Turbulence, Condensation, and Liquid Water Transport in Numerically Simulated Nonprecipitating Stratocumulus Clouds

a. The q_l budget

An excellent balance exists between the flux divergence (including both resolved and SGS fluxes) and the condensation terms, while the subsidence advection term is negligibly small (Fig. 2a). Between 500 and 580 m near the cloud base (∼530 m), the net evaporation is compensated by the flux convergence as shown in Fig. 2a. This evaporation and the negative liquid water flux (Fig. 1e) are largely a result of turbulent transport of liquid water in the downdrafts. The net condensation is balanced by the flux divergence between 580 and 640 m. In the middle of the cloud layer (700 m), both the flux divergence and CE terms are small, indicating that there are no source and sink terms at those levels. Just below the cloud top (between 720 and 780 m) both the divergence and the net condensation are significantly increased by radiative cooling. Finally, at the cloud top, the turbulent eddies transport the droplets to the warmer and drier inversion where significant evaporation occurs.

Figure 2b shows various terms of the CE term in (6). The mean saturation term is almost always negative from the cloud base up to 750 m except at the level of 600 m where the mean condition is saturated due to considerably supersaturated updrafts. The negative S clearly results from considerable subsaturation in the downdrafts despite the water vapor surplus in the updrafts. Near the cloud top (∼780 m), this term is positive because of strong radiative cooling. At cloud top, the warmer and drier air in the inversion leads to large negative values of the mean saturation term.

In contrast to the mean saturation term, contributions from the two turbulence covariance terms are generally positive; a larger S tends to result in more condensation/activation or less evaporation, leading to positive covariance N′S′ and q^′_lS′. In addition, N′S′ dominates R′S′ term due to the activation in updrafts and the evaporative loss in downdrafts near the cloud base.

Recall from (3) that these two turbulence terms represent the correlation between R and S and therefore, R′S′ > 0. Furthermore, Fig. 2c shows that for either positive S or negative S, R′S′ is almost always positive. This means that the effects of the turbulence are to enhance the mean CE rate by increasing condensation in the updrafts (usually S > 0) as well as decreasing the evaporation in the downdrafts (usually S < 0). Therefore, without the turbulence covariance terms, the mean CE rate not only ignores the condensation in the updrafts, but also overestimates the evaporation in the downdrafts. This is particularly true at cloud base and cloud top, as shown in Fig. 2b. This analysis shows that the mean supersaturation should not be used as the only factor to drive the droplet growth and the covariance R′S′ needs to be represented in the CE term. Note that these turbulence terms have been ignored in previous coupled turbulence closure-bin-microphysical models (e.g., Ackerman et al. 1995; Bott et al. 1996).

b. The w′q^′_l budget

The most striking feature in the w′q^′_l budget shown in Fig. 3a is a close balance below 700 m between the gradient (G) and the microphysical terms in (7). The mean gradient term represents the traditional downgradient transport (or the diffusive mixing). Its large negative values are a direct result of the large gradient of q_l with height, while the positive values of the microphysics term are due to the fact that condensation mainly occurs in the updrafts while evaporation is likely to occur in the downdrafts, leading to a positive correlation between w and CE perturbations. Therefore, it is condensation (evaporation) in the updrafts (downdrafts) that results in positive (negative) q_l fluctuations and leads to a mostly positive liquid water flux for both updrafts and downdrafts.

The buoyancy term is positive below 760 m in the cloud layer where the buoyancy fluctuations are positively correlated with vertical velocity, and it becomes strongly negative at the cloud top due to the entrainment of more buoyant and drier air. The pressure correlation term is negative below 760 m and becomes strongly positive in the inversion, therefore it tends to cancel the contribution of the buoyancy term. This is consistent with the general description of the corresponding terms in other scalar flux budgets (Moeng 1986).

The first three components of the microphysical part in (7) are presented in Fig. 3b. The fourth is negligibly small due to the small radius (∼1.5 μm) at which droplets are activated, and thus is not included. The contribution of the w′S′ term dominates the other two from the cloud base up to 770 m, above which it tends to cancel out the negative contribution from the others. The covariance w′S′ (Fig. 3c) represents a fundamental element of the interaction between the turbulence and microphysics, and thus is essential to the microphysical term. The maximum value just above the cloud base is clearly associated with large positive (negative) S and w in the updrafts (downdrafts) at those levels as discussed by many authors (e.g., Stevens et al. 1996a). At the very top levels of the cloud layer, the warmer and drier inversion air (S′ < 0) is entrained into the downdrafts (w′ < 0), leading to the second maximum of the w′S′ profile.

Recently, Khairoutdinov (2000, personal communication) showed analytically that w′S′ is the term needed to approximately balance the mean gradient term. To demonstrate this, we consider two major source terms for w′S′ in the steady-state form of the S tendency equation (e.g., Clark 1973):

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where γ = Lc⁻¹_pT θ⁻¹∂q_s/∂T and is considered a known function of the ensemble mean atmospheric pressure and temperature. Assuming R in (10) remains the ensemble mean value, multiplying by w′, and performing Reynolds averaging, we have

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where dq_sa/dz is the gradient of saturation water vapor mixing ratio following the mean moist adiabat. Note that (11) only applies to the levels where all the grid points are cloudy. Equation (11) states that w′S′ is always positive, and is equivalent to the correlation between w and the moist adiabatic CE rate (−wd q_sa/dz). The estimated w′S′ based on (11) is shown in Fig. 3c for comparison. Replacing w′S′ in (7) with (11) and neglecting SGS and other microphysical terms, leads to

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Therefore, as long as q_l follows the mean moist adiabat, which is approximately true for stratocumulus clouds, the mean gradient and w′S′ terms nearly cancel each other. It is interesting to note that w′S′ is inversely proportional to R, which depends on the droplet distribution and the droplet number N. This point is discussed further in section 7.

This analysis shows that the w′S′ term balances the negative contribution of the gradient term to produce the positive w′q^′_l. Therefore, this term must be considered in the parameterization of w′q^′_l when a bin-microphysical model is coupled to a turbulence parameterization.

c. The N budget

As in the case of the q_l budget, the flux divergence (T) and microphysical terms (A + E) in (8) are closely balanced in the N budget (Fig. 4a). Particularly, both terms have large magnitudes in the lower part of the cloud layer, implying significant activation/evaporation and turbulent transport of the droplets there. At the cloud top, both terms reach their local maximum although the magnitudes are significantly lower than those at the cloud base.

Figure 4b shows the activation (A) and the evaporation (E) components in the microphysical term. Below cloud base, the evaporation is likely to have resulted from the downward transport of small cloud droplets, which leads to the w′N′ convergence. Both the activation rate and the flux divergence reach a maximum at 575 m, implying that the newly activated cloud droplets at the cloud base are effectively transported upward to the upper levels of the cloud layer. At the cloud top, both activation and the evaporative loss attain second local maxima, which may be strongly influenced by the spurious production of cloud drops due to the LES-BM inability to represent SGS cloud field at cloud edges as discussed by Stevens et al. (1996b). The droplets are generally larger at the cloud top than those at the cloud base (in the absence of drizzle processes), leading to a reduced loss of the droplets due to evaporation at cloud top compared with that at cloud base.

d. The w′N′ budget

As shown in Fig. 5a, the gradient and the microphysical terms again have opposite signs and are dominant in the budget in the lower part of the cloud. In the inversion layer, the buoyancy and pressure perturbation terms are in close balance. Figure 5b shows that both components of the microphysical term, the activation and evaporation fluxes as defined by “A” and “E” terms in (9), are generally positive, because the activation is most likely to occur in the updrafts and the evaporation in the downdrafts. Similar to the budget of w′q^′_l, the results here indicate that w′N′ should not be modeled as a diffusive flux, because the activation and the evaporation contribute significantly and positively to the flux. Without these positive contributions, w′N′ would be largely negative and not be able to transport the newly activated droplets at the cloud base to upper levels to maintain a uniform profile of N in the middle of the cloud layer.

a. Liquid water flux w′q^′_l

Two types of parameterizations have been frequently used for the stratocumulus cloud parameterizations. The first, and the most popular one, is the statistical approach of Sommeria and Deardorff (1977) and Mellor (1977); (or some variation thereof) and can be written as

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where C is cloud fraction, q_t total water mixing ratio, and θ_l liquid water potential temperature. Note that (13) is not related to the cloud droplet spectrum because liquid water is diagnosed by any excess above the saturation mixing ratio (saturation adjustment scheme); that is, S = 0 at any cloudy grid point. Wang and Wang (1999) analyzed (13) and concluded that the turbulence-generated condensation is dominant in defining the flux profile in the parameterization. The second parameterization is the mass flux scheme in which w′q^′_l is parameterized as a product of a mass flux and the difference between updraft and downdraft liquid water content. Wang and Albrecht (1986) used this method to compute the buoyancy flux in their modified mixed layer model. Randall (1987) discussed this approach in detail; it can be expressed in general as

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where subscripts “u” and “d” represent updraft and downdraft variables, σ is the fractional coverage of either updrafts or downdrafts, w_u the average updraft velocity, and φ any variable (φ = q_l for liquid water flux), respectively.

Based on the w′q^′_l budget analysis presented in section 5b, a new dynamic and microphysical approach is proposed here. Following the common practice of second-order flux parameterization, the tendency of (7) is set to zero for the equilibrium solution to w′q^′_l. In addition, the pressure correlation term is split into a buoyancy and a return-to-isotropy term (e.g., Moeng and Wyngaard 1986); that is, −1/ρ₀ × q^′_l∂p′/∂z = −0.5g/θ₀ × θ^′_υq^′_l − w′q^′_l/τ_R, where τ_R is return-to-isotropy timescale. Furthermore, the transport term and all the third moments in the microphysical term are neglected. The two second-order turbulence–microphysics covariance terms of (7) are assumed to be related by

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Using the definition of q_t(q_t = q_υ + q_l) and S, we have S′ = q^′_t − q^′_l − q^′_s (T, p). Further expressing q^′_s in terms of θ^′_l and q^′_l, we can approximate S′ as

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where the effects of pressure fluctuations are neglected. An expression of w′S′ can be derived from (16), and is then substituted into (7) to obtain

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where τ_R is of the same order as the large eddy turnover time (Moeng and Wyngaard, 1986), and τ_CE is the CE timescale defined by

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Note that τ_CE is the same as the commonly used supersaturation absorption or phase relaxation timescale defined by an e-folding decrease of the initial surplus water vapor discussed by many authors (e.g., Squires 1952; Kogan et al. 1995).

Parameterization (17) represents a balance among three dynamic and one microphysical terms in (7); these terms are the mean gradient, buoyancy, pressure correlation, and surplus vapor flux terms. An interesting feature of (17) is that w′q^′_l is directly related to τ_CE, which is defined by the droplet spectrum through (18). A smaller τ_CE (or larger R) would tend to reduce the effect of the mean gradient term, which can be explained as follows. The effective timescale over which the turbulence can affect w′q^′_l is (1/τ_R + 1/τ_CE)⁻¹ (rather than τ_R), as shown by (17). Therefore, when τ_R/τ_CE ≫ 1, the turbulence has a response timescale that is equivalent to τ_CE to counteract the microphysical change. Consequently, a smaller τ_CE would result in a smaller turbulence diffusion flux and a larger w′q^′_l. Considering both the turbulence and condensation timescales, we would have two limiting cases. If τ_R/τ_CE → 0, w′q^′_l can be represented by the turbulence part only, which is simply the traditional downgradient formulation. If τ_R/τ_CE → ∞, w′q^′_l is determined by the microphysical part only, which is similar to (13). These two formulations represent only two extreme cases in which either turbulence or microphysical contributions dominate.

For real clouds, both τ_R and τ_CE are finite. For example, given the generalized convective velocity scale (Deardorff 1980) w∗ ≅ 0.9 m s⁻¹ and the boundary layer height z_i ≅ 800 m, we obtain τ_R ∼ z_i/w∗ ≅ 880 s. Figure 6a shows that τ_CE is between 3 and 10 s. Clearly, τ_CE ≪ τ_R, demonstrating that the microphysical processes are always important and cannot be ignored. Since the gradient of q_l is significant for stratocumulus clouds, the turbulence diffusion term is large and must also be included for a supersaturation-based cloud model as well.

Assuming the equilibrium condition in (12), using the same parameterization for the pressure correlation term and ignoring the resultant buoyancy term, we obtain the following approximation:

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This equation states that w′q^′_l can be treated as a downgradient flux, as long as the fluctuation of q_l resulting from vertical motion includes both the condensation/evaporation and the gradient parts. However, a major weakness of (19) is that the microphysical contribution is independent of τ_CE due to the approximations made in deriving (11). Consequently, both the turbulence and CE contributions depend only on the turbulence timescale τ_R. [See Eq. (17) for comparison.] Note that (19) has been obtained by Khvorostyanov and Curry (1999) and Sigg (2000). The development in this paper differs from theirs mainly in that (17) and (19) are derived from the dynamic balance among the turbulence and microphysical processes.

Figures 6b and 6c show the comparison between the LES-BM resolved and the parameterized fluxes. The parameterized w′q^′_l from (17) compares well with the LES-BM resolved flux except in the inversion where the negative values result, probably from the buoyancy term. For this reason, w′q^′_l is again calculated from (17) but without the buoyancy term. The new flux performs better in the inversion. The parameterized w′q^′_l from (13) and (14) are also shown (Fig. 6c). Because the mass flux scheme (14) ignores the variability within updraft or downdraft plumes, it understandably underestimates w′q^′_l. The statistical approach is based on a Gaussian distribution of the moisture departure from the mean saturation, and based on the saturation adjustment practice. Therefore, if C = 1 at certain levels (i.e., all the grid points are cloudy at that level), the value calculated from (13) should be almost equal to the LES resolved flux if the saturation adjustment were used in the model. The flux from (13) is, however, significantly larger than the LES-BM resolved flux. This implies that an LES coupled with a saturation adjustment (SA) cloud scheme gives larger w′q^′_l than an LES coupled with a supersaturation-based cloud model. This result will be discussed further in section 7.

b. Droplet number density flux w′N′

Using the equilibrium and return-to-isotropy assumptions, we can obtain the following diagnostic equation from (9):

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where the transport term is included because it is important in the middle of the cloud layer as shown in Fig. 5. The mass flux formulation for w′N′ can be obtained by letting φ = N in (14). As shown in Fig. 7, the flux without the transport term derived from (20) does not even provide solutions of reasonable magnitude. However, the inclusion of the transport term significantly improves the parameterization, although its performance is still less satisfactory than (17) for w′q^′_l. The main reason lies in the fact that all the terms are small and of the same order as that of the tendency term in the middle cloud layer. The simple relationship (15) compares well with the resolved w′N′ at the cloud base and in the upper part of the cloud layer. However, it does not obtain the maximum above the cloud base because the activation contributes little to w′q^′_l as discussed in the budget analysis. In contrast, the mass flux formulation (14) obtains the correct positive maximum because the activation and evaporation contribution is automatically included in N_u and N_d, respectively, while it considerably underestimates the negative values near the cloud base.

Although the parameterizations of w′q^′_l and w′N′ are derived for the ensemble mean fluxes, some of them may also be used for the SGS fluxes in LES-based microphysical parameterizations like those of Feingold et al. (1998) and Khairoutdinov and Kogan (2000). Normally, the SGS cloud water mass and number fluxes are parameterized in terms of the diffusion formulation as those of q_t and θ_l. However, the major difference between q_l and q_t is that the former typically increases significantly with height due to condensation, while the latter is conserved during condensation and is therefore usually well mixed in the boundary layers. For an LES, the subgrid-scale diffusion timescale τ_R can be estimated as Δz (SGS TKE)^−0.5 ∼ 100 s using the simulation parameters, and is still significantly larger than τ_CE. Therefore, the microphysical impact should be considered in formulating the SGS liquid water fluxes.

c. Ensemble mean CE rate

At the heart of any microphysical parameterization is the mean CE rate, which must include the turbulence contribution as discussed in the last section. One possibility is to use a higher-order turbulence closure model to directly predict or diagnose q^′_lS′ and N′S′ in (6). This can be accomplished by deriving the budgets of q^′_lS′ and N′S′ following the approach of Wang et al. (1998) and Wang and Wang, (2000).

Another approach is to recognize that the turbulent updraft and downdraft circulation is responsible for most of the turbulent transport and that vertical motion is the main driving force for the microphysics. Therefore, we could compute CE rates in each of the updraft and downdraft branches; that is, we could rewrite (3) as

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The supersaturation and the integral radius of updrafts and downdrafts demonstrate normal profiles as shown in Fig. 8 (and derived from S1). Profiles of the parameterized updraft and downdraft CE rates closely follow those from direct conditional sampling of LES-BM model resolved CE rates in most of the cloud layer. This consistency reflects the coherence between w and S at those levels, which is the basis for the parameterization. The consistency does not, however, exist in the inversion layer because the circulation is ill-defined there due to strong influences of small-scale turbulence. Therefore, even though (21) can accurately describe the CE process in most of the cloud layer, a special treatment of the entrainment zone at the cloud top must be formulated to include the effects of small-scale turbulence. A similar framework was used by Considine (1997) to study diurnal variability in microphysical properties of stratocumulus clouds. Note that (21) has the same form as the mass flux representation of the turbulence statistics in many models (e.g., Wang and Albrecht 1986; Lappen and Randall 2001). Therefore, it may be particularly useful in a coupled mass flux–microphysics parameterization.

a. Dynamic feedback of the CE timescale

The results in section 6a (Fig. 6) suggest that w′q^′_l resolved by an LES coupled with a saturation adjustment cloud scheme is larger than that resolved by a LES coupled with a supersaturation-based cloud model. The reason lies with the fact that S = 0 at a cloudy grid point for a LES-SA, while it can be greater or smaller than zero at a cloudy grid for a LES-BM. This can be expressed in the following relationship obtained from (16):

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where w′S′, while always positive for the LES-BM model, is zero for the LES-SA at a level at which all the grid points are cloudy. Equation (22) shows that w′q^′_l derived from LES-SA is larger than that from LES-BM when all the grids are cloudy. In other words, LES-SA produces more condensation in updrafts and evaporation in downdrafts than LES-BM does because S must be zero for a saturation adjustment when cloud is present. Consequently, the magnitudes of q^′_l and w′ from LES-SA are larger than those from LES-BM.

The above results suggest that the CE timescale τ_CE, to some extent, regulates the turbulence field. Note that droplet number N is related to τ_CE through the integral radius R. When N is large, R is large, leading to a small τ_CE as seen from (18). Then, larger CE fluctuations are produced and result in larger w′q^′_l. Therefore, the larger N is, the smaller τ_CE is, and the closer the turbulence structure derived from LES-BM is to that derived from LES-SA. More specifically, when τ_CE decreases (i.e., R increases), w′S′ decreases as shown by (11), resulting in an enhanced w′q^′_l as shown by (22). This feedback can also be seen from the w′q^′_l parameterization (17). Since the negative contribution from the gradient term decreases with decreasing τ_CE, w′q^′_l increases with increasing N and approaches (13) when τ_CE → 0. Because the extra q^′_l produced by larger N (or smaller τ_CE) in up- and downdrafts are likely to cancel for the ensemble mean due to their different signs, and because instantaneous CE rate is strongly coupled to the vertical motion of turbulent eddies, the direct impact of this feedback is on the turbulence field.

To test these ideas, two more simulations are performed. One (S2) uses the same LES-BM model, but with a background CCN number of 1000 mg⁻¹. The other (S3) uses the same LES dynamic model but with a saturation adjustment scheme. The high CCN number concentration used in S2 is chosen to facilitate discussion, but it may not be unrealistic for a polluted air mass. All the simulation procedures are identical to those of the previous simulation (S1) and listed in Table 1.

Figure 9 shows the comparison of some statistics among three simulations over the last hour (hour 5–6). Both S2 and S3 result in significantly larger resolved w′q^′_l than S1, as expected from the above discussion. Particularly, w′q^′_l from S2 is closer to S3 than to S1, despite the fact that S3 is run with the LES-SA model, while S1 and S2 are run with the LES-BM. The integral droplet radius R in S2 is about four times that of S1, leading to a reduction in w′S′ from S2. This result is consistent with the earlier discussion in which we argued that an increase in R will decrease w′S′ and thus increase w′q^′_l. Since local CE rates in turbulent eddies increase with R (or decreasing τ_CE), the cloud-top entrainment is enhanced, which is demonstrated by the larger total water flux near the cloud top, and consequently q_l and cloud fraction from S3 are reduced as shown in Fig. 9d. Because LES-SA results in the largest instantaneous local CE rate among the three simulations, the entrainment from S3 is strongest and q_l is the smallest, as can be seen in both Figs. 9c and 9d.

The above results suggest that different CCN number concentrations may lead to different turbulence fields due to the different CE timescales. Consequently, the impact of variable CCN number density is not only due to the drizzle process, but also due to the different CE timescale.

b. Impact of drizzle on the parameterizations

The parameterizations (17) and (20) have been derived based on the formulation without drizzle processes. It is natural to ask how drizzle will affect these parameterizations. To answer this question, we perform two simulations (S4 and S5) that include the processes of droplet coalescence–collection and sedimentation as shown in Table 1. In S5, the higher water vapor mixing ratio above the cloud leads to a stronger drizzle rate as seen in Fig. 10a. The flux w′q^′_l, calculated from parameterization (17) without the buoyancy term compares well with the LES-BM solution. This is because the drizzle contribution to the w′q^′_l budget is at least one order of magnitude smaller than the major terms (not shown here), and thus can be neglected. Therefore, (17) should apply as long as the dynamics are primarily driven by the cloud-top radiative cooling (rather than by convective instability in the shallow cumulus regime). The fluxes based on the saturation adjustment scheme (13) considerably overestimates the flux value in the lower half of the cloud layer where τ_CE is relatively large (Fig. 6a) and S has a positive maximum value in updrafts.

The inclusion of drizzle may affect parameterization (20) through the correlation between w′ and droplet collection, and sedimentation, respectively. We have found in particular that the contribution from the collection tendency flux is significant and highly variable, and thus the quasi-equilibrium condition is not met for the w′N′ budget. Consequently, the validity of (20) is degraded. The flux calculated from (15) is compared with the LES-BM resolved in Figs. 10c and 10f, and the discrepancies are relatively large compared to those for w′q^′_l. Therefore, the formulations (15) and (20) do not provided satisfactory solutions to the w′N′ parameterization, although they are probably useful steps toward that goal.

The mass flux representation of w′q^′_l, w′N′, and the ensemble CE rate should not be significantly influenced by the drizzle processes, because the formulations, (14) and (21), only involve the conditional sampling of updraft–downdraft properties. They should apply in drizzle conditions so long as the convective circulation remains the major turbulent transport mechanism.