; instead it will evaporate. Substituting (3.2) into (3.1) and seeking the condition for which @(ΔE)/@R ¼ 0, we obtain an expression for the critical radius Rc at which this equilibrium condition holds. This expression is Rc ¼

2σ vl nl kB T lnðe=es Þ

(3.3)

and is referred to as Kelvin’s formula.4 This radius is evidently crucially dependent on the relative humidity (defined as e/es  100%). Air is said to be saturated whenever the relative humidity is 100% (e/es ¼ 1). However, it is clear from (3.3) that it is impossible for a cloud droplet to form under saturated conditions since Rc ! 1 as e/es ! 1. Rather, the air must be supersaturated (e/es > 1) for Rc to be positive. The greater the supersaturation [defined, in percent, as (e/es  1)  100%], the smaller the size of the drop that need be exceeded by the initial chance collection of molecules. It should be noted that Rc is also a function of temperature. Not only does T appear in the denominator of (3.3) explicitly, but σ vl and es are functions of T. However, at atmospheric temperatures, the dependence of Rc on temperature is comparatively weak. In view of the primary dependence of Rc on ambient humidity, it is not surprising that the rate of nucleation of drops exceeding the critical size Rc is a strong function of the degree of supersaturation. The rate at 4. Named after Lord Kelvin, who first derived the expression.

which the vapor molecules collide to form aggregates of various sizes can be computed using principles of statistical quantum mechanics applied to an ideal gas whose molecules are in a state of random motion.5 This rate of formation of drops exceeding the critical size is the nucleation rate. It is found to increase from undetectably small values to extremely large values over a very narrow range of e/es. The value of e/es at which this rise occurs is in the range of 4–5. Thus, the air must be supersaturated by 300–400% in order for a drop of pure water to be nucleated homogeneously. Since supersaturation in the atmosphere seldom exceeds 1%, one concludes that homogeneous nucleation of water drops plays no role in natural clouds. However, the physics of the process are nonetheless relevant, as will become evident below. Heterogeneous nucleation is the process whereby cloud drops actually form. The atmosphere is filled with small aerosol particles, and molecules of vapor may collect onto the surface of aerosol particles as illustrated ideally in Figure 3.2. If the surface tension between the water and the nucleating surface is sufficiently low, the nucleus is said to be wettable, and the water may form a spherical cap on the surface of the particle. A particle onto which the molecules collect in this manner is referred to as a CCN. If a CCN is insoluble in water, the physics governing the survival of an embryonic cloud droplet are the same as in the case of homogeneous nucleation. It can be shown that Equation (3.3) still applies, but Rc has the more general interpretation in that it refers to the critical radius of curvature of the embryonic drop. Since the radius of curvature of the droplet forming on a particle is greater than what it would be if the same number of molecules were to aggregate in the absence of the particle (Figure 3.2), the aggregation of the vapor molecules has a greater chance of producing a drop exceeding the critical radius. If the aggregated water molecules form a film of liquid completely surrounding a particle, then a complete droplet is formed whose radius is larger than it would be in the absence of the nucleus. Clearly, the larger

5. See Chapter 2 of Fletcher (1966).

Cloud Microphysics Chapter 3

such a nucleus is, the more likely is the survival of a drop formed by a film around it. For this reason, the larger the aerosol particle, the more likely it is to be a site for drop formation in a natural cloud. If the CCN happens to be composed of a material that is soluble in water, the efficacy of the nucleation process is further enhanced. Since the saturation vapor pressure over the liquid solution is generally lower than that over a surface of pure water, e/es is increased. According to (3.3), the critical radius is then reduced, and nucleation is easier to achieve at the ambient vapor pressure. There are generally more than enough wettable aerosol particles in the air to accommodate the formation of all cloud droplets. However, the physics of the nucleation process just described indicate that the first droplets in a cloud will tend to form around the largest and most soluble CCN. The sizes and compositions of the aerosol particles in a sample of air thus have a profound effect on the size distribution of particles nucleated in a cloud.

3.1.2

Condensation and Evaporation

Once formed, water drops may continue to grow as vapor diffuses toward them. This process is called condensation. The reverse process, drops decreasing in size as vapor diffuses away from them, is called evaporation. Particle size change owing to condensation or evaporation may be represented quantitatively by assuming that the flux of watervapor molecules through air is proportional to the gradient of the concentration of vapor molecules.6 In this case, the vapor density ρv (defined as the mass of vapor per unit volume of air) is governed by the diffusion equation @ρv ¼ rðDv rρv Þ ¼ Dv r2 ρv @t

(3.4)

where Dvrρv is the flux of water vapor by molecular diffusion and Dv is the diffusion coefficient (assumed constant) for water vapor in air. The concentration of vapor around a spherical pure water drop of radius R is assumed to be symmetric about a point located at the center of the drop, and the diffusion is assumed to be in a steady state. Under these assumptions, ρv depends only on radial distance r from the center of the drop, and (3.4) reduces to
 1 @ 2 2 @ρv ¼0 (3.5) r ρv ð r Þ ¼ 2 r @r r @r The vapor density at the surface is ρv(R). As r ! 1, the vapor density approaches the ambient or free air value ρv(1). The solution to (3.5) satisfying these boundary conditions is 6. This assumption is called Fick’s First Law of Diffusion.

R ρv ðr Þ ¼ ρv ð1Þ  ½ρv ð1Þ  ρv ðRÞ r

49

(3.6)

If the drop has mass m, the flux of molecules causes its mass to increase or decrease at a rate given by  dρv  2 _ dif ¼ 4πR Dv  (3.7) m dr R where Dv dρv/drjR is the flux of vapor in the radial direction across a spherical surface of radius R. Substitution of (3.6) into (3.7) yields _ dif ¼ 4πRDv ½ρv ð1Þ  ρv ðRÞ m

(3.8)

Since m∝R3 , there are two unknowns in (3.8), ρv(R) and either m or R. Conditions in the environment (r ¼ 1) are assumed to be known. To obtain a solution for m or R, other relationships are needed. First, a heat-balance equation is introduced. In the condensation of water vapor _ dif , where L is on a drop, latent heat is released at a rate Lm the latent heat of vaporization. Assuming that heat is conducted away from the drop as rapidly as it is being released, we have by analogy to (3.8) _ dif ¼ 4πκ a R½T ðRÞ  T ð1Þ Lm

(3.9)

where κa is the thermal conductivity of air and T is temperature. The equation of state for an ideal gas applied to water vapor under saturated conditions over a plane surface of pure water is es ¼ ρvs Rv T

(3.10)

where Rv is the gas constant for a unit mass of water vapor, and es and ρvs are the saturation vapor pressure and density over a planar surface of water. Since es depends only on temperature,7 it is evident from (3.10) that ρvs is a known function of T. If it is, then assumed that the vapor density at the drop’s surface is given by the saturation vapor density, we may write ρv ðRÞ ¼ ρvs ½T ðRÞ

(3.11)

and (3.8), (3.9), and (3.11) can be solved numerically for _ dif , T(R), and ρv(R). These equations can, moreover, be m combined analytically for the special case of a drop growing or evaporating in a saturated environment (i.e., for the case in which e(1) ¼ es[T(1)]). In this special case, use is made of the Clausius–Clapeyron equation8: 1 des L ffi es dT Rv T 2 Combination of (3.10) and (3.12) yields

7. See pp. 80–81 of Wallace and Hobbs (2006). 8. See pp. 221–222 of Wallace and Hobbs (2006).

(3.12)

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PART I Fundamentals

dρvs L dT dT ¼  ρvs Rv T 2 T

(3.13)

equilibrium vapor pressure above the drop. The factor b^ is given by

Then (3.8), (3.9), (3.11), and (3.13) may be combined9 under saturated environment conditions to obtain _ dif ¼ m

4πRSe F κ þ FD

(3.14)

where Se depends on the humidity of the environment, Fκ on the heat conductivity, and FD on the vapor diffusivity. More specifically, Se is the ambient supersaturation (expressed as a fraction): e ð 1Þ 1 Se e s ð 1Þ

(3.15)

The other factors are given by Fκ

L2 κ a Rv T 2 ð 1 Þ

(3.16)

Rv T ð1Þ Dv es ð1Þ

(3.17)

and FD

From (3.14) to (3.17), it is evident that the diffusional growth rate of a drop depends on the temperature and humidity of the environment and on the radius of the drop. The relation (3.11) used in deriving (3.14), assumes that saturation at the drop’s surface may be approximated as if it obtained over a plane surface of water (i.e., that the growing drop was large enough for the curvature of the drop’s surface to have negligible influence upon the equilibrium vapor pressure). The drop has also been assumed to be sufficiently dilute with respect to dissolved nuclei or other impurities that the drop may be regarded as being composed of pure water. For very small drops, however, curvature and solution effects must be included. If a drop is growing on a water soluble nucleus, ρv(R) becomes ! a^ b^ ρv ðRÞ ¼ ρvs ½T ðRÞ 1 þ  3 (3.18) R R

3ivH ms Mw b^¼ 4πρL Ms

(3.20)

where ivH is the van’t Hoff factor,10 ms and Ms are the mass and molecular weight of the dissolved salt, respectively, and Mw is the molecular weight of water. Replacing (3.11) with (3.18) leads, following steps similar to those leading to (3.14), to the equation ! ^ ^ 4πR b a _ dif ¼ Se þ 3 (3.21) m F κ þ FD R R which applies when the air is saturated. When the air is unsaturated, (3.8), (3.9), and (3.18) must be solved _ dif for the evaporation rate of numerically to obtain m the drop. When drops are falling relative to the surrounding air, the diffusion of vapor and heat is altered. To account for this process, the right-hand sides of (3.8) and (3.9) may be multiplied by a ventilation factor VF. In this case, (3.14) and (3.21), the growth/evaporation rate under saturated conditions become _ dif ¼ m

4πRVF Se F κ þ FD

and _ dif m

4πRVF e a^ b^ S þ 3 ¼ F κ þ FD R R

(3.22) ! (3.23)

respectively.11

3.1.3

Fallspeeds of Drops

where σ vl is the surface tension of liquid–vapor interface, ^ 3 repreand ρL is the density of liquid water. The term b=R sents the effect of salt dissolved in the drop on the

Cloud droplets are subject to downward gravitational force, which can lead to their fallout. However, as a particle is accelerated downward by gravity, its motion is increasingly resisted by frictional force. Its final speed is called the terminal fallspeed V. For drops of water in air, V is a function of the drop radius R. Cloud droplets, which are < 100 μm (Section 3.1, Figure 3.1), have very low, barely detectable fallspeeds, < 1 m s1. Cloud droplets are the small particles that form all the familiar visible nonprecipitating clouds described in Chapter 1 (stratus, stratocumulus, cumulus, altostratus, altocumulus, and cirriform clouds). Although their fallspeeds are slight, cloud droplets do undergo very slow sedimentation, and this slow sedimentation of cloud droplets must be accounted for in accurately

9. See pp. 99–102 of Rogers and Yau (1989) for details of the derivation.

10. This factor is equal to the number of ions into which each molecule of salt dissociates. See p. 213 of Wallace and Hobbs (2006). 11. See pp. 546–567 of Pruppacher and Klett (1997).

^ represents the effect of drop curvature where the term a=R on the equilibrium vapor pressure above the drop. The factor a^ is given by a^¼

2σ vl ρ L Rv T

(3.19)

Cloud Microphysics Chapter 3

describing certain types of clouds.12 Precipitation particles are those that fall more rapidly, generally at speeds > 1 m s1. The smallest liquid precipitation particles are called drizzle, and as noted in Section 3.1, Figure 3.1, are generally considered to be  50 to a few hundred microns in diameter in radius. Drops greater than a few hundred microns in dimension are called rain. Drizzle and raindrops have terminal fallspeeds that increase monotonically with increasing drop radius. We will represent this function as V(R). For drops < 500 μm in radius, V increases approximately linearly with increasing drop radius (Figure 3.3). For larger drops, V(R) increases at a slower rate (Figure 3.4), becoming constant at a radius of about 3 mm. This asymptotic behavior is associated with the fact that a drop becomes increasingly flattened, into the shape of a horizontally oriented disc, at larger sizes (see Figure 4.5).

3.1.4

51

FIGURE 3.3 Fall velocity of water drops <500 μm in radius for various atmospheric conditions. From Beard and Pruppacher (1969). Republished with permission of the American Meteorological Society.

Continuous Collection

Cloud drop growth by coalescence with other drops can be envisioned in terms of a drop of mass m falling through a cloud of particles of mass m0 . The water contained in the particles of mass m0 is assumed to be distributed uniformly through the cloud with liquid water content ρqm0 ðg m3 Þ, where qm0 is the cloud water mixing ratio (mass of cloud water per mass of air). As it falls, the particle of mass m is assumed to increase in mass continually at a rate given by the continuous collection equation, _ col ¼ Am jV ðmÞ  V ðm0 Þjρqm0 Σ c ðm, m0 Þ m

(3.24)

where Am is the effective cross sectional area swept out by a particle of mass m, V represents the fallspeeds of the drops of masses m and m0 (Figures 3.3 and 3.4), ρ is the density of the air, and Σ c ðm, m0 Þ is the collection efficiency. For the purpose of calculating collectional growth, water drops are usually assumed to be spherical. In that case, the factor Am in (3.24) is given by A m ¼ π ð R þ R0 Þ

2

(3.25)

where R and R0 are the radii of drops of mass m and m0 , respectively. This area is based on the sum of the drop radii since any drop centered within a distance R þ R0 of the center of the drop of radius R can be intercepted by that drop. The absolute value notation is used in (3.24) since it is only the relative motion of the particles that matters for collectional growth. For the case of a large drop collecting smaller drops, the absolute value symbol is redundant since the fall velocity of the larger drop always 12. For example, Bretherton et al. (2007) showed that sedimentation of cloud droplets is important in correctly predicting the behavior of marine stratocumulus.

FIGURE 3.4 Fall velocity of water drops >500 μm in radius. From Beard (1976). Republished with permission of the American Meteorological Society.

exceeds that of the smaller drops. However, (3.24) may also be used to calculate the increase of mass of a smaller drop coalescing with larger drops. If the absolute value were not used in that case, negative growth would be calculated. Moreover, as will be seen below, (3.24) is applied also to cold clouds where in some special cases (e.g., an ice particle collecting water drops) the fall velocity of the larger particle may not be the greater of the two.

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PART I Fundamentals

A more general version of (3.24) may be written for the case in which a particle of mass m is falling relative to a population of particles of varying size. For that case, the generalized continuous collection equation is ð1 _ col ¼ Am jV ðmÞ  V ðm0 Þjm0 N ðm0 ÞΣ c ðm, m0 Þdm m 0

(3.26) where N ðm0 Þdm0 is the number of particles per unit volume of air in the size range m0 to m0 þ dm0 . The collection efficiency Σ c ðm, m0 Þ in the above expressions is the efficiency with which a drop intercepts and unites with the drops it overtakes. It is the product of a collision efficiency and a coalescence efficiency. The collision efficiency (Figure 3.5) is determined primarily by the relative airflow around the falling drop. Smaller particles may be carried out of the path of a larger particle (efficiency < 1), or small particles not in the direct path of a large particle may collide with the large particle if they are pulled into its wake (efficiency > 1). The coalescence efficiency expresses the fact that a collision between two drops does not guarantee coalescence. A common practice in theoretical or modeling studies is to assume a coalescence efficiency of unity so that the collection efficiency reduces to the collision efficiency. However, this assumption should be made with caution. Laboratory results show that under

certain conditions, especially in heavy rain, colliding drops may remain united only temporarily.13

3.1.5

Stochastic Collection

Cloud drop growth by coalescence is actually not a continuous process, as assumed in (3.24), but rather proceeds in a discrete, stepwise, probabilistic manner. In a time interval Δt, drops of a given initial size do not grow uniformly. Some may undergo more than the average number of collisions and thus grow faster than others. Consequently, a drop-size distribution develops. The probabilistic nature of collection may be accounted for straightforwardly if the coalescence efficiency is taken to be 100%. Laboratory results show that coalescence is not necessarily 100% at all drop sizes. When drops  300–500 μm (small raindrops) collide with larger drops, then ensuing coalescence exhibits small “satellite” droplets breaking off from two coalesced drops, so that the combined mass of the coalescence is not necessarily the sum of the masses of the two combining drops.14 Here, we will temporarily ignore this fact to illustrate the probabilistic nature of coalescence without the complication of satellite drop production. We will return to the problem of satellite offshoots during collisions of drops in Section 3.1.6. To begin, we consider the size distribution N ðm, tÞ, where N ðm, tÞdm is the number of particles per unit volume of air in mass range m to m0 at time t. The change in N ðm, tÞ with time is computed as follows. The rate at which the space within which a particle of mass m0 is located is swept out by a particle of mass m is given by the collection kernel, defined as

Collision efficiency

K^ðm, m0 Þ Am jV ðmÞ  V ðm0 ÞjΣ c ðm, m0 Þ

(3.27)

The probability that a particular drop of mass m will collect a drop of mass m0 in time interval Δt is ^ P^ N ðm0 , tÞdm0 KΔt

(3.28)

where it is assumed that Δt is small enough that the probability of more than one collection in this time is negligible. Making use of (3.27) and (3.28) we note that the mean number of drops of mass m that will collect drops of mass m0 at time Δt is ^ ðm, tÞdm ¼ K^ðm, m0 ÞN ðm0 , tÞN ðm, tÞdmdm0 Δt (3.29) PN Rearranging this expression we obtain ^ ðm, tÞ PN ¼ K^ðm, m0 ÞN ðm0 , tÞN ðm, tÞdm0 Δt

(3.30)

which expresses the rate at which the number of drops of mass m is reduced as a result of coalescence with drops

FIGURE 3.5 Collision efficiency for collector drops of radius R1 with droplets of radius R2. The dashed portions of the curve represent regions of dubious accuracy. From Wallace and Hobbs (1977). Copyright Elsevier.

13. Brazier-Smith et al. (1972, 1973) demonstrated the production of satellite drops during coalescence of raindrops in a laboratory setting. 14. This size range is that where Brazier-Smith et al. (1972, 1973) found the production of satellite droplets to be most likely.

FIGURE 3.6 Example of the evolution of a drop-size distribution as a result of stochastic collection. The function gm is the mass distribution function and R is the drop radius. The dashed lines show the radii (Rn and Rg) corresponding to the means of the number and mass distributions, respectively. From Berry and Reinhardt (1974). Republished with permission of the American Meteorological Society.

53

m

Cloud Microphysics Chapter 3

of mass m0 per unit volume of air. It follows that the rate of decrease of the number concentration of drops of mass m as a result of their coalescence with drops of all other sizes is given by the integral ð1 (3.31) K^ðm, m0 ÞN ðm0 , tÞN ðm, tÞdm0 I1 ðmÞ ¼ 0

By reasoning similar to that given above, we may express the rate of generation of drops of mass m by coalescence of smaller drops as ð 1 m ^ K ðm  m0 , m0 ÞN ðm  m0 , tÞN ðm0 , tÞdm0 I2 ð m Þ ¼ 2 0 (3.32) where the factor of 1/2 is included to avoid counting each collision twice. The net rate of change in the number density of drops of mass m is obtained by subtracting (3.32) from (3.31) and may be written as
 @N ðm, tÞ ¼ I2 ðmÞ  I1 ðmÞ (3.33) @t col This result is referred to as the stochastic collection equation. Computations may be made with (3.33) starting with some arbitrary initial drop-size distribution N ðm, 0Þ. The result obtained by integrating (3.33) over time yields the drop-size distribution altered by the stochastic collection process. In addition to the initial distribution, one must also assume reasonable values of the collection efficiencies and fall velocities appearing in (3.31) and (3.32). For realistic conditions, it is generally found that a large portion of the liquid water accumulates in the tail of the distribution. An example of such a calculation is shown in Figure 3.6. The drop-size distribution at successive times is plotted as mass distribution gm mN ðmÞ, rather than number distribution N ðmÞ, so that the area under each curve is proportional to the total liquid water content in the distribution. The mass distribution is plotted versus the radius of a drop of mass m on a logarithmic scale. This plotting convention emphasizes the result that a large portion of the liquid water becomes concentrated in

the large drops as time progresses. The two peaks in the mass distribution after 30 min correspond to the amount of water contained in cloud droplets (radii  103 cm) and raindrops (radii  101 cm). The two dashed lines following the centers of the two peaks correspond to the means of the number and mass concentrations. The mean of the number distribution follows the cloud droplet peak. This result illustrates that the cloud droplets are far more numerous than the raindrops, but that the latter nonetheless contain a large part of the liquid water after half an hour of stochastic collection. Stochastic collection can thus quickly convert cloud water to rainwater.

3.1.6 Spontaneous and Collisional Breakup of Drops and Modification of the Stochastic Collection Formulation When raindrops achieve a certain size, they become unstable and break up spontaneously into smaller sized drops. Laboratory experiments have led to empirical functions to describe breakup quantitatively.15 For example, laboratory data show that the probability PB ðmÞ that a drop of mass m breaks up per unit time is nearly zero for drops less than about 3.5 mm in radius and increases exponentially with size for radii greater than this value (Figure 3.7). The function shown in the plot is PB ðmÞ ¼ 2:94  107 expð3:4 RÞ

(3.34)

where R is the radius in millimeters of a drop of mass m and PB ðmÞ is in s1. A second empirical function is QB ðm0 , mÞ, which is defined such that QB ðm0 , mÞdm is the number of drops of mass m to m þ dm formed by the breakup of one drop of mass m0 . QB ðm0 , mÞ is approximately exponential. It is given by QB ðm0 , mÞ ¼ 0:1R0 expð15:6 RÞ 3

(3.35)

15. The formulation of spontaneous breakup presented in this subsection was developed by Srivastava (1971).

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PART I Fundamentals

drops, the size of the larger drop produced by their coalescence and the sizes of the satellite drops splitting off from the coalesced drop. The number and sizes of the satellite drops are based on laboratory experiments. One simplification that seems to produce reasonable results is to assume that exactly three satellite drops are produced and that they are all the same size, with the size being made consistent with laboratory results.18 With that assumption, at each time step, drops are removed from the size bins m0 and m00 , representing the masses of the colliding drops, and added to the size bins m000 representing small satellite drops and m representing the large coalesced drop.

3.2

MICROPHYSICS OF COLD CLOUDS

3.2.1 Homogeneous Nucleation of Ice Particles FIGURE 3.7 The probability PB that a drop of radius R breaks up per unit time. Based on the empirical formula of Srivastava (1971).

where the radii are in cm. The empirical functions PB ðmÞ and QB ðm0 , mÞ indicate the net effect of spontaneous breakup on the drop-size distribution N ðm, tÞ as
 @N ðm, tÞ ¼ N ðm, tÞPB ðmÞ @t break ð1 N ðm0 , tÞQB ðm0 , mÞPB ðm0 Þdm0 þ m

(3.36) In addition to breaking up spontaneously, drops that collide with each other can produce satellite drops as a byproduct of an imperfect coalescence process when drops exceed  300 μm,16 and calculations indicate that when collisional breakup occurs it dominates over the spontaneous breakup process.17 It is therefore sometimes important to incorporate collisional breakup into calculations. To account for satellite production quantitatively, the formulation of stochastic collection (3.33) must be replaced by a more general scheme, since this type of breakup is a byproduct of coalescence. When satellite drops form during coalescence, the stochastic formulation becomes much more complicated since the mass of the drop formed by coalescence is no longer equal to the sum of the masses of the two colliding drops. The probability P^ in (3.28) must be replaced by a more general probability that a collision of drops of mass m0 and m00 result in the formation of a drop of mass m, where m ¼ 6 m0 þ m00 . As a result, at least four size categories must therefore be considered in each interaction: the sizes of the two colliding

16. Brazier-Smith et al. (1972,1973) demonstrated and measured collisional breakup and formation of satellite drops in laboratory experiments. 17. Shown by Young (1975)

In principle, an ice particle may be nucleated directly from the vapor phase in the same manner as a drop. The critical size for homogeneous nucleation of an ice particle directly from the vapor phase is given by an expression similar in form to (3.3). In this case, the critical size depends strongly on both temperature and ambient humidity. Theoretical estimates of the rate at which molecules in the vapor phase aggregate to form ice particles of critical size indicate, however, that nucleation occurs only at temperatures below 65
C and at supersaturations  1000%. Such high supersaturations do not occur in the atmosphere, which indicates that homogeneous nucleation of ice directly from the vapor phase is not a process leading to natural ice clouds. Homogeneous nucleation of ice particles in real clouds occurs directly from the liquid phase. Homogeneous nucleation of ice from the liquid phase is analogous to nucleation of drops from the vapor phase. An embryonic ice particle can be considered a polyhedron of volume αi4πR3/3 and surface area βi4πR2, where R is the radius of a sphere that can just be contained within the polyhedron, and αi and βi are both greater than unity, but approach unity as the polyhedron tends toward a spherical shape. By reasoning analogous to that leading to (3.3), the expression for the critical radius Rci of the inscribed sphere is Rci ¼

2βi σ il αi ni kB T lnðes =esi Þ

(3.37)

where σ il is the free energy of an ice–liquid interface, ni is the number of molecules per unit volume of ice, and esi is the saturation vapor pressure with respect to a plane surface of ice. The saturation vapor pressures of liquid and ice in the denominator and the free energy in the numerator are all functions of temperature. The critical radius is thus a function of temperature.

18. Suggested by Brazier-Smith et al. (1972,1973).

Cloud Microphysics Chapter 3

Theoretical and empirical results indicate that homogeneous nucleation of liquid water occurs at temperatures lower than about 35 to  40
C. The exact temperature at which the homogeneous nucleation occurs in a natural cloud depends on the ambient supersaturation relative to the droplet undergoing freezing. The droplets nucleate on CCN, and at these small sizes, the concentration of dissolved CCN affects the droplet’s equilibrium vapor pressure, and a higher concentration of dissolved salt in the droplet enhances the probability of freezing by homogeneous nucleation.19 Drop size also affects the supersaturation, and larger drops freeze homogeneously at slightly higher temperatures than smaller ones.20 Generally at temperatures above about 38
C, homogeneous nucleation rates are sufficiently low that generally only water droplets are activated. At lower temperatures, solution droplets homogeneously freeze before water saturation is reached.21 It is therefore possible for a natural cloud to have unfrozen liquid (i.e., supercooled) drops in the temperature range of  0 to 38
C. However, wherever the temperature in the cloud is below about 38
C, the cloud is usually composed entirely of ice particles and is said to be glaciated.

3.2.2 Heterogeneous Nucleation and Other Processes Forming Small Ice Particles in Clouds Observations show that clouds commonly contain ice crystals at temperatures between 0 and about 38
C. Since homogeneous nucleation does not occur in this temperature range, the crystals must form by a heterogeneous process. As in the case of heterogeneous nucleation of liquid drops, the foreign surface on which an ice particle nucleates reduces the critical size that must be attained by chance aggregation of molecules. However, in the case of drops nucleating from the vapor phase, the atmosphere has no shortage of wettable nuclei. In contrast, ice crystals do not form readily on many of the particles found in air. The principal difficulty with the heterogeneous nucleation of the ice is that the molecules of the solid phase are arranged in a highly ordered crystal lattice. To allow the formation of an interfacial surface between the ice embryo and the foreign substance, the latter should have a lattice structure similar to that of ice. Figure 3.8 illustrates schematically an ice embryo that has formed on a crystalline substrate with a crystal lattice different from that of the ice. There are two ways in which such an embryo could form. Either the ice could retain its normal lattice 19. The physics and chemistry of homogeneous nucleation of solution drops to form ice-cloud particles are discussed by Heymsfield and Sabin (1989) and Koop et al. (2000). The latter study provides formulations for calculating the nucleation. These studies also provide evidence that the ice particles in cirrus clouds are consistent with theory. 20. See Rogers and Yau (1989, p. 151). 21. See Heymsfield and Sabin (1989).

55

FIGURE 3.8 Schematic illustration of an ice embryo growing upon a crystalline substrate with a slight misfit. Dislocations of the interface are indicated by arrows. From Fletcher (1966). Republished with permission of Cambridge University Press.

dimensions right to the interface, with dislocations in the sheets of molecules, or the ice lattice could deform elastically to join the lattice of the substrate. The effect of dislocations is to increase the surface tension of the ice-substrate interface. The effect of elastic strain is to raise the free energy of the ice molecules. Both of these effects lower the ice nucleating efficiency of a substance. There are several modes of action by which an ice nucleus can trigger the formation of an ice crystal. An ice nucleus contained within a supercooled drop may initiate heterogeneous freezing when the temperature of the drop is lowered to the value at which the nucleus can be activated. There are two possibilities in this case. If the CCN on which the drop forms is the ice nucleus, the process is called condensation nucleation. If the nucleation is caused by any other nucleus suspended in supercooled water, the process is referred to as immersion freezing. Drops may also be frozen if an ice nucleus in the air comes into contact with the drop; this process is called contact nucleation. Finally, the ice may be formed on a nucleus directly from the vapor phase, in which case the process is called deposition nucleation. These heterogeneous nucleation and freezing processes are illustrated in the left-hand portion of Figure 3.9. As indicated in middle and right-hand portions of the illustration, other mechanisms can also lead to the formation of small ice particles in clouds. These additional sources of ice particles involve the pre-existence of ice in the cloud or environment, as opposed to nucleation directly from the vapor or liquid phase. These processes will be discussed later (Section 3.2.6) because they require knowledge of certain physics yet to be described. As the case of drop formation, the efficacy of the nucleation depends on the ambient water-vapor content and the surface area of nucleus material exposed to the air. In addition, however, the surface-tension effects are highly temperature dependent in the sense that the higher the temperature, the more the surface tension and elastic strain are increased. The temperature dependence is so strong, that it dominated the thinking behind early work in cloud physics. The probability of ice-particle nucleation increases with decreasing temperature, and substances possessing a crystal lattice structure similar to ice provide best nucleating surface. One important

56

PART I Fundamentals

FIGURE 3.9 Ice formation processes distinguishing ice nucleation occurring on aerosol particles from other processes affecting cloud particle composition, such as the formation of ice from secondary processes involving the collision of pre-existing ice crystals (Vardiman, 1978), the splintering of ice off of graupel during the accretion of cloud droplets (Hallett and Mossop, 1974), and the fracture of ice crystals exposed to dry air layers (Oraltay and Hallett, 1989). Adapted from DeMott et al. (2011), with permission of the American Meteorological Society.

A

C

G

consequence of the latter fact is that ice itself provides the best nucleating surface; whenever a supercooled drop at any temperature 0
C comes into contact with a surface of ice it immediately freezes.22 Non-ice particles possessing a crystal lattice structure similar to ice and on which ice particles may form are called ice nuclei. Other than ice, the most common natural substances possessing a crystal lattice similar to ice are mineral dust (primarily clay minerals, accounting for about half of natural ice nuclei) and biological particles (mostly decayed leaf material and bacteria23 from vegetation). The clay mineral nucleate ice at temperatures of 10 to 37
C in concentrations of 105 to 1 cm3. At warmer temperatures, 1 to 10
C, bacteria from leaves appear to be the primary natural ice nuclei, which generate ice particles in much lower concentrations, 1014 to 105 cm3 (see Figure 3.10).24 The low concentrations of natural ice nuclei at high temperatures (low supercooling) motivated the use of non-naturally occurring nuclei such as silver iodide (which nucleates ice at about 7
C) to seed clouds for the purpose of converting liquid to ice. Such conversion can produce temporary improved visibility in supercooled fogs, although turbulent mixing will quickly fill in the hole. Silver iodide seeding has been frequently tried but never proven effective for generating enough latent heat of fusion in cumuliform clouds to enhance updrafts and hence precipitation via increased buoyancy. Specialized instruments are used in the laboratory and aircraft to determine how many ice nuclei can be activated under various ambient temperature and humidity and with different types and populations of aerosol (e.g., the asterisks 22. An exception to this instantaneous freezing of liquid collected by an ice particle occurs when the rate of collection becomes large, as in the case of the growth of some hailstones. This “wet growth” process is discussed in Section 3.2.5. 23. Specifically, Pseudomonas syringae. 24. For an in-depth review of laboratory studies of the various types of ice nuclei found in the atmosphere, see Murray et al. (2012)

in Figure 3.10).25 It has long been known that to a rough approximation the number of ice nuclei increases exponentially as temperature decreases.26 However, the natural variability of aerosol populations and ambient humidity lead to large uncertainties if only temperature is considered. A more complete theory of heterogeneous nucleation takes into account the specific composition and surface area of the ice nucleus as well as relative humidity and temperature.27 All of these details of the aerosol characteristics are difficult to know in practice, and a simplified version of the theory is based on the fact that the most important ice nucleus property is the surface area of the particles. The simplified theory builds on the fact that measurements show that the total concentration of aerosol particles larger than 0.5 μm in diameter is well correlated with the net aerosol surface area. In addition, the total aerosol concentration is a readily measurable property of the environment, and the ice nucleus concentration is well approximated by using 25. Early work used an expansion chamber to determine the frequency of ice-particle formation as the ambient temperature was lowered. More recent laboratory studies of ice-particle nucleation have used the Continuous Flow Diffusion Chamber (CFDC), which can distinguish the effects of temperature, humidity and particle characteristics. See DeMott et al. (2011) a comprehensive history of these instruments and a description of the CFDC. 26. The exponential variation ice nucleus concentration with temperature shown by early expansion chamber measurements was given by Fletcher (1966) as ln NI ¼ aI(253 K  T), where aI varies with location but has values in the range of 0.3–0.8. Note that according to this relationship, there is only about one ice nucleus per liter at 20
C. For a value of aI ¼ 0.6, the concentration increases by approximately a factor of ten for every 4
C of temperature decrease. As noted in the text, the experimental data represented by this best fit formula have a very large spread because of not accounting for variability in humidity and aerosol characteristics. 27. Phillips et al. (2008) introduced a theory leading to parameterization of heterogeneous nucleation of ice particles suitable for inclusion in models of cloud formation. Nucleation in this model depends on aerosol type and surface area as well as temperature and humidity.

Cloud Microphysics Chapter 3

57

FIGURE 3.10 Potential immersion mode ice nuclei concentrations as a function of temperature for a range of atmospheric aerosol species. Also shown are ice crystal number concentrations from DeMott et al. (2010), taken using a continuous flow diffusion chamber at water saturation within a 500-m altitude layer. From Murray et al. (2012). Republished with permission of the Royal Society of Chemistry.

the concentration of particles > 0.5 μm in diameter as a proxy for the net exposed nucleus surface area.28 For a wide range of atmospheric conditions, the observed ice nucleus concentration NI is reasonably well predicted (or “parameterized”) by this simplification. Figure 3.11 shows concentration NI versus that predicted by the simplified theory for a wide range of atmospheric conditions computed by the empirical formula NI ¼ að273:17  T Þb ðNaer,0:5 Þcð273:16T Þþd

(3.38)

where NI is ice nucleus number concentration per standard liter, Naer,0.5 is the number concentration per standard cubic meter29 of aerosol particles with diameters larger than 0.5 μm, T is cloud temperature in K, and a ¼ 0.0000594, b ¼ 3.33, c ¼ 0.0264, d ¼ 0.0033. The solid diagonal line Figure 3.11 represents perfect prediction. A total of 62% of the points lie within the dashed lines, which represent a factor of two range of uncertainty. This graph illustrates the vast range of active ice nucleus concentrations that occur in the atmosphere. They range over three orders of magnitude, from  0.1 to 100 per liter.

3.2.3

Vapor Deposition and Sublimation

Growth of an ice particle by diffusion of ambient vapor toward the particle is called deposition. The loss of mass of an ice particle by diffusion of vapor from its surface into the environment is called sublimation. These processes are the 28. See DeMott et al. (2010) for details of this modification of the theory. 29. Standard cubic meter and standard liter refer to these volumes at standard atmospheric temperature and pressure.

FIGURE 3.11 Comparison of predicted versus observed ice nucleus concentrations. Predictions are based on a parameterization that depends on both the aerosol concentration exceeding 0.5 mm and temperature. Uncertainties (one standard deviation) are shown on selected data points. Dotted lines outline a range of a factor of two about the 1∶1 line. From DeMott et al. (2010). Republished with permission of the National Academy of Sciences.

ice-phase analogs of condensation and evaporation. However, since ice particles take on a variety of shapes, the spherical geometry assumed in evaluating the growth and evaporation of drops by vapor diffusion (Section 3.1.2) may not always be assumed in calculations of the change of mass of ice particles. Diffusion of vapor toward or away from nonspherical ice particles is accounted for by replacing R in (3.8), and thus in

58

PART I Fundamentals

FIGURE 3.12 Schematic representation of the main shapes of ice crystals: (a) columnar or prism-like, (b) plate, and (c) dendrite. Adapted from Rogers and Yau (1989). Copyright Elsevier.

(c)

(a)

(b)

e which is analogous to (3.14) and (3.22), by a shape factor C, electrical capacitance.30 Thus, the analog to (3.8) is   e v ρv ð1Þ  ρv sfc _ dif ¼ 4π CD m (3.39) where ρv sfc is the vapor density at the particle’s surface. It follows that the analogs to (3.14) and (3.22) are _ dif ¼ m

4π CeSei Fκi þ FDi

(3.40)

_ dif ¼ m

e F Sei 4π CV Fκi þ FDi

(3.41)

and

e Fκ, and FD in respectively. Sei , Fκi, and FDi are the same as S, (3.15)–(3.17) except that L is replaced by the latent heat of sublimation Ls in (3.16), and es(1) is replaced by the saturation vapor pressure over a plane surface of ice esi(1) in (3.15) and (3.17). The relations (3.40) and (3.41), like (3.14) and (3.22), apply only when the air is saturated (in this case with respect to ice). As in the case for drops, _ dif must be obtained numerically if the air is unsaturated. m The shape, or habit, adopted by an ice crystal growing by vapor diffusion is a sensitive function of the temperature T and supersaturation Sei of the air.31 These growth modes are known from observations in the laboratory and in clouds themselves. The basic crystal habits exhibit a hexagonal face. Let a crystal be imagined to have an axis normal to its hexagonal face. If this axis is long compared to the width of the hexagonal face, it is said to be columnar (sometimes also referred to as prism-like). If this axis is short compared to the width of the hexagonal face, the crystal is said to be platelike. The basic crystal habits are illustrated schematically in Figure 3.12. The habits change back and forth

30. The analogy between the vapor field around an ice crystal and the field of electrostatic potential around a conductor of the same size and shape was first applied by Houghton (1950). See Hobbs (1974) for further notes on the origin of the analogy. 31. See Chapters 8 and 10 of Hobbs (1974).

between columns and plates as the ambient temperature changes (Table 3.1). The effect of increasing the ambient supersaturation is to increase the surface to volume ratio of the crystal. The additional surface area gives the increased ambient vapor more space on which to deposit. The multi-armed, fern-like crystals that appear at temperatures of 12 to 16
C have six main arms and several secondary branches (Figure 3.12c). They may be thought of as hexagonal plates with sections deleted to increase the surface to volume ratio of the crystal. They occur in the temperature range where the difference between the saturation vapor pressure over water (an approximation to the actual vapor pressure in many cold clouds) and the saturation vapor pressure over ice (an approximation to the condition at the surface of the crystal) is greatest.

3.2.4

Aggregation and Riming

If ice particles collect other ice particles, the process is called aggregation. If ice particles collect liquid drops, which freeze on contact, the process is called riming. The continuous collection equation (3.24) may be used to describe the growth of ice particles by aggregation or riming. Aggregation depends strongly on temperature. The probability of adhesion of colliding ice particles becomes much more likely when the temperature increases to above 5
C at which the surfaces of ice crystals become sticky. Another factor affecting aggregation is crystal type. Intricate crystals, such as dendrites, become aggregated when their branches become entwined. These facts are known from laboratory experiments and observations of natural snow. The sizes of collected snow aggregates are shown as a function of the temperature at which they were observed in Figure 3.13. The sizes increase sharply at temperatures above 5
C, while aggregation does not appear to exist below 20
C. A secondary maximum occurs between 10 and 16
C, where the arms of the dendritic crystals growing at these temperatures apparently become entangled. Little quantitative empirical information exists

Cloud Microphysics Chapter 3

59

TABLE 3.1 Variations in the Basic Habits of Ice Crystals with Temperature (Wallace and Hobbs, 2006) Temperature (
C)

Basic Habit

Types of Crystal at Slight Water Supersaturation

0 to 4

Platelike

Thin hexagonal plates

4 to 10

Prism-like

Needles (4 to 6
C) Hollow columns (5 to 10
C)

10 to 22

Platelike

Sector plates (10 to 12
C) Dendrites (12 to 16
C) Sector plates (16 to 22
C)

22 to 50

Prism-like

Hollow columns

FIGURE 3.13 Maximum dimensions of natural aggregates of ice crystals as a function of the temperature of the air where they were collected. Points designated by  are for crystals observed aboard an aircraft. Circles represent crystals collected on the ground. From Hobbs et al. (1974). Republished with permission of the American Geophysical Union.

to accompany these qualitative facts based on observations of ice-particle shapes and sizes. Yet another factor that can affect aggregation is the degree of supersaturation, which can affect the wetness and hence sticking efficiency of the ice particles. One approach has been to assume that the aggregation efficiency is an exponentially increasing function of temperature in calculations using (3.24) or (3.26).32 However, that approach ignores the effects of supersaturation and particle type on aggregation efficiency.

32. See Lin et al. (1983)

A more ambitious method uses a postulated look-up table that expresses the aggregation efficiency as a function of temperature, particle type, and degree of supersaturation (Table 3.2). This table shows higher values for more complex crystal types and supersaturated conditions. For simplicity, many applications, simply assume a constant value of 0.1.33 33. For example Reisner et al. (1998) and Morrison and Grabowski (2008a). Field et al. (2006) found that this simplification in stochastic collection calculations leads to realistic particle-size distribution for cirrus anvil clouds.

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PART I Fundamentals

TABLE 3.2 Weighting Factors for the Coalescence Probability Based on Ice Crystal Habits (Mitchell, 1988) Category

Temperature (
C)

Crystal Habits Near or Above Water Saturation

1

T < 20

Columns Sideplanes Bullets

0.25

Columns Bullets

0.10

2

20 T <  17

Plates

0.40

Plates

0.10

3

17 T <  12.5

Dendritic crystals

1.00

Plates

0.10

4

12.5 < T < 9

Plates

0.40

Plates

0.10

5

9 T < 6

Sheaths Columns

0.10

Columns

0.10

6

6 T < 4

Needles

0.60

Columns

0.10

7

T  4

Plates

0.10

Plates

0.10

3.2.5

Weight

Crystal Habits Below Water Saturation

Weight

Hail

The collection efficiency for riming varies primarily with the size of the supercooled drops being collected by an ice particle. Cloud droplets are mostly 10–100 μm in radius. Theoretical calculations show that the collision efficiency for ice crystals falling through a cloud of drops in this size range is near zero for tiny droplets < 10 μm in radius but rises quickly as the drop sizes approaches 10 μm (Figure 3.14). The efficiency drops again to near zero for the largest cloud droplets. The efficiency also depends strongly on the Reynolds number, Re 2VD/ν, where V is the terminal fallspeed of the ice particle, D is its diameter (or characteristic long dimension), and ν is the viscosity of the air. The Re variability indicates that the efficiency increases with the size and/or fallspeed of the collecting ice particle. The mechanism and rate of growth by riming depends on the type of ice particle undergoing growth, and the mechanism can change as the particle grows and changes from a lightly rimed crystalline nature to a solider more heavily rimed structure. Branched particles appear to grow first by “filling in.”34 This process results in an increase of the particle mass, but not the particle’s long dimension. Eventually, filling in of the unoccupied volume of the branched particle results in denser quasi-spherical particle (incipient graupel). At this point, further growth of the particle increases both mass and dimension. Once filled in, they grow as spherical particles. The growth rate increases smoothly as the particles evolve toward more heavily rimed structure. Lightly to moderately rimed crystals retain vestiges of the collector’s original crystal habit (Figure 3.15a–d). Under heavy riming, the identity of the collector is lost, and the particle is called graupel, which may be in the form of lumps or cones (Figure 3.15e and f).

_ col is the rate of increase of the mass of the The factor m hailstone as a result of collecting liquid water. It is given by (3.26). The hailstone is assumed to be spherical with radius R. Lf is the latent heat of fusion released as the droplets freeze on contact with the hailstone. The second term in the curly brackets is the heat per unit mass gained as the collected water drops of temperature Tw come into temperature equilibrium with the hailstone. The factor cw is the specific heat of water. If the air surrounding the particle is subsaturated, the temperature Tw is approximated by the wet bulb temperature of the air, which is the equilibrium temperature above a surface of water undergoing evaporation at a given air pressure.35 This temperature may be several degrees less than the actual air temperature when the humidity of the air is very low. If the air surrounding the particle is saturated Tw ¼ T(1).

34. The “filling in” conceptual model of ice-particle growth by riming was introduced by Heymsfield (1982).

35. See pp. 83–84 of Wallace and Hobbs (2006).

Extreme riming produces hailstones. These particles are commonly 1 cm in diameter but have been observed to be as large as 10–15 cm. They are produced as graupel or frozen raindrops collect supercooled cloud droplets. So much liquid water is accreted in this fashion that the latent heat of fusion released when the collected water freezes significantly affects the temperature of the hailstone. The hailstone may be several degrees warmer than its environment. This temperature difference has to be taken into account in calculating the growth of hail particles, which is determined by considering the heat balance of the hailstone. The rate at which heat is gained as a result of the riming of a hailstone of mass m is   _ col Lf  cw ½T ðRÞ  Tw  (3.42) Q_ f ¼ m

Cloud Microphysics Chapter 3

61

The rate at which heat is lost to the air by conduction is obtained from a modified version of (3.9), which may be written as

(a)

Q_ c ¼ 4πRκa ½T ðRÞ  T ð1ÞVFc

(3.44)

where VFc is a ventilation factor for conduction. In equilibrium, we have Q_ f þ Q_ s ¼ Q_ c

(3.45)

which upon substitution from (3.42)–(3.44) may be solved for the hailstone equilibrium temperature as a function of size. As long as this temperature remains below 0
C, the surface of the hailstone remains dry, and its development is called dry growth. The diffusion of heat away from the hailstone, however, is generally too slow to keep up with the release of heat associated with the riming (depositional growth is much less than the riming). Therefore, if a hailstone remains in a supercooled cloud long enough, its equilibrium temperature can rise to 0
C. At this temperature, the collected supercooled droplets no longer freeze spontaneously upon contact with the hailstone. Some of the collected water may then be lost to the warm hailstone by shedding. However, a considerable portion of the collected water becomes incorporated into a water–ice mesh forming what is called spongy hail. This process is called wet growth. During its lifetime, a hailstone may grow alternately by the dry and wet processes as it passes through air of varying temperature. When hailstones are sliced open, they often exhibit a layered structure, which is evidence of these alternating growth modes.

(b)

(c)

3.2.6

FIGURE 3.14 Collision efficiencies of ice particles colliding with supercooled droplets computed for (a) hexagonal plates, (b) branched crystals, and (c) columns. From Wang and Ji (2000). Republished with permission of the American Meteorological Society.

The rate that the hailstone gains heat by deposition (or loses heat by sublimation) is obtained from a modified form of (3.8) Q_ s ¼ 4πRDv ½ρv ð1Þ  ρv ðRÞVFs Ls

(3.43)

where VFs is a ventilation factor for sublimation, and Ls is the latent heat of sublimation.

Ice Enhancement

As indicated in Figure 3.9, small ice particles may form by processes other than nucleation, with the result that clouds may have higher concentrations of ice particles than would be expected from the observations and theory of homogeneous or heterogeneous nucleation alone. These additional sources of cloud ice particles are referred to collectively as ice enhancement.36 One type of ice enhancement represented in Figure 3.9 requires the pre-existence of ice particles in the cloud. This type of enhancement occurs in at least three possible ways, illustrated schematically in the middle portion of Figure 3.9. (i) Ice particles undergoing collisions or thermal shock may fracture and break into pieces. (ii) During the riming process, small splinters of ice are produced when the temperature is 3 to  8
C and 36. It was once thought that ice enhancement was more extreme than it actually is. However, as discussed by Korolev et al. (2011) it has been found that the high concentrations of ice particles traditionally seen in aircraft observations were due mainly to shattering of ice caused by impact with the measurement probes themselves and that this problem could largely be eliminated by a redesign of the instrument aerodynamics.

62

PART I Fundamentals

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 3.15 (a) A lightly rimed needle, (b) densely rimed column, (c) densely rimed plate, (d) densely rimed stellar, (e) lump graupel, and (f) cone graupel. From Wallace and Hobbs (1977). Copyright Elsevier.

supercooled droplets > 23 μm in dimension impact an ice particle at a speed of  1.4 m s1.37 (iii) Dendritic crystals exposed to dry air breakup as they evaporate.38

process increases the ice-particle concentration of a cloud by incorporation of ice from outside the cloud, either by sedimentation of particles from clouds at higher altitudes or perhaps by some form of lateral entrainment.

Another type of ice enhancement, indicated in the far right portion of Figure 3.9, is called ice redistribution. This

3.2.7 37. The laboratory experiments were performed by Hallett and Mossop (1974). This ice-enhancement process is often called the Hallett–Mossop mechanism. 38. This breakup process was found in laboratory experiments by Oraltay and Hallett (1989).

Fallspeeds of Ice Particles

The fallspeeds of ice particles encompass a wide range. Observations show that these speeds depend on particle type, size, and degree of riming and that the more heavily rimed a particle, the more its fallspeed depends on its size. Pristine snow crystals drift downward at speeds of

Cloud Microphysics Chapter 3

63

FIGURE 3.16 Terminal fallspeeds of unrimed snow crystals as a function of their maximum dimensions. Boxes indicate the median value and whiskers indicate 1 standard deviation. This plot was created from past data sets for this book by Andrew Heymsfield and is published here with his permission.

FIGURE 3.17 Terminal velocity and maximum dimension of aggregates of ice particles. The “early aggregates” observed by Kajikawa (1982) were composed of 2–3 crystals and basically two dimensional. The particles seen by Locatelli and Hobbs (1974) were complex combinations of ice particles and more three dimensional in nature. Boxes indicate the median value and whiskers indicate 1 standard deviation. This plot was created from past data sets for this book by Andrew Heymsfield and is published here with his permission.

0.1–0.7 m s1 (Figure 3.16).39 Some aggregates fall at a similar speed if they consist of only 2–3 crystals and are fairly two dimensional, presenting only a small cross sectional area to the air through which they are falling. These particles are the “early aggregates” in Figure 3.17. Most aggregates are more complex and three dimensional and fall at speeds of  1–1.5 m s1 (dots in the upper part of Figure 3.17). The fallspeeds of graupel particles, whose sizes build up in proportion to the degree that they have undergone riming sharply from  1–3 m s1 as their diameters increase from 1–5 mm (Figure 3.18). 39. The empirical data in Figures 3.16 present the same basic information as was reported in the early work of Nakaya and Terada (1935) at Hokkaido University and is also consistent with the fallspeed determination method of Mitchell (1996).

The empirical data in Figures 3.16–3.18 are presented graphically with fallspeed as a function of the particle’s diameter. However, the data indicate that the mass of the particle (especially as determined by the degree of riming) is also a factor determining fallspeed, as is evident from the fact that fallspeed is greater for rimed than unrimed particles in Figures 3.16–3.18. To account for both mass and size, we note that the drag force on a falling particle of mass m may be expressed as 1 Fdrag ¼ ρV 2 Am CD 2

(3.46)

where ρ is the density of air, V is the terminal fallspeed of the particle, Am is the cross sectional area of the particle normal to the flow, and CD is the drag coefficient. Then assuming that the particle is at its terminal velocity, the drag

64

PART I Fundamentals

FIGURE 3.18 Terminal velocity versus dimension of graupel particles. Boxes indicate the median value and whiskers indicate 1 standard deviation. This plot was created from past data sets for this book by Andrew Heymsfield and is published here with his permission.

force is equal to the gravitational force mg, and the fallspeed in (3.46) becomes


2mg V¼ ρAm CD

1=2 (3.47)

Theory and empirical data show that the drag coefficient CD is a function of the Best Number, which is the particle’s mass divided by its cross sectional area normal to the airflow around the particle.40 Substituting such a function into (3.47) expresses the fallspeed as a function of the Best Number. Since the resulting fallspeed relationship is based on the physical characteristics of geometric size and mass, it is applicable to any type of falling particle, thus making it possible to calculate fallspeed of most particles without having to make a qualitative decision about the category of the ice particle being considered (needle, dendrite, rimed crystal, etc.). However, when an ice particle reaches the dimensional size and mass of a snow aggregate, a “pillow” of air forms around the boundary of the particle, making its effective area larger. Correcting for this effective increase in area leads to the result that aggregates reach a maximum velocity at which increase of size leads to no further increase of fallspeed, and may even reduce the fallspeed, as illustrated in Figure 3.19. Hailstones are so large and dense that their fall velocities predicted by the above theory as well as empirical data are an order of magnitude larger than those for snow and graupel. Figure 3.20 shows fallspeeds of 8–19 m s1 for hailstones in the diameter range of 5–25 mm.

40. The relationship of the drag coefficient to particle mass and dimension has been explored by Mitchell (1996), Khvorostyanov and Curry (2002, 2005), and Mitchell and Heymsfield (2005).

Hailstones can be even larger, up to  80 mm in diameter.41 Fallspeeds for these gigantic stones can approach 50 m s1.42 These large values imply that updrafts of the order of tens of meters per second must exist in the cloud to support the hailstones long enough for them to grow. Consistent with this reasoning, hail is found only in very intense thunderstorms, of the types considered in Chapters 8 and 9.

3.2.8

Melting

Ice particles can change into liquid water when they come into contact with air or water that is above 0
C. A quantitative expression for the rate of melting of an ice particle of mass m can be obtained by assuming heat balance for the particle during the melting. According to (3.45), this balance may then be written as _ mel ¼ 4πRκa ½T ð1Þ  273 KVFc  Lf m _ col cw ðTw  273 KÞ þ Q_ s þm

(3.48)

_ mel is the rate of change of the mass of the ice parwhere m ticle as a result of melting. The first term on the right is the diffusion of heat toward the particle from the air in the environment. The second term is the rate at which heat is transferred to the ice particle from water drops of temperature Tw that are collected by the melting particle. The third term, Q_ s 41. The size distribution of hail as determined by Auer (1972). 42. Pruppacher and Klett (1997) suggest an empirical formula based only on hailstone size. At a pressure of 800 mb and a temperature of 0
C, this formula predicts a hailstone fallspeed given by V(m s1) ¼ 9D0.8 h , where Dh is the diameter of the hailstone in cm. It is valid for hailstones in the size range 0.1–8 cm and yields fallspeeds of  1050 m s1 for an 8 cm hailstone.

Cloud Microphysics Chapter 3

65

3.3 TYPES OF MICROPHYSICAL PROCESSES AND CATEGORIES OF WATER SUBSTANCE IN CLOUDS From the foregoing review of cloud microphysics (Sections 3.1 and 3.2), it is evident that water substance can take on a wide variety of forms in a cloud, and that these forms develop under the influence of seven basic types of microphysical processes: 1. 2. 3. 4. 5. 6. 7. FIGURE 3.19 Terminal velocities of unrimed aggregates predicted theoretically by Khvorostyanov and Curry (2002, dotted curve) and Mitchell and Heymsfield (2005, heavy solid curve). Also shown are fall velocities from measurement derived power laws for aggregates of unrimed dendrites (Locatelli and Hobbs, 1974, long dashed) and unrimed needles (Hanesch, 1999, light solid). Republished with permission of the American Meteorological Society.

These individual processes may sometimes be isolated for study in numerical models or in the laboratory. However, in a natural cloud several or all of these processes occur simultaneously, as the entire ensemble of particles comprising the cloud forms, grows, and dies out. Thus, the various forms of water and ice particles co-exist and interact within the overall cloud ensemble. It is the behavior of the overall ensemble that is of primary interest in cloud dynamics, and it is often unnecessary to keep track of every particle in the cloud in order to describe the cloud’s gross behavior. At the same time, it is also impossible to ignore the microphysical processes and accurately represent the cloud’s overall behavior. One way to retain the essentials of the microphysical behavior is to group the various forms of hydrometeors in a cloud into several broad species: l

Matson and Huggins (1983) Lozowski and Beattie (1979) Knight and Heymsfield (1983)

l

FIGURE 3.20 Hailstone fallspeed versus particle dimension from theory (solid curve) and three sets of observations. From Mitchell (1996). Republished with permission of the American Meteorological Society.

is the gain or loss of heat by vapor diffusion [given by (3.43) in the case of a spherical hailstone with Ls replaced by L since the hail is melting]. If both the air and drop temperatures exceed 273 K, then both the first and second term in (3.48) contribute to the melting.

Nucleation of particles Vapor diffusion Collection Breakup of particles Fallout Ice enhancement Melting

l

Cloud liquid water is in the form of small liquid-phase droplets that are too small to have any appreciable terminal fallspeed and therefore are generally carried along by the air in which they are suspended. They do settle slightly, which can sometimes be a significant factor in determining the structure of certain thinly layered clouds. Cloud droplets grow primarily by vapor diffusion. Precipitation liquid water (rain) is in the form of liquidphase drops that are large enough to have an appreciable fallspeed toward Earth. This water may be subdivided into drizzle (drops 0.1–0.25 mm in radius) and rain (drops > 0.25 mm in radius), as defined in Section 3.1. Drizzle and raindrops can (and often do) grow by collection/coalescence as well as by vapor diffusion. Cloud ice is composed of particles that have relatively small fallspeeds compared to the fallspeeds of precipitation particles but large enough to affect the structure of cirrus clouds. Cloud ice particles may be in the form of pristine crystals nucleated directly from the

vapor or water phase, or they may be tiny particles of ice produced in some form of ice-enhancement process. Precipitation ice refers to ice particles that have become large and heavy enough to have a terminal fallspeed  0.3 m s1 or more. These particles may be pristine crystals, larger fragments of particles, rimed particles, aggregates, graupel, or hail. To simplify the description of the gross behavior of a cloud, these particle types are sometimes grouped into categories according to their density or fallspeed. Such groupings are arbitrary; however, a commonly used scheme (employed in discussions below) is to divide these particles into snow, which has lower density and falls at speeds of  0.3–1.5 m s1 (see Figures 3.16 and 3.17), graupel which falls at speeds of  1–3 m s1 (see Figures 3.16 and 3.18), and hail which falls at speeds of  10–50 m s1 [see Equation (3.46)].

According to these species, the water substance in a sample of air may be represented by eight mixing ratios: qv mass of water vapor=mass of air qc mass of cloud liquid water=mass of air

(3.49)

One approach to modeling the evolution of a cloud is to keep track of these particular mixing ratios. However, in doing so, it must be recognized that the subdivision of the water substance in an air parcel into these various species is somewhat subjective, and that the species are highly interactive. For example, drops grow at the expense of vapor during nucleation and condensation, precipitation ice grows at the expense of cloud and precipitation liquid water during riming, rainwater is produced at the expense of precipitation ice during melting, and so on. Figure 3.21 illustrates the large number of possible conversions of water substance from one form to another for a cloud whose water substance is divided into water vapor, cloud liquid water, rainwater, cloud ice, snow, and graupel. Also indicated are the possible gains or losses by precipitation fallout. The six species in Figure 3.21 are a subset

Evaporation during melting Initiation Deposition

WATER VAPOR

Riming Melting

Deposition

of ice by rain

Collection

Collection

Collection

Cloud ice

Collection

Cloud water Autoconversion

Evaporation / condensation

Deposition / evaporation during melting

Condensation

qd mass of drizzle=mass of air qr mass of rainwater=mass of air qI mass of cloud ice=mass of air qs mass of snow=mass of air qg mass of graupel=mass of air qh mass of hail=mass of air

Collection

l

PART I Fundamentals

Conversion

66

Riming Rain

Snow

Collection Shedding Melting Melting

Collection Graupel

Precipitation fallout FIGURE 3.21 Conversions of water substance from one form to another in a bulk water-continuity model in which there are six categories of water substance. Dashed lines indicate various interactions leading to production of graupel: collection of cloud ice by rain freezes the ice by contact nucleation and produces either snow or graupel; or riming of snow by collection of either cloud water or raindrops can also lead to production of graupel. In the model, the mass of ice produced by these two processes passes temporarily through the snow category. Adapted from Rutledge and Hobbs (1984). Republished with permission of the American Meteorological Society.

Cloud Microphysics Chapter 3

of the eight listed above, drizzle and hail being omitted. This six species scheme has been used frequently in cloud dynamics (with hail rather than graupel sometimes used as the sixth category). However, because of the many interactions illustrated by Figure 3.21, it can be advantageous to combine these categories. The liquid species, drizzle and rain, combine naturally into a single liquid water mixing ratio. In the case of the ice-hydrometeor species listed above, the degree of riming determines their density and fallspeed characteristics more than any other factor, and a useful approach for the ice species is not to track the variation of the mixing ratios corresponding to the above described particle species but rather to predict the portions of the ice mass produced by vapor deposition (qdep) and riming (qrim).43 Instead of (3.49), we then have the variables qv mass of water vapor=mass of air qdep mass of vapor  grown ice=mass of air qrim mass of rime ice=mass of air

3.4

(3.50)

WATER-CONTINUITY EQUATIONS

In cloud dynamics, one is concerned with the overall development of a cloud, in which any combination of the hydrometeor species may be occurring in the context of a particular set of air motions. It is therefore necessary to have a way to keep track of all of the hydrometeor mixing ratios and processes over time in some systematic way. For this purpose, the water-continuity equations (2.21) are used. They allow one to account numerically for the amount of water contained in the form of vapor and in the form of particles of different types and sizes throughout a cloud, as it evolves. This system of equations is referred to as a water-continuity model. In (2.21), each category of water substance represented by a mixing ratio qi, where the subscript i refers to a particular category of water and the mixing ratio is defined as the mass of water of category i per unit mass of air. The categories of water substance represented by the mixing ratios qi include the vapor mixing ratio as well as mixing ratios of water in the forms of drops and ice particles of different types and/or sizes. The total water substance in a parcel of air is then given by the sum of the water contained in each of the categories: qT ¼

n X

qi

(3.51)

i¼1

where n is the total number of categories into which the total water qT in the air parcel has been divided. The source terms 43. Morrison and Grabowski (2008a) developed the approach of separately calculating accumulation of ice mass by deposition and riming. We will examine their methodology further in the last section of this chapter.

67

on the right-hand side of (2.21) must be formulated to allow for all possible interactions among the different forms of water represented by the qi ’ s. The source terms are formulated in terms of the seven basic microphysical mechanisms mentioned earlier (nucleation, vapor diffusion, collection, particle breakup, fallout, ice enhancement, and melting). There are two general strategies for formulating the source terms. In bulk models, it is assumed that we know the form of the particle-size distribution function (e.g., exponential, gamma function, monodisperse) within one or more of the hydrometeor types represented by a mixing ratio qi in (2.21), and that we need only to predict the moment(s) of that distribution function to obtain the hydrometeor content of that species. In bin models, the shapes of the size distribution functions are not assumed a priori. Within one or more hydrometeor species the size distribution function is allowed to evolve naturally by first lumping similar species together and then subdividing the water substance into size categories. For example, cloud, drizzle, and rain are combined into a single liquid water mixing ratio qL. Then the water is subdivided into a finite number of drop-size ranges, and the drop-size distribution function for each size range N(Di) is computed, where N(Di)dD yields is the number concentration (per volume of air) of drops ranging in size from diameter Di to Di þ dD.44 Similarly, we may combine various icehydrometeor species into one or more broad categories and compute the size distribution function for the ice particles constituting each broad category. Below we describe the fundamentals of bin models before we describe bulk models because the microphysical source terms in the water-continuity model equations of the form (2.21) in bin models are simply the particle-growth process formulations arrived at from first principles and described above. Bulk models require the microphysical processes to be parameterized in ways that make them less transparent and require further explanation. The bin and bulk options for representing the water continuity provide for a wide variety of possibilities for modeling clouds, even including hybrid approaches using combinations of both bin and bulk options. The options chosen for any particular study are usually tailored to the

44. In this text we express number concentration in the traditional way, in units of number of particles per unit volume of air. However, it is becoming increasingly common for models to compute number mixing ratio (number of particles per unit mass of air) instead of number concentration. Whether the number of particles is computed per unit volume or per unit mass is a matter of preference and does not affect the conceptual understanding of the processes, which is the focus of this book. The primary difference in the formulation of the predictive equations is that the effect of the wind is accounted for in a number convergence term when calculating the volume concentration while an advection term is used when calculating the number mixing ratio.

68

PART I Fundamentals

type of cloud being represented. For example, at one extreme, if the clouds being modeled occur in the lower troposphere at temperatures > 0
C, then all the ice-hydrometeor categories and processes would simply be ignored. On the other hand, if the goal were to understand the effects of different ice nucleus populations on deep clouds extending from near the ground to the tropopause, then a bin or hybrid model allowing for the water substance to be subdivided into numerous liquid and ice-particle types and sizes would probably be necessary. The general philosophy of most modeling studies is to choose options leading to the least amount of computational intensiveness and complication of physical insight while accounting fully for the physics of interest in the clouds under study. The following sections cannot describe every possible combination. Rather we will examine the basic characteristics of each approach in a general way that provides the reader a rudimentary background from which it will be possible to appreciate the approach chosen in any particular modeling study.

(increase) in concentration associated with the expansion (contraction) of the volume of air within which the population of drops is located. The second term expresses the convergence of particles of a given size falling into and out of a volume air as a result of their terminal velocities Vi (defined to be positive downward), given by empirical and/or theoretical relations such as those discussed in Section 3.1.3. The terms on the lower line of (3.52) bin represent the microphysical processes that can change the number concentration of particles of a given size in a volume of air. They are, in order: nucleation, condensation/evaporation, collection, and breakup. These microphysical processes correspond to the first five of the seven basic microphysical mechanisms listed in Section 3.3. The remaining two on that list, ice enhancement and melting, do not apply in a warm cloud. To complete the water continuity in the bin model, the computation of the water-vapor mixing ratio is given by "

 # k Dqv 1X @Ni @Ni ¼ þ mi ðΔmÞi (3.53) Dt @t nucl @t c=e ρ i¼1

3.5

where k is the number of drop-size bins. Thus, the complete water continuity is expressed by the k þ 1 equations, contained in (3.52) and (3.53). By calculating the drop-size number distribution function Ni via the bin model, we seemingly should be able to calculate the liquid water content in each drop-size category by simply multiplying the mass mi of a drop in the size range i times the number of particles in that size category,

3.5.1

BIN WATER-CONTINUITY MODELS General

As noted above, bin water-continuity models predict the particle-size distribution in each of several narrow size ranges for a given category of water, such as all the liquid water contained in a unit mass of air. We will illustrate this technique by examining in detail how it is applied to clouds containing only liquid hydrometeors and then discuss some of the complications that arise when the method is extended to cold clouds.

3.5.2

Bin Modeling of Warm Clouds

In considering clouds without ice, none of the ice categories listed in (3.49), and the cloud and precipitation liquid water categories are combined into a single total liquid water mixing ratio qL qc þ qd þ qr. This total liquid water content is viewed as a continuous spectrum of drops, ranging from small, freshly nucleated cloud droplets to large raindrops. The size distribution function for the drops is represented by N ðmÞ, where N ðmÞdm is the number of particles per unit volume of air in mass range m to m þ dm. We may discretize this distribution function into k small but finite size ranges and calculate the distribution function for each size range as DNi @Vi Ni ¼ Ni rv þ Dt


@z 
 (3.52) @Ni @Ni @Ni @Ni þ þ þ þ @t nucl @t c=e @t col @t break where i refers to a particular particle-size bin. The first term on the right-hand side of (3.52) represents changes in Ni associated with air motions. It expresses the decrease

qLi mi Ni ðΔmÞi

(3.54)

where ðΔmÞi is the width of size category i. Such an approach has often been used. However, a more accurate approach, numerically, is to simultaneously predict the number distribution function Ni and the mass distribution function mi Ni by employing a stochastic collection equation for the latter as well as for the number distribution.45 The nucleation term (@Ni/@t)nucl in (3.52) is calculated by first making assumptions about the characteristics of the condensation nuclei present in the air. Then the appropriate concentration of drops is nucleated according to the supersaturation of the air and the relations governing nucleation (Section 3.1.1). Nucleation is the result of a diverse spectrum of CCN of various sizes and compositions. In a bin model, the goal is for the term (@Ni/@t)nucl to produce

45. Tzivion et al. (1987) showed that, in addition to the number distribution function Ni, stochastic collection concepts lead to predictive equations for higher moments of the size distribution function. The mass distribution is the third moment of the drop-size distribution, and these concepts therefore lead to a predictive equation for miNi that can be solved alongside that for Ni. This “method of moments” and other techniques for solving the stochastic collection equation are discussed by Khain et al. (2000).

Cloud Microphysics Chapter 3

the correct number of drops in each drop-size bin as a result of heterogeneous nucleation; that is, as many new drops of the critical size Rc that can be formed from the excess vapor, where Rc is calculated from (3.3) with the supersaturation determined relative to the saturation mixing ratio over a solution droplet made up of the CCN dissolved in the drop. These new droplets enter into the drop-size distribution function Ni according to their nucleated size at or near the low end of the drop-size distribution. Various numerical strategies exist for this nucleation calculation. Some models simply postulate that when supersaturation occurs, a certain size distribution of drops appears in the smallest size bins. More ambitious models predict a CCN number distribution function NCCNi where CCN are removed from size category i whenever they nucleate and are moved into the cloud drop-size distribution Ni. Such a scheme also must keep track of the regeneration of CCN when cloud droplets evaporate.46 Early attempts to simultaneously keep track of CCN in a bin model

predicted a joint number distribution function N mi , nj , where mi and nj represent mass size ranges for the drop and CCN, respectively. This approach was applied many years ago to a few highly specialized cases but not pursued further, likely because it encounters conceptual difficulties when drops coalesce and when more than one type of CCN is present.47 The vapor-diffusion term (@Ni/@t)c/e represents the rate of change in the number concentration of drops resulting from differing condensation or evaporation rates of drops in adjacent size categories. This rate of change is expressed by 


 @Ni @ _ dif N  m ¼ (3.55) @t c=e @m m¼mi


 @Ni ¼ I2 ðmi Þ  I1 ðmi Þ @t col

69

(3.56)

and the rate of production of drops of mass mi by spontaneous breakup can be expressed according to (3.36) as
 @Ni ¼ N ðmi , tÞPB ðmi Þ @t break ð1 N ðm0 , tÞQB ðm0 , mi ÞPB ðm0 Þdm0 (3.57) þ mi

However, as discussed in Section 3.1.6, when collisional breakup occurs, it dominates over spontaneous breakup, and when collisional breakup is included, its effects must be accounted for within the stochastic collection. Specifically, Equation (3.56) must be modified or replaced to include effects of the production of satellite drops.48 In situations where collisional breakup is occurring, the spontaneous breakup may be neglected.

3.5.3

Bin Modeling of Cold Clouds

A bin water-continuity model for cold clouds is an extension of that used for warm clouds. The basic idea is the same, but number concentration distribution functions must be predicted for more particle-type categories.49 The basic predictive equation is

 @Nij @Vij Nij @Nij @Nij ¼ Nij rv þ þ þ @t @z @t @t f =m


nucl 
 @Nij @Nij @Nij @Nij þ þ þ þ @t break @t col @t c=e @t d=s (3.58)

_ dif is the diffusional growth (or evaporation) rate of where m an individual drop of mass m. If the air is saturated with _ dif is given by (3.22). If not, it respect to liquid water, m is obtained by solving (3.8), (3.9), and (3.11) numerically. The term (@Ni/@t)c/e may be thought of as the convergence of number concentration of drops, where the growth rate _ dif plays the role of a velocity. m If collisional breakup (Section 3.1.6) is ignored, the collection term (@Ni/@t)col in (3.52) is given by the stochastic collection equation (3.33), which may be written for the i-th drop-size category as

where i is an index indicating the type of particle. For example: i ¼ 1 for water drops; 2 for plates; 3 for columns; 4 for branched crystals; 5 for graupel; 7 for frozen drops or hail (i.e., high density ice); and 8 for CCN.50 In addition to these traditional hydrometeor species, the ice-particle hydrometeors may be subdivided simply by their degree of riming.51 Subdivision of the ice spectrum by degree of riming will be discussed further in Section 3.7. Insofar as CCN and liquid particles are concerned, the computation of the microphysical terms on the lower row of (3.58) is as discussed in connection with (3.52). However, the

46. Xue et al. (2010) have formulated and applied a scheme for predicting the number distribution function for CCN and applied it to warm orographic clouds.

47. The prediction of a joint number distribution function N mi , nj was ´ rnason and employed by Silverman (1970), Tag et al. (1970), A Greenfield (1972), Clark (1973), and Silverman and Glass (1973). Further details of the technique can be found in these articles.

48. See Brazier-Smith et al. (1973) for the development of a stochastic collection scheme including collisional breakup. 49. The cold cloud bin methodology described here is presented in detail by Khain et al. (2004). 50. These are the particle-type categories used by Khain et al. (2004). 51. Morrison and Grabowski (2008a) introduced the idea of classifying the ice hydrometeors by degree of riming since the main quantitative differences of the different ice-hydrometeor categories relate to their densities, which in turn are largely determined by degree of riming.

70

PART I Fundamentals

possible presence of ice particles introduces additional effects that must be represented by these terms. The nucleation term (@Nij/@t)nucl expands to include not only the nucleation of droplets on CCN but also the generation of ice crystals by homogeneous nucleation and heterogeneous nucleation on CCN (Sections 3.2.1 and 3.2.2), as well as the generation of tiny ice particles by splintering during riming (Section 3.2.6.). Nucleation by immersion freezing is included in the freezing/melting term (@Nij/@t)f/m, in which an empirical relation is assumed for the function Nim(T), defined as the number density of ice nuclei suspended in liquid water at a given temperature. As the temperature decreases, the concentration of drops in the j-th size range is assumed to change by an amount 4 3 dNim dT N1j dN1j ¼  πr1j dT 3

(3.59)

where r1j is the drop radius. Melting of ice particles is computed as part of (@Nij/@t)f/m according to (3.48) or some similar formula, and the melted water enters the appropriate size category of the liquid drop spectrum. The breakup term (@Nij/@t)break in principle should include breakup of ice particles, as from thermal shock or shattering of branched crystals during collisions. But little is known about how to express these effects quantitatively. Computation of the collection term (@Nij/@t)col is the same as discussed in connection with (3.52) for the liquid drop category. However, this term must now include analogous stochastic collection expressions for ice–ice and ice–liquid collisions. These collection kernels for these other types of collection processes have considerable uncertainties associated with the collection efficiencies, fall velocities, and densities of ice particles (Section 3.2.4). For hailstones, the collection process must take into account shedding, which is a process that occurs if the latent heat released when accreted liquid freezes (Section 3.2.5) prevents all of the water from freezing. In that case, the unfrozen water separates from (i.e., is shed by) the hailstone and enters back into the drop-size spectrum. The condensation/evaporation term is the same as in (3.52), and the deposition/sublimation term is analogous but uses the vapor-diffusion equation for ice (Section 3.2.3) instead of that for liquid.

3.6

BULK WATER-CONTINUITY MODELS

The bin models are computationally demanding because the mixing ratio corresponding to each particle size must be predicted as a separate variable. The basic idea of bulk water-continuity models is to represent a cloud accurately by predicting as few variables as possible. This simplification requires that physics be more parameterized; that is, the physics are lumped together in ways that allow fewer degrees of freedom. However, the bulk method is used extensively in cloud dynamics and the following subsections outline its essential features.

3.6.1 The Classic Kessler Approach to Bulk Water-Continuity Modeling of Warm Precipitating Clouds The simplest type of cloud is a warm, nonprecipitating cloud. The minimum number of hydrometeor categories to describe it is two: vapor, represented by qv, and cloud liquid water, represented by qc. The total water-substance mixing ratio, qT ¼ qv þ qc, is conserved, and the watercontinuity model (2.21) consists simply of the two equations Dqv ¼ C Dt Dqc ¼C Dt

(3.60) (3.61)

where C represents the condensation of vapor when C > 0 and evaporation when C < 0. If a cloud is precipitating so that parcels of air lose hydrometeors by having them exit across the bottom of the parcel or gain particles as a result of them falling into the parcel from above, then the bulk water continuity becomes much more complicated. In the 1960s, Edwin Kessler devised a scheme for computing the bulk water continuity in a warm cloud by making several bold assumptions. His methodology has been used for decades, and its simplicity serves as an instructive introduction to bulk water-continuity modeling. However, his method has limitations that have led to modern improvements in the accuracy of this approach. In this section, we will briefly review the Kessler approach to introduce this topic. In the next subsection, we will discuss how this approach has been extended to remove these limitations. Kessler’s basic trick was to split the liquid water into two categories: cloud and rain. (Drizzle was either ignored or considered to be part of the rainwater.) The watercontinuity model (2.21) then expands into the three equations (instead of the two equations above that describe a nonprecipitating cloud): Dqv ¼ Cc þ Ec þ Er Dt

(3.62)

Dqc ¼ Cc  Ec  Ac  Kc Dt

(3.63)

Dqr ¼ A c þ Kc  E r þ F r Dt

(3.64)

where qr is the mixing ratio of rainwater, as defined in (3.49), and Kessler’s source and sink terms are the condensation of cloud water (Cc), the evaporation of cloud water (Ec), and the evaporation of rainwater (Er), and the autoconversion (Ac). The autoconversion is the rate at which cloud water content decreases as particles grow to precipitation size by coalescence and/or vapor diffusion. Kc is the collection (accretion) of cloud water, which is the rate at

Cloud Microphysics Chapter 3

which the precipitation content increases as a result of the large falling drops intercepting and collecting small cloud droplets lying in their paths. Fr is the sedimentation of the raindrops in the air parcel; it is the net convergence of the vertical flux of rainwater relative to the air. All of the terms on the right of (3.62)–(3.64) are defined to be positive quantities, except for Fr, which is positive or negative depending on whether more rain is falling into or out of the air parcel. According to this model, cloud water qc first appears by condensation Cc. Vapor is not condensed directly onto raindrops. Once sufficient cloud water has been produced, microphysical processes can then lead to the autoconversion (Ac) of some of the cloud water to rain. After autoconversion has begun to act, the amount of precipitation can then increase further through either Ac or Kc, or both. Once sufficient rainwater qr has been produced, the additional microphysical processes Er and Fr can become active.52 To calculate the microphysical sources and sinks of rainwater, Ac, Kc, Er, and Fr, in terms of the mixing ratios qv, qc, and qr, Kessler made several bold assumptions. To avoid calculating the drop-size distributions of cloud and rain, he disregarded the particle-size distribution for cloud droplets and assumed that raindrop sizes follow an exponential distribution, N ¼ No expðλDÞ

(3.65)

where N has the units number of particles per unit volume of air per unit diameter (i.e., m4), and the parameters No and λ control the magnitude and dispersion of the distribution. This function is called the Marshall–Palmer distribution.53 Kessler took the value of No to be an empirical constant. No has a value often taken to be  8  106 m4 for rain situations, but data show that this quantity has a lot of natural variability, sometimes changing by an order of magnitude from one part of a storm to another. Nevertheless, many studies have used a constant No, with the parameter λ being predicted such that it is consistent with the computed value of qr from the integral: ð1 1 qr ¼ ρ mðDÞNo expðλDÞdD (3.66) 0

Kessler’s assumptions about drop-size distributions have enabled rough approximations to the bulk water

52. The bulk warm cloud water-continuity model, consisting of the three categories vapor, cloud water, and rain interrelated by autoconversion and collection, was proposed by Kessler (1969). Virtually all bulk watercontinuity models used today are an outgrowth of the concepts introduced in Kessler’s seminal monograph. 53. Marshall and Palmer (1948) analyzed images of a large sample of manually sampled raindrops. They collected the images by exposing treated filter paper in the rain and timing each sample with a stopwatch. Their article describing the results of this experiment was slightly over one page in length. It has been cited over 1600. Today drop-size distributions are measured by a variety of automated methods.

71

continuity of clouds leading to many informative results. However, to obtain accuracy, one cannot ignore the cloud droplet size distribution, nor can one assume that No is constant. To remove these restrictive assumptions, Kessler’s approach has been extended to include the prediction of multiple moments (e.g., both the number concentration and the mass mixing ratio) of the drop-size distribution. This extension of the bulk approach will be discussed in the next subsection. Another major assumption of the Kessler methodology is that all of the raindrops constituting qr fall with the same mean velocity. For example, a common choice is to use the mass weighted fall velocity of the rain ð1 V ðDÞmðDÞN ðDÞdD (3.67) V^ 0 ð 1 mðDÞN ðDÞdD 0

where mðDÞ is the mass of a drop of diameter D, N(D)dD is the number of drops per unit volume of air with diameter D to D þ dD, and V(D) is the terminal velocity as a function of drop size given by a relation such as that in Figure 3.4. The above assumptions allow the microphysical source/ sink terms Kc, Er, and Fr to be calculated straightforwardly by integration over the raindrop-size distribution. According to the continuous collection equation (3.24), the rate of increase of the mass m of an individual raindrop of diameter D is given by _ col ¼ m

πD2 V ðDÞρqc Σ rc 4

(3.68)

where Σ rc is the collection efficiency of raindrops collecting cloud drops. The variables Σ rc and V(D) are assumed to be known empirically. The fall velocity of the cloud liquid water is assumed to be zero, in accordance with the definitions given in Section 3.3. Σ rc is often taken to be a constant  1, which is an approximation that is representative of the majority of rainfall situations (Section 3.1.4, Figure 3.5). The depletion of cloud water by accretion Kc by all of the raindrops in a parcel of air is computed from the integral ð1 _ col N ðDÞdD (3.69) Kc ¼ ρ1 m 0

where N(D) is assumed to have the exponential form (3.70). Carrying out this integration leads to the accretion term being roughly proportional to the products of the cloud and rainwater mixing ratios, specifically Kessler obtained Kc ∝qc q7=8 r qc qr

(3.70)

which is qualitatively reasonable since the accretion should jointly depend on the amounts of both cloud and rainwater.

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PART I Fundamentals

The bulk rate of evaporation of rainwater mass from all raindrops Er is computed in an analogous manner from the integral ð1 _ dif N ðDÞdD (3.71) Er ¼ ρ1 m 0

_ dif is the rate of evaporation by diffusion of vapor where m mass away from a single raindrop of diameter D falling _ dif as a function through unsaturated air. A relationship for m of D must be obtained by numerical solution of (3.8), (3.9), and (3.11). It is not given in general by (3.22), which applies only when the air is saturated (i.e., in cloud). Carrying out this derivation, Kessler found the term Er to be proportional to the rain mixing ratio qr and the saturation deficit of the environment. Often the rain is falling below cloud base, where the air is quite unsaturated. The mass weighted fallspeed of the rain, given by (3.67), can be used to calculate the sedimentation of raindrops in the air parcel according to Fr ¼

@ ^

Vqr @z

(3.72)

Thus the accretion, evaporation of rain, and sedimentation terms can all be expressed physically under the assumptions of the Kessler scheme. However, the autoconversion term has no straightforward parameterization under Kessler’s physical assumptions. So to complete the scheme, Kessler formulated the autoconversion rate Ac entirely intuitively. Specifically, he made the assumption that Ac should be proportional to the amount by which the cloud liquid water mixing ratio exceeds a user selected threshold; that is, αðqc  aT Þ Ac ¼ e

widely used as a simple parameterization method in many modeling applications. However, to achieve greater accuracy, bulk modeling of warm cloud water continuity has moved toward schemes that eliminate the more problematic and limiting assumptions of the classic Kessler scheme. One major improvement has been to replace the intuitive Kessler autoconversion with a physically based scheme that takes into account the size distribution of cloud drops, which was disregarded in the Kessler scheme. This improvement is especially important when considering the relationship of the condensed cloud water to the ambient concentration of CCN. One approach is to assume that the cloud drop-size distribution follows a gamma distribution N ¼ No Dμ expðλDÞ

(3.74)

where μ ¼ η2  1, and η is the ratio of the standard deviation to the mean of the particle radius. For cloud droplets, the value of η is empirically related to the number concentration of cloud droplets and is generally less than about 0.6. This empirical relation is used in (3.74). For larger drops, that is, drizzle and rain, the value of η tends toward unity, in which case the gamma distribution reduces to the Marshall–Palmer distribution (3.65).54 The Kessler approach described above is referred to as a single moment method of representing the water continuity. The methodology for eliminating assumptions of the Kessler method is a multimoment method, which is obtained from a generalization of the stochastic collection concept discussed in Section 3.1.5. If the k-th moment of the drop-size distribution is defined as ð1 ðkÞ mk N ðmÞdm, k 2 ℕ (3.75) M 0

(3.73)

where aT is the autoconversion threshold (often assumed to α is a positive constant when qc > aT and be 1 g kg1) and e 0 otherwise. Thus, whenever cloud water exceeds the threshold amount, it is converted to rainwater at an exponential rate. Although this crude representation of autoconversion has been much used, it is unsatisfying physically and disallows considerations of the role of different cloud droplet size distributions, especially insofar as these distributions are in turn related to ambient concentrations of CCN. This autoconversion assumption, as well as the assumptions of constant No and uniform fall velocity V^ are removed in the multimoment extension of Kessler’s approach discussed in the next section.

3.6.2 Multimoment Bulk Water-Continuity Modeling of Warm Clouds The Kessler approach described above has led to many groundbreaking papers on cloud modeling and is still

then the stochastic collection equation (3.33) can be generalized55 to ð ð @MðkÞ 1 1 1 ¼ N ðmÞN ðm0 ÞK^ðm, m0 Þ @t 2 0 0 (3.76) h i 0 k k 0k 0 ðmþm Þ  m  m dmdm It has been shown that the collection kernel K^ defined in (3.27) can be written as a polynomial of the form 8  < kc m2 þ m0 2 , m ^ m0 < m  (3.77) K^ ¼ : kr m2 þ m0 2 , m∨m0  m where kc and kr are constants and the mass m separates the cloud and precipitation portions of the drop-size

54. For discussion regarding η 1 for drizzle, see Wood (2005), and for rain Seifert and Beheng (2001). 55. This generalization of the “coagulation equation” was derived by Drake (1972).

Cloud Microphysics Chapter 3

distribution.56 The notation in (3.77) means that if both m and m0 are of cloud size then the coefficient kc applies, but if one of the drops or both are raindrops then kr applies. Consistent with Figure 3.1 and results such as those in Figure 3.6, this drop-size threshold is usually taken to be  40 μm. Using this threshold, the moments of the dropsize distribution can be partitioned into cloud and precipitation particle moments by integrating over subdivision of the drop-size distribution: ð m ðkÞ mk N ðmÞdm, k 2 ℕ, for cloud droplets (3.78) Mc 0

and MrðkÞ



ð1 m

mk N ðmÞdm, k 2 ℕ, for rain drops:

(3.79)

The 0th and 1st moments of these subdomains of the drop-size distribution are the number concentrations (Nc and Nr) and mixing ratios (qc and qr) of cloud and raindrops, respectively. These moments are the predictands in the multimoment method of bulk water-continuity modeling. This method thus predicts both the number concentration and mixing ratios of the two categories of liquid water, in contrast to the Kessler method, which predicts only the mixing ratios and makes rigid assumptions about the drop concentrations. The cloud water and rainwater mixing ratios are given by



 @qc @qc @qr @qr ¼ vrqc þ   @t @t c=e @t au @t rc (3.80) and



 @qr @qr @qr @qr ¼ vrqr þ þ þ @t @t c=e @t au @t rc
 @qr þ @t fall (3.81) As in the Kessler scheme, the autoconversion and accretion exchange water between the two hydrometeor species. The number concentrations of cloud droplets and raindrops are predicted by



 @Nc @Nc @Nc @Nc ¼ rNc v þ þ þ @t @t c=e @t au @t rc (3.82) and

56. The theoretical basis for this polynomial formulation was introduced by Long (1974).

73





 @Nr @Nr @Nr @Nr ¼ rNr v þ þ þ @t @t c=e @t au @t rc

 @Nr @Nr þ þ @t rr @t fall (3.83) The term (@Nr/@t)rr in (3.83) represents the self collection of raindrops, which systematically move raindrops into larger size categories when they coalesce. This term does not appear in (3.81) since self collection does not remove mass from the rain species; rather it only redistributes the mass by size. In the case of cloud droplets, self collection is manifest as autoconversion, which not only affects the number concentration but also moves mass from the cloud to rain species. The prediction of the number concentrations, represented by (3.82) and (3.83), is the primary advance over the Kessler scheme, which relies on rigid constraints on the particle concentration. The above Equations (3.80)–(3.83) are for a two moment approach, where particle concentration and mass are the 0th and 1st moments of the mass distribution defined by (3.75). A three moment scheme can also be employed that uses the radar reflectivity factor (Section 4.4) as an additional variable. The radar reflectivity is the 6th moment of the particle size [Equations (4.23) and (4.32)]. Thus, the reflectivity factor can be regarded as the 3rd moment of the mass distribution according to (3.75). Using the reflectivity as the third moment predictand has two major advantages. First, it is no longer necessary to prescribe the shape of the particle mass distribution function a priori (e.g., as a Gamma function). Second, since radar is the primary mode of detailed cloud and precipitation observation, the watercontinuity model results are more easily relatable to observations.57 In addition to the predictive equations for hydrometeor species, the water-vapor mixing ratio must be computed from


 @qv @qr @qc ¼ vrqv   (3.84) @t @t c=e @t c=e This calculation is crucial, and much work has been done to make this calculation numerically and physically accurate.58 To carry out the predictions of the number concentrations and mixing ratios of cloud and rain, parameterizations of the autoconversion, accretion, and self collection terms are required. That is, just as in the classic Kessler scheme, these terms must be computed from bulk properties of the drop-size distribution. Substitution of (3.77)–(3.79) into

57. See Milbrandt and Yau (2005) for details of the three moment approach to water continuity modeling. 58. See for example Morrison and Grabowski (2007, 2008b).

74

PART I Fundamentals

(3.76) leads to these required parameterizations.59 The autoconversion terms take the form
 @Nc ∝ No2 q2c (3.85) @t au and




@qc @t

 ∝ ð qc Þ a ð m c Þ b

(3.86)

au

where mc is the mean mass of cloud droplets, and the exponents a and b are in the range of 1.5–3.5, depending on the choice of assumptions in the derivation. The accretion and self collection terms have the forms
 @Nc

kr Nc qr (3.87) @t rc
 @qc

kr qc qr (3.88) @t rc and


 @Nr ¼ kr Nr qr @t rr

(3.89)

These relations are qualitatively reasonable since the accretion terms depend on the products of the bulk amounts of cloud and hydrometeors present, and the self collection depends on both the number and mass concentrations of rainwater. The accretion term is of the same form as that obtained in the Kessler scheme [recall Equation (3.70)].

3.6.3 Bulk Modeling of Cold Clouds By Extending the Kessler Scheme One approach to the bulk modeling of cold clouds is to extend the Kessler methodology to predict mixing ratios of cloud ice, snow, and graupel (or, alternatively, hail). This approach is single moment and therefore suffers the same limitations as the Kessler warm cloud water continuity, exacerbated by the multiplicity of forms of ice that must be represented. Nevertheless, this approach has been so widely used in the literature that we will first describe it and then follow in Section 3.7 with a discussion of the multimoment approach to bulk cold cloud water continuity. The Kessler bulk water-continuity scheme described in Section 3.6.1 can be extended to cold clouds by adding the hydrometeor species of cloud ice, snow, and graupel, represented by the mixing ratios qI, qs, and qg defined in (3.49).60 (One could alternatively include hail instead of graupel.) 59. See Beheng and Doms (1990) and Seifert and Beheng (2001) for details. 60. This extension of the bulk water-continuity method to cold clouds was developed by Lin et al. (1983).

We thus obtain the six category water-continuity scheme illustrated in Figure 3.21. The water-continuity equations (2.21) may then be written as Dqv ¼ ðCc þ Ec þ Er Þδ4 þ Sv Dt Dqc ¼ ðCc  Ec  Ac  Kc Þδ4 þ Sc Dt Dqr ¼ ðAc þ Kc  Er þ Fr Þδ4 þ Sr Dt DqI ¼ SI Dt Dqs ¼ Fs þ S s Dt Dqg ¼ Fg þ S g Dt

(3.90) (3.91) (3.92) (3.93) (3.94) (3.95)

where the S terms represent all of the sources and sinks associated with ice-phase microphysical processes, except for the sedimentation of snow and graupel, which are represented by terms Fs and Fg, respectively. The term δ4 is defined as

0 if T < 40
C (3.96) δ4 ¼ 1 otherwise Thus, it is assumed that, if the air temperature drops below 40
C, all supercooled water freezes by homogeneous nucleation (Section 3.2.1) and hence all the terms in the liquid water part of the model are set to zero. The terms on the right in (3.90)–(3.95) include all of the possible interactions among the six categories of water, as illustrated in Figure 3.21. Among these interactions are several bulk collection terms of the form (3.69). These represent graupel collecting cloud water and rainwater, snow collecting cloud ice, etc. There are also several evaporation terms of the form (3.71). These include the sublimation and depositional growth of snow, graupel, and cloud ice. In addition, there are melting terms representing the increase of rainwater mixing ratio as a result of the melting of snow and graupel. The process of shedding liquid water collected by but not frozen to the surface of graupel or hail particles is also included. There are also three way interactions that can occur, such as rain collecting cloud ice to produce graupel or hail. To obtain mathematical expressions for the Fg, Fs, and S terms in (3.90)–(3.95), the same types of basic assumptions are made about the precipitating ice particles as were made for raindrops in the warm cloud scheme. The knowledge of collection efficiencies of ice particles (summarized in Section 3.2.4) is less certain than for liquid drops. The collection efficiencies of ice particles collecting other ice particles is sometimes assumed to be a function of temperature that drops off exponentially from a value  1 at 0
C to zero

Cloud Microphysics Chapter 3

at lower temperatures. This assumption mirrors the observation of more frequent aggregation of falling particles as they near the melting level (Figure 3.13). The collection efficiency in riming is often assumed to be unity, but this assumption can be refined according to Figure 3.14. As in the Kessler warm cloud scheme, rigid assumptions are made about the particle-size distributions. The precipitation particles are assumed to be exponentially distributed, as in (3.65), but with different values of No. For example, No might be assumed to be  8  106–2  107 m4 for snow,61  4  106 m4 for graupel,62 and  3  104 m4 for hail.63 Various assumptions can be made about the size distribution of the cloud ice species, for example, it is often assumed to be monodisperse. The fallspeeds of snow and high density ice particles as a function of size are assumed so as to be consistent with the relationships discussed in Section 3.2.7, and each of the precipitation species in a parcel of air is assumed to fall with a mass weighted fall velocity, similar to that expressed for rain by (3.67).64

3.7 WATER-CONTINUITY MODELING OF COLD CLOUDS USING GENERALIZED MASS-SIZE AND AREA-SIZE RELATIONS65 The extended Kessler approach described above has the same limitations as the classic Kessler warm cloud watercontinuity model. The neglect of the size distribution of the cloud species, the rigid assumptions about the constancy of No in the precipitating hydrometeor species, and the vagueness of the autoconversion function, which plague warm cloud Kessler approach, are greatly increased when applied to the multiple hydrometeor species present in cold clouds (Figure 3.21). Unlike warm clouds, the distinction between cloud and precipitation sized hydrometeors is less distinct. For drops, the distribution quickly bifurcates by stochastic collection into cloud and raindrop populations that behave differently (Figure 3.6). Ice particles may grow to precipitation size by both vapor deposition and collection and several types of precipitation particles may develop (pristine crystals, aggregates, rimed crystals, graupel, hail, etc.) and, unlike cloud versus rain, the type and size 61. This value exhibits temperature dependence. See Houze et al. (1979). 62. See Rutledge and Hobbs (1983, 1984). 63. See Lin et al. (1983). 64. See Lin et al. (1983) for further details of how the cold cloud bulk parameterization terms may be formulated. Rutledge and Hobbs (1983) give a concise summary of the technique. 65. This section is based on the work of Morrison and Grabowski (2008a), who introduced this approach to bulk cold cloud water continuity. Their paper includes an informative introductory section elaborating on the history of bulk and bin microphysical modeling.

75

boundaries of these species are often indistinct. Typically one type of ice particle evolves into another slowly, not by the sudden transitions from one category to another in the extended Kessler formulation. These conceptual and quantitative difficulties have inspired the development of a methodology to allow particle-size distributions to evolve naturally. Accomplishing this improvement requires redefining ice-hydrometeor species to allow for gradual natural transitions from on particle-type category to the next. The redefinition of the ice-hydrometeor species is based on the fact that although ice particles have a myriad of shapes and textures, the main factors needed for predicting the number and mass concentrations of ice particles are the dependence of particle mass and cross sectional area on the particle’s long dimension. The mass-size relations allow the calculation of moments of the particle-size distributions from equations of the form (3.75) for ice particles. Processes such as the collection of liquid drops by ice particles depend on expressions of the form (3.24), which demand knowledge of the area-size relation for the ice particles. The literature on ice-particle microphysics provides enough information to determine the required mass-size and areasize relations if we recognize that two main factors determine these relations: (i) whether the particles are quasi-spherical or quasi-platelike, and (ii) the degree of riming that the particles undergo. Figure 3.22 illustrate mass-size and area-size distributions for ice particles obtained using these two factors and the properties of ice particles gleaned from the literature. Particle mass and size increase first by vapor deposition, then as the particles become larger, they particle grow by riming to fill in crystals and finally by particles becoming more spherical and accumulating water mass as dense balls of ice (graupel or hail). These distinctions happen gradually as the particles get bigger, without having to make sudden changes of species, as in the extended Kessler approach. The change in the nature of particles occurs more naturally and without having to track the behaviors of different types of particles. Using the continuous functions in Figure 3.22, the moments of the ice spectrum and the physical processes determining the number and mass moments of the ice-particle spectrum (number concentration and mixing ratio) can be predicted. Since the mass-size and area-size relations in Figure 3.22 depend on the degree to which the particles have undergone riming vis-a`-vis depositional growth, separate mixing ratios are computed for the mass accumulated by vapor deposition qdep and riming qrim. Note that this redefinition of ice categories in terms of their three-dimensional geometry (platelike versus spherical) and degree of riming can be applied in any water-continuity scheme (single or multimoment bulk scheme or bin scheme). To illustrate how this approach might be used, the equations below indicate how a two moment bulk water-continuity scheme might be formulated with this new approach. The predictive

76

PART I Fundamentals

(b)

(c)

(d)

FIGURE 3.22 (a) Mass–dimension relationships in unrimed conditions for solid spherical ice and unrimed nonspherical ice and boundary Dth separating the two particle-type regimes. (b) Schematic diagram of a gamma particle-size distribution N(D) divided into two regions based on Dth. (c) Mass–dimension relationships in rimed conditions for solid spherical ice, graupel, dense nonspherical ice, and partially rimed ice and critical dimensions Dth, Dgr, and Dcr. (d) Schematic diagram of the gamma particle-size distribution N(D) divided into four regions based on Dth, Dgr, and Dcr. From Morrison and Grabowski (2008a). Republished with permission of the American Meteorological Society.

log N(D)

log N(D)

(a)

equations in that case (neglecting subgrid-scale turbulence effects) would be:



 @N @N @N @N @N þ þ þ ¼ @t @t nucl @t sub @t frz @t mlt


 @N @N @N @N V^N þ þ þ þ rvN þ @z @t enh @t agg @t mltc (3.97)


 @qdep @qdep @qdep @qdep @qdep ¼ þ þ þ @t @t nucl @t dep @t sub @t frz

 @qdep @qdep @qdep V^q þ þ þ rvqdep þ @t mlt @t mltc @z (3.98)



 @qrim @qrim @qrim @qrim @qrim ¼ þ þ þ @t @t frz @t mlt @t sub @t accc

 @qrim @qrim @qrim V^q þ þ þ rvqrim þ @t accr @t mltc @z (3.99)




where v is the wind vector, V^N and V^q are the number and mass weighted mean particle fallspeeds, respectively, analogous to V^ in (3.67). The symbolic terms on the right-hand side of (3.97)–(3.99) represent the source/sink terms for N, qrim, and qdep. These include primary nucleation on aerosol through deposition or condensation freezing (nucl), vapor deposition (dep), sublimation (sub), freezing of cloud droplets and rain (frz), melting (mlt), ice enhancement (enh; affecting N only), aggregation of ice (agg; N only), collection of cloud droplets (accc; qrim only), freezing of rain due to ice–rain collisions in subfreezing conditions (accr; qrim only), and melting of ice due to ice–rain collisions in above freezing conditions (mltc). Formulas for these terms are based on the processes discussed in previous sections of this chapter.