1. Introduction
The characteristics and evolution of the cloud droplet size spectra determines cloud radiative properties and the formation of precipitation. To account for these processes in cloud-resolving models and climate models requires correct understanding and then parameterization of cloud microstructure and its dependence on predicted atmospheric parameters.
Condensation theory for a constant updraft has been shown to narrow the droplet spectrum with time (e.g., Mordy 1959; Neiburger and Chien 1960; Buikov 1961). Since broad spectra are observed in clouds and are required to initiate collision and coalescence growth of precipitation, various hypotheses and approaches have been developed to explain and to describe broad spectra in clouds. Warner (1969), Baker and Latham (1979), and Telford and Chai (1980) have attempted to explain the broadening of the size spectra and occurrence of the bimodal spectra and large droplets (before coagulation) by mixing of cloud parcels with environmental air. Additional mechanisms that have been proposed include horizontal inhomogeneities in vertical velocity (e.g., Mazin 1968) and in cloud condensation nuclei (CCN) (Kabanov et al. 1970), microscopic supersaturation associated with random distribution of drops (Srivastava 1989), and sedimentation of the cloud droplets (Baker et al. 1984; Considine and Curry 1996).
The theory of stochastic condensation attributes the presence of broad drop size spectra in clouds to the occurrence of condensation in a turbulent medium. This theory was developed intensively during the period 1960–80. While considerable research was devoted to the explanation of the various mechanisms of the broadening of the droplet size spectra (see reviews in Cotton and Anthes 1989; Pruppacher and Klett 1997), the treatment that is mathematically most promising describes stochastic condensation in terms of kinetic equations that allow direct calculations of the evolution of the size spectra. These equations have been derived for two limiting cases: the low-frequency regime in which the characteristic Lagrangian timescale of turbulent fluctuations, τL, is much greater than the supersaturation absorption time, τf, that is, τL ≫ τf; and the high-frequency regime in which τL ≪ τf. Since τf ∼ 1–10 s in a developed cloud, and τL ∼ 5–10 min, it was believed in earlier studies that the high-frequency approximation might be representative of rapidly changing conditions such as might occur in an entrainment zone or during the initial phase of condensation, while the low-frequency approximation might be representative of a developed cloud that is quasi-steady. More detailed analysis of the turbulence in clouds showed that there are pronounced maxima in the turbulent power spectra of vertical velocity at the frequences ω ∼ 10−3–10−2 s−1 (e.g., Curry et al. 1988; Sassen et al. 1989). The supersaturation relaxation (absorption) time in the cloud regions with smaller particles concentrations can be τf ∼ 30–60 s for liquid clouds and can exceed τf ∼ 30–180 min for crystalline clouds (e.g., Khvorostyanov and Sassen 1998). Thus the timescales τf and τL can be quite comparable and their relation can change during cloud evolution; hence more general kinetic equations for arbitrary relations between τf and τL are required.
The method of kinetic equations was introduced in cloud physics by Buikov (1961, 1963), who wrote the kinetic equation by analogy with the diffusion equation but incorporated an additional term describing condensational growth. When deriving analytical solutions to the kinetic equations, the additional covariances that appear due to turbulent fluctuations of supersaturation and the size distribution function were neglected. It will be shown below that this approach is equivalent to the high-frequency approximation, τL ≪ τf. Another version of the kinetic equations that corresponds to the low-frequency approximation was derived by Levin and Sedunov (1966) and Sedunov (1974), who developed the theory of stochastic condensation, which attributes the presence of broad drop size spectra in clouds to turbulent fluctuations. The stochastic method treats supersaturation and droplet growth rate as stochastic variables and replaces the usual operator of turbulent diffusion K∂/∂x by the operator K(∂/∂x + A∂/∂s) for a nonconservative substance s (here, the droplet surface) where A is proportional to the vertical velocity. The solution to this equation predicts the growth of the peak radius with time and dispersion of the drop size spectra that asymptotes with time toward a constant value.
More rigorous and detailed derivations of the equation of stochastic condensation using Reynolds averaging and more careful evaluation of covariances were performed by Stepanov (1975), Voloshchuk and Sedunov (1977), and Manton (1979) using the method of perturbation theory, which is valid for weak turbulence (fluctuations are smaller than the mean values). This derivation was generalized by Merkulovich and Stepanov (1977) to account for curvature and solution effects in droplet growth.
The kinetic equations derived in the aforementioned studies are valid in the low-frequency approximation, that is, when τL ≫ τf. In this case, the fluctuations of supersaturation and droplet growth rate are highly correlated with the vertical velocity. Although the low-frequency stochastic kinetic equations can explain some features of the drop size spectral dispersions in stratiform clouds, Manton (1979) showed that the stochastic approach leads to dispersions that are still narrower than those observed in convective clouds. By hypothesizing a negative correlation between the fluctuations in mean droplet radius and vertical velocity, Manton (1979) derived a modified version of the kinetic equation that produced broader (or bimodal) size spectra with dispersions increasing for a longer time. In a subsequent discussion of Manton’s theory (Merkulovich and Stepanova 1981; Manton 1981), it was clarified that the condition of mass balance imposes some additional limitations on the basic assumptions and analytical solutions to the kinetic equation; in particular, it was shown that the bimodal spectra predicted by this theory should be monomodal in many cases, and the dispersions should be smaller. Austin et al. (1985) showed that the basic assumption of Manton’s theory on the negative correlation between the fluctuations in integral radius and vertical velocity is not observed in the continental cumuli, while Curry (1986) found some evidence of negative correlation in Arctic stratus. Thus further verification or another hypothesis is required to break the link between supersaturation and vertical velocity, which appears to be the primary requirement for deriving a sufficiently broad drop spectra using the stochastic kinetic equations.
The first cloud models with explicit microphysics based on the kinetic equations were developed in the early 1970s (e.g., Clark 1973; Buikov and Pirnach 1973, Buikov and Khvorostyanov 1976). However, for many years explicit microphysical models were still a rather rare phenomenon among the much more frequently used bulk microphysical models. During 1980s and 1990s, the intensive development of cloud models with explicit microphysics was made possible by an increase in computational speed (e.g., Khvorostyanov 1982, 1995; Flossman et al. 1985; Marchuk et al. 1986; Tzivion et al. 1987; Kogan 1991; Feingold et al. 1994; Khvorostyanov and Sassen 1998). Additional impetus for explicit microphysics models has been provided by the Global Energy and Water Experiment (GEWEX) Cloud System Study, designed for development of the parameterizations for climate models (Browning 1994).
However, a gap remains between the theory of stochastic condensation and the practical needs of modeling. In applying the kinetic equation for stochastic condensation in numerical cloud models, we are faced with the following dilemmas.
A few attempts have been made to use the more complete stochastic kinetic equations in the low-frequency approximation of the kinetic equations in numerical cloud models (e.g., Vasilyeva et al. 1984), but the small characteristic timescale for the condensation process and presence of cross derivatives makes the numerical solution very computationally intensive. Hence, only the simplest versions of the kinetic equations in the high-frequency approximation are used now in most of the cloud models with explicit microphysics, neglecting terms that may influence the numerical solutions and physical effects.
Analytical solutions to stochastic kinetic equations in the low-frequency regime have been obtained of the Gaussian distribution type, which does not agree very well with observations that are typically in better agreement with a gamma distribution. This suggests that there are deficiencies in the formulation of the kinetic equations for stochastic condensation.
To address these concerns, we derive a new version of the kinetic equation for stochastic condensation for arbitrary relative values of τL and τf that is suitable for numerical cloud models. The principal assumptions that differentiate the new version of the kinetic equation from previous versions are 1) consideration of supersaturation as a nonconservative substance with differentiation between the macroscale and microscale supersaturation, and 2) consideration of the supersaturation fluctuations of various frequencies over the whole turbulent spectrum without assuming proportionality of supersaturation and vertical velocity. These assumptions break the link between the fluctuations in supersaturation and vertical velocity and lead to a solution of the gamma distribution type (see Part II of this paper, Khvorostyanov and Curry 1999a).
2. A model of the condensation process in a turbulent cloud
a. Basic equations
b. Stochastic equations
Henceforth we shall omit the bars over the mean values, other than for the covariance terms. The nucleation term
c. Supersaturation fluctuations
As noted in the introduction, the basic assumption used previously in derivations of the stochastic kinetic equation, that
Supersaturation fluctuations can be associated with turbulent fluctuations in temperature and humidity, which may or may not be associated with vertical velocity. There is a hierarchy of spatial scales of turbulent motions in the cloud, with the larger eddy sizes having a greater coherence with supersaturation. Additional supersaturation fluctuations may occur on the drop microscale that are unrelated to vertical velocity fluctuations but might be induced by other turbulent fluctuations such as those caused by the mixing of the cloudy parcels with different properties that act to break the link between vertical velocity and supersaturation.
Suppose a supersaturation fluctuation S′ arises in a cloudy parcel due to a turbulent fluctuation. Vapor begins to flow to the droplets and unless the equilibrium is attained in the volume VL in the vicinity of each droplet, there will be a nonstationary vapor concentration field ρυi(r). To calculate these vapor fields, we need to solve a complex system of many diffusion equations for ρυi(r) from all drops that influence each other in order to calculate a self-consistent vapor field formed by the superposition of the overlapping fields from individual drops (Sedunov 1974; Srivastava 1989). The exact solution to this complicated diffusion problem has not been found; thus various approximate methods are used.
Many models of condensation growth were considered over the last two decades that may help to break the link between S′ and w′. The common feature of several of them is that they predict the local growth of supersaturation fluctuations in proportion to the droplet radius, S′ ∼ r. Srivastava (1989) introduced the concept of a microscopic supersaturation fluctuations, which can vary from drop to drop due to randomness in the droplet spatial distribution; the main part of this microscopic supersaturation S′ is proportional to the droplet radius.
The inhomogeneous mixing theory of Baker and Latham (1979) and Baker et al. (1984), based on the chamber mixing experiments by Latham and Reed (1977), predicts the local increase of supersaturation in the vicinity of the larger drops that are less affected by mixing with entrained dry air, because the newly activated smaller droplets (that form later in the areas of complete evaporation) cannot compete as effectively for the available water vapor. Thus the effective supersaturation in the inhomogeneous mixing theory also increases with the drop radius.
Further evidence for the proportionality S′ ∼ r for small timescales comes from the detailed calculations of droplet growth that account for the kinetic correction (Fukuta and Walter 1970) and the concept of the modified diffusion coefficient, Dm(r), which accounts for this correction (Pruppacher and Klett 1997, p. 506). Calculations of droplet growth show that the magnitude of the kinetic correction depends on time and drop mass. Over short time periods comparable to τf (10–20 s), the kinetic correction causes a strong suppression of the growth of the smaller droplets. The value of Dm(r) ∼ r increases almost linearly with radius for small droplets (Pruppacher and Klett 1997, Table 13.1). Over periods much longer than ∼20 s, the kinetic correction becomes negligible and Maxwellian growth occurs. We performed similar calculations of supersaturation during relaxation (∼3–10 s) with account for the curvature and kinetic corrections with various accomodation coefficients (0.04–1), which also showed that the effective supersaturation, S′(r, t), increases with radius such that it can be roughly approximated by a linear dependence, S′(r, t) ∼ α(t)r. The slope α(t) decreases with time; thus the dependence of S′ on r weakens and the usual Maxwellian growth takes place for the larger times. Thus accounting for the kinetic and curvature corrections leads to the quasi-kinetic regime of growth for short timescales comparable to the phase relaxation time τf and the diffusion regime during long timescales.
The physical meaning of (2.16)–(2.18) is that due to suppressed growth of the smaller drops, the effective relaxation rate of supersaturation is faster for the larger droplets, so during relaxation the larger droplets absorb the larger fraction of the initial supersaturation fluctuation S′. Equation (2.18) shows that fluctuation of the growth rate described by this model is independent of radius, which is equivalent to kinetic regime in fluctuations. Another important feature of (2.18) is that we do not assume here S′ ∼ w′ as in previous theories, but simply consider S′ as a stochastic variable.
Support for our model of microscale condensation is provided by several previous studies, as discussed above. After substituting into (2.13) the effective diffusivity Dm(r) ∼ r, which can approximate the results by Fukuta and Walter (1970) and Pruppacher and Klett (1997, Table 13.1), we also arrive at a formulation (2.18) of ṙ′ that is independent of radius.
From (2.18), we can infer that the droplet spectrum during Δt in the kinetic regime would be displaced as a whole by Δr =
Hence, we have developed a model of condensation in a turbulent medium that is naturally separated by scales: 1) the usual Maxwellian diffusion regime controls condensational growth for longer timescales as for the regular condensation by using (2.1) for ṙ [with use of calculated values of supersaturation or equilibrium supersaturation Sq (2.9) for the steady state]; and 2) droplet growth occurs in the kinetic regime (2.17), (2.18) for small timescales in fluctuations. Therefore fluctuations in supersaturation, S′, and growth rate, ṙ′, have a stochastic meaning, but they are not proportional to the vertical velocity fluctuation w′ as in the low-frequency approximation. This model therefore breaks the link between the fluctuations in S′ (or growth rate) and w′.
3. Evaluation of correlation functions
a. Supersaturation as a nonconservative variable
Determination of covariances involving supersaturation is complicated by the fact that supersaturation is nonconservative because of turbulent fluctuations of temperature and humidity, and of phase changes when S ≠ 0. Nonconservativeness of supersaturation requires introduction of nonconservative covariances and turbulence coefficients.
If we assume in (3.6b) that ω ≪ ωp (low-frequency approximation), the amplitude dS′(ω) becomes proportional to the vertical velocity du′(ω) at all frequencies, and we come to the previous formulations of stochastic theory by Levin and Sedunov (1966), Voloshchuk and Sedunov (1977), and Manton (1979). Thus our approach generalizes the previous theories by use of (3.6b), which breaks the links between supersaturation and vertical velocity. In deriving (3.6) we assumed for simplicity that τf =
b. Covariances with supersaturation
c. Covariances with drop size distribution function
4. General kinetic equations of stochastic condensation
If the scale of averaging (e.g., over a grid box of a numerical model) is chosen to be sufficiently large so that the characteristic time of averaging is much larger than τf, then the mean supersaturation
This
The representation (4.1a) may be suitable for models with grid boxes a few tens or hundred meters or larger and/or for processes where the mean supersaturation is close to the quasi-steady value, while the more complete form (4.1) is preferable for models with finer resolution (e.g., on the smaller scales of LES models), or for processes or cloud layers where supersaturation is not quasi-steady. Note that the minimum scales of averaging and conditions when S ≈ Sq can be determined probably only with the use of the LES models with fine resolution after detailed analysis of the entire supersaturation field.
The term (∂/∂r)[(c/r)wf] describes “advection” in the space of radii with effective speed (c/r)w, inversely proportional to the radius; thus this advection is faster for the smaller droplets and this term alone causes narrowing of the size spectra. Analogous to the term −kij∂f/∂xi which represents the turbulent flux of the droplets in the usual space, the term −kijG∂f/∂r represents the turbulent flux in the space of radii. So the term kr = Gkij is the effective diffusion coefficient in the space of radii, and the parameter G is the scaling factor between the diffusion coefficients in xi and r spaces (G ∼ 10−9 to 10−8). Since the derivative −∂f/∂r < 0 for the radii less than the modal radius, r < rm, the flux of droplets is directed along the gradient toward the smaller radii to the left of the mode, at r < rm, and vice versa, −∂f/∂r > 0 at r > rm, and the flux is directed to the larger radii. So this “diffusion” in the space of radii tends to smooth the gradients and broaden the size spectrum. The resulting shape of the size spectra is determined by the relative speed of the advection and diffusion in the space of radii and in the usual space. If the effects of advection dominate (e.g., in the vigorous updrafts but with weak turbulence), the spectra may narrow even in the presence of turbulence, which alone might be insufficient to produce the broad spectra.
In most cloud models with explicit microphysics, the high-frequency form of the kinetic equation (4.4) is used and only a few attempts have been made to use the more complete kinetic equations in the low-frequency approximation (e.g., Vasilyeva et al. 1984). Incorporation into cloud models of the more complete kinetic equations of stochastic condensation in the forms (4.3), (4.5), or (4.6) requires development of efficient economical numerical algorithms for solutions of equations with cross-derivatives by coordinates and radii and the small characteristic time- and spatial scales that determine the condensation process.
5. Summary
A new look at the kinetic equations of stochastic condensation has been motivated by the increasing use of explicit microphysics in cloud models and the gap that presently exists between the theory of stochastic condensation and the practical needs of modeling. On one hand, stochastic theory produced several versions of the kinetic equations, which are rather general but perhaps still with some deficiencies as their solutions were of the Gaussian form, while a gamma distribution proves a better representation of observations of drop size spectra. On the other hand, because of the complexity of numerically solving the stochastic kinetic equations, most cloud models use a version of the kinetic equations that is incomplete. Toward addressing these issues, we have derived a new kinetic equation for stochastic condensation that is more suitable for the numerical models and with some simplifications has an analytical solutions of the gamma distribution type.
Relative to the earlier versions of kinetic equation of stochastic condensation, we introduce two new features:1) consideration of supersaturation as a nonconservative substance with differentiation between the macroscale and microscale supersaturation, and 2) consideration of the supersaturation fluctuations of various frequencies over the entire turbulent spectrum without the assumption of proportionality of supersaturation and vertical velocity. We consider supersaturation as a stochastic field of a nonconservative scalar and relate it to the velocity field by means of spectral expansions in Fourier–Stieltjes integrals. Supersaturation fluctuations S′ are related to the vertical velocity fluctuations w′ using the spectral analog of the supersaturation equation; thus S′ are not proportional to w′, breaking the link between the S′ and w′ and the reversibility of the condensation process typical of the some previous stochastic theories and the parcel models.
Following Srivastava (1989), we distinguish between supersaturation on the microscale and the macroscale. The supersaturation on the macroscale (i.e., generated by the mean vertical velocities and radiation) is typically treated as resulting in the Maxwellian diffusion growth of droplets. We showed that supersaturation variations on the microscale give rise to a droplet growth rate that is independent of drop radius. This regime is valid only for high-frequency fluctuations; the dependence of the effective microscale supersaturation on radius weakens with time as the supersaturation relaxes, and the growth rate transforms gradually into the Maxwellian diffusional law for low-frequency fluctuations.
The kinetic equation derived in this paper can be recommended for use in numerical cloud models with explicit microphysics. To apply this equation beyond the high-frequency regime will require economical numerical algorithms for parabolic differential equations with cross derivatives by space and droplet radius. The splitting method can be effectively used for these purposes (Marchuk 1974; Marchuk et al. 1986), which allows reduction during each time step of the multidimensional problem to the sequence of the one-dimensional problems (substeps), and condensation growth can be calculated at the final substep as a zero-dimensional problem by evaluating the integrals of supersaturation over the time step.
The exact and approximate asymptotic analytical solutions to this kinetic equation of the gamma-distribution type are obtained and analyzed in Part II of this paper. These analytical solutions can provide an appropriate basis for parameterization of the cloud processes in the bulk cloud models and the large-scale models.
Acknowledgments
This research was supported by the Department of Energy Atmospheric Radiation Measurement Program and the NASA FIRE program. The assistance of Jody Norman, Olga Melnik, and Alexander Kats in preparation of the manuscript. Oleg Melnik and Dmitry Khvorostyanov are thanked for their help in calculations. Comments on the manuscript from Branko Kosovic and Geoffrey Considine are also appreciated.
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APPENDIX
List of Symbols
A Coefficient in supersaturation equation, (2.7)
Bij(t − t′) Velocity correlation function
Nonconservative correlation function of velocity and supersaturationBnij(t − t′) Nonconservative autocorrelation function of supersaturationBnnij(t − t′) D Water vapor diffusion coefficient
c Coefficient of regular condensation, (4.1a)
cp Specific heat capacity
Fij Spectral function of turbulence
f Droplet size distribution function
G Parameter defined in (3.13)
ĤL, ĤR Left and right operators in (4.2)
J Source term on the rhs of kinetic equation
kij, kz Components of turbulence coefficient
(τf) Nonconservative turbulence coefficient of velocity and supersaturationknij (τf) Nonconservative turbulence coefficient of supersaturationknnij L Latent heat of condensation
Prandtl’s mixing lengthl′j Mixing length in space of radiil′r m Mass of a droplet
N Droplet concentration
p Indices of gamma distribution
Qrad Radiative temperature change
q, qs Specific humidity, saturation specific humidity
qL Liquid water content
Rυ Gas constant of water vapor
r,
Droplet radius, mean radiusr ref Effective optical radius
ṙ Droplet growth rate
ṙ′ Fluctuations in growth rate
S Supersaturation
S′ Supersaturation fluctuations
Sq Quasi-steady-state supersaturation, (2.9)
S̃q Relative quasi-steady-state supersaturation, (2.9a)
t Time
T Temperature
U, ui Vector and components of wind speed
υ(r) Terminal velocity
w Vertical velocity
wrad Radiative-effective velocity, (2.8)
wef Effective velocity
x, y, z Coordinates
γd, γs Dry and moist adiabatic lapse rates
Operator of turbulent diffusionΔ̂ Γ Psychrometric correction in equation for growth rate
δ(x) Dirac’s delta function
δij Kronecker’s symbol
εc Condensation rate
ρυ, ρw, ρa Densities of vapor, water, and air
σr Relative spectral dispersions
θ(x) Heaviside step function
τf Supersaturation relaxation (absorption) time
τL Lagrangian turbulent time
ω Turbulent frequency
ωp Frequency of supersaturation relaxation