Part I of this two-part series presents a method to account for the effects of subgrid variability on average microphysical process rates. The method involves upscaling a local microphysics scheme, that is, computing a grid-box average by integrating over an assumed probability density function (PDF). In this paper (Part II), the method is tested. The test case is based on research flight two (RF02) of the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field experiment. The upscaled microphysics scheme is incorporated into a single-column model (SCM). Then two SCM simulations of the RF02 test case are performed. They are identical except that one upscales the local microphysics scheme and one does not. Both SCM simulations are compared with a benchmark large-eddy simulation (LES) of the same test case using the same local microphysics scheme.

Large-scale models of the atmosphere cannot resolve small-scale variability in moisture and temperature. Neglecting this variability can lead to errors in grid-box-averaged microphysical process rates. To avoid these errors, we may use the subgrid variability to drive local microphysical rate formulas that are valid at a point in space. By accounting for the effects of subgrid variations on local microphysical rates, the average rate over an extended volume may be estimated. That is, a local microphysical formula may be upscaled to estimate grid-box average rates.

In Part I of this two-part series (Larson and Griffin, 2012), we upscale the Khairoutdinov and Kogan (2000) microphysics scheme (hereafter, KK scheme) by use of the assumed probability density function (PDF) method. In the present paper, we apply the upscaled microphysics to a simulation of a drizzling marine stratocumulus cloud. Specifically, the case is based upon the second research flight (RF02) of the DYCOMS-II field experiment (Stevens et al., 2003). We choose a stratocumulus cloud because the KK scheme was developed for drizzling stratocumulus clouds.

In order to assess the effects of upscaling, we perform two single-column model (SCM) simulations. The two SCMs are identical, except that one uses local KK microphysics and the other upscales the same local microphysics. The two SCM simulations are then compared to a three-dimensional large-eddy simulation (LES). The LES model simulates the RF02 case using the local KK microphysics. Because the LES model uses an identical case configuration and microphysics as the SCM, the LES serves as a benchmark with which to compare the SCM's representation of spatial variability. We note that this paper does not assess the accuracy of the local KK microphysics; rather it is an assessment of the SCM's ability to account for means and subgrid variability, and thereby accurately drive the local microphysical processes.

An outline of this paper is as follows. Section 2 presents details of the SCM and LES model, the KK microphysics scheme, and the RF02 Sc case. Section 3 presents plots of rain and other relevant variables. Section 4 explores the question of why the use of a local microphysics scheme in the SCM underestimates rain at the ocean surface. Section 5 outlines a general method to estimate the importance of upscaling processes that can be represented by power laws. Finally, section 6 provides concluding remarks.

2. Models, microphysics scheme, and case set-up

2.1. LES model: SAM

The LES model used is the System for Atmospheric Modeling (SAM) (Khairoutdinov and Randall, 2003). SAM is a non-hydrostatic model that solves the anelastic equations of fluid flow on a Cartesian grid. It prognoses moist static energy, total non-precipitating water (the sum of water vapour and cloud condensate), and total precipitating water. The amount of cloud condensate is diagnosed based on a saturation adjustment. Scalars are transported using a monotonic flux limiter in order to avoid spurious numerical oscillations. The time-stepping scheme is the third-order (multi-step) Adams–Bashforth scheme. Periodic boundaries are used in the horizontal and a rigid lid is used at the top of the domain. Subgrid-scale fluxes are computed using a 1.5-order closure based on a prognostic subgrid-scale turbulent kinetic energy (Deardorff, 1980). SAM has successfully performed LES of a variety of boundary layer cases, such as the DYCOMS-II RF01 marine stratocumulus case (Stevens et al., 2005), a trade-wind cumulus case (Siebesma et al., 2003), and a stable boundary layer case (Beare et al., 2006). Of special relevance is the fact that SAM produced comparable output to the other LES models in the recent DYCOMS-II RF02 intercomparison (Ackerman et al., 2009).

2.2. SCM: CLUBB

The SCM we use is the Cloud Layers Unified By Binormals (CLUBB) model (Golaz et al., 2002a, 2002b; Larson and Golaz, 2005). The SCM is a higher-order closure model that achieves closure by prediction of a PDF of subgrid variability. In earlier work (Golaz et al., 2002a), the functional form of the PDF was assumed to be a mixture of two multivariate normal distributions. The PDF was a function of liquid water potential temperature θ_{l}, total water mixing ratio r_{t}, and vertical velocity w. The use of a multivariate PDF tightly integrates the dynamics, such as turbulence and updraughts/downdraughts, with the thermodynamics. In order to incorporate the KK microphysical scheme in a consistent, unified way, Part I extends the PDF to include rainwater mixing ratio r_{r}, raindrop number concentration N_{r}, and cloud droplet number concentration per mass of air, N_{c}. All three of these variates are assumed to follow single log-normal distributions. Cloud water is diagnosed from the aforementioned PDF using a saturation adjustment scheme. In our simulation, the mean of N_{c} is prescribed rather than prognosed, but fluctuations in N_{c} about the mean do occur and are distributed according to the PDF outlined in Part I. Guo et al. (2010) have allowed the mean of N_{c} to vary and integrated over a PDF in order to compute droplet activation.

2.3. Khairoutdinov and Kogan microphysics

The KK microphysics scheme uses power laws of two or three variables to empirically approximate bin microphysical calculations. Namely, KK optimized the power laws to fit bin microphysics output from one case: a drizzling marine stratocumulus cloud. The KK scheme is a warm-rain scheme; it does not account for ice. A key advantage of the KK scheme is that it is simple, which makes it possible to analytically integrate the microphysical process rates over a PDF. The KK scheme is a full double-moment scheme that includes predictive equations for r_{r}, N_{r}, cloud water mixing ratio r_{c}, and N_{c}. For simplicity, however, our SCM includes only the prognostic equations for r_{r} and N_{r}. The SCM diagnoses r_{c}, rather than prognosing it. The SCM sets the grid-box average of N_{c} to a constant value.

We call the KK scheme as presented in Khairoutdinov and Kogan (2000) the ‘local’ KK scheme. We use this scheme in both SAM and CLUBB. However, in some (but not all) of our CLUBB simulations, the local KK scheme is driven by subgrid variability, and in these cases we call the resulting scheme the ‘upscaled’ KK scheme.

2.4. Configuration of the DYCOMS-II RF02 case

The set-up of the SCM is based on the specifications for the DYCOMS-II RF02 intercomparison. Large-scale subsidence is imposed, and the surface latent and sensible heat fluxes are held constant. Long-wave radiative cooling is computed using a simple analytic approximation (Wyant et al., 2007; see also Stevens et al., 2005; Larson et al., 2007). The intercomparison used a constant value of cloud droplet concentration. The simulations lasted 6 h. As previously stated, either the local or the upscaled version of the KK microphysics scheme is used in all simulations. A complete description of the SCM intercomparison can be found in Wyant et al. (2007).

The SAM LES model is set up identical to the SCM, so far as possible. Both models have the same initial profiles, forcings, boundary conditions, and prescribed values of cloud droplet concentration. Nevertheless, there are some slight differences between the LES and SCM grids. The LES uses a time step of 0.5 s, while the SCM uses a time step of 60 s. The LES uses a stretched grid that has a minimum vertical grid spacing of 5 m at the surface and at the inversion, and a maximum vertical grid spacing below the inversion of 25.1 m (at about 400 m in altitude). In contrast, the SCM employs a constant vertical grid spacing of 10 m. The LES is set up with a constant horizontal grid spacing of 50 m × 50 m and a horizontal domain of 6400 m × 6400 m. For both models, the top of the domain is at approximately 1460 m.

This study neglects sedimentation of cloud droplets (Ackerman et al., 2004; Bretherton et al., 2007). The reason is that the value of the geometric standard deviation of the cloud droplets, σ_{g}, is controversial. The LES models in the intercomparison were run with σ_{g} = 1.5 or with no cloud droplet sedimentation. However, Wyant et al. (2007) mentions that σ_{g} = 1.5 may represent an unrealistically broad cloud droplet distribution, and that σ_{g} = 1.2 was estimated by the RF02 observations. Most SCMs in Wyant et al. (2007) ran without cloud droplet sedimentation. Therefore, in order to remove this source of uncertainty in our simulations, cloud droplet sedimentation has been shut off.

3. Model results

In this section, we present simulations of the DYCOMS-II RF02 case using three models: the SCM with upscaled microphysics, the SCM with local microphysics, and SAM (the benchmark LES model). The LES uses the local KK microphysics, but it includes information on horizontal variability of the microphysical fields because it solves the governing equations on a fine-scale three-dimensional mesh. Over an entire horizontal slab of the LES domain, variability occurs. This allows the LES to drive the local microphysical process rates using the spatially varying values of microphysical fields such as N_{c} and r_{r}. The SCM with local microphysics contains information about the horizontal variability of r_{t}, θ_{l}, and w, but not the microphysical fields such as r_{r}. The upscaled microphysics represents this microphysical variability but in the approximate form of a PDF whose functional form is assumed. The goal of this section is to assess whether or not the upscaled microphysics matches the rain fields simulated by the LES better than the local microphysics. Because both LES and SCM use KK microphysics, a poor match can arise only from differences in variability between LES and SCM, which in turn may arise from a variety of errors in the SCM, including vertical turbulent transport.

3.1. Turbulence, thermodynamic, and cloud fields

Before we turn to the rain fields, we first assess whether or not other important fields produced by the SCM simulations match the LES, such as those related to cloud water. Such fields influence rain production, and if they are not simulated accurately they can lead to inaccurate prediction of rain mixing ratio even if accurate local or upscaled rain processes are used.

Production of rain is defined here to consist of two processes: autoconversion of (small) cloud droplets to (large) raindrops, and accretion (i.e. collection) of cloud droplets onto raindrops. Production of rain within cloud and evaporation below clouds is influenced by 〈θ_{l}〉 and 〈r_{t}〉. For instance, 〈θ_{l}〉 influences saturation mixing ratio, and 〈r_{t}〉 is related to cloud water mixing within cloud and relative humidity below cloud. Here 〈〉 denotes a grid-box mean value.

The profiles of 〈θ_{l}〉 and 〈r_{t}〉 for all three simulations are broadly similar, as shown in Figure 1. However, there are slight differences. Both SCM profiles of 〈θ_{l}〉 are slightly cooler than SAM LES below the inversion. The SCM profiles of 〈r_{t}〉 are not as well mixed as that of SAM LES. That is, the SCM profiles are moister below cloud as compared to SAM LES.

The values of 〈w^{′2}〉 in all three models are fairly similar (see Figure 1(c)). However, both SCMs exhibit a minimum in 〈w^{′2}〉 near cloud base at about 450 m, a minimum that is not produced by the LES. The existence of such a minimum hints that the SCM produces a slightly ‘decoupled’ boundary layer in which vertical turbulent transport is inhibited between the cloud layer and the sub-cloud layer. This may explain why the SCM leaves excessive moisture in the sub-cloud layer as compared with SAM LES (see Figure 1(b)). Nevertheless, both SCM profiles of 〈w^{′2}〉 lie within the range of the LES models (grey shaded region in Figure 1(c)) that participated in the GCSS intercomparison (Ackerman et al., 2009). The vertical velocity variance 〈w^{′2}〉 indirectly influences rain, but because our simulations specify cloud droplet number it does not directly influence rain.

The profiles of cloud water, 〈r_{c}〉, and cloud fraction are similar for all three models (see Figure 2). Cloud water 〈r_{c}〉 has the same peak magnitude for all three models, and all three clouds are overcast with about the same thickness. However, both cloud top and cloud base are located slightly lower in the SCM simulations than in the LES. Presumably, the reason is that the SCM simulations do not entrain as much above-cloud air as the LES.

Not only do all three SCM simulations produce similar profiles of mean cloud water, but they also produce similar profiles of horizontal variance in cloud water mixing ratio, (see Figure 2(c)). Variability in r_{c}, in addition to its horizontal average, influences rain because rain production is a nonlinear process.

The similarity of the cloud water fields among the models suggests that the differences in rainwater fields discussed below are not related to differences in the cloud water itself but rather to the neglect or inclusion of subgrid variability within the microphysics.

3.2. Rain

Differences between SAM, the SCM with upscaled microphysics, and the SCM with local microphysics first appear in the rain-related fields. The upscaled microphysics produces more rain at all altitudes than does the local microphysics. For instance, depending on altitude, the upscaled 〈r_{r}〉 is a factor of 1.1–3.5 times larger than the local 〈r_{r}〉 (see Figure 3). Furthermore, the upscaled raindrop number concentration, 〈N_{r}〉, and the precipitation flux are larger at all altitudes than their local counterparts (Figure 4). The upscaled 〈r_{r}〉, 〈N_{r}〉, and precipitation flux better match the LES near cloud top and especially near the ocean surface (see Figure 4).

Near the ocean surface, the local 〈r_{r}〉, 〈N_{r}〉, and precipitation flux are all an order of magnitude less than SAM LES. The upscaled SCM still underestimates SAM LES, but at later times lies within the range of the LES that participated in the GCSS intercomparison.

The time series of liquid water path and surface precipitation flux are presented in Figure 5. The liquid water path of all three models is similar, indicating that the microphysics of all three models experience similar environments. Nevertheless, the local microphysics underestimates the surface precipitation flux, whereas the upscaled microphysics mostly lies within the range of LES in the GCSS intercomparison. The fact that LWP is similar among the models but the surface precipitation differs suggests that rain does not strongly deplete liquid water in these simulations. As an aside, we also note that the time evolution of the SCM precipitation fields are free of spurious numerical noise.

3.3. What processes contribute to rain formation and evaporation?

In order to assess which physical processes generate or deplete rain, Figure 6 presents a budget of rain mixing ratio, 〈r_{r}〉. We divide the budget into four terms: (1) evaporation, which depletes 〈r_{r}〉; (2) production, which is defined as the sum of autoconversion and accretion, and which generates 〈r_{r}〉; (3) Gravitational sedimentation, which transports 〈r_{r}〉 downward and allows some of it to strike the ocean and leave the atmosphere; and (4) transport, which is defined as the sum of mean vertical advection and eddy (turbulent) diffusivity, and which transports 〈r_{r}〉 to different altitudes but does not change its vertical average.

The shapes of the process rate profiles are physically plausible. Evaporation of raindrops occurs below cloud because that region is subsaturated. The evaporation rate is greatest just below cloud base, where raindrops are still large and numerous. Production due to autoconversion and accretion is positive everywhere within cloud. Gravitational sedimentation removes raindrops from the upper part of the cloud layer and deposits them below. Turbulent transport removes rain from the cloud layer and deposits it near cloud top and below cloud. All process rates are small near the ocean surface, because few raindrops survive the fall to those altitudes.

Let us examine the differences in Figure 6 between the SCM process rates produced by the upscaled microphysics (dashed line) and the local microphysics (dotted line). All four process rate profiles are largely similar in shape and magnitude. The most prominent difference is that the upscaled microphysics produces greater production (i.e. autoconversion plus accretion) than does the local microphysics. Autoconversion is underpredicted by the local microphysics because the autoconversion process rate formula is convex (i.e. has upward curvature) nearly everywhere and hence often produces underpredictions if subgrid variability is neglected (Larson et al., 2001). Accretion is underpredicted because it depends on the correlation of r_{c} and r_{r}, which is ignored in the local microphysics. The upscaled microphysics still underestimates the LES production rate but nevertheless significantly improves on the local microphysics.

Now consider the differences between the process rates predicted by the SCM simulations and LES (again, see Figure 6). The largest differences occur near cloud top. The SCM significantly underestimates the magnitudes of the rates of all three processes that occur strongly near cloud top: production, sedimentation, and transport. The root of the problem appears to be insufficient transport of rain to cloud top. Near cloud top, this leads to a diminished magnitude of rain mixing ratio itself and of the vertical gradient of rain mixing ratio (see Figure 4(a)). The lack of rain inhibits the collection rate of cloud droplets by raindrops (Figure 6) because the collection rate is nearly proportional to rain mixing ratio. The diminished gradient of rain leads to diminished sedimentation, because sedimentation rate is nearly proportional to the gradient of rain.

Differences in process rates also occur below cloud. Again, the largest error in the SCM is insufficient transport (Figure 6), which allows the rain mixing ratio to become too small near the ocean surface (see Figure 4(a)).

However, improving the transport of rain in the SCM would be difficult. The SCM represents transport of rain via an eddy diffusivity formulation. A higher-order closure treatment could be envisaged, but it would entail a greater computational cost. Therefore, we have not pursued it here.

4. Why does the SCM with local microphysics underpredict surface rain?

Figure 5 has shown that the upscaled microphysics produces more rain at the ocean surface than does the local microphysics. Which microphysical processes (autoconversion, accretion, or evaporation) are most responsible for the increased rain in the upscaled simulation? A hint is provided by the terms in the rain budget that are plotted in Figure 6. For instance, the upscaled microphysics exhibits larger production rates (autoconversion and accretion rates) than does the local microphysics. However, the rain budget is not conclusive. Is the upscaled production rate larger because r_{r} is larger, or is it larger because the upscaled production rate is more efficient per unit of r_{r} present? It is difficult to distinguish these possibilities because of the presence of internal feedbacks, which confound cause and effect.

To help separate the possibilities, we will isolate the efficiency of the upscaled process rate formulas from the magnitudes of the inputs to these formulas. To do so, we compare local and upscaled formulas using the same inputs. Specifically, we analyse the simulation with upscaled microphysics, and we compute the ratio of [the local process rates that would have occurred if we had fed in the relevant fields from the upscaled simulation] to [the upscaled process rates themselves]. For instance, for autoconversion, we compute (see Eq. (28) of Part I):

(1)

In both numerator and denominator, we feed in values of r_{c} and N_{c} from the upscaled simulation. Likewise, for accretion, we compute (see Eq. (38) of Part I):

(2)

Finally, for evaporation, we compute (see Eq. (52) of Part I):

(3)

Here N_{r} is the number concentration of raindrops per mass of air in kg^{−1}, and S is supersaturation, which is dimensionless.

Each of these three normalized process rates is a non-dimensional ratio. If the ratio is less than one, then it means that upscaling the microphysics increases the process rate relative to the local formula, which neglects subgrid variability. If the ratio is greater than one, then the upscaled rate is less than the local rate.

The ratios of autoconversion, accretion, and evaporation are displayed in Figure 7. Figure 7(a) shows that the upscaled autoconversion rate is greater than the local rate at all altitudes. At the altitude where both versions autoconvert the most, which is just above 800 m (see Figure 8(a)), the upscaled autoconversion rate is approximately 20% larger than the local autoconversion rate. The difference at that altitude occurs because autoconversion depends sensitively on cloud water variance, , and the variance is greatest at that altitude, as shown in Figure 2(c). Additionally, the vertical average of the upscaled autoconversion rate is calculated to be about 20% larger than the vertical average of the local autoconversion rate.

Figure 7(b) shows that the upscaled version accretes cloud water faster than does the local version at all altitudes. At the altitudes of greatest accretion rate, the upscaled accretion rate is 10–15% larger. The vertical average of the upscaled accretion rate is 13% larger than the vertical average of the local accretion rate. Thus the percentage increase of the upscaled version over the local version is less for accretion than for autoconversion. Nevertheless, the absolute increase for accretion and autoconversion is comparable. The reason is that accretion is larger than autoconversion. Specifically, Figure 8 shows that the upscaled microphysics produces a peak autoconversion of about 8 × 10^{−9} kg kg^{−1} s^{−1} but a peak accretion rate of 15 × 10^{−9} kg kg^{−1} s^{−1}.

Figure 7(c) shows that the normalized evaporation ratio is greater than one below an altitude of approximately 500 m. That is, below 500 m the upscaled evaporation rate has a lesser magnitude than the local rate. The difference between the two ranges from about 3% to 11% and is greatest where evaporation is greatest, between 400 m and 500 m (see Figure 6(a)), which is near cloud base. In contrast, in the region above 500 m, the upscaled evaporation rate has a much greater percentage magnitude than the local rate.

The large difference in evaporation between the upscaled and local microphysics arises because the upscaled version accounts for the effect of partial cloudiness on evaporation, whereas the local version does not. Specifically, the local version drives the evaporation formula with the grid-box mean saturation. The region near cloud base is, by necessity in a model with non-zero vertical grid spacing, a region of partial cloudiness. Figure 2 shows that the region of partial cloudiness extends from an altitude of approximately 600 m to an altitude of approximately 400 m.

In the region above 500 m, the cloud fraction is large, and 〈r_{t}〉 is either greater than or extremely close to the mean saturation mixing ratio. In that region, the local version assumes saturated conditions at all altitudes and times, and therefore does not produce much, if any, evaporation. However, the upscaled equation accounts for subgrid variability and hence accounts for the pockets of clear air. Thus, upscaled evaporation rate is greater than that of the local version.

In the region below 500 m, the cloud fraction is small, and the opposite occurs. That is, 〈r_{t}〉 is less than the mean saturation mixing ratio. The local version assumes that this region is entirely clear, whereas the upscaled equation recognizes that some cloud is present. Therefore, the upscaled version produces less evaporation than the local version.

Although the upscaled version evaporates more than the local above 500 m and less below 500 m, the vertical average of the upscaled evaporation rate has a magnitude 10 larger than the local evaporation rate. Thus, over the entire vertical domain, the net effect of upscaling the evaporation equation is an increase in evaporation.

In summary, the upscaled microphysics has more rain at the surface than the local microphysics because, when we normalize for the values of r_{c}, r_{r}, N_{r}, N_{c}, and S, the upscaled microphysics produces larger rates of autoconversion and accretion, which outweighs the larger upscaled rate of evaporation. Calculating vertical averages yields the following numbers. The upscaled autoconversion rate exceeds the local autoconversion rate by about 1.0 × 10^{−10} kg kg^{−1} s^{−1}. The upscaled accretion rate exceeds the local accretion rate by about 2.45 × 10^{−10} kg kg^{−1} s^{−1}. On the other hand, the magnitude of the upscaled evaporation rate exceeds the local evaporation by about 2.33 × 10^{−10} kg kg^{−1} s^{−1}. Therefore, given identical variable inputs, the upscaled method increases the vertical average of rainwater tendency (due to microphysics) by a net amount of 1.12 × 10^{−10} kg kg^{−1} s^{−1}.

In summary, the analysis in this section suggests that upscaled microphysics produces more rain than the local microphysics per unit of r_{c}, r_{r}, N_{r}, N_{c}, and S present.

5. When is upscaling important?

In comparison with the use of local microphysical formulas, upscaling the microphysics is more complex and entails greater computational cost. Therefore, before undertaking the labour of upscaling a microphysics scheme for a particular application, one may ask: in what cases and for what processes is upscaling important?

In the case study that we present here–DYCOMS-II RF02–the use of upscaling influences drizzle near the ocean surface (Figures 4 and 5). When we plot the ratios of upscaled to local process rates (see Figure 7), we find that upscaling has significant effects on autoconversion, accretion, and evaporation rates. However, the magnitude of the effects depends on both the nonlinearity of the process rate formula and the size and nature of the subgrid variability. Therefore, the magnitude of the errors varies from case to case. For instance, the DYCOMS-II RF02 case has moderate variability in cloud water, such that the value of is approximately 0.3.

When one is confronted with a new case or process, it would be convenient to have a simple estimate of the potential importance of upscaling. In principle, one could estimate the ratios of upscaled to local formulas, such as those presented in Figure 7. However, depending on the complexity of the formula for the process rate, this may require significant calculations. In such cases, it is convenient to form an estimate by expanding the process rates in a Taylor series. Furthermore, the Taylor series provides insight into why upscaling influences the results and not merely when upscaling matters.

As examples, we Taylor-expand the KK formulas for autoconversion and accretion, both of which are power laws of two variables. We now write down a general expansion for such two-dimensional power laws. The mean of the power law is of the form

(4)

where x and y denote the two variables involved (e.g. r_{c} or N_{c}), and α and β are their respective exponents. Each variable is divided into a mean component and a perturbation component (denoted by a prime symbol), so that x = 〈x〉 + x′ and y = 〈y〉 + y′. We Taylor-expand x^{α}y^{β} about x = 〈x〉 and y = 〈y〉, truncate it at the second order, and average over both sides. This yields

(5)

This equation can be used to estimate the errors involved in using the local microphysics formula, 〈x〉^{α} 〈y〉^{β}, instead of the upscaled formula, 〈x^{α}y^{β}〉. In Eq. (5), the local formula is given by the pre-factor in front of the parentheses on the right-hand side; the upscaled formula is given by the left-hand side. Therefore, the non-unity terms in the parentheses are an estimate of the fractional error in the local microphysics.

Additionally, we Taylor-expand the KK formula for evaporation, which is a power law of three variables. For reference, we now list the general expansion for such three-dimensional power laws. The mean of the power law is of the form

(6)

where x, y, and z denote the three variables involved (e.g. S, r_{r}, or N_{r}), and α, β, and γ are their respective exponents. Each variable is divided into a mean component and a perturbation component, so that x = 〈x〉 + x′, y = 〈y〉 + y′, and z = 〈z〉 + z′. We Taylor-expand x^{α}y^{β}z^{γ} about x = 〈x〉, y = 〈y〉, and z = 〈z〉, truncate at the second order, and average over both sides. This yields

(7)

This equation can be used to estimate the errors involved in using the local microphysics formula, 〈x〉^{α} 〈y〉^{β} 〈z〉^{γ}, instead of the upscaled formula, 〈x^{α}y^{β}z^{γ}〉 . Once again, the non-unity terms in the parentheses are an estimate of the fractional error in the local microphysics.

Equation (5) can be applied to both the autoconversion and accretion equations. The KK autoconversion equation is proportional to (Khairoutdinov and Kogan, 2000, Eq. 29). If, in Eq. (5), we substitute x = r_{c} and y = N_{c}, then we find

(8)

where α = 2.47, β = −1.79, and .

Similarly, the KK equation for accretion is proportional to (Khairoutdinov and Kogan, 2000, Eq. 33). If we let x = r_{c} and y = r_{r} in Eq. (5), then we find

(9)

where α = 1.15, β = 1.15, , and .

Furthermore, the KK equation for evaporation is proportional to (Khairoutdinov and Kogan, 2000, Eq. 22). In Eq. (49) of Part I, we relate S to extended liquid water mixing ratio, s, where s is given in Eq. (5) of Part I. Thus evaporation is proportional to . If we let x = s, y = r_{r}, and z = N_{r} in Eq. (7), then we find

(10)

where α = 1, β = 1/3, γ = 2/3, s = 〈s〉 + s′, , and .

Although Eqs (8), (9), and (10) provide simple estimates of the error that arises from neglecting subgrid variability, a key assumption is that the third-order terms are negligible. To test this assumption, Figure 8 overplots the full upscaled formulas for autoconversion, accretion, and evaporation rates (Part I, Eqs (33), (41), and (60), respectively), and the Taylor series approximations implied by Eqs (8), (9), and (10). The Taylor series underestimates the upscaled evaporation rate but matches the autoconversion and accretion rates closely. The good fit for accretion probably arises because both exponents are near unity. For instance, if both exponents are exactly unity, then the covariance (AR3) term is the only non-zero term, and all higher-order terms in the Taylor series vanish. When one or both exponents are far from unity, then the second-order power series fit may not be as accurate, particularly when there is large variability and/or the PDF is skewed. In such cases, the exact analytic integrations–e.g. Eqs (33), (41), and (60) from Part I–are useful.

In order to shed light on what terms contribute most to the autoconversion, accretion, and evaporation formulas, we plot each second-order term found in each Taylor-expanded equation (see Figure 9). Of the second-order autoconversion terms, the (AU1) term contributes most. Accretion has significant contributions from all three second-order terms (AR1, AR2, and AR3).

The single largest second-order contributor to the evaporation equation is the term (EV6), which increases the rate of evaporation. However, we recall that the net effect of subgrid variability below the altitude of 500 m is to reduce the evaporation rate. The reason is that the sum of the other second-order evaporation terms, all of which reduce the evaporation rate, outweigh the enhancement of evaporation rate by EV6 (see Figure 9(c)). In the region that is below 400 m, the chief positive contributors are the (EV2) and (EV3) terms. In the partly cloudy layer between approximately 400 m and 550 m, which is where the greatest amount of evaporation occurs, the chief positive contributors are (EV4) and (EV5). This layer, however, is where the Taylor series is least accurate (see Figure 9), and therefore the relative contributions of the terms are suspect.

6. Conclusions

This two-part series of papers has presented a method for incorporating the effects of subgrid variability on microphysical processes. The method thereby ‘upscales’ local microphysical formulas so that they yield an average over an extended grid box.

Part I (Larson and Griffin, 2012) discussed the theory underlying the upscaling method. It involved analytically integrating the local microphysical formulas over the probability density function (PDF) of spatial variability within a grid box. A key point was that the PDF includes variability in hydrometeor fields such as raindrop mixing ratio, r_{r}, and number concentration, N_{r}. Another key point was that variability in cloud fields, e.g. r_{c}, was used to drive microphysical process rates.

The present paper, Part II, evaluates the analytic upscaling method. To do so, we implement upscaled microphysics into the single-column model (SCM) of Golaz et al. (2002a). Then we simulate a drizzling marine stratocumulus cloud that was observed during Research Flight 2 (RF02) of the DYCOMS-II field experiment (Stevens et al., 2003). We simulate the RF02 case using (1) the SCM with upscaled microphysics; (2) the same SCM except that subgrid variability in clouds does not drive microphysics, and subgrid variability is neglected in the hydrometeor fields (but not the thermodynamic or turbulence fields); (3) and a three-dimensional benchmark large-eddy simulation (LES) model. The three simulations are configured identically so far as possible, thereby allowing direct comparison of the representation of variability among them. Furthermore, all three simulations produce similar fields of cloud water mixing ratio (r_{c}), horizontal variance of r_{c}, and humidity below cloud. Therefore, the cloud and humidity fields in all three simulations drive the drizzle processes similarly.

In the RF02 case, the SCM with upscaled microphysics produces nearly four times as much precipitation flux at the ocean surface as does the SCM with local microphysics. The profile of upscaled precipitation flux lies mostly within the range simulated by LES models participating in the RF02 GCSS intercomparison, but the local precipitation flux does not (Figure 4). Relatedly, upscaling the microphysics increases 〈r_{r}〉 by about 20% within cloud and about 75% at the surface (Figure 3). Differences in rain of this magnitude can also arise from changes in the formulation of local microphysical formulas (e.g. Cotton and Anthes, 1989). Nevertheless, our results demonstrate that accounting for subgrid variability has a non-negligible effect in the RF02 case. The effects of upscaling may differ in other cases.

Why does accounting for subgrid variability increase rain? The upscaled rain is enhanced by increased autoconversion and accretion, which outweighs the diminishment of rain by enhanced evaporation. Within cloud, accounting for subgrid variability increases the autoconversion rate by an average of 20%, increases the accretion rate by an average of 13%, and increases the evaporation rate by an average of 10% (Figure 7).

The increased rain simulated by the upscaled microphysics agrees better with LES than does the local microphysics (Figures 4 and 5). However, the SCM with upscaled microphysics still underestimates r_{r}. One reason for the discrepancy appears to be the fact that the SCM fails to transport enough r_{r} to the upper part of cloud (Figure 6). The lack of r_{r}, in turn, leads to a lower accretion rate (Figure 6).

The power laws used in the KK microphysics are fairly general and may serve as adequate models in other applications. If so, it may be of interest to assess whether accounting for subgrid variability is worthwhile in other cases. Such estimates are non-trivial because they depend, in part, on both the values of the exponents and the degree of variability in the case. To this end, we provide approximate estimates of the terms that are neglected in the local formula (Eqs (5) and (7)). However, we caution that this formula is based on a second-order Taylor series expansion and therefore ignores terms that may be large in cases with skewed PDFs, such as cumulus clouds.

We now cite one example of the use of Eqs (5) and (7). Equations (5) and (7) suggest that upscaling is important when the standard deviation of a variable is significant as compared to its mean. This may occur, for instance, for vertical velocity (w) in a wide grid box. Such a grid box may contain strong updraughts and downdraughts yet have a small horizontal mean velocity. If this vertical velocity is used to drive other processes, such as activation of cloud condensation nuclei (CCN), then subgrid variability in w should be taken into account. For instance, if 〈w〉 = 0, and subgrid drafts are ignored, then CCN may spuriously fail to activate (Lohmann et al., 2007; Ivanova and Leighton, 2008; Guo et al., 2010).

We believe that the analytic upscaling method presented here may be suitable for future implementation in a large-scale model, if the microphysics formulas are sufficiently simple. The horizontal domain of our LES is 6.4 km × 6.4 km, and for a large-scale model with a grid column of those dimensions the presented results are directly relevant. For wider grid columns or for cloud types other than stratocumulus, further testing is needed.

Acknowledgements

The authors are grateful for financial support provided by Grant 04-062 from the UWM Research Growth Initiative and by Grants ATM-0618818 and ATM-0936186 from the National Science Foundation. The authors thank Dr Andrew Ackerman for providing LES output from the GCSS LES intercomparison for use in the plots in this paper.