The preceding section describes where microphysical tendencies, of any kind, enter CLUBB's equation set. This section describes how the microphysical tendencies are related to the specific processes of autoconversion, accretion, and evaporation.

The source terms for the model predictive equations require microphysical process rates from a microphysics scheme. The scheme used here is the warm microphysics scheme described in hereafter KK. KK is a two-moment scheme that predicts rr and rain drop concentration (per unit mass), Nr. It was developed by using the least squares method to find a “best-fit” curve through microphysical rate data that were generated by simulating a drizzling stratocumulus case using an explicit (or “bin”) microphysics scheme.

The KK scheme was chosen because of its simplicity. It expresses microphysical rates as power laws of two or three variables, which means that the product of a microphysical rate and the corresponding PDF is always integrable. More recently, the coefficients and exponents in the KK scheme have been tailored to cumulus clouds . The Kogan scheme is covered by the analytic integrals presented in this paper because they are generalized for arbitrary coefficients and exponents. However, this paper uses the original KK coefficients and exponents because the KK scheme has been widely used for a variety of cloud types and is adequate for our idealized purposes.

The KK warm microphysics scheme produces rr through the processes of autoconversion (collision) and accretion (collection). These processes produce rainwater, deplete cloud water, and leave water vapor unchanged. As a result, these processes increase the value of θl, as shown by Eqs. () and (), and decrease the value of rt. Evaporation reduces rr as rain falls through subsaturated air. Condensational growth does not apply to rainwater in CLUBB. Instead, all supersaturation is automatically applied to cloud water. When rainwater evaporates, cloud water remains unchanged, and rt increases due to the increase in water vapor. Meanwhile, evaporative cooling decreases θl due to the decrease in temperature.

The relationship of all three KK microphysics tendencies to the rt microphysics tendency can be written as ∂rt∂tmc=-∂rr∂tauto-∂rr∂taccr-∂rr∂tevap, where ∂rr∂tauto is the rate of change of rr due to the process of autoconversion, ∂rr∂taccr is the rate of change of rr due to the process of accretion, and ∂rr∂tevap is the rate of change of rr due to the process of evaporation. Note that when evaporation occurs, ∂rr∂tevap<0. The relationship of all three tendencies to θl microphysics tendency can be written as ∂θl∂tmc=LvCpdpp0-RdCpd∂rr∂tauto+∂rr∂taccr+∂rr∂tevap. The decrease in temperature from the evaporation of a unit of rainwater is the same as the decrease in temperature from the evaporation of the same amount of cloud water.

The Reynolds-averaged microphysics term in the predictive equation for w′rt′‾, as found in Eq. (), is rewritten as w′∂rt∂tmc′‾=-w′∂rr∂tauto′‾-w′∂rr∂taccr′‾-w′∂rr∂tevap′‾. Likewise, the Reynolds-averaged microphysics term in the predictive equation for w′θl′‾, as found in Eq. (), is rewritten as w′∂θl∂tmc′‾=LvCpdp‾p0-RdCpd×w′∂rr∂tauto′‾+w′∂rr∂taccr′‾+w′∂rr∂tevap′‾. Any variability of p within the grid box is ignored for simplicity. Additionally, the -Rd/Cpd exponent would greatly limit the effects of variability of p on the solution. As a result, p‾ is used in the equation. In the predictive equation for rt′2‾, Eq. (), the microphysics term becomes rt′∂rt∂tmc′‾=-rt′∂rr∂tauto′‾-rt′∂rr∂taccr′‾-rt′∂rr∂tevap′‾. In the predictive equation for θl′2‾, Eq. (), the microphysics term becomes θl′∂θl∂tmc′‾=LvCpdp‾p0-RdCpd×θl′∂rr∂tauto′‾+θl′∂rr∂taccr′‾+θl′∂rr∂tevap′‾. The Reynolds-averaged microphysics terms in the predictive equation for rt′θl′‾, as found in Eq. (), are rewritten as θl′∂rt∂tmc′‾=-θl′∂rr∂tauto′‾-θl′∂rr∂taccr′‾-θl′∂rr∂tevap′‾,and

rt′∂θl∂tmc′‾=LvCpdp‾p0-RdCpd×rt′∂rr∂tauto′‾+rt′∂rr∂taccr′‾+rt′∂rr∂tevap′‾. The above equation set contains nine individual microphysical covariance terms, each involving one of w, rt, or θl with one of autoconversion, accretion, or evaporation rate. These terms can be parameterized through use of the PDF method.

The multivariate PDF used by CLUBB consists of w, rt, θl, all hydrometeor species used by the selected microphysics scheme (in the case of KK microphysics, rr and Nr), and an extended cloud droplet concentration, Ncn, which is equal to cloud droplet concentration, Nc, within cloud, but has a positive value outside of cloud . CLUBB's PDF is a weighted mixture, or sum, of two multivariate normal/lognormal functions. Each multivariate function is known as a PDF component.

When variables are integrated out of the multivariate PDF, a marginal PDF consisting of fewer variables remains. When all variables but one are integrated out of the PDF, the result is a univariate marginal or individual marginal. The individual marginal for each of w, rt, and θl is a two-component normal (also known as a binormal) distribution. The two-component shape allows skewness to be included in model fields. The individual marginal for Ncn is assumed to be a (single) lognormal distribution.

The individual marginal for each of rr and Nr is delta-lognormal within each PDF component . Each PDF component can contain precipitating and precipitationless regions. The fraction of each PDF component that contains any hydrometeor species (other than cloud liquid water) is known as the component's precipitation fraction. The precipitationless region is represented by a delta at 0 for all hydrometeor species. Within precipitation, a lognormal distribution is used to represent a hydrometeor species. The lognormal distributions can differ between the two components, so that when the components are summed to form the overall distribution, a delta double lognormal (DDL) distribution results.

The PDF method for parameterizing the nine microphysics covariance terms requires analytic integration over the multivariate PDF. As listed in , the general form of a multivariate PDF of n components and D variables, where D can be all the variables involved in the PDF or any subset of those, is given by Px1,x2,…,xD=∑i=1nξiPix1,x2,…,xD, where ξi is the mixture fraction, or relative weight of the ith PDF component. The sum of the mixture fractions is equal to 1.

The D variables listed are categorized, and the first J variables are normally distributed in each PDF component (w, rt, and θl), the next K variables are lognormally distributed (Ncn), and the last Ω variables are the hydrometeor species that are distributed delta-lognormally in each PDF component (rr and/or Nr). The equation for the ith PDF component is Pix1,x2,…,xD=fpiPJ,K+Ωix1,x2,…,xD+1-fpiPJ,Kix1,x2,…,xJ+K∏ϵ=J+K+1Dδxϵ, where fpi is the precipitation fraction in the ith PDF component. The subscripts in the ith component, PJ,Ki or PJ,K+Ωi, denote the number of normal variates, J, and the number of lognormal variates, K or K+Ω, used in Eq. ().

Both the precipitating and precipitationless portions (subcomponents) of Eq. () contain a hybrid normal/lognormal distribution of m variables, where the first j variables are normally distributed and the remaining k variables are lognormally distributed. The general form of this multivariate normal/lognormal PDF is given by Pj,kix1,x2,…,xm=12πm2Σi12∏τ=j+1m1xτ×exp⁡-12x-μiTΣi-1x-μi, where x is an m×1 vector of the variables (in normal space) in the PDF and μi is an m×1 vector of the (normal-space) PDF subcomponent means. The transpose of the vector is denoted T. The m×m (normal-space) covariance matrix is denoted Σi and its determinant is denoted Σi .

Equation () lists the general functional form for the subgrid PDF, but specific examples of marginals for a single mixture component are written out in the Supplement to this article. These examples help provide intuition about the shape of the PDF. For instance, a univariate normal marginal of the PDF is written in Eq. (S7), and a univariate lognormal is written in Eq. (S8). A normal distribution is symmetric, extends from (-∞,+∞), and has short tails. A lognormal distribution, on the other hand, has a skewed shape that is useful for representing the distribution of a quantity such as rainwater mixing ratio. Such distributions are non-negative and often have a peak at low values and a long tail of larger values extending to the right. They are not well represented by normal distributions.

Also useful for gaining intuition are the bivariate marginals listed in Sect. S3 of the Supplement. A normal–normal bivariate form is listed in Eq. (S4), a lognormal–lognormal form is listed in Eq. (S6), and a hybrid normal–lognormal form is listed in Eq. (S5). Where a lognormal variate appears, the corresponding axis takes on only non-negative values and has a long tail. Which bivariate form is used depends on which functional forms are used to represent the variates of interest, e.g., rainwater mixing ratio (lognormal) or vertical velocity (normal).

Using a two-component PDF requires a method to divide one overall (grid-box) mean value of a variable into two PDF component mean values of that variable. Likewise, one overall variance needs to be split into two PDF component standard deviations. The multivariate PDF also requires information on the correlations between variables.

The PDF component means, standard deviations, and correlations involving w, rt, and θl, as well as the mixture fractions, are calculated according to the Analytic Double Gaussian 1 (ADG1) PDF presented in Sect. (d) of the Appendix of . The overall (grid-box) precipitation fraction is set to the maximum cloud fraction found at or above that grid level . The calculation of the component precipitation fractions fp(i) from the overall precipitation fraction are outlined in . Also described there is the calculation of the PDF component means and standard deviations involving Ncn, rr, and Nr. Interactive CLUBB runs prescribe a constant ratio of the in-precipitation variance to the square of the in-precipitation mean for rr and Nr. Additionally, all remaining correlations between variables are prescribed constants.

The covariance of PDF variables x1 and x2 can be calculated by x1′x2′‾=∫∫x1-x1‾x2-x2‾Px1,x2dx2dx1. The covariance of a PDF variable and a microphysics function (written in terms of PDF variables) can be calculated in the same manner. For example, the covariance of θl and KK evaporation rate found in Eqs. () and () can be rewritten as θl′∂rr∂tevap′‾=(θl-θl‾)∂rr∂tevap-∂rr∂tevap‾‾, where mean evaporation rate, ∂rr∂tevap‾, is also calculated by integrating over the PDF Supplement to. The KK evaporation rate can be written as a function of θl, rt, rr, and Nr, so here it will be referred to as EVθl,rt,rr,Nr. The covariance of θl and KK evaporation rate is calculated by

θl′∂rr∂tevap′‾=∫∫∫∫(θl-θl‾)EVθl,rt,rr,Nr-EVθl,rt,rr,Nr‾×Pθl,rt,rr,NrdNrdrrdrtdθl. The remaining eight covariances involving microphysical functions are calculated in the same manner. Further and more detailed description of this method can be found in Appendix  and the Supplement.

In order to assess which physical processes are most important, the LES budget terms for turbulent fields are analyzed for the RICO precipitating cumulus case. Additionally, the LES budgets and CLUBB's budgets are compared in order to assess the accuracy of CLUBB's budget terms.

Unlike the LES, the CLUBB budget terms for turbulent fields are taken directly from the predictive equation set. The anelastic predictive equations for the turbulent fluxes w′rt′‾ and w′θl′‾ are given by ∂w′rt′‾∂t=-1ρs∂ρsw‾w′rt′‾∂z-1ρs∂ρsw′2rt′‾∂z︸advection-w′2‾∂rt‾∂z-w′rt′‾∂w‾∂z︸production-1ρsrt′∂p′∂z‾︸pressure+gθvsrt′θv′‾︸buoyancy+εwrt︸diffusion+w′∂rt∂tmc′‾︸microphysics,and ∂w′θl′‾∂t=-1ρs∂ρsw‾w′θl′‾∂z-1ρs∂ρsw′2θl′‾∂z︸advection-w′2‾∂θl‾∂z-w′θl′‾∂w‾∂z︸production-1ρsθl′∂p′∂z‾︸pressure+gθvsθl′θv′‾︸buoyancy+εwθl︸diffusion+w′∂θl∂tmc′‾︸microphysics, where g is gravity and θv is virtual potential temperature. The dry, anelastic base-state values of air density, ρs, and θv, denoted θvs, vary only with altitude. The higher-order turbulent advection terms, w′2rt′‾ and w′2θl′‾, are closed using the PDF . The pressure terms are parameterized following (see also ). The slow (return-to-isotropy) term is approximated by Newtonian damping. The buoyancy terms are closed by linearizing and then integrating over the PDF . The terms denoted εwrt and εwθl are background numerical vertical diffusion terms .

As in CLUBB, the SAM LES budgets for the horizontally averaged turbulent fluxes contain advective transport terms and turbulent (gradient) production terms, which both ultimately arise from the 3-D advection of w, rt, and θl. The turbulent production terms generate variability when the vertical derivative of the mean field is non-zero. SAM also records the effects of pressure, buoyancy, and microphysics on the turbulent fluxes. SAM's budget term for diffusion of w′rt′‾ and w′θl′‾ records the effects of diffusion associated with the subgrid TKE scheme. In Figs.  and , following , the SAM LES budget terms for buoyancy and pressure are combined because they are both large compared to other terms, yet are in close equilibrium because of the quasi-hydrostatic balance of perturbation buoyancy and perturbation pressure gradient. The CLUBB buoyancy and pressure terms have been combined in an analogous manner.

The SAM LES turbulent flux budgets show that the largest terms are pressure + buoyancy, which usually acts as a net sink of turbulent flux, and turbulent production, which acts as a source of turbulent flux (see Figs. a and a). Another major term in the budget is the (turbulent) advection term. The turbulent advection term (e.g., -(1/ρs)∂(ρsw′2rt′‾)/∂z) has a mass-weighted vertical integral of zero. That is, averaged in the vertical, it is neither a net source nor a net sink. Instead, it takes the excess variability at some altitudes and transports it to regions with a deficit of variability. The microphysics term is a sink of turbulent flux in the cloudy layer, a layer which spans the altitude range from 500 to 3000 m. The microphysics term is more significant for w′θl′‾ than for w′rt′‾, but even for w′rt′‾, it is non-negligible.

CLUBB's turbulent flux budgets usually agree qualitatively with those from LES (Figs. b and b). CLUBB's advection terms have approximately the correct shape, although they are usually too small in magnitude. In CLUBB, the buoyancy + pressure and turbulent production terms are dominant, as in SAM LES, but in CLUBB's RICO simulation their magnitudes are larger than in SAM LES.

The microphysics terms in both the w′rt′‾ and w′θl′‾ budgets have the same signs and close to the same peak magnitudes as their counterparts in the LES. However, in CLUBB, the range of altitudes where the microphysics budget terms have significant values is shifted lower than in SAM LES. This occurs because rr‾ peaks at a lower altitude in CLUBB than in SAM LES. The lower-altitude peak in rain, in turn, occurs because there is too much evaporation near cloud top, as shown in Fig. 7a of . As noted there, the excessive evaporation is caused by an excessively long-tailed marginal subgrid PDF of saturation deficit, which extends to unrealistically dry values. The excessive evaporation near cloud top also causes a similar problem in the microphysical terms in the other budgets presented below. See for more details.

The CLUBB anelastic predictive equations for the scalar variances rt′2‾ and θl′2‾, and the covariance rt′θl′‾, are given by ∂rt′2‾∂t=-1ρs∂ρsw‾rt′2‾∂z-1ρs∂ρsw′rt′2‾∂z︸advection-2w′rt′‾∂rt‾∂z︸production+εrtrt︸diss+diff+2rt′∂rt∂tmc′‾︸microphysics,∂θl′2‾∂t=-1ρs∂ρsw‾θl′2‾∂z-1ρs∂ρsw′θl′2‾∂z︸advection-2w′θl′‾∂θl‾∂z︸production+εθlθl︸diss+diff+2θl′∂θl∂tmc′‾︸microphysics,and∂rt′θl′‾∂t=-1ρs∂ρsw‾rt′θl′‾∂z-1ρs∂ρsw′rt′θl′‾∂z︸advection-w′rt′‾∂θl‾∂z-w′θl′‾∂rt‾∂z︸production+εrtθl︸diss+diff+rt′∂θl∂tmc′‾+θl′∂rt∂tmc′‾︸microphysics. As in the predictive equations for the fluxes, the higher-order turbulent advection terms, w′rt′2‾, w′θl′2‾, and w′rt′θl′‾, are closed using the PDF . The terms denoted εrtrt, εθlθl, and εrtθl each contain a dissipation term (parameterized in CLUBB as Newtonian damping) that reduces the magnitude of the turbulent field, as well as a background numerical vertical diffusion term .

The SAM LES budgets for the horizontally averaged turbulent (co)variances contain advective transport terms and turbulent (gradient) production terms, as well as microphysics terms. In Figs. , , and , the diffusion and dissipation terms are combined for both SAM and CLUBB. Both SAM and CLUBB contain vertical diffusion, with SAM's associated with TKE. However, SAM's subgrid TKE is also used to diffuse fields horizontally. Horizontal diffusion smooths out a model field across the grid level, reducing the variances and covariances of model fields. In CLUBB, this effect is parameterized by the dissipation (Newtonian damping) term.

The SAM LES budgets for rt′2‾, θl′2‾, and rt′θl′‾ show that microphysics is a dominant term in the upper half of the cloud layer. At those levels, microphysics is balanced by turbulent production and turbulent advection (at higher altitudes) (Figs. a, a, and a). Near cloud base, the budget is predominantly a balance of advection and production. The dissipation/diffusion terms are smaller, but not negligible.

The time-averaged CLUBB SCM budgets found in Figs. b, b, and b show that the CLUBB scalar (co)variance budgets are qualitatively similar to the LES budgets. The microphysics term in the rt′2‾ budget has the correct sign and is a significant sink term, and the shape of the profile of the advection and production terms qualitatively resemble the LES. CLUBB's dissipation term is too large, but the microphysics terms in the θl′2‾ and rt′θl′‾ budgets are dominant terms in the cloudy layer, just as in the LES. The production terms largely balance the microphysics terms. The advection terms are too small in magnitude relative to the other terms, but have approximately the right shape.

The figures show that the microphysics terms are sink terms in the cloudy layer, reducing the variances and the magnitudes of the covariances, for all five of these turbulent fields. Physically, this happens because cumulus clouds arise in the regions of the horizontal domain that are moister than average. Additionally, cloudy regions are usually associated with updrafts (where vertical velocity is greater than average) in a cumulus regime. Within cloud, the moistest regions contain the greatest amount of cloud (liquid) water. The microphysics processes of autoconversion and accretion occur only in cloud and at greater rates in regions with a greater amount of cloud water. When autoconversion and accretion occur, rainwater is produced at the expense of cloud water. The local value of rc decreases, which decreases rt and increases θl preferentially in the moistest portions of domain. As a result, scalar variances rt′2‾ and θl′2‾ are reduced, and the (negative) covariance rt′θl′‾ is reduced in magnitude. Similarly, since moister regions of cloud are associated with stronger updrafts, the covariance w′rt′‾ is reduced by microphysics and the (negative) covariance w′θl′‾ is reduced in magnitude by microphysics.

In the region below cloud, a different microphysical process occurs: rain falls into clear air below cloud and evaporates. Evaporation increases water vapor at the expense of rainwater and also cools the air. Hence, where evaporation occurs, rt is increased and θl is decreased. If rain preferentially falls through regions of air that have already been cooled by evaporation, then cool air is further cooled. In a partly rainy case such as RICO, rain cools the rain shafts but not other portions of the domain, increasing variability in θl. In RICO, the positive tendency of subcloud θl′2‾ by microphysics is significant, as shown in Fig. . Parameterizing a positive subcloud microphysics tendency of θl′2‾ in CLUBB requires prescribing the within-component correlations such that rain tends to fall in cool air below cloud. In a PDF-based model such as CLUBB, a cold pool would be represented by an increase in θl′2‾ in the subcloud layer (owing to microphysics). CLUBB's ability to parameterize this effect (Fig. b) opens the door to future parameterization of the effects of cold pools on convection.

The aforementioned results show that CLUBB's formulas are able to qualitatively approximate the microphysical (co)variance terms for a boundary-layer case with small cloud fraction (RICO). Can CLUBB's formulas approximate the microphysical (co)variance terms produced by SAM LES in other cloud cases? To begin to address this question, we simulate a boundary-layer case that has a layer-averaged rainwater mixing ratio that is comparable to that of RICO but that has a cloud fraction and precipitation fraction of nearly one. The case we simulate is the DYCOMS-II RF02 marine stratocumulus case.

In the SAM LES budgets of DYCOMS-II RF02 for rt′2‾ (Fig. a) and θl′2‾ (Fig. a), the microphysics term is negligible in comparison to the other terms. While CLUBB's dissipation and production terms are overestimated, CLUBB's microphysics term is also negligible for both rt′2‾ (Fig. b) and θl′2‾ (Fig. b), in agreement with SAM LES. In addition, both SAM and CLUBB show similarly negligible microphysics terms in the w′rt′‾, w′θl′‾, and rt′θl′‾ budgets (not shown). This agreement suggests that CLUBB's microphysical (co)variance formulas are applicable for shallow cloud cases with either small or large values of cloud fraction.

Why are the microphysics budget terms so much less significant in DYCOMS-II RF02 than they are in RICO? Consider the covariance of a field and a microphysics process rate – for example, the covariance of θl and accretion rate. The magnitude of this covariance is related, in part, to the magnitude of θl′2‾ and the magnitude of the variance of accretion rate. (We set aside the issue of the correlation between the two fields.) Comparing SAM LES results at altitudes where precipitation is large, both θl′2‾ and rt′2‾ are smaller in DYCOMS-II RF02 than in RICO (not shown). This is because marine stratocumulus cloud layers are well mixed by turbulence. DYCOMS-II RF02 also exhibits less variability in microphysical process rates. The variance of warm-rain microphysics process rates is related, in part, to the variance of rainwater mixing ratio. In RICO, precipitation is found over a small region of the horizontal domain, while in the overcast DYCOMS-II RF02 case, precipitation is found over almost the entire horizontal domain. The in-precipitation mean of rr is much larger in RICO than it is in DYCOMS-II RF02 (not shown). Additionally, in RICO the ratio of the in-precipitation variance to the square of the in-precipitation mean for rr (≈5) is much larger than the corresponding ratio (≲1) in DYCOMS-II RF02. As a result, both the in-precipitation variance and the layer-mean variance of rr is much greater in RICO than in DYCOMS-II RF02. In summary, the microphysical (co)variance terms are smaller in marine stratocumuli than in cumuli partly because both the thermodynamic and microphysical fields are more homogeneous in marine stratocumuli.

If the microphysical terms in the (co)variance equations are omitted, how large are the resulting errors? To address this, a second CLUBB simulation of RICO was run that is identical to the original simulation with the one exception that the microphysical (co)variance terms are turned off.

When the microphysical (co)variance terms are removed, compensating errors in other terms must be induced in order to restore balance in the budgets. A large compensation occurs in the budgets for the scalar (co)variances because the microphysical terms in those budgets are large. The budgets of scalar variances rt′2‾ and θl′2‾ are shown for the CLUBB simulation with microphysics feedback disabled in Fig. a and b, respectively. When compared to the same budgets from the CLUBB simulation with microphysics feedback enabled in Figs. b and b, respectively, the terms from the simulation with microphysics feedback disabled are all much larger in magnitude. Within the cloudy layer, microphysics is a dominant sink of scalar variances. In order to compensate for the loss of that sink term, both dissipation and advection (below 3000 m) increase in (negative) magnitude. Since the integral of the (turbulent) advection term over the vertical profile must have a mass-weighted vertical integral of 0, it becomes an excessive source of rt′2‾ and θl′2‾ above 3000 m. In essence, when the microphysical sink of variance is removed, the layer becomes more variable, develops more turbulence, and grows deeper. Similar characteristics are exhibited in the budget of rt′θl′‾ (not shown).

The budgets of turbulent fluxes w′rt′‾ and w′θl′‾ are shown for the CLUBB simulation with microphysics terms disabled in Fig. c and d, respectively. When compared to the same budgets from the CLUBB simulation with microphysics terms enabled in Figs. b and b, respectively, the buoyancy + pressure terms, the advection terms, and the w′rt′‾ production term all increase in magnitude. This increase is relatively small, however, which is expected because microphysics is a less significant term in the turbulent flux budgets. The terms from the simulation with microphysics terms disabled extend much higher in altitude, again because the layer has more vigorous turbulence.

The errors induced by the loss of the microphysical (co)variance terms propagate throughout the model solution, infecting, for instance, the mean fields. Figure  shows profiles of mean fields in a three-way comparison between (1) SAM LES, (2) CLUBB with microphysical effects on (co)variances disabled, and (3) CLUBB with microphysical effects on (co)variances enabled. In Fig. a, when the microphysical effects on (co)variances are turned off, CLUBB's θl‾ becomes too warm at lower altitudes and too cool aloft. As a result of the cooler temperatures and excessive turbulence aloft, Fig. c shows that rc‾ extends too high in altitude when compared to SAM LES. Omitting the microphysical (co)variance terms would constitute a significant model error.