Cirrus Cloud Simulation Using Explicit Microphysics and Radiation ...

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Cirrus Cloud Simulation Using Explicit Microphysics and Radiation. Part I: Model Description VITALY I. KHVOROSTYANOV Central Aerological Observatory, Dolgoprudny, Moscow Region, Russian Federation

KENNETH SASSEN Department of Meteorology, University of Utah, Salt Lake City, Utah (Manuscript received 26 March 1996, in final form 5 September 1997) ABSTRACT A mesoscale 2D/3D cloud model complex with explicit account for the water and ice cloud microphysics and radiative processes is described. The model has several versions suitable for the simulation of various cloud types, including, of particular concern in the current version, cirrus clouds. Model computations are based on the two kinetic equations for droplet and crystal size distribution functions, with division of droplet and crystal size spectra into 30 bins from 1 mm to 3.5 mm, and with account for the various mechanisms of cloud condensation and ice nuclei activation, condensation–deposition, and coalescence–accretion growth. These equations are solved along with supersaturation and radiative transfer equations for longwave and solar radiation. Simple yet accurate analytical expressions are presented for the scattering and absorption coefficients, and the single-scattering albedo. This allows detailed calculations of the optical and radiative characteristics of clouds (i.e., fluxes, divergence, and albedo) to be made at each grid point of the computational domain, thus yielding the spatial and time evolution of these properties along with the evolving cloud microstructure and phase state. Results of a simulation for a typical midlatitude cirrus cloud using the current model are presented in Part II of this paper.

1. Introduction One of the most significant factors that modulates the climate of the earth–atmosphere system is the globally widespread occurrence of upper-level cirrus clouds (see Liou 1986). Various numerical studies have indicated that the radiative effects of these clouds depend critically on their physical and microphysical properties (e.g., Stephens et al. 1990), and according to reasonable scenarios employed in general circulation models, even the sign of their net effect on climate (e.g., from heating to cooling) is still open to question. Other major uncertainties that hamper attempts to improve our treatment of cirrus clouds in general circulation models (GCMs) involve the characterization of their formation, vertical composition variations, and mesoscale structures. Since GCM results depend so sensitively on the cloud parameterization schemes employed, a major focus of current atmospheric research is to refine the numerical treatment of high clouds in key weather and climate model components. Along with focused experimental research, findings from numerical cirrus cloud

Corresponding author address: Kenneth Sassen, 819 Browning Bldg., University of Utah, Salt Lake City, UT 84112. E-mail: [email protected]

q 1998 American Meteorological Society

simulations offer considerable promise to help improve this situation. Numerical cloud models can be divided into two basic classes with respect to the treatment of microphysical processes: those with parameterized microphysics, usually based on the Kessler parameterization that characterizes the hydrometeors into 3–6 different types with prescribed shapes for the size spectra, and those with the detailed simulation of microphysical processes, usually based on solutions to the kinetic growth equations for populations of cloud droplets and/or ice crystals. The latter method is, of course, more computationally intensive, but it is also more physically well founded since it allows the calculation of the temporal and spatial variations in hydrometeor spectra, which can strongly influence the cloud characteristics of concern to climate research. Such properties include cloud optical properties like scattering and absorption coefficients, singlescattering albedo, and emissivity, as well as the rates of phase transition, and precipitation formation and fallout. Although models with explicit microphysics are still few in comparison to the number of bulk models, a rapid development of computer capabilities and numerical techniques, along with a better understanding of cloud microphysics, have recently stimulated new efforts in the development of such models. The task of properly describing the synoptic-scale

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KHVOROSTYANOV AND SASSEN

(1000–5000 km) and mesoscale (1–200 km) structures of cirrus clouds, which requires the simulation of the dynamic fields over these scales, is at present a very difficult one. Currently, modeling efforts are being directed either to reproducing the gross structure of cirrus and the environment on synoptic scales (Heckman and Cotton 1993), or the simulation of mesoscale properties at scales of 0.5–100 km (e.g., Starr and Cox 1985a,b; Cotton et al. 1986; Levkov et al. 1992; Demott et al. 1994; Mitchell 1994; Mitchell and Arnott 1994; Jensen et al. 1994; Khvorostyanov 1994, 1995). At even shorter cloud microphysical scales, the parcel modeling of cirrus clouds has shown value for serving as the basis for improving the parameterization of clouds and radiation in larger-scale models (Sassen and Dodd 1989; Heymsfield and Donner 1990; DeMott et al. 1994; Khvorostyanov and Khvorostyanov 1994). Here in Part I we describe our approach for simulating the 2D mesoscale properties of cirrus clouds, and in Part II (Khvorostyanov and Sassen 1998) we shall present the model findings in terms of microphysics, vapor and ice mass budgets, and optical and radiative properties for a rather typical midlatitude cirrus cloud. 2. General model description The 2D version of our mesoscale 2D/3D cloud model with explicit microphysics and radiation, which has been under development for more than 20 years, has been applied to various cloud types (see Khvorostyanov 1994, 1995). Although this model has been described previously, it is necessary to consider here the revisions required to better account for the specific processes involved in cirrus cloud formation and development. In general, computations are based on two kinetic equations for the droplet and crystal size distribution functions, along with vapor saturation and radiative transfer equations. The model contains six basic kernels: 1) cloud microphysics and thermodynamics, 2) mesoscale dynamics, 3) longwave radiation, 4) solar radiation, 5) radiation exchange with the underlying surface, and 6) the transport of cloud condensation nuclei (CCN) and ice nuclei (IN), and their interaction with clouds and radiation. This approach allows detailed calculations of the phase transformations and dynamics, precipitation, and the optical and radiative characteristics of the simulated cirrus clouds. The model is capable of simulating the transport of aerosols of up to 20 different types (natural CCN and IN, volcanic aerosols, aircraft emissions, etc.), providing the opportunity to investigate various processes of anthropogenic and natural impact on the microphysical and radiative properties of clouds, and on their chemical processes. Figure 1 provides a schematic representation of the framework of the model. The number of model grid points is variable in various model versions; in the horizontal from 33 to 61 in the 2D version, and 21 3 21 in the 3D version; in the vertical, 31 levels for the main units and up to 64–151

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for dynamics and radiation. The horizontal domain (grid length) can range up to 300–500 km (3–5 km). Both the horizontal and vertical domains and grid lengths are varied in order to simulate various cloud types. The time step is 10 s for dynamics and transport of heat, humidity, droplets, crystals, and aerosols, and 0.1–10 s for microphysics. The time step value is chosen for conditions of computational stability. 3. Model microphysics and thermodynamics a. Governing equations The system of equations for microphysics and thermodynamics in mixed-phase clouds contains two kinetic equations for the droplet f d and crystal f c size or mass distribution functions, which depend on space variables x, y, and z, time t, and masses m d and m c of the individual droplets and crystals: f d (x, y, z, t, m d ) and f c (x, y, z, t, m c ) or, in terms of their equivalent radii r d and r i , f d (x, y, z, t, r d ) and f c (x, y, z, t, r i ). Equivalent radius of a crystal is defined as the mass-equivalent radius, that is, the radius of a spherical crystal with the same mass with account for the ice density (see below). According to the scheme of Magono and Lee (1966), there are up to 80 various forms of ice crystal habits. The most frequent forms in cirrus clouds are bullet rosettes, columns, plates, and irregular aggregates, although the presence of other shapes is possible (e.g., Heymsfield et al. 1990; Sassen et al. 1989). Crystal shapes may be very complicated, and crystal habits and axis ratios can vary during the growth process. Strictly speaking, each of these shapes should be described by the distribution function depending on the three variables representing the three crystal axes. Such a description would be extremely complicated, so various simplifications are employed. One of the most important simplifications is based on the fact that ice crystals are often axially symmetric, such that their shapes can be described by two characteristic dimensions (i.e., length along axis r and width l), or, alternatively, their shapes can be approximated by spheroids, cylinders, or plates. One can then treat either the crystal mass growth rate dm/dt, or the crystal axes growth rate r˙c (e.g., Hall and Pruppacher 1976; Stephens 1983; Sassen and Dodd 1988, 1989; Heymsfield and Sabin 1989; Mitchell 1994; Jensen et al. 1994). Although some of these studies consider the crystals as an ensemble of monodispersed particles, others consider collective effects during the growth–evaporation of the polydispersed media, which is described by some a priori specified function like the gamma distribution (e.g., Starr and Cox 1985a; Cotton et al. 1986; DeMott et al. 1994; Mitchell 1994), or by a size distribution function evaluated from kinetic equations (e.g., Khvorostyanov 1994, 1995; Jensen et al. 1994). Both of these approaches still require some prescription of the crystal axes relation, which can be either fixed (i.e., independent of size) for a particular shape,

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FIG. 1. Schematic representation of framework of the 2D/3D model, showing the required input and the predicted output information.

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or parameterized as a function of size with regard to empirical relations between mass and size. These latter are retrieved from measurements or geometrical relations for various shapes like hexagonal columns and plates, bullet rosettes, cylinders, or spheroids (see, e.g., Mitchell and Arnott 1994). We employ the second approach and describe the crystal size distributions over the one (major or minor, depending on the shape in question) axis r i , which can be called the equivalent radius, or simply ‘‘radius,’’ for the case when we consider the volume equivalent sphere, as in Part II. The use of the size distribution function f i (x, y, z, t, r i ) allows us to account for the crystal shapes by specifying the axes ratio, j f 5 r i /l, functions of actual radius r i or major (minor) axis (see below). In the general 3D case, the kinetic equations for the size distributions can be written in the form (see Cotton and Anthes 1989; Jensen et al. 1994; Khvorostyanov 1994, 1995): ] f i (x, y, z, t, r i ) ] 1 = · (U f i ) 2 [y i (r i ) f i ] ]t ]z

1 2

ˆ fi 1 ] fi 5D ]t 1

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1 ]t 2 ] fi

1

nucl

1

freez

1 ]t 2 ] fi

1

cond /evap

1 ]t 2 ] fi

1 melt

1 ]t 2

1 ]t 2

] fi

] fi

coag

.

(1)

mult

Here r i is the crystal radius in the sense discussed ˆ the operator above, y i (r i ) denotes its terminal velocity, D ˆ 5 (]/]x)k x(]/]x) 1 (]/]y)k y(]/ of turbulent diffusion, D ]y) 1 (1/r a )(]/]z)r a k z(]/]z), and U 5 (u, y , w) is the vector of the wind speed. The last six terms on the righthand side of Eq. (1) describe the following microphysical processes: nucleation, condensation–deposition (or evaporation), coagulation of crystals, freezing, melting, and the multiplication of particles, respectively. These processes are evaluated or parameterized as described below. Fields of the potential temperature u and humidity q are calculated with the equations: ]u ˆ u 1 L w« cw 1 L i« ci 1 L f «freez 1 = · (Uu) 5 D ]t cp cp cp 1

(2)

1 2 1T2 ,

Lf ]T « 1 c p melt ]t

]q ˆ 2 « cw 2 « ci ; 1 = · (Uq) 5 Dq ]t

rad

u

(3)

where « ci is the integral deposition–sublimation rate, « cw the integral condensation–evaporation rate, (L f /c p )« ci and L f /c p )« cw the latent heating rates of deposition and condensation (evaporation), (L f /c p )«freez and (L m /c p )«melt the freezing–melting heating rates, and (]T/]t)rad 5

rad rad (]T/]t)rad is the radiative long 1 (]T/]t)short; here (]T/]t) temperature change, and the last two terms are longwave and shortwave radiative heating–cooling rates. The integral deposition–sublimation rate « ci can be expressed via individual growth–evaporation rates and size distribution function:

E1 `

« ci 5

0

2

dm i (r i ) f i (x, y, z, r i , t) dr i . dt

(4)

This quantity is related later to the supersaturation, crystal number density, and mean size. b. Ice nucleation The mechanisms of nucleation and formation of pristine crystals are of key importance for cirrus clouds, as they determine the subsequent deposition–coagulation growth rate of crystals and their optical properties. It is also one of the most difficult questions of cirrus cloud physics to treat because neither the dominant mechanisms under different conditions nor the exact values of parameters are satisfactorily known. With the importance of correctly treating the ice nucleation process, we review and discuss our current knowledge below. For many years, it was believed that the process of initial ice crystal formation was dominated by the heterogeneous nucleation of ice nuclei (IN), and nucleation rates were parameterized by use of a Fletcher-type temperature dependence in the IN activity spectrum. However, observations of highly supercooled liquid water at cirrus altitudes and temperatures as low as 2378C (Sassen and Dodd 1988; Sassen 1992; Heymsfield and Miloshevich 1993) indicated that heterogeneous nucleation may not always be dominant at high altitudes, especially within vigorous updrafts. It was apparent that if heterogeneous nucleation was dominant, exceedingly fast crystallization should occur at these temperatures, and so no liquid water could be observed. An explanation for the existence of highly supercooled water near cirrus cloud base was given by Sassen and Dodd (1988, 1989) and Heymsfield and Sabin (1989), who hypothesized that within intense updrafts, the process of ice nucleation below ;2358C would proceed via the formation and growth of haze particles (consisting of ammonium sulfate solutions) and their subsequent homogeneous freezing. After condensational growth and freezing, the haze particles give rise to a numerous crystal population. Sassen and Dodd (1988, 1989), using a 1D model with explicit microphysics, suggested that the freezing rate of haze particles (after correcting for solution effects) was considerably greater than predicted by classic homogeneous nucleation theory, and with an updraft speed of ;1 m s21 , the number density of crystals formed from haze particles could reach the exceptionally high values of 3 3 10 3 L21 (i.e., without any interference from the heterogeneous nucleation of IN).

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These findings stimulated further intensive studies of homogeneous nucleation and comparisons with heterogeneous nucleation in cirrus clouds. A very detailed study of homogeneous nucleation and comparison with heterogeneous nucleation was performed by DeMott et al. (1994) using the results of laboratory cloud chamber experiments, a parcel model, and a 2D model version of the Colorado State University CSU/RAMS mesoscale model. Under similar assumptions (i.e., CCN consisting of ammonium sulfate), they produced several interesting findings. First, the homogeneous nucleation crystal production rate depended strongly on the value of vertical velocity w: the crystal number density formed at typical cirrus temperatures (2388 to 2468C) increased from 100 to 200 L21 at w 5 5 cm s21 to 10 4 L21 at w 5 50 cm s21 , which is comparable to the values obtained by Sassen and Dodd (1988) and Heymsfield and Sabin (1989) at w 5 1 m s21 . When both the homogeneous and heteorogeneous mechanisms were allowed to act in the parcel model at w 5 5 cm s21 , and the vertical profile of IN was chosen to decrease with height, heterogeneous nucleation could act at all levels and be dominant at lower levels, while homogeneous freezing dominated at higher levels. Under similar conditions, except without an IN height dependence, the changes were dramatic: heterogeneous activation of IN and their subsequent growth lowered the saturation ratio and completely switched off homogeneous nucleation. So DeMott et al. (1994) concluded that for this particular case, ‘‘the homogeneous process becomes irrelevant for clouds with vertical motions less than 30 cm s21 .’’ Thus, it follows that the dominant ice nucleation mechanism is determined by the vertical CCN and IN profiles and the value of vertical velocity in updrafts. These characteristics are strongly variable in space and time, so we can conclude from the results of DeMott et al. (1994) that both homogeneous and heterogeneous mechanisms may act during cirrus formation: the homogeneous mode seems to dominate within strong updrafts (i.e., embedded convective cells), while the heterogeneous one dominates within weak updrafts (i.e., cirrostratus with weak synoptic-scale uplift and lacking pronounced convection). A comprehensive study of cirrus microphysics sensitivity to the mechanisms of nucleation was carried out by Jensen et al. (1994) with use of a 1D model with explicit microphysics. They modeled an event of cirrus formation intensively studied during the 1986 First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment (FIRE) (IFO-I) field campaign and assumed that the dynamic forcing in their 1D framework could be simulated by choosing a vertical velocity profile in the cirrus layer with a maximum vertical velocity wmax near 8–9 km. Their results indicated the following: 1) using only homogeneous crystal nucleation via haze particles freezing, crystal concentrations N I less than 10 2 L21 were produced at wmax 5 6 cm s21 , which increased to ;10 3 L21 at wmax 5 20 cm

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s21 ; 2) simulation with heterogeneous nucleation gave almost the same values of integral cloud microphysical and optical properties; and 3) comparison of the crystal size spectra simulated with use of the homogeneous nucleation scheme with those measured during the FIRE IFO-I case study showed general agreement, but there was a lack of the small crystals predicted by the model (,50 mm), which were probably not sampled effectively by the particle measuring systems (PMS) probes. Jensen et al. (1994) noted that with homogeneous nucleation, ‘‘appreciable number densities of small particles should only occur near the top of the cloud where ice crystal nucleation occurs and only for a short time after nucleation occurs.’’ However, this is in contradiction with growing numbers of measurements in cirrus showing the presence of numerous small crystals at various positions within cirrus clouds, especially using newer counters capable of detecting small particles (see Part II). Thus, if the measurements are correct, homogeneous nucleation may explain the high concentrations of crystals nucleated in generating regions but is alone unable to explain the correct shape and time evolution of the size spectra in all cloud regions. Additional mechanisms may exist that could provide a more continuous influx of newly activated crystals and better account for the behavior of measured size spectra. One possible mechanism is heterogeneous nucleation, which does not need threshold humidity and is simply proportional to the cooling rate. As discussed below, both the homogeneous and heterogeneous nucleation mechanisms are accounted for in the present model. The term (] f i /]t)nucl in Eq. (1) describes the nucleation of crystals, that is, crystal generation rate, J c . This process can be divided into two main modes, homogeneous 1 and heterogeneous freezing with the rate J c 5 J hom c . The first term includes both the two processes of J het c the homogeneous freezing of droplets and haze partihom cles, J hom 5 J hom c c,drop 1 J c,haze. Homogeneous freezing of droplets might be essential for cirrus formation within the vigorous updrafts, where the liquid phase may form and freeze, for example, in orographic cap and lee waves, convective anvils, and cirrus uncinus cells. As the process of cirrus formation described in Part II proceeds within weak updrafts without liquid phase, we do not describe here homogeneous freezing of droplets used in the model. Heterogeneous crystal generation at the rate J het c proceeds through several modes (Pruppacher and Klett 1978): deposition freezing, condensation freezing, and heterogeneous drop freezing caused by contact nuclei J het c,cont. Meyers et al. (1992) noted that it is difficult in nature to separate the processes of pristine crystal nucleation through the deposition freezing and condensation freezing modes, and suggested describing them as a single mode. Thus it is appropriate to consider the combined deposition–condensation–freezing rate and

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denote it as J het c,df. Then, the total crystal nucleation rate can be described as

1 ]t 2 ] fc

hom het het 5 J hom c,drop 1 J c,haze 1 J c,df 1 J c,cont

(5)

nucl

Below, we examine in more detail our treatment of the two main ice crystal nucleation mechanisms in cirrus clouds. 1) HOMOGENEOUS

FREEZING OF HAZE PARTICLES

An important process of cirrus ice crystal formation is the freezing of haze particles. This process differs from the freezing of nearly pure cloud droplets because solution effects can become quite important. Detailed studies by Sassen and Dodd (1988, 1989), Heymsfield and Sabin (1989), and DeMott et al. (1994) showed that the process of haze particle deliquescence and freezing occurs in an environment increasingly subsaturated with respect to water with decreasing temperature. We use in the present model a parameterization developed by DeMott et al. (1994) especially for mesoscale models. The total concentration of haze droplets freezing over the time step Dt is

E

`

N hf 5

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[1 2 exp(2J hom c,haze V hDt) f h (r h )] dr h ,

(6)

0

where V h is the volume of a haze particle, f h (r h ) is the size distribution function, and J hom c,haze is homogeneous freezing rate that can be evaluated based on homogeneous nucleation theory (Pruppacher and Klett 1978), which yields the expression J hom c,haze 5 (N l kT/h) exp[2(DGcr 1 DFact)/kT ], DGcr 5 (4p/3)s i2w rcr2 ,

(7)

rcr 5 (2s i2w /r i L f ) ln(T m /T ), (8)

where DGcr is the critical Gibbs energy of formation of an ice embryo, rcr the radius of the embryo, DFact the activation energy, k and h the Boltzmann and Planck constants, s i-w the surface tension at the ice–water interface, T and T m the droplet and melting point temperatures, and N l the number of water molecules per unit volume of the liquid. The most uncertain parameters here are the values of s i-w and DFact . Equation (7) could be directly used for the parameterization of homogeneous freezing with the values of constants that were determined from the earlier laboratory experiments. However, Sassen and Dodd (1988) showed that the measured value of the homogeneous droplet freezing rate in high-level clouds is about 1 3 10 6 times less than that predicted by the classical homogeneous nucleation theory (Pruppacher and Klett 1978), which stimulated new studies of this phenomenon. Starting from Eq. (7), DeMott et al. (1994) accounted for the corrections to the freezing temperature T m caused

by solution effects in a manner similar to Sassen and Dodd (1988). Based on these detailed evaluations of freezing rates, an approximation for the fractions of haze particles freezing F hf for a given diameter D s and water saturation ratio S w was found: F hf 5 1 2 exp(2a DD sc 3), a D 5 (pr s ) 2/6 3 10 c11c 2(12S w) ,

(9)

where r s is solute density for ammonium sulfate, parameters c1 and c 2 are functions of solution droplet temperature T lc 5 T l 2 273.16 as c1 5 214.65 2 1.045T lc , c 2 5 2492.35 2 8.34T lc 2 0.061T lc2 , c 3 5 6, and Eq. (9) is valid for 0.82 , S w , 0.99. Finally, DeMott et al. (1994) integrated over the haze size spectrum with the assumption that it displays an exponential form and offered a parameterization of the total number concentration DN hf of haze particles freezing during Dt, which we write in the form J hom c,haze 5 DN hf /Dt

E

`

5 (N h /Dt)

exp(2y){1 2 exp[2y c3 (a D D nc3 )]} dy.

0

(10) This expression is used in the simulation of cirrus clouds described in Part II. Using a model time step of a few seconds, we assume that a crystal formed from a frozen haze particle is prescribed to the first bin size of the model (here, 2.0 mm). Its subsequent growth is governed by the collective effects in the field of supersaturation according to Eq. (1). 2) HETEROGENEOUS

ICE NUCLEATION

Parameterization of the deposition–condensation– freezing rate J het c,df are usually based on various results of laboratory and in situ measurements, as typically expressed by the exponential temperature dependence of Fletcher (1962), or the combination of Fletcher’s and power-law supersaturation dependence as suggested by Cotton et al. (1986). More advanced parameterizations have been discussed by Meyers et al. (1992), Sassen (1992), and DeMott et al. (1994). Sassen (1992), employing a large amount of polarization lidar observations of supercooled liquid water clouds collected during project FIRE extended time observations, in the temperature range from 228 to 2388C and heights from 3.6 to 10.2 km, proposed a height-dependency correction to the Fletcher dependence of the IN temperature activation spectrum, as N(T) 5 A S0 exp(2A z z) exp[B S (T S 2 T)].

(11)

This modification to the Fletcher equation (where A s0 and B s vary according to IN conditions, and T s 5 2738C) consists of introducing the height z dependent term exp(2A z z), which attempts to account for the vertical decline in surface-derived ice nuclei with height. The

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value of parameter A z that fits the observational data of supercooled water (i.e., does not cause too rapid crystallization and thus allows homogeneous nucleation to take place) was found in Sassen (1992) as A z 5 0.75 km21 . In the simulation presented in Part II, we use this parameterization after conversion into a nucleation rate for use in the kinetic equation [Eq. (1)] as

1 2 ]f ]t

dN c dN c dT 5 ; dt dT dt

het

5 J het c,df (r c , T ) 5

nucl

dN c d(r I ), dt (12)

where d(r I ) is Dirac’s delta function that describes the appearance of crystals in the first size bin. From Eqs. (11) and (12), we have, finally, J het c,df (r c , T ) 5 A s0 exp(2A z z) exp(B s (T s 2 T ))B s (2dT/dt)d(r I )u (d i ) (13) where u(d i ) is the Heaviside step function, which means that activation is allowed when d I . 0 (i.e., the threshold humidity for activation is dependent on temperature). c. Deposition–sublimation The third term on the rhs of Eq. (1) is related to the growth rates by vapor diffusion of the mass (dmi /dt) and size r˙i of the individual particles, and can be written as

1 ]t 2 ] fi

52

cond /evap

1 2

] dm i f , ]m i dt i

dm i 5 4pC e Fcond Fkin Fven . dt

(14) (15)

Here, C e denotes the shape factor of a crystal, and the factors on the rhs account for the condensation– deposition and growth–evaporation (Fcond ), the kinetic effects on the diffusion growth mechanism due to the vapor pressure jump near the crystal surface (Fkin ), and the enhancement of growth–evaporation due to ventilation effects (Fven ), which are evaluated similar to that in Hall and Pruppacher (1976) and Stephens (1983), through Schmidt and Reynolds numbers. The term Fcond can be written as Fcond 5

Dd ir a , r i Qi

d i 5 q 2 q si (T ),

(16)

where D is the water vapor diffusion coefficient, r i the density of ice, r a the air density, d i the supersaturation with respect to ice, q si the saturation humidity over ice, and Q i 5 1 1 (L i /c p )(]q si /]T) is the psychrometric correction. The shape factor C e is usually defined from the electrostatic analog (Fletcher 1962). Hall and Pruppacher (1976) pointed out that unfortunately, analytical expressions for this quantity are only available for simple

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geometrical shapes as that of a sphere, an oblate and prolate spheroid, and a disk. Fortunately, these shapes approximate geometrically more complex shapes as columnar crystals (prolate spheroid), and hexagonal plates and bullet rosettes (oblate spheroid) that are common in cirrus clouds. This allows us to relate mass growth rate (dm/dt) and size growth rate r˙c , and follow the evolution of the size spectrum from Eq. (1), which can then be compared directly to the size spectra measured in situ. We consider a spheroid with axes, (r i , l c , l c ), and an axes ratio, j i 5 l c /r i . The mass of the spheroid can be written as m i 5 (4p/3)r il c2 r i 5 (4p/3)r ir i3 /j i2 . Differentiating m i by time and using Eq. (15) we obtain for the growth rate r˙ c of the axis of a crystal radius r c : r˙ c 5

Dd ir a k fi (j i ) F F , r i Q i kin ven j i2

(17)

where k fi 5 C e /r i is the electrical capacity factor. For a spherical particle, C e 5 r i , so, k fi 5 j i 5 1. For a prolate spheroid, j i , 1, k fi 5 2E e /[ln(l 1 E e )/(l 2 E e )], where E e 5 (1 2 j i2)1/2 . For an oblate spheroid, j i . 1, k fi 5 1/2 (1 2 j22 /arccos(j22 i ) i ). Several crystal types, such as plates, bullet rosettes, and columns, can be simulated with these equations by variation of j i . In Part II, we present results for volume equivalent spheres, since then the crystal size spectra and optical properties can be recalculated for any other shape using mass-size relations (e.g., Mitchell and Arnott 1994). Equations (1)–(17) explicitly include the value of supersaturation, which is calculated in the following way. The system of equations for thermodynamics and microphysics is solved by the splitting method (Marchuk 1974), which is based on the ‘‘splitting’’ of a multidimensional equation with the sources on the rhs describing the evolution of any 3D field as a consequence of the one-dimensional equations describing transport along the axes (splitting by the directions) and zerodimensional equations describing adjustment processes (splitting by processes) as illustrated below. In this model, the fields of temperature, humidity, and particle size spectra at one model time step, Dt, are calculated (split) in five substeps, thus the variables at each substep are denoted by the corresponding fractional indices. The first three substeps account for the transport of the substances due to turbulent and wind transport along x, y, and z: ]w (1/5) ](uw (1/5) ) ] ]w (1/5) 1 5 kx , ]t ]x ]x ]x ]w (2/5) ](yw (2/5) ) ] ]w (2/5) 1 5 ky , ]t ]y ]y ]y ]w (3/5) ](w ˜ w (3/5) ) ] ]w (3/5) 1 5 kz . ]t ]z ]z ]z

(18)

Here w j denotes any of the parameters u, q, or f c ; w˜ 5 w for u, q and w˜ 5 w 2 y i for f c ; the index (1/5) means

15 MAY 1998

the first substep of 5, etc. These differential equations are solved by using the sweeping technique (Samarsky and Nikolaev 1977), that is, they are approximated by the finite-difference equations that represent three-point recurrent relations in space and are implicit with respect to time. This essentially permits an increase in the time step (as compared to the explicit time schemes) without violation of computational stability. In this model, the second-order approximation was chosen. During the fourth substep, the growth–evaporation of crystals by deposition–sublimation are calculated and we also treat the effects caused by supercooled cloud droplets, which may contribute to cirrus ice particle nucleation under some conditions. This is done after a transition to the equations for supersaturation with respect to ice, d i 5 q 2 q si (T), and supersaturation with respect to water, d w 5 q 2 q sw , which determines also mechanisms of nucleation. After the removal of space variables, in fact, a Lagrangian zero-dimensional problem is solved and the equations are similar to those used in a parcel model (e.g., Sassen and Dodd 1988, 1989). In order to clarify the difference between liquid clouds, where high supersaturation cannot exist, and crystalline clouds where it may exist (see Part II), we consider now a three-phase cloud containing vapor, droplets (these can be haze droplets in cirrus), and crystals. Equations for the temperature and humidity can be written in the form dT L L 5 w« cw 1 i« ci 2 wg a 1 Rrad , dt cp cp dq 5 2« cw 2 « ci . dt

(19)

Differentiating supersaturation d k 5 q 2 q sk (T) by time (index k means here both water and ice), and after substituting, we obtain: dd i dq dq dT 5 2 si dt dt dT dt 5 2(« cw 1 « ci ) 2 3

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[

E1 E `

« ck 5

0

2

dm k (r k ) f k (x, y, z, r k , t) dr k dt

`

5

(4pDr k k fir ad k /Q k ) f k (r k ) dr k

0

5 (4pDk fid k /Q k )

E

r k(r a f k ) dr k

5 (4pDN k r k k fid k )/Q k ,

5

k5

d, droplets c, crystals

(21)

We can now introduce the phase relaxation times for crystals t fc and for droplets t fd

t fd 5 (4pDN dr d )21 ,

t fc 5 (4pDN Ir I k fi )21 , (22)

and with account for the fact that the values of Q w and Q i are close to unity (i.e., at cirrus altitude and temperatures between 2308 and 2508C, they differ from 1 by 3%–7%, and their relation gives a correction of 2%– 3%), we can rewrite the deposition and condensation rates with use of Eqs. (21) and (22) as

« cw 5 d w /t fd ,

« ci 5 d i /t fc .

(23)

Thus, Eq. (23) means that the deposition and condensation rates are equal to the mean rates of supersaturation relaxation. Substituting Eqs. (22) and (23) into (20), and neglecting again Q mn we obtain the following equations for supersaturations with respect to water and ice: dd w 21 21 5 2(t 21 fd 1 t fc )d w 2 t fc (q sw 2 q si ) dt 1

[

1 2

[

1 2

]q sw ]T gaw 2 ]T ]t

]

rad

,

(24a)

dd i 21 21 5 2(t 21 fd 1 t fc )d i 1 t fd (q sw 2 q si ) dt ]q si ]T

1

1 2

Lw L ]T « cw 1 i« ci 2 g a w 1 cp cp ]t

[

rad

]

5 2« cw Q wi 2 « ci Q ii 1 (]q si /]T ) g a w 2

1 ]t 2 ]T

]

,

rad

(20) where Q mn 5 1 1 (L m /c p )(]q sn /]T), and indices m 5 (w, i) and n 5 (w, i) denote water and ice. Expressions for deposition and condensation rates can be obtained from Eq. (15) and (16), using the equation for the individual mass growth rate (dm k /dt) 5 4pDr k k fi r a d k (here we neglect for simplicity the corrections Fven and Fkin , which will be introduced in the final formulas):

]q si ]T g w2 ]T a ]t

]

rad

.

(24b)

These equations, along with the temperature equation, describe the kinetics of the three-phase cloud system containing water vapor, droplets, and crystals. Note that supersaturations can represent small differences between two large values (i.e., humidity and saturated humidity) such that the approach based on the evaluation of supersaturation seems to be more precise than the direct calculation of humidity, while Eqs. (24a,b) provide a direct physical basis to choose an optimum time step for integration, which should be reasonably less (but not much less) than the minimum in relaxation times. To illustrate the physical meaning of relaxation times, let us consider a lifting parcel before nucleation takes place. The third terms on the rhs of Eqs. (24a,b) describe

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JOURNAL OF THE ATMOSPHERIC SCIENCES

growth (generation) of supersaturation due to positive vertical velocity and radiative cooling. At this time, crystals and droplets are still absent, hence t 21 fd 5 0, t 21 fc 5 0, and the first two terms are switched off. After formation of liquid, haze, or ice particles by either the homogeneous or heterogeneous mode, these two terms differ from zero and begin to slow down supersaturation growth: the first term describes vapor absorption by liquid particles and crystals. The second term on the rhs describes the increase of supersaturation d i (or decrease of d w ) due to the diffusion of water vapor from droplets to crystals (i.e., it acts in mixed clouds only). Since saturated humidity over ice is lower than that over water (the Bergeron–Findeisen mechanism), this term is proportional to the difference (q sw 2 q si ). In this study, we consider processes without vigorous updrafts and without the liquid phase. So, after the freezing of haze particles and the formation of pristine crystals, t 21 fd 5 0, and we can rewrite Eq. (24b) for the pure crystalline cloud:

[

1 2

dd i d ]q ]T 5 2 i 1 si g a w 2 dt t fc ]T ]t

]

rad

.

(25)

If the first term becomes much larger than the second one, that is, the rate of supersaturation depletion due to absorption by crystals exceeds the generation of supersaturation by vertical velocity and radiative cooling described by the second term (e.g., t fc becomes much smaller by that time), the solution to Eq. (25) is

d i (t) 5 d i (tmax ) exp(2t/t fc ).

(26)

Thus, during the time period t fc, supersaturation is decreasing (‘‘relaxing’’) by e-times due to the vapor-toice phase transition. Due to this reason, it is called the ‘‘phase relaxation time.’’ Its value characterizes the capability of a crystalline cloud to absorb vapor. It should be noted that in a mesoscale model t fc is a local characteristic of cloud microstructure and there is generally a 3D field of t fc (see Part II). Also, relaxation time itself depends on time and usually decreases in a growing cloud. Because characteristic times of t fc variations are usually much larger than those of supersaturation, t fc can serve as a useful characteristic of supersaturation variability and the rate of accumulation of cloud mass. The third terms on the rhs of Eqs. (24a,b) describe the continuing generation of supersaturation in a cloud due to upward vertical velocities and radiative cooling (]T/]t)rad , or the occurrence of negative supersaturation and evaporation of the cloud, if [g a w 2 (]T/]t)rad ] , 0. After some time, an equilibrium state may be reached in cirrus when both terms on the rhs compensate for each other, leading to the concept of equilibrium supersaturation (see Part II). Note that t fc is a very important quantity not only for mesoscale models but also for GCMs and climate models. In both mesoscale and large-scale models, after steps of horizontal and vertical transport, an imbalance exists

VOLUME 55

between the fields of humidity and temperature (i.e., saturated humidity). Thus, supersaturation or subsaturation occur at each grid point, and an ‘‘adjustment’’ step is needed to evaluate the amount of condensate produced or evaporated at each time step. This adjustment may be the instantaneous condensation of all the vapor excess into the condensed phase (or instantaneous evaporation under subsaturation) or only some portion of the vapor excess may be condensed during a finite time. In this respect, the phase time t fc is probably the most informative quantity because it serves as a characteristic time of the phase transitions, and characterizes the apparently limited rate of vapor absorption by cirrus cloud particles. In a mixed-phase cloud, the first terms on the rhs of Eqs. (24a,b) describe relaxation of supersaturation due to the vapor absorption by droplets and crystals during 21 21 the time t f,eff 5 (t 21 , which is the ‘‘effective fd 1 t fc ) relaxation time.’’ Since the value of t f 5 t fd for a pure droplet cloud is typically of the order 1–10 s, the process of phase transformations in liquid clouds is rapid compared to the model time step, and the residual supersaturation is usually a fraction of a percent. However, as will be shown in Part II, t f 5 t fc in a pure crystalline cloud varies from a few minutes up to a few hours, and so the process of phase transition is relatively slow and only a small portion of supersaturation could be transformed into ice during the usual model time steps. Analyses of field experiments often confirm that a residual supersaturation with respect to ice of 5%–10% exists even in developed cirrus clouds; see, for example, Sassen et al. (1989). So, during the slow process of largescale weak ascent (e.g., ‘‘large-scale condensation’’ in GCMs), a quasi-equilibrium value of residual supersaturation may exist such that the amount of uncondensed vapor may actually be comparable with the ice water content (see appendix, Part II). This may strongly influence cirrus cloud optical and radiative properties, and, in particular, slow down precipitation formation, increase cirrus lifetime, and decrease cloud optical thickness in both the solar and infrared regions, thereby influencing the relation between cloud albedo and greenhouse effects. In this model, during the step of adjustment, the evolution of the supersaturation field is being evaluated self-consistently along with particle growth– evaporation processes. The spatial and temporal evolution of t fc will be analyzed in Part II for a simulated cirrus cloud and can be used directly in the most advanced GCMs (e.g., Fowler et al. 1996). d. Coalescence and accretion of crystals During the fifth and final step, the growth by coalescence and accretion of crystals is calculated according to the stochastic collection equations (Cotton and Anthes 1989). After calculating the distribution functions f c , such quantities as ice water content (IWC, q LI ), number concentrations of crystals N I , their mean radii r I ,

15 MAY 1998

effective radii reff , radar and lidar reflectivities, and other characterisitcs can be calculated as the integrals (moments) over the size spectra in each grid point at each time, as

E

`

NI 5

E

`

f c (r i ) dr i;

r I 5 (1/N I )

0

q LI 5

1 3 2E

5. Longwave and solar radiation

`

4pr i

r i3 f i dr i .

(27)

e. Ice mass budget The evolution of condensed ice is governed by gravitational settling and by wind and turbulent transport. The characteristics of this transport can be evaluated with use of the previously calculated size distribution function f i (r i ); that is, the gravitational flux of crystals (precipitation rate) Pgrav 5 ∫ `0 m i (r i )y i(r i ) f i (r i ) dr i , where m(r i ) is the mass of an individual crystal and y (r i ) is its terminal velocity. In the same manner, the regular flux Pver is caused by vertical velocities, and the turbulent flux of crystals Ptur is caused by the vertical gradient of the distribution function 2k z(] f i /]z). The total budget of crystal mass (dM/dt)tot , which determines the rate of change of IWC, consists at each grid point of four components: 1) the deposition–sublimation rate (dM/dt)dep , which is equal to, but of opposite sign to the supersaturation depletion; 2) the influx (dM/dt)grav due to the different gravitational settling at different levels; 3) the influx (dM/dt)ver caused by the gradient of the regular vertical flux Pver ; and 4) the turbulent influx of crystals (dM/dt)tur caused by the gradient of the turbulent flux Ptur :

1 2 1 2 tot

dM 5 dt

dep

1 2

dM 1 dt

grav

1 2

dM 1 dt

with use of equations for the vorticity h and the Poisson equation for the streamfunction c. After that, vertical and horizontal velocities are calculated. A more detailed description for the various versions of dynamics is given in Khvorostyanov (1994, 1995).

r i f i (r i ) dr i;

0

0

dM dt

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KHVOROSTYANOV AND SASSEN

ver

1 2

dM 1 dt

. tur

(28)

These components of the mass budget can be calculated as the gradients of corresponding fluxes defined above, for example, (dM/dt)grav 5 2(]Pgrav /]z), etc. The fluxes and components of the mass budget will be analyzed in Part II. 4. Dynamics The mesoscale dynamics in the 3D versions of the model is determined by solving the seven equations of motion and continuity for the horizontal (u, y ) and vertical (w) components of the wind speed along with the equation of turbulence energy, the similarity and dimensional relations of the mixing length l, vertical turbulence coefficient k z , and dissipation rate of turbulence energy e b . This system of equations is closed with a first- or second-order closure hypothesis for the mixing length of stratified flows. Mesoscale dynamics in the 2D model versions is calculated in the vertical X–Z plane

The radiative fluxes were calculated by solving the radiative transfer equations in the two-stream approximation with account for the rapid variations in the cloud optical properties (absorption and scattering coefficients). Two well-known major problems arise here: integration over broad wavelength bands and the evaluation of optical properties. The usual methods of integration over wavelengths are based on the use of integral transmission functions, which are rather time consuming and may take 30%– 50% of the computational time. Therefore, a simplified but sufficiently accurate method for the calculation of longwave fluxes and cooling–heating rates was chosen. It is based on the schematization of Kondratyev for the longwave spectrum, which represents a simplified version of the k-distribution method (see Liou 1992). According to Kondratyev (1969), the absorption spectrum of water vapor for longwave radiation (LWR) can be described well enough by dividing the spectrum into n (4–6) regions with different transparency and vapor absorption coefficients a y n . Accounting for absorption, but neglecting scattering in the LWR range, fluxes in each nth region F ↑,↓ and total longwave upward and downn ward fluxes F ↑,↓ can be calculated according to the usual l two-stream equations:

O

1

216P B 7 F 2 ,

dF ↑,↓ n (n) 5 b lr a a y n q 1 a Lj q Lj dz j51,N F ↑,↓ 5 l

OF

↑,↓ n

.

n

↑,↓ n

(29)

n

Here, b l 5 1.66 is the diffusivity, B(T) the blackbody radiative flux; P n the relative fraction of the radiative flux for the nth part of the LWR spectrum (with S P n 5 1); and the summation over j represents all absorbing substances including vapor, droplets, crystals, and aerosols (with a Lj(n) being the mass absorption coefficients in the nth region). The spectral divisions and coefficients were tuned by comparison with the results of line-byline calculations from the Intercomparison of Radiative Codes for Climate Models (ICRCCM; Ellingson and Fouquart 1990). After tuning, an error in fluxes does not exceed 3–8 W m22 through the whole troposphere, and an error in the LWR balance is less than 2–5 W m22 , which is quite sufficient for the cloud model with a rather short time of integration. The spectral fluxes F ↑sl , F ↓sl , and the flux divergence of solar radiation are calculated in the two-stream approximation with account for the cloud microstructure and phase state:

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JOURNAL OF THE ATMOSPHERIC SCIENCES

1 dF ↑,↓ v l ↑,↓ V l ↑,↓ sl 5 F ↑,↓ (F sl 1 F ↓,↑ (F sl 2 F ↓,↑ sl 2 sl ) 7 sl ), 2 2 Ï3 dt i

aabs Li 5

O s˜ @O s˜ , V 5 O (s˜ ^cosu &)@O s˜

1

N

vl 5

sc lLi

i51

at lLi

3 (1 1 c i )2( p i11) 2

N

sc lLi

i

i51

sc lLi

,

3 pi 1 1 4r i r I p i 1 3

[

1

p i 1 1 l 2 (m li 2 1) 2 2 k l2i p i 1 2 4pr I2 [(m li 2 1) 2 1 k l2i ] 2

]

2

8pk li r I 1 11 l ( p i 1 1)

2( p i13)

,

2

(31)

where the summation in (30b) is taken over all absorbing and scattering substances; v l is the single-scattering albedo; t l the optical thickness; s SlLi , a lS L , a ly the mass scattering and absorption coefficients of droplets, crystals, aerosols, and vapor at the wavelength l; and ^cosu i & is the scattering asymmetry factor of the ith substance. In this model, the spectral solar fluxes were calculated for 31 wavelengths, to resolve in the 0.4–4-mm interval eight basic absorption bands: a, 0.8m, rst , f, c, V, X, 3.2m. This enables the study of the spectral and vertical dependence of absorption, transmission, and albedo in the cloud layer. Note that Eq. (30a) is written for the total SW fluxes (direct plus diffuse); the source term (direct solar flux) is absent in (30a) and is used as an upper boundary condition. Since cloud optical properties may vary strongly at each grid point and change rapidly along with microstructure, it is a very difficult task to evaluate them with the use of time-consuming Mie theory in each grid point at each time step. Therefore, the following approach is used in this model. First, calculated size spectra are approximated by gamma distributions in the radiative units of the model, and then the simple but accurate formulas for the mass scattering coefficients s sclLi and absorption coefficients aabs Li of crystals are derived from van de Hulst (1957) anomalous diffraction theory (ADT). These formulas were revised in this study to include the modified anomalous diffraction theory (MADT) and the effects of refraction, as described by Stephens et al. (1990). We obtained these coefficients by integrating ADT absorption and scattering efficiencies with gamma distributions in a simplified form that relates these coefficients to the mean radius and the index of gamma-distribution p I :

3 11

c i ( p i 1 2)

2( p i12) 3 (1 1 c iÏ1 2 m22 i )

i51

s˜ atlLi 5 s sclLi q Li 1 a lLi q Li 1 a yl q,

2m iÏ1 2 m22 i

(30b)

s˜ sclLi 5 s sclLi q Li ,

s sclLi 5

2m i2 c ( p i 1 1)( p i 1 2) 2 i

i51

N

l

[

3 2m i2 11 (1 1 c i )2( p i12) 4r i r i c i ( p i 1 2)

(30a) N

VOLUME 55

(32)

2m i2 c ( p i 1 1)( p i 1 2) 2 i

]

2( p i11) 3 (1 1 c iÏ1 2 m22 , i )

(33)

where the mean radius r I is related to the effective radii reff,i often used in radiative calculations as reff,i 5 ( p i 1 3)/( p i 1 1)r I , c i 5 8pk lr I /l( p i 1 1); m li , k li are the real and imaginary parts of the refractive index of ice (or water). The absorption coefficient a Li has different asymptotic behavior for small and large r I :

a Li 5

6pk l (for small r I or small k l ), l

a Li 5

3 pi 1 1 3 5 (for large r i ). 4r i r I p i 1 3 4r i reff

(34)

The first limit does not depend on r i and applies in the atmospheric 8–12-mm window for small particles; a Li 5 3100 cm 2/g at l 5 10.8 mm, k l 5 0.177 for ice [channel 4 of Advanced Very High Resolution Radiometer (AVHRR)], and a Li 5 6490 cm 2/g at l 5 11.9 mm, k l 5 0.410 (channel 5 of AVHRR). This strong dependence on wavelength shows a good potential for the detection with remote sensing techniques of small droplets and crystals if their contribution to the cloud emissivity dominates. The second limit in Eq. (34) shows an inverse dependence of absorption coefficient on mean radius a Li ; r21 eff , which agrees with a parameterization proposed by Stephens et al. (1990) for l 5 11 mm based on Mie calculations. This limit is valid for large particles but can overestimate a Li for the small particles present, for example, in the upper layers of cirrus clouds. Formula (34) can be considered as a generalization of the parameterization of Stephens et al. (1990) because it is valid over a wide range of various relations between radii and wavelengths. The accuracy of this approach for typical cloud particles with size spectra expressed in the form of gamma distributions is within 1%–5% as compared to the Mie calculations (e.g., Khvorostyanov and Khvorostyanov 1994). The parameterization of scattering and absorption coefficients of the form

s 5 a 1 b/reff

(35)

15 MAY 1998

KHVOROSTYANOV AND SASSEN

was proposed by Slingo (1989) for liquid clouds and by Ebert and Curry (1992) for crystalline clouds. One can see that the form of Eq. (35) is similar to (34), thus following from ADT. Equations (34), (35) can serve as a basis for such parameterizations and for the initial fitting of parameters. The gamma distribution, which is used to approximate the calculated size spectra of droplets and crystals, is determined by three parameters, N I , r i , and p i : f c (r i ) 5 cN Ir pi exp[2( p i 1 1)r i / r I ].

(36)

These three parameters can be determined after the calculation of distribution functions f c (r i ) from the model. According to Eq. (27), N I , r i , q LI are first evaluated. Then the dimensionless parameter n i can be introduced, and the parameter p i of the gamma distribution can be expressed as

ni 5

q Li ; (4/3)pr i r i3 N i

pi 5

2n i 2 5 6 Ï1 1 8n i 2(1 2 n i )

. (37)

These quantities are used in the radiation units of the model. Note that although the mass scattering and absorption coefficients depend on the mean radius and index p i , their dependence on p i is rather weak, as shown by Stephens et al. (1990). Thus the main variation of these coefficients is determined by the mean radius, that is, by that part of the size spectrum to the right of the modal radius, and not by the fraction of the smallest particles. Calculation of mean radius is reliable if the size spectra are regular enough. It will be shown in Part II that they are, and so the error introduced in optical coefficients by approximating the size spectra calculated from the kinetic equation with gamma distributions is relatively small. 6. Conclusions The model described here represents one of the first attempts to apply a microphysical cloud model for the simulation of cirrus clouds (see Part II). This mesoscale cloud model with explicit water and ice cloud microphysics is characterized by three distinguishing features. 1) The model does not use prescribed or parameterized hydrometeor size spectra as found in many cloudresolving bulk models and some advanced GCMs, but is based on the two kinetic equations for droplet and crystal size distribution functions, with division of both size spectra into 30 bins from 1 mm to 3.5 mm, and accounting for the various mechanisms of particle activation and growth. 2) The model does not assume 100% humidity with respect to water or ice (that is, saturation adjustment is not used); rather the kinetic equations are solved along with the supersaturation equation in an interactive manner. Thus, supersaturation excess is not transformed immediately into the condensate, allowing for the possibility that humidity may locally exceed the saturated

1819

humidity, perhaps significantly. With this approach, the processes of supersaturation generation by updrafts, advection, and radiative cooling are evaluated in competition with supersaturation depletion due to ice crystal vapor absorption. It will be shown in Part II that these effects are especially strong in cirrus clouds and may result in a quasi-equilibrium residual supersaturation (i.e., uncondensed ice) that is comparable to, or larger than, the amount of condensed ice. In GCMs, this residual supersaturation should be subtracted from the amount of condensed ice. Therefore, it can be seen that it is not only the vapor field that influences particle growth rates, but, in turn, the crystal or droplet populations also influence the supersaturation field, and a negative feedback occurs in clouds with respect to the external forcing (cooling rate). The stronger the cooling rate (proportional to the crystal size and concentration), the weaker the residual supersaturation (inversely proportional to size and concentration), which governs particle growth. 3) The model includes detailed calculations of the optical and radiative characteristics of clouds (i.e., scattering and absorption coefficients, fluxes, divergence, and albedo) at each grid point of the computational domain, thus accounting for the spatial and temporal evolution of these properties. It will be shown in Part II that optical cloud properties (i.e., scattering and absorption coefficients) are strongly variable in space, and this causes the nonlocal character of radiative transfer through the cloud. This nonlocality is well understood for gaseous absorption but is still insufficiently understood for cloud absorption and scattering, and, as shown in Part II, has major consequences for radiative flux calculations in models using optical coefficients that are constant with height. This finding is the result of the treatment of detailed microphysics in our model. Hence, this model belongs to the type of so-called microphysical models that are still very few in number compared to bulk cloud models, which consider a cloud as a continuous medium and describe it with use of three to five sorts of some size distribution functions with an a priori chosen shape. Both of these approaches have their own advantages and deficiencies. Although the microphysical approach seems to be more detailed and accurate, it is far more time consuming than the bulk approach. In comparison, bulk cloud models can support simulations using larger domains, higher resolutions, and longer time periods. Thus, they can provide information on cloud development that might be lacking in microphysical models. In these respects, both approaches can be considered as complementary to each other and should be used in tandem. Despite rapid developments in computer speed capabilities, we can hardly expect that GCMs will use explicit microphysics in the near future, while the bulk microphysics now used in climate models are based on mesoscale cloud models with parameterized microphysics (e.g., Fowler et al. 1996). In this regard, inter-

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JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 55

action among microphysical and bulk cloud models, and (dM/dt)dep , components of ice mass budget climate models might proceed as follows: 1) in-depth (dM/dt)grav , studies of spatial–temporal microphysical cloud prop(dM/dt)ver , erties with microphysical models, with recommenda(dM/dt)tur , tions for improved cloud parameterizations of both me(dM/dt)tot soscale cloud models and GCMs; 2) in-depth studies of m c mass of a crystal dynamical fields on large scales (with account for the m i , m lI real part of refractive index of ice effects of cloudiness) with bulk cloud models and/or N I crystal concentration GCMs, with recommendations for microphysical mod- Pgrav , Ptur , P y e fluxes of crystals els on the selection of space–timescales and dynamical p i indices of gamma distribution features; and 3) direct intercomparison of the results q, q LI specific humidity, IWC obtained by cloud models with explicit and parameterq w , q i saturated humidity over water and ice ized microphysics for the same simulated cases, yielding r I crystal mean radius, refinements in microphysical treatments and recomreff effective optical radius mendations for GCMs. r˙ c crystal axes growth rate In Part II of this paper, we will illustrate the extent T kinetic temperature to which our modeling approach can provide new in(]T/]t)rad radiative temperature change sights into the formation and maintenance of cirrus a Lj(n) longwave mass absorption coefficient clouds, and on this basis, we point out where improvea y n water vapor absorption coefficient ments in large-scale model parameterizations may be a slL shortwave mass absorption coefficient made. In future numerical research we will provide more g a dry adiabatic lapse rate detailed recommendations for GCMs concerning a range d w 5 q 2 q w supersaturation with respect to water d I 5 q 2 q i supersaturation with respect to ice of cirrus cloud optical and radiative properties, and va« cw , « ci , «freez , condensation, deposition, freezing, and por–ice budgets. «melt melting rates, respectively k lI imaginary part of refractive index of ice Acknowledgments. The interest of one of the authors r i , r w , r a densities of ice, water, and air (V.K.) to simulate cirrus clouds was stimulated during s slLi, a slLi spectral mass scattering and absorption his stay at the University of Hamburg and GKSS Recoefficients of SW radiation search Center in Germany, and the Laboratoire Optique u potential temperature Atmospherique (LOA) of the Lille University in France. t fc , t fd crystal and droplet phase relaxation time V.K. is grateful to H. Grassl (WCRP), S. Bakan (Hamt l spectral optical thickness burg University), E. Raschke, M. Quante, and R. Stuhlj k crystal axes ratio mann (GKSS), and Y. Fouquart, M. Herman, G. Brogv l single-scattering albedo niez, J. Descloitres, and V. Giraud (LOA) for their kind hospitality and numerous useful discussions. Recent model improvements and simulations have been supREFERENCES ported by Grant DE-FG03-94ER61747 from the Department of Energy Atmospheric Radiation Measure- Cotton, W. R., and R. A. Anthes, 1989: Storm and Cloud Dynamics. ment program, and by NSF Grant ATM-9528287 (for Academic Press, 883 pp. , G. J. Tripoli, R. M. Rauber, and E. A. Mulvihill, 1986: NuK.S.). APPENDIX

List of Symbols Ce ^cosu& D F ↑l , F ↓l F ↑s , F ↓s , F ↑ls, F ↓ls fc hom J hom , J c,haze c,drop, het J het c,d, J c,haze k fi Lw, Lc, Lf , Lm ,

shape factor of crystal growth rate scattering asymmetry factor water vapor diffusion coefficient integral upward and downward fluxes of longwave radiation integral and spectral upward and downward fluxes of shortwave radiation crystal size distribution function homogeneous and heterogeneous crystal nucleation rates electrical capacity factor of a crystal specific heat of condensation, deposition, freezing, and melting, respectively

merical simulation of the effects of varying ice crystal nucleation rates and aggregation processes on orographic snowfall. J. Climate Appl. Meteor., 25, 1658–1680. DeMott, P. J., M. P. Meyers, and W. R. Cotton, 1994: Parameterization and impact of ice initiation processes relevant to numerical model simulation of cirrus clouds. J. Atmos. Sci., 51, 77–90. Ebert, E. E., and J. A. Curry, 1992: A parameterization of ice cloud optical properties for climate models. J. Geophys. Res., 97, 3831–3836. Ellingson, R. G., and Y. Fouquart, 1991: The intercomparison of radiation codes in climate models: An overview. J. Geophys. Res., 96, 8925–8927. Fletcher, N. H., 1962: The Physics of Rainclouds. Cambridge University Press, 386 pp. Fowler, L. D., D. A. Randall, and S. A. Rutledge, 1996: Liquid and ice cloud microphysics in the CSU general circulation model. Part I: Model description and simulated microphysical processes. J. Climate, 9, 489–529. Hall, W. D., and H. R. Pruppacher, 1976: The survival of ice particles falling from cirrus clouds in subsaturated air. J. Atmos. Sci., 33, 1995–2006. Heckman, S. T., and W. R. Cotton, 1993: Mesoscale numerical sim-

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