1. Introduction

Despite recent progress in cloud physics, the understanding of key issues such as spatial distribution of cloud particles, turbulent mixing of clouds with the environment, or interaction of turbulence and microphysics, is still far from being complete. These issues are important—beyond fundamental understanding—in applications such as radiative transfer through clouds, initiation of precipitation, and parameterization of small-scale and microscale processes¹ in models resolving larger scales. The present work is an attempt to simulate directly the interfacial microscale mixing (i.e., mixing between cloudy and clear air occurring at submeter scales) in order to study the impact of the evaporative cooling and cloud droplets' sedimentation on the generation of the turbulent kinetic energy (TKE) and other properties of microscale turbulence.

Cloud observations at submeter scales are scarce and incomplete. A new generation of aircraft instrumentation, such as the fast forward scattering spectrometer probe (FFSSP; Brenguier 1993) are capable of providing data on microphysical properties of clouds at spatial scales down to a fraction of a centimeter. Studies of a microscale variability of the temperature field in clouds (Haman and Malinowski 1996; Korolev and Isaac 2000; Haman et al. 2001) show, in certain regions of clouds, temperature fluctuations as large as a few kelvins over distances as small as a centimeter. High-resolution liquid water content (LWC) data show that the variance of LWC exceeds the −5/3 law—in contrast to the dry turbulence scenario—at spatial scales smaller than a few meters (Davis et al. 1999; Gerber et al. 2001). In situ measurements aiming at interactions among the temperature, water vapor, and cloud condensate, and their impact on the microscale dynamics, have never been documented.

Strong assumptions about cloud turbulence are typically made in small-scale studies concerning interactions among cloud microphysics, thermodynamics, and dynamics. Usually, it is assumed that cloud turbulence is generated at larger scales (say, 100 m or more) and a well-developed inertial range of turbulent eddies exists down to the dissipation scale (cf. sections 5 and 6 in Grabowski and Clark 1993a; section 2b in Vaillancourt and Yau 2000). At small scales, turbulence is assumed homogeneous and isotropic, and it is described using statistical distributions that fit measurements (laboratory, wind tunnel, atmospheric boundary layer, etc.) or results from direct numerical simulation (DNS). Also, it is typically assumed that temperature and moisture are merely passive scalars that do not influence small-scale dynamics through buoyancy effects (e.g., Pinsky and Khain 1997; Shaw et al. 1998; Shaw and Oncley 2001). In general, however, it is unclear how results based on such assumptions correspond to natural processes in clouds (Grabowski and Vaillancourt 1999).

The concept of larger scales driving the microscale moist turbulence is not universal, however. For instance, Lilly (1968), Deardorff (1980), and Randall (1980) argued that cloud-top entrainment of dry air into stratocumulus can lead to its destruction (i.e., “cloud-top entrainment instability;” see also Siems and Bretherton 1992; Krueger 1993). These studies are concerned with the impact of buoyancy reversal on global cloud dynamics, but not with the impact on the microscale turbulence. At cloud microscale, a dramatic impact of the mixing between cloudy and cloud-free air on the buoyancy field, and thus possibly on the microturbulence, has been documented in the context of an elementary 1D model for moist homogenization (Fig. 3 in Grabowski 1993). Our goal is to quantify the impact of evaporative cooling and buoyancy reversal on the dynamics of turbulent mixing. The coupling between cloud turbulence and cloud microphysics is poorly understood and it is likely the key in explaining some cloud observations, such as the width of the cloud droplet spectrum or rapid development of precipitation (see extensive review in Shaw 2003). Perhaps the central issue is whether these microscale processes can be represented (“parameterized”) in traditional cloud models that apply resolutions of tens of meters at best. This especially applies to cloud microphysical properties, such as the spectrum of cloud droplets. Microscale mixing has also been discussed in the context of combustion and two-phase flows. Some ideas from these studies have been adopted in cloud physics to represent subgrid-scale evaporation and condensation in numerical models (Krueger 1993; Margolin et al. 1997).

Because in situ microscale cloud data are unavailable, the only tools capable of revealing microscale cloud structures and processes are laboratory experiments and numerical modeling. Early laboratory experiments on turbulent mixing in fluids (e.g., Broadwell and Breidenthal 1982; Sreenivasan et al. 1989) motivated development of conceptual models of cloud–clear air mixing (e.g., Jensen and Baker 1989; Malinowski and Zawadzki 1993; Malinowski et al. 1994). Recently, laboratory experiments focused explicitly on the turbulent mixing of cloudy and clear air were conducted by Malinowski et al. (1998). In these experiments, cloud–clear air mixing at scales from about 1 mm to almost 1 m was investigated in a laboratory chamber using the light-sheet technique. The images of planar sections through the mixing volume show complicated structures with sharp boundaries between cloudy and clear-air filaments. Geometric analysis of these images suggests that small-scale structures in laboratory clouds have distinct properties at scales larger and smaller than 2 cm. At scales larger than 2 cm, the interface separating cloudy and clear-air volumes is convoluted in a self-similar manner, resembling isoconcentration surfaces in turbulent mixing at scales larger than the Batchelor scale (see Sreenivasan et al. 1989). At scales smaller than 2 cm, no self-similarity is observed. Further analysis of the experimental data shows that filaments are elongated in the vertical, suggesting that gravity/buoyancy effects are important in the investigated range of scales (Banat and Malinowski 1999). Moreover, gradients of light intensity at the edges of filaments depend on the orientation of the interface (i.e., vertical versus horizontal) indicating a possible importance of droplet sedimentation across the interface (Malinowski and Jaczewski 1999).

A number of numerical studies have addressed various aspects of the dynamics of cloud–clear air mixing on scales in the range O(1)–O(100) m (e.g., Klaassen and Clark 1985; Grabowski 1989; Grabowski and Clark 1991, 1993a,b; Carpenter et al. 1998a,b,c); however, we are aware of only a few studies concerned with the smaller scales (Krueger et al. 1997; Malinowski and Grabowski 1997; Su et al. 1998). Our numerical model is set up to simulate the crux of the laboratory experiment described in Malinowski et al. (1998). Our motivation is to bridge between the processes relevant to our laboratory measurements and those occurring in natural stratocumulus and small cumulus clouds. Because of the limitations of numerical technology, our study can only address the final stages of the entrainment– mixing process, where the filamentation by large-scale eddies is already less important, and where mixing– evaporative cooling takes place (Jensen and Baker 1989). Insofar as the initial and boundary conditions are concerned, we do not strive to mirror the laboratory case—the latter would be both computationally demanding and mathematically complex—but tend rather to capture key physical features of the problem at hand. In particular, to assess the role of larger-scale (than submeter) flow inhomogeneities, we sample three different levels of the TKE input. This is in the spirit of DNS turbulence studies, which assume heavily idealized initial and boundary conditions, while focusing on an accurate representation of the evolving flow (cf. Moin and Mahesh 1998, for a review). Although care has to be taken when extrapolating the results of our simulations to natural clouds, the quantitative nature of this study provides a guidance for future research of small-scale and microscale processes in clouds.

The paper is organized as follows. The next section provides an overview of the mathematical model employed with an emphasis on the representation of moist thermodynamics. Details of numerical simulations are presented in section 3. Model results are discussed in section 4 (dynamical aspects) and in section 5 (cloud microphysics). Concluding remarks are presented in section 6.

2. Mathematical formulation

The theoretical formulation assumes incompressible-Boussinesq approximation—a standard in atmospheric turbulence studies—with the governing model equations

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where D/Dt ≡ ∂/∂t + v · ∇ is the material derivative with v = (u, υ, w) denoting the velocity vector; π is the pressure perturbation (from a static environment) normalized by the reference Boussinesq density (ρ_o = 1 kg m⁻³); k is the unit vector in the vertical; B denotes the buoyancy; ν = 1.50 × 10⁻⁵ m² s⁻¹ is the kinematic viscosity of air; T and q_υ are the temperature and the water vapor mixing ratio, respectively; L = 2.5 × 10⁶ J kg⁻¹ and c_p = 1005 J kg⁻¹ K⁻¹ are the latent heat of condensation and the specific heat at constant pressure; C_d is the condensation rate; and μ_T = μ_υ = 2.14 × 10⁻⁵ m² s⁻¹ are molecular diffusivities of the temperature and water vapor in air, respectively. The buoyancy is defined as

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where g = 9.81 m s⁻² is the acceleration of gravity, T₀ and q_υ 0 are the reference temperature and water vapor mixing ratio (selected as 293 K and 9.9 g kg⁻¹), ε + 1 ≡ R_υ/R_d is the ratio of the gas constants for water vapor and dry air, and q_c is the mixing ratio for the cloud condensate (the cloud water mixing ratio).

The system (1) is supplemented with the equation(s) governing evolution of the cloud condensate. Two different approaches are used to quantify the microscale transport across the cloud–clear air interface via molecular diffusion of heat and water vapor versus droplet sedimentation and to quantify the impact of mixing on the size distributions (spectra) of cloud droplets. The standard bulk approximation assumes equilibrium thermodynamics such that condensation–evaporation of cloud water maintains the in-cloud saturation—thus, defining implicitly the condensation rate C_d (cf. Grabowski and Smolarkiewicz 1990, for a discussion)—while cloud water follows the air motions without droplet sedimentation. The resulting evolution equation for the cloud condensate is

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Contrasting with the simplicity of the bulk approach, the detailed microphysics approximation² relaxes the equilibrium assumption by admitting in-cloud departures from the saturation, and divides cloud water into size categories each subject to its own sedimentation rate (see Grabowski 1989 for a discussion). In effect, the cloud water equation (3) is replaced with the evolution equation for the number density function f(x, r, t), where f(x, r, t)dr is the number of cloud droplets in a unit mass of air (viz. the mixing ratio) of the radius between r and r + dr, at a given point (x, t) in space and time. The equation governing the evolution of the number density function takes the form

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where D*/D*t ≡ ∂/∂t + (v − kυ_t) · ∇ is the material derivative along a droplet-class trajectory, with υ_t denoting the size-dependent sedimentation velocity. The droplet growth rate dr/dt = AS/r is proportional to the supersaturation S = q_υ/q_υs − 1, where q_υs denotes the saturated water vapor mixing ratio, and A = 10⁻¹⁰ m² s⁻¹. The source–sink term η symbolizes the nucleation or deactivation of cloud droplets. In the radius space, it represents the boundary condition at the edge of the smallest size category: in undersaturated conditions, η represents the “outflow” of droplets; whereas in the presence of the supersaturation, η specifies the “inflow.” The latter is parameterized by relating the total concentration of cloud droplets to the concentration, N_n, of activated cloud condensation nuclei under a given supersaturation (see section 2b of Grabowski 1989, for further discussion). Here, the standard empirical formula employed, N_n = aS^b, assumes a = 300 cm⁻³ and b = 0.5, that approximate the conditions of the cloud chamber experiment of Malinowski et al. (1998). In contrast to the bulk-microphysics approach, the condensation rate derives explicitly from the growth of cloud droplets in all size categories

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where m = 4πr³ρ_w/3 is droplet mass and ρ_w = 10³ kg m⁻³ is the density of water. Finally, the sedimentation velocity is prescribed according to the Stokes law υ_t(r) = Cr², where C is selected to satisfy υ_t(r = 10 μm) = 10⁻² m s⁻¹.

The Boussinesq system adopted describes isobaric mixing (e.g., Rogers 1979) between cloudy and cloud-free air, appropriate for laboratory experiments of Malinowski et al. (1998). As far as natural clouds are concerned, vertical displacement of the volume during mixing introduces changes of its thermodynamic properties due to adiabatic expansion or compression of the mixing volume. These effects need to be considered if the vertical displacement of the air parcel during the course of the turbulent mixing results in significant changes of the temperature and moisture variables in cloudy and clear-air volumes. Since the mixing considered in this paper is completed in less than half a minute, the vertical displacement of the mixing volume (assuming typical cloud vertical velocity of a few meters per second) would be less than 100 m. Consequently, the nonisobaric effects are of secondary importance and are neglected.

3. Numerical model setup

The analytic equations of section 2 are solved by means of finite-difference approximations using the semi-Lagrangian/Eulerian nonhydrostatic anelastic model EULAG, broadly documented in the literature (Smolarkiewicz and Margolin 1997, 1998; Grabowski and Smolarkiewicz 1996, 2002; Smolarkiewicz and Prusa 2002a,b; and references therein). For this work, the Eulerian (flux form) second-order-accurate option of the model is chosen.

Small-scale turbulent mixing in clouds invokes several competing processes, each dictating its own computational demands. To optimize the design of numerical experiments a reasonable compromise is in order. Representing mixing features down to the dissipation scale and sharp cloud–clear air interfaces (Malinowski and Jaczewski 1999) calls for a fine computational mesh. On the other hand, a detailed microphysics approach becomes ill posed at very small scales (below a few millimeters) where tracking of individual cloud droplets may be required (Vaillancourt et al. 2001, 2002). Simultaneously, the computational domain should be large (as large as technically feasible) in order to represent the input of TKE far upscale from the Kolmogorov– Batchelor scales. Finally, computer simulations with detailed microphysics spawn another spatial dimension (in the radius space), whereupon they are an order of magnitude more computationally demanding than their dry and bulk-microphysics analogs. Taking into account all these factors, we compromised on using a relatively coarse grid length 0.01 m and modest computational grid 64³, for all simulations reported in this paper. In all detailed-microphysics simulations, 16 classes of cloud droplets are used with the droplet sizes linearly distributed in the range from 0.78 to 24 μm.

From the experimental viewpoint, the 1-cm grid increment corresponds to the observed break in scaling properties at the cloud–clear air interface (around 2 cm; Malinowski et al. 1998; Banat and Malinowski 1999). Such a grid length, however, hardly corresponds to a well-resolved DNS, because the estimated Kolmogorov microscale is O(10⁻³) m (see Table 1). Experiments discussed in this paper employ the nonoscillatory forward-in-time (NFT) approach based on the finite-volumewise transport scheme MPDATA (see Smolarkiewicz and Margolin 1998, for a review) that supplements an implicit subgrid-scale model when needed, in the spirit of large-eddy simulation (LES). This adaptive LES property of the NFT–MPDATA approach has been broadly documented/quantified in the literature for a variety of turbulent flows (Margolin et al. 1999, 2002; Margolin and Rider 2002; Smolarkiewicz and Prusa 2002a,b; Domaradzki et al. 2003). The simulations reported in this paper may be viewed as falling into the gray area between poorly resolved DNS and well-resolved LES.

The dynamical design of our experiments follows Herring and Kerr's (1993) DNS study of the dry decaying turbulence. In their simulations, the initial velocity field is constructed from a few low-wavenumber Fourier modes to mimic the instantaneous large-scale input of the TKE. As the simulation progresses, small scales develop and TKE cascades down-scale, until it is dissipated near the Kolmogorov microscale. In addition, in the moist case, TKE is also buoyantly generated by the evaporation of the liquid water. We complement all moist experiments with corresponding reference simulations of a dry turbulence—where only (1a) and (1b) are solved assuming B ≡ 0—to quantify the buoyant production of the TKE and small-scale anisotropy as well as to contrast the role of a dry and moist turbulent mixing in clouds.

The initial velocity perturbations, identical in dry and moist simulations, are formulated after Herring and Kerr (1993), in the Fourier space as follows:

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where υⁱ denotes u, υ, and w velocity components, respectively, for i = 1, 2, 3; k = (2π/L)n²_x + n²_y + n²_z is the magnitude of the wavenumber; n_x, n_y, n_z are integers from 0 to N/2 + 1; N = 64 is the number of grid points; L = 0.64 m is the size of the computational domain; k₀ = (2π/L) 4.7568; A_υ = const denotes the amplitude; and ψ ∈ [0, 1] is the white-noise random phase, fixed for each k. To allow TKE input at low wavenumbers only, velocity components (6) are taken for n²_x + n²_y + n²_z ≤ 4.

Three different amplitudes A_υ are selected to simulate three different inputs of TKE(≡E = 1/2 〈u · u〉, where 〈 · 〉 ≡ 1/V ∫ dV and integration is over the entire computational domain): 2.16 × 10⁻², 5.4 × 10⁻³, and 2.16 × 10⁻⁴ m² s⁻²; hereafter referred to as high-, moderate‐, and low-intensity levels of TKE input. The initial velocity field is depicted in Fig. 1, for the case of the low-TKE input. The initial velocity for the moderate- and high-TKE input has the same spatial distribution, but the velocity scale (shown in Fig. 1) has to be multiplied by a factor of 5 and 10, for the moderate- and high-TKE case, respectively. In general, the magnitude of the input velocity fluctuations is in the range of a few centimeters per second (for the low-TKE input) to a few tens of centimeters per second (for the high-TKE input). These values are consistent, assuming inertial-range scaling, with aircraft observations of turbulence associated with convective clouds (MacPherson and Isaac 1977) that show typical in-cloud velocity fluctuations between a few tenths to a few meters per second at scales of tens of meters. Furthermore, the assumed initial velocity fluctuations result in the eddy dissipation rates (cf. Table 1) consistent with low-to-moderate cloud turbulence levels (see discussion in Vaillancourt and Yau 2000 and Vaillancourt et al. 2001, 2002), thus corroborating the assumed initial conditions a posteriori. In the low-TKE-input case, initial flow velocities are of the same order as the sedimentation velocities of cloud droplets; whereas in the high-TKE case, they are one order of magnitude larger.

The initial air temperature is set to 293 K everywhere (i.e., to T₀) and the water vapor mixing ratio outside the cloud is 9.9 g kg⁻¹ (i.e., q_υ0; this corresponds to the relative humidity of 65%). Such conditions are representative of warm summertime cumuli and resemble conditions in the cloud mixing chamber (Banat and Malinowski 1999; Malinowski and Jaczewski 1999). The initial cloud water is specified to form filaments, with the temperature T₀, 100% relative humidity, and the cloud water mixing ratio (around 3.2 g kg⁻¹) defined such that cloudy filaments are neutrally buoyant. The filaments are collocated with ϕ̂(x) > 0, where ϕ̂ is the inverse Fourier transform of ϕ in (6). The initial distribution of the cloud water (the same for all TKE inputs) is shown in Fig. 1.

For the detailed-microphysics simulations, the cloud water is distributed among three classes (bins) of droplets (corresponding sizes about 7, 8.5, and 10 μm). The initial number of droplets is prescribed according to the requirement that bins 5, 6, and 7 contain 25%, 50%, and 25% of the total cloud water, respectively.

The model time step varies in the range between 4 × 10⁻³ and 2 × 10⁻². Boundary conditions are periodic in all three directions. High-TKE-input simulations are run for 20 s of physical time, whereas moderate- and low-TKE-input simulations are run for 25 s. After these times, the homogenization of all thermodynamic fields is statistically accomplished.

4. Dynamics of turbulent mixing

Table 1 summarizes model simulations, listing representative characteristic parameters in the three flow configurations considered. The entries in the table are the Taylor microscale Reynolds number Re_λ discussed later in this section; initial value of the TKE; the maximum TKE dissipation rate ϵ_max = −minĖ; the maximum velocity; the minimum Kolmogorov length scale η_min = (ν³/ϵ_max)^1/4; and the eddy mixing time scale, after Baker and Latham (1979), τ_T = (X²/ϵ_max)^1/3 with X denoting the characteristic length scale (taken as the Taylor microscale of spatial fluctuations of the moisture field calculated at the time of the maximum dissipation rate, approximately equal to 0.03 m; cf. section 5). In addition, the diffusive mixing time scale τ_D = X²/μ_υ (Baker and Latham 1979), and the evaporation time scale τ_r = r²/AS (section 5) are 43 and 2 s, respectively, for all three cases.

The impact of the moist processes on the turbulent mixing is best illustrated at the low-TKE input. Figures 2 and 3 show snapshots of model results at two times, respectively, t = 3.6 s and t = 7.2 s, for both detailed and bulk model simulations. At t = 3.6 s, the mixing has barely started and the thermodynamic fields are still dominated by the initial conditions, the same in both simulations. However, the temperature field already shows substantial difference between detailed and bulk simulations. In the bulk case, the evaporation-induced temperature undershoots at cloud–clear air interfaces are symmetric below and above horizontally oriented cloudy filaments as seen, for example, in the centers of the left panels in Fig. 2. In the detailed microphysics case, however, the undershoots are about 1 K colder below the filaments—compare the centers of the right panels in Fig. 2—an apparent effect of droplet sedimentation. Furthermore, comparing left and right panels shows that sedimentation tends to dilute sharp interfacial gradients of thermodynamic fields. At t = 7.2 s (Fig. 3), the mixing is well under way, spatial patterns of the temperature and cloud water are distinct, documenting the loss of memory of the initial condition. In contrast to the earlier time, this solution is dominated by small-scale structures. The smoothness of the detailed-microphysics solution is accentuated even further. In both detailed- and bulk-microphysics cases, cloudy filaments appear anisotropic, elongated in the vertical; we shall return to this point later in this section.

To highlight the evolution of thermodynamic properties from the initial condition to the homogenized state, Fig. 4 displays the histories of the mean cloud water q_c and its standard deviation, for all moist simulations (three TKE inputs and two microphysics schemes). Plots for the temperature and water vapor mixing ratio, not shown, exhibit similar evolution. As the mixing progresses, in all simulations, the average cloud water mixing ratio approaches about half of the initial value—expected from elementary thermodynamics of isobaric mixing—of ∼0.8 g kg⁻¹. In all simulations, the standard deviation of cloud water mixing ratio decreases gradually toward zero—characteristic of a fully homogenized state. As expected, homogenization is rapid in the high-TKE case, and slow in the low-TKE case. The evolution of the mean values is close for bulk- and detailed-microphysics models.

Figures 5 and 6 show evolutions of the TKE and enstrophy (Ω ≡ 1/2 〈ω · ω〉, where ω ≡ ∇ × u) for all simulations, including dry reference runs. In simulations with high- and moderate-TKE input, the overall evolution of TKE mimics dry simulations with (i) a period of slow decrease during the first second of the simulation, (ii) a few-seconds-long period with rapid decrease during which most of the TKE is dissipated, and (iii) a slow decrease in the final 10–15 s. The enstrophy, dominated by vorticity of the finest eddies, rapidly increases at the early stages and peaks at about the same time when TKE decreases at the maximum rate. These features are consistent with the familiar picture of a decaying turbulence, where the development of small-scale structures from imposed large-scale perturbations is followed by the TKE dissipation (Herring and Kerr 1993).

For high-TKE input (Figs. 5 and 6, top), moist processes have negligible impact on the dynamics, as measured by the energy and enstrophy of the flow. As far as the TKE evolution is concerned, only after about 10 s (when over 90% of the initial TKE has dissipated) can small differences between dry and moist cases be seen. For moderate-TKE input (Figs. 5 and 6, middle), kinetic energy produced by evaporation already starts affecting TKE after 5 s. After 15 s, moist flows are several times more vigorous (in terms of TKE magnitude) than their dry counterpart. In both cases, there is not much difference between bulk and detailed microphysics.

For low-TKE input (Figs. 5 and 6, bottom) dramatic differences are observed between dry and moist cases. While gradual dissipation of TKE governs the dry reference flow, production of TKE due to phase changes dominates during the initial 5 s of moist simulations. This is followed by a long period when the TKE (of moist simulations) is approximately constant in time (cf. Figs. 7, 12, and 23 in Siems and Bretherton 1992). Similar differences between dry and moist simulations are also observed in the enstrophy evolution. Toward the end of the moist simulations, both enstrophy and TKE are higher than in all high- and moderate-TKE input cases. An explanation for such a longevity of the turbulence is a relatively slow homogenization of the thermodynamic fields in moist simulations when the large-scale input of TKE is low, evident in Fig. 4. The evolution of TKE differs between simulations applying bulk and detailed microphysics. In particular, the detailed-microphysics simulation yields larger TKE values between 4 and 20 s, and the peak value (around 7 s) is about 50% higher than in the bulk case. This, together with the accompanying enstrophy excess in Fig. 6, suggests that droplet sedimentation invigorates microscale mixing—consistent with laboratory results of Banat and Malinowski (1999) and Malinowski and Jaczewski (1999). The enstrophy budget documents that this invigoration is initially a response to the negative buoyancy production due to the evaporation of sedimenting droplets, followed by the enhanced stretching of vortex tubes. This is substantiated in Fig. 7, which shows the evolution of the buoyancy production and stretching terms in the enstrophy evolution equation (cf. section 5 in Grabowski and Clark 1993a, for a discussion).

Figure 8 presents temporal evolution of the Taylor microscale Reynolds number [cf. (5) in Herring and Kerr 1993],

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The Taylor microscale Reynolds number is the Reynolds number obtained using rms flow velocity and the Taylor microscale, the smallest resolvable scale in the flow. For high- and moderate-TKE inputs, Re_λ yields a typical evolution with a rapid decrease from its initial value as the turbulence develops and a gradual leveling-off in the later stage as TKE is dissipated (cf. Fig. 1 in Herring and Kerr 1993, Fig. 10 in Grabowski and Clark 1993a). For low-TKE input, Re_λ increases after the initial drop—consistent with the slow homogenization argument earlier.

Figures 9 and 10 show the evolution of horizontal and the vertical Taylor microscales

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For high- and moderate-TKE input, all simulations show similar evolution of the horizontal microscale. In the low-TKE-input case, dry and moist simulations differ considerably, yet the final values for all nine simulations are about the same (∼5 cm). The most spectacular difference between dry and moist simulations is seen for the vertical Taylor microscale (Fig. 10). While final values for dry simulations are around 5 cm, the same as for the horizontal microscale, the values for the moist cases are about twice as large (8 and 10 cm, respectively, for high- and low-TKE input). This result is in accord with the findings of Banat and Malinowski (1999) who show that filaments created by turbulent mixing of cloudy and clear air in the mixing chamber are elongated in the vertical. The anisotropy of the moist turbulence is further supported by Figs. 11 and 12, that compare the horizontal and vertical velocity derivative skewness

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For dry isotropic turbulence, laboratory experiments with low Reynolds number flows yield values of ∼0.4, whereas direct numerical simulations at moderate Reynolds numbers give ∼0.5 (Lesieur 1997). In all dry simulations, the velocity skewness settles at the value of about 0.5. In moist simulations, on the other hand, the horizontal velocity skewness tends to be lower than its dry counterpart, whereas the vertical skewness tends to be higher. Interestingly, the anisotropy of the simulated moist turbulence does not depend on the level of TKE input. Moreover, the differences between bulk and detailed microphysics appear insignificant.

5. Implications for cloud microphysics

Entrainment and mixing have long been postulated as crucial processes shaping the spectrum of cloud droplets (e.g., Su et al. 1998; Lasher-Trapp et al. 2003, manuscript submitted to Quart. J. Roy. Meteor. Soc., and references therein). There are two conceptual models illustrating possible impacts of microscale cloud–clear-air mixing on a spectrum of cloud droplets: the homogeneous mixing model, and the extremely inhomogeneous mixing model [cf. Baker and Latham (1979), Baker et al. (1980)]. The homogeneous model assumes that all cloud droplets are exposed to the same environmental conditions during homogenization. It follows that all droplets experience some evaporation and the number of cloud droplets does not change. In the extremely inhomogeneous model, on the other hand, some droplets evaporate completely whereas others do not change their sizes at all. In such a case, the number of droplets decreases, but their size remains constant. The two models represent two limits of possible microphysical realizations of a mixing event with prescribed bulk properties, such as temperatures, water vapor and cloud water mixing ratios of the two air parcels.

Baker and Latham (1979) and Baker et al. (1980), who introduced the concept of inhomogeneous mixing, argued that the effect of mixing in natural clouds depends on the two relevant time scales. The first one, related to the fluid dynamics, defines the time needed to homogenize the volume under consideration through the process of turbulent diffusion. This time scale can be estimated as τ_T = (X²/ϵ)^1/3 where ϵ is the dissipation rate of turbulent kinetic energy (taken herein as the maximum dissipation rate ϵ ≡ ϵ_max) and X is the horizontal length scale of moist filaments (taken as the Taylor microscale of spatial fluctuations of the moisture field calculated at the time of the maximum dissipation rate and approximately equal to 0.03 m). This time scale changes from 0.5 to 1.3 s for simulations discussed in this paper (cf. Table 1). The second time scale, related to cloud microphysics, is the time needed to evaporate a cloud droplet for a given relative humidity of the cloud-free undersaturated volume. The time scale for droplet evaporation can be estimated as 1/τ_r = (1/r)(dr/dt) where r is droplet size and dr/dt is the simplified droplet growth formula applied in this study [cf. the text beneath (4)]. It follows that τ_r = r²/AS ≈ 2 s for the droplet size of about 8 μm and relative humidity of 65% (i.e., S = −0.35).

The cornerstone of Baker and Latham's argument is that when τ_T ≪ τ_r, turbulent mixing is completed before significant evaporation of cloud droplets can occur, consistent with the concept of homogeneous mixing. When the opposite is true, that is, τ_T ≫ τ_r, complete evaporation of some droplets occurs as the mixing progresses and extremely inhomogeneous mixing model applies. In the case when τ_T ∼ τ_r, which is exactly the case in our simulations, microphysical transformations are in between the two limiting cases. The same seems to apply for the case of entrainment and mixing in natural clouds (e.g., Jensen and Baker 1989; Brenguier and Burnet 1996; Su et al. 1998).

Results of the detailed-microphysics-model simulations discussed in this paper are summarized in Table 2, which lists the initial and final number of cloud droplets N, the mean size r and the mean standard deviation σ of the cloud droplet spectrum, and the mean volume radius r_υ defined as the radius required to obtain observed cloud water content given the number of cloud droplets N [cf. Eq. (6) in Brenguier and Burnet 1996]. As the table demonstrates, the mixing results in reduction of both the number and the size of cloud droplets and thus it fits the τ_T ∼ τ_r picture of the mixing process discussed earlier. Moreover, mixing increases the width of the cloud droplet spectrum.

These results are further illustrated in Fig. 13 using a diagram similar to that applied by Brenguier and Burnet (1996, cf. Figs. 1–4 therein). It shows the relationship between the number of cloud droplets N and the mean volume radius cubed r³_υ. Brenguier and Burnet applied the diagram to cloud observations and used droplet concentration rather than number of droplets. In idealized simulations considered here, however, number of cloud droplets is more useful.³ The number of cloud droplets and the radius cubed are scaled by their initial values N₀ and r_υ;th0. In the diagram, final states of all possible microphysical realizations for given initial conditions reside on a hyperbola defined by Nr³_υ = βN₀r³_υ0 = const where β, the ratio of the mean cloud water before and after the mixing, is given by the bulk thermodynamic properties of cloudy and cloud-free air and the ratio between initial cloudy and cloud-free volumes. In our case, these dictate that the final cloud water is slightly less than a half of the initial mean value (i.e., β ≈ 0.49).

Figure 13 shows model results for high, moderate, and low levels of input TKE (marked as h, m, and l, respectively), together with the hypothetical limiting cases of homogeneous (H) and extremely inhomogeneous (EI) mixing scenarios. The figure illustrates that the three simulations are quite far from the limiting cases, and that the input of the TKE is an important factor for shaping the final spectrum of cloud droplets. The fact that the high-TKE-input case is the closest to the homogeneous mixing scenario and the low-TKE-input case is the most distant from it agrees with the argument put forward by Baker and Latham (1979). Since the case with high-TKE input results in a rather moderate turbulence levels when compared to cloud observations, our results seem consistent with idealized model results of Jensen and Baker (1989) who claim that, for the high-turbulence levels commonly found in cumuli, the mixing between cloudy air and entrained subsaturated air appears not far from homogeneous.

6. Concluding remarks

We have discussed the results from an exploratory series of idealized numerical simulations of decaying moist turbulence. An adopted modeling setup follows DNS studies of dry turbulence of Herring and Kerr (1993), in an attempt to bridge between the processes relevant to our laboratory measurements and those occurring in natural stratocumulus and small cumuli. Our simulations focus on the role of buoyancy production at the edges of cloudy filaments and its impact on the microscale turbulence, and on the role of droplet sedimentation across the cloud–clear air interface—in the final stages of the entrainment–mixing process, where the filamentation by large-scale eddies is already completed and where mixing–evaporative cooling takes place. Consequently, three sets of numerical simulations have been performed, respectively, for three intensities of initial large-scale eddies that provide high, moderate, and low large-scale TKE input. For the high-TKE input, initial velocities are one order of magnitude larger than the sedimentation velocities of cloud droplets, whereas for the low-TKE input they are of the same order. In each set, a reference dry simulation has been performed as well as two moist simulations using either a bulk- or detailed-microphysics approximation of the moist thermodynamics.

Our simulations suggest that, at high initial values of the large-scale TKE, moist processes have minor impact on the microscale dynamics of mixing and homogenization. In particular, the evolutions of enstrophy and TKE differ (slightly) between moist and dry simulations only at the final stage of mixing, when most of the TKE has already been dissipated. There are, however, significant differences related to the small-scale anisotropy. Our results show that moist turbulence is anisotropic even for high values of the TKE input, with turbulent filaments elongated in the vertical, consistent with the laboratory experiment of Banat and Malinowski (1999). For moderate and low initial values of TKE, the influence of moist processes on microscale mixing and homogenization is substantial. Especially, for low initial TKE, the energy–enstrophy evolutions are dominated by moist processes, and the impact of cloud droplet sedimentation is the largest.

Impact of entrainment and mixing on the spectrum of cloud droplets is well recognized but still poorly understood. There is no doubt, however, that transformations of cloud droplet spectra as a result of turbulent mixing between cloudy and cloud-free air at submeter scales are essential (e.g., Baker and Latham 1979; Baker et al. 1980; Jensen and Baker 1989; Brenguier and Grabowski 1993; Brenguier and Burnet 1996; Su et al. 1998; Lasher-Trapp et al. 2003, manuscript submitted to Quart. J. Roy. Meteor. Soc.). Simulations applying detailed microphysics discussed in this paper allow one to look at transformations of cloud droplet spectra as a result of microscale mixing and homogenization. Predicted cloud droplet spectra at the end of model simulations correspond to neither extremely inhomogeneous nor homogeneous mixing scenarios, the two possible limits. Model results indicate that the spectra are somewhere between the two limits, that is, both the mean droplet size and the number of droplets change as a result of microscale homogenization. The shift from low to high intensity of initial large-scale eddies results in the shift of the mixing scenario toward the homogeneous case. The fact that high-TKE-input case is the closest to the homogeneous mixing and the low-TKE input is the most distant from it corroborates the time-scale argument put forward by Baker and Latham (1979).

Relative differences between simulations with the bulk and detailed microphysics appear moderate even for low-TKE input. However, coarse spatial resolution afforded in this study tends to overestimate the role of the temperature and moisture diffusion compared to the cloud droplet sedimentation, across the cloud–clear air interface. This warrants numerical experimentation at higher resolutions (e.g., 128³, 256³, etc.) whenever possible. Also, present simulations consider only a particular set of initial conditions. It is uncertain, therefore, to what extent our conclusions are universal, that is, independent of the initial conditions. This needs to be addressed in an expanded future study.