1. Introduction
Formation and subsequent growth of cloud droplets in atmospheric clouds is a subject of a significant interest to a wide range of atmospheric science disciplines, such as atmospheric chemistry, aerosol science, cloud microphysics and dynamics, precipitation processes, weather, and climate. The theory of droplet formation on soluble atmospheric aerosol particles acting as the cloud condensation nuclei (CCN) was developed in the early twentieth century (Kohler 1936) and subsequently extended to include variations in aerosol chemical composition and hygroscopicity (Petters and Kreidenweis 2007 and references therein). The diffusional growth of a single cloud droplet as well as a population of cloud droplets, including impacts of the surface tension, dissolved salt, and molecular effects, has been extensively studied over the last half-century and it is described in various levels of detail in cloud physics textbooks (e.g., Rogers and Yau 1989; Pruppacher and Klett 1997).
Numerous idealized (e.g., rising air parcel) and realistic (i.e., multidimensional numerical model) modeling studies clearly demonstrate that supersaturations inside clouds are small—typically below 1%. Although impossible to measure with high accuracy from an instrumented aircraft, estimates of the supersaturation from the combination of measured in-cloud vertical velocities and cloud droplet spectral characteristics confirm relatively small values of the supersaturation (e.g., Warner 1968; Politovich and Cooper 1988 and references therein). Considering complications of the accurate modeling of CCN activation (some to be discussed in this paper), one may wonder if it is possible to neglect the supersaturation and simply assume that a cloud is always at saturation. Such reasoning is the basis of the bulk scheme of condensation often used in microphysical schemes [e.g., Soong and Ogura (1973); see also Grabowski and Smolarkiewicz (1990) and references therein]. Although the representation of cloud microphysics is extremely simple in bulk condensation schemes when compared to bin schemes, the key question in this paper is whether the bulk scheme is accurate enough to faithfully represent cloud dynamics.
One of the first studies suggesting significant differences between bulk and bin approaches is that of Árnason and Greenfield (1972). However, the highly idealized modeling setup, with the saturated atmosphere and a bubble as the initial condition, makes the results uncertain from the point of view of natural clouds. Kogan and Martin (1994) compared condensation rates from bulk and bin schemes and noted significant differences in low droplet concentrations (below 100 cm−3; see Figs. 4 and 5 therein). Those simulations also involved an idealized initialization of a cloud (a set of bubbles) as described in Kogan (1991). The issue of bulk versus bin microphysics—or more precisely vanishing versus finite cloud supersaturation—attracted recently some attention in the context of the so-called convective invigoration—that is, presence of more buoyant clouds with stronger updrafts in polluted environments. Originally proposed for the deep convection and ice processes (e.g., Rosenfeld et al. 2008), the invigoration has also been suggested to occur in warm nonprecipitating clouds through the impact on the condensation rate. For instance, Igel et al. (2015) discuss the impact of the droplet concentration and droplet size on the condensation rate and suggest that the condensation rate is significantly affected by changes in these parameters. A polluted cloud has indeed a larger total droplet surface area when compared to a pristine cloud for the same liquid water content, and this might result in a higher condensation rate when compared to a pristine cloud. The major flaw in such an argument lies in the lack of recognition of the interactive nature of the condensation process: polluted clouds have larger droplet surface area, but the supersaturation is smaller because it comes from the interplay between the source due to updraft strength and the sink due to condensation [Squires (1952); Politovich and Cooper (1988); see discussion in section 3.1 of Grabowski and Wang (2013)]. The quasi-equilibrium supersaturation Sqe represents an exact balance between the source and the sink, and it gives condensation rate that depends only on the vertical velocity (and meteorological variables such as the temperature and pressure). This is because Sqe for a parcel rising with the vertical velocity w and carrying cloud droplets with the mean radius r and concentration N is Sqe ~ w/(Nr) [see, e.g., Grabowski and Wang (2013)]. Since the condensation rate C ~ SNr, assuming S = Sqe gives C ~ w regardless of N and r. Thus, the results in Kogan and Martin (1994) can be explained by larger departures of the local supersaturation from Sqe for clouds with lower droplet concentrations. However, the main variable affecting cloud dynamics is not the condensation rate, but the cloud buoyancy as given by the difference in the potential density temperature between the cloud and its mean environment.
The above considerations provide motivation for the current study. We address the impact of the finite supersaturation on the cloud dynamics by first analyzing the impact of the supersaturation on the cloud buoyancy. Then a 1D constant-speed updraft is considered with bulk and bin microphysical schemes. Although conceptually similar to the adiabatic-parcel framework, the 1D advection–condensation test allows investigating the impact of model numerics (e.g., the vertical resolution) by mimicking CCN activation and droplet growth taking place in the cloud updraft. The issue of model numerics, although to some extent a sidetrack for the main thrust of this paper, needs to be taken into account while assessing advantages and drawbacks of bulk and bin approaches. The vertical resolution is especially important for the bin scheme that requires accurate prediction of the maximum cloud-base supersaturation and droplet activation. Subsequently, cloud field simulations with bulk and bin schemes are executed applying the piggybacking methodology (Grabowski 2014, 2015) to assess the impact of finite super- and subsaturations on the cloud dynamics with high confidence.
The main idea behind microphysical piggybacking is to apply two sets of thermodynamic variables: one coupled to the dynamics and driving the simulation, and the second one piggybacking the simulation—that is, responding to the simulated flow but not affecting it. Microphysical piggybacking allows assessing the impact of cloud microphysics with unprecedented fidelity. This is especially true for convective processes where even small differences in initial conditions (e.g., due to small-amplitude random perturbations added during model initialization) lead to solution trajectories that diverge after a relatively short time (say, after a few cloud lifetimes, an hour or two in the shallow convection case). This was highlighted in the LES study of precipitating shallow convection reported in Grabowski (2014). The impact of cloud microphysics is difficult to assess in traditional shallow convection simulations (i.e., applying parallel simulations, each with different microphysics) as illustrated, for instance, in Wyszogrodzki et al. (2013). Moreover, replacing the scheme that drives the simulation with the one that piggybacks the simulated flow, and vice versa, allows assessing the effect on cloud dynamics. This was demonstrated in simulations of shallow-to-deep convection transition discussed in Grabowski (2015).
The paper is organized as follows. The next section provides a quantitative estimate of the impact of the finite supersaturation on the cloud buoyancy. Section 3 provides a brief description of the microphysical schemes applied in the current study (with details given in the appendix) as well as the specific shallow convection case considered in numerical simulations. Section 4 presents selected results from the 1D advection–condensation simulations. Results from simulations of a shallow convection cloud field applying the microphysical piggybacking are discussed in section 5. Discussion of simulation results and conclusions are presented in section 6.
2. Theoretical considerations
Figure 1 shows the difference 
Difference between potential density temperatures using saturation adjustment 
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
Overall, the theoretical analysis suggests that the effects of a finite supersaturation are small for the cloud dynamics. This is because 1% supersaturation (subsaturation) reduces (increases) the density temperature by around 0.1 K when compared to the bulk scheme. However, as a preamble to the discussion of model results in section 5, we note that for the cloud water evaporation resulting from entrainment and mixing, one expects subsaturations significantly larger in amplitude than supersaturations inside cloudy updrafts, several percent or higher, and thus more significant impact on the cloud buoyancy.
3. Numerical simulations: Microphysical schemes and the shallow convection case
This section provides a brief description of bulk and bin microphysical schemes. The mathematical formulation as well as details of the numerical implementation are presented in the appendix. The bulk condensation scheme assumes that the supersaturation inside a cloud is not allowed and, if present, it is instantaneously removed through condensation. Cloud water cannot exist in subsaturated conditions and it has to evaporate partially or completely to bring the air as close to saturation as possible. Besides the potential temperature, the bulk condensation scheme employs only two variables: the water vapor and cloud water mixing ratios. We will refer to the bulk condensation scheme and to the set of thermodynamic variables it predicts as BLK.
We apply the Barbados Oceanographic and Meteorological Experiment (BOMEX; Holland and Rasmusson 1973) shallow convection modeling case of Siebesma et al. (2003). Figure 2 shows the BOMEX temperature and moisture profiles together with profiles of selected variables within the adiabatic parcel rising from the surface across the lower troposphere. Temperature and moisture profiles feature 1-km-deep trade wind convection layer overlaying the 0.5-km-deep mixed layer near the ocean surface, covered by the 0.5-km-deep trade wind inversion and free troposphere aloft. The adiabatic cloud water mixing ratio profile shows that the cloud base is slightly above the mixed-layer top, at about 0.6 km. The potential density temperature difference between the adiabatic parcel and its environment Δθd increases from close to zero near the cloud base to its maximum, close to 2 K, near the bottom of the trade wind inversion, and subsequently decreases to zero at around 2 km. The potential density temperature difference determines the buoyancy of the adiabatic parcel; its magnitude, typically reaching maximum of a couple degrees for shallow convection, can be used to gauge the significance of the difference in the density temperature between bulk and bin schemes. The bottom-right panel of the figure shows the profile of the cumulative CAPE (cCAPE) defined as 
(top) BOMEX sounding and (bottom) its adiabatic-parcel analysis. (bottom left) Profiles of the adiabatic water mixing ratio, (bottom center) the potential density temperature difference between the parcel and environment, and (bottom right) cCAPE. See text for details.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
4. One-dimensional constant cloud updraft simulations
As an initial comparison between the bulk scheme (i.e., without the supersaturation) and the bin scheme (i.e., predicting the supersaturation) we apply a 1D constant-updraft setup as in section 4.2 of GAW11 [see also section 4a in Morrison and Grabowski (2008)]. One can also employ an adiabatic-parcel model (i.e., a box model) with bin microphysics [e.g., as in Grabowski and Wang (2009)], but when the bin microphysics is used in a multidimensional cloud model, the numerical solutions depend on the spatial (as well as temporal) discretization. The bulk and bin schemes, the latter assuming SPRI, PRI, POL, and SPOL conditions, are run side by side in 1D advection–condensation test. The 1D version of the multidimensional positive-definite advection transport algorithm (MPDATA; Smolarkiewicz 2006) facilitates advection of model variables. The updraft velocity is assumed as either 0.5 or 5 m s−1. The inflow boundary conditions assume constant-in-time temperature and moisture values that correspond to those at the 500-m height of the BOMEX sounding. These values are advected into the 1200-m-deep vertical domain with activation of cloud droplets and their growth by the diffusion of water vapor taking place as in the cloud updraft of the multidimensional cloud model. The domain is covered by a regular grid with the grid length varying between 1 and 50 m in various tests. The model time step is 0.1 s, which is much smaller than dictated by the stability of the advection scheme but appropriate for the bin microphysics because of the Euler forward time integration as detailed in the appendix.
Figure 3 shows profiles of the supersaturation and the difference of the cloud water and potential density temperature between bulk and bin schemes for two simulations, BLK with either SPRI and the updraft of 0.5 m s−1 or SPOL with 5 m s−1 updraft, both applying 1-m grid length. The three variables have similar profiles, which is in agreement with the condensed water and density temperature dependence on the supersaturation as discussed in section 2. The differences are relatively small, reaching the maxima of less than 0.1 g kg−1 for the cloud water and around 0.1 K for the potential density temperature and significantly smaller values beyond the region of the CCN activation. Note that these values represent significant fractions of the total cloud water and potential density temperature close to the cloud base and thus affect cloud-base updraft strength in LES simulations.
Example of results from 1D advection–condensation test applying BOMEX sounding for (left) SPRI and (right) SPOL. The vertical distance starts at 500-m height of the BOMEX sounding. Profiles of (top) the supersaturation, (middle) the difference between the bulk and bin cloud water (
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
Figure 4 documents convergence of the numerical solutions when the vertical grid length is increased applying two updraft strengths: 0.5 and 5 m s−1. As the figure shows, a relatively high spatial resolution, with the grid length smaller than 10 m, is needed to accurately simulate the number of activated cloud droplets. The concentration of activated droplets with the Twomey activation is set by the maximum supersaturation experienced by the volume of air rising across the cloud base. The maximum is poorly resolved with the vertical grid length larger than a few meters [see also the discussion in section 4a in Morrison and Grabowski (2008)].
Concentration of activated cloud droplets as a function of the vertical grid length in 1D advection–condensation tests for (clockwise from top left) SPRI, PRI, POL, and SPOL applying 0.5 (lower lines) and 5 m s−1 (upper lines) updraft speed.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
In summary, the 1D advection–condensation tests suggest that (i) bin microphysics requires high vertical resolution, typically impossible in 3D LES simulations, and (ii) the buoyancy differences between bulk and bin schemes are relatively small but, perhaps, sufficient to affect the cloud field simulations. The latter aspect is investigated in the next section.
5. Cloud field simulations with microphysical piggybacking
a. The model and simulation setup
The numerical model, babyEULAG, is the same as applied in Grabowski (2014, 2015). It applies nonoscillatory forward-in-time integration scheme based on MPDATA (e.g., Smolarkiewicz 2006) and uses an elliptic pressure solver for the anelastic dynamics [see Prusa et al. (2008) for a review and comprehensive list of references]. The babyEULAG does not have any subgrid-scale turbulence scheme but it employs the implicit large-eddy simulation (ILES) approach (Margolin et al. 2006; Grinstein et al. 2007).
We apply the microphysical piggybacking methodology (Grabowski 2014, 2015) to unambiguously extract the difference between results applying bulk and bin microphysics schemes. Microphysical piggybacking is about using two sets of thermodynamic variables (bulk and bin in the current study), with the first one coupled to the dynamics and driving the simulation, and the second one driven by the predicted airflow but not affecting it. The bin schemes apply CCN conditions as in the previous section—that is, SPRI, PRI, POL, and SPOL. As before, the bulk set of thermodynamic variables will be referred to as BLK. In the first set of four simulations, BLK drives the flow and the bin microphysics piggyback the predicted flow. The flow evolution is exactly the same in these simulations and this allows direct comparison of bulk and bin results. In the second set of four simulations, the bin microphysics scheme (either SPRI, PRI, POL, or SPOL) drives the simulation, and BLK scheme piggybacks the simulated flow. These simulations develop different flow patterns, but—as documented in Grabowski (2015)—comparing the results from corresponding pairs of the two sets of simulations (e.g., comparing BLK driving and SPOL piggybacking, referred to as D-BLK/P-SPOL, with SPOL driving and BLK piggybacking, D-SPOL/P-BLK, etc.) allows assessing the impact of microphysics on the dynamics.
The doubly periodic horizontal domain is 5 km by 5 km with the horizontal grid length of 50 m. The horizontal domain is only slightly smaller than the one used in the BOMEX intercomparison study of Siebesma et al. (2003), but the horizontal grid length is half as big. In the vertical, the 3-km-deep domain is covered with a stretched grid. The smallest vertical grid length of 20 m is applied near the cloud base to better represent CCN activation. Although still relatively coarse according to results presented in the previous section, it is sufficient to provide contrasting droplet concentrations between bin schemes. The grid length increases away from the cloud base to reach around 60 m near the rigid-lid lower and upper boundaries. The entire vertical domain is covered with 81 levels. The time step is 2 s for the bulk model and substepping is used for the bin model with a 0.25-s time step (see appendix for details). As in Siebesma et al. (2003), model simulations are run for 6 h, with the analysis covering 4 h (from 120 to 360 min) applying snapshots of the 3D model data saved every 6 min.
b. Results
Figure 5 shows profiles of the mean droplet concentration for the bin set of variables from all eight simulations including those driven by the bulk scheme and piggybacking with bin schemes (i.e., P panels; bottom row) and those driven by bin schemes (D panels; top row). The panels also show the standard deviation of the concentration. Note that the standard deviation of the droplet concentration near the cloud base reflects variability of the supersaturation during CCN activation. The mean and standard deviation are calculated for every snapshot including cloudy points with the cloud water mixing ratio larger than 0.01 g kg−1 and then averaged over all snapshots. The figure shows that the mean droplet concentration increases from SPRI to SPOL schemes, as one should expect. The mean droplet concentration is around 5 mg−1 (i.e., 5 cm−3 for the air density of 1 kg m−3) for the SPRI, around 40 mg−1 for PRI, around 300 mg−1 for POL, and around 4000 mg−1 for SPOL. However, these numbers significantly depend on the threshold used to select cloudy points. For instance, selecting less diluted cloudy volumes (the threshold of 0.1 g kg−1) gives the mean concentration around 10, 60, 600, and 5000 mg−1 for SPRI, PRI, POL, and SPOL, respectively. For each simulation, the mean droplet concentration is approximately constant with height. There is, however, a consistent trend in both P and D sets, with a slight increase with height of the mean concentration in SPOL and a slight decrease in PRI and SPRI. The approximately constant-with-height mean droplet concentration was argued in Slawinska et al. (2012) to result from additional activation of cloud droplets above the cloud base. The standard deviation is typically about a half of the mean concentration, which documents a significant spatial variability of the droplet concentration at all heights, which is in agreement with numerous in situ observations of convective clouds [e.g., Prabha et al. (2011) and references therein].
Profiles of the mean cloud droplet concentrations in BOMEX simulations applying bin schemes. The thick horizontal lines show twice the standard deviation of the concentration at a given height. Shown are simulations in which bin microphysics (top) drives and (bottom) piggybacks (D vs P) the simulated flow.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
The droplet-concentration profiles suggest that similar cloud fields are simulated regardless of which scheme drives the simulation. This is further supported by the inspection of the cloud fraction profiles shown in Fig. 6. The cloud fraction is defined as a fraction of grid points at a given height with the cloud water mixing ratio larger than 0.01 g kg−1. Each panel in the figure shows profiles from bin and bulk set of thermodynamic variables. The bulk profiles at bottom four panels are exactly the same because the bulk scheme drives the simulations. Small differences in upper panels (e.g., the largest cloud-base fraction for SPRI) come from different realizations of the cloud field. The figure shows that the profiles are similar among all simulations, with bin schemes having larger cloud fractions than the bulk scheme. Overall, SPOL is close to BLK and the difference between bulk and bin profiles increases with the reduction of the droplet concentration. The largest cloud fractions are for SPRI. This contradicts the hypothesis of convective invigoration for the high droplet concentration because the invigoration should result in more efficient condensation and transport of the cloud water into the middle and upper parts of the cloud field. Apparently, the opposite is true as the cloud fractions are the smallest for SPOL. However, the simulated impact can be explained by another mechanism, already investigated by Xue and Feingold (2006) and related to the evaporation of cloud water near cloud edges. Saturation adjustment in the bulk scheme leads to an instantaneous evaporation in subsaturated conditions, whereas bin schemes simulate finite-time-scale cloud water evaporation. The time scale depends on the mean droplet size and it is short (long) for small (large) cloud droplets. Such considerations explain the largest bin–bulk difference for SPRI evident in the figure.
Cloud fraction profiles in BOMEX simulations. Solid (dashed) lines are for cloud water predicted by the bin (bulk) scheme. Shown are simulations in which bin microphysics (top) drives and (bottom) piggybacks (D vs P) the simulated flow.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
Figures 7 and 8 show histograms of the cloud-top height for simulations driven by either the bulk or the bin scheme (D panels) and the other scheme piggybacking the simulations (P panels) for SPRI and SPOL, respectively. As in Grabowski et al. (2015), the cloud-top height is defined on a column-by-column basis as the level at which the liquid water path integrated downward from the upper-model boundary reaches 10 g m−2. Note that such a definition typically leads to several values of the cloud-top height for a single cloud rather than just a single value. Since the flows driven by the bulk scheme are exactly the same in each simulation, the D-BLK panels in the two figures are exactly the same. Comparing the two figures, one notices that all panels are similar in Fig. 8 (i.e., for BLK and SPOL), but there are significant differences between BLK and SPRI panels in Fig. 7, with a larger contribution of higher cloud tops in the SPRI case. This is inconsistent with the invigoration of polluted clouds but consistent with a slower evaporation of larger droplets near cloud edges in SPRI, which is in agreement with the cloud fraction profiles in Fig. 6. Moreover, for a given microphysical scheme, the results from simulations where the scheme either drives or piggybacks the flow are similar. This suggests that there is little (if any) dynamical effect when using different microphysical schemes to drive the model.
Histograms of the cloud-top height calculated applying cloud water field from either D or P sets of variables for SPRI and BLK schemes. The simulation is driven by (left) BLK and (right) SPRI. Shown are (top) piggybacking and (bottom) driving microphysical schemes. Horizontal dashed lines are included to better expose differences between panels. The height bin is 100 m. Numbers in each panel show the number of cloudy columns detected in the analysis.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
As in Fig. 7, but for BLK and SPOL simulations.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
The above discussion is supported by the 4-h-averaged cloud cover and liquid water path (LWP) data for eight piggybacking simulations shown in Table 1. The simulations with BLK driving the flow (i.e., D-BLK) show the same values because the flow evolves exactly the same regardless of the piggybacking bin thermodynamics. With the bin thermodynamics driving the flow (i.e., D-SPRI, D-PRI, D-POL, and D-SPOL), each simulation represents a different realization of the shallow convection field. Nevertheless, the results are consistent among all simulations. First, the difference between bulk and bin schemes decreases with the increase of the droplet concentration (i.e., from SPRI to SPOL) regardless which scheme drives the simulation (see percentage changes shown in columns 4 and 7). When bin microphysics drives the simulation, the reduction of the cloud cover and LWP from D-SPRI to D-SPOL is again inconsistent with the invigoration of polluted clouds. However, it is consistent with the suppressed cloud evaporation near cloud edges in low-droplet-concentration conditions due to the increase of the droplet mean radius. One has to keep in mind, however, that each simulation driven by the bin scheme evolves differently and this explains why the cloud cover is slightly larger in D-POL than in D-PRI. Finally, comparable effects in corresponding simulations (e.g., D-BLK/P-SPRI and D-SPRI/P-BLK, D-BLK/P-PRI and D-PRI/P-BLK) suggest again a small impact on the cloud dynamics.
Cloud cover and domain-averaged liquid water path averaged for hours 3–6 for eight piggybacking simulations discussed in this paper.
Figures 9 and 10 compare histograms of the vertical velocity (including cloudy and cloud-free grid points) near the cloud base and at 1.5 km, respectively, for simulations driven by either bulk or bin schemes. Each panel includes the same data from the simulation driven by the bulk scheme and data from the simulation driven by the bin scheme, from SPOL to SPRI. The figures highlight increasing departures of bin simulation statistics from BLK with the decreasing droplet concentration. The distributions are narrower near the cloud base, demonstrating weaker updrafts and downdrafts, as one might expect. Since the data used to construct the two figures do not involve piggybacking, one has to keep in mind statistical limitations of such an analysis. This arguably explains why occasionally there is a histogram bin with a larger number of occurrences in the bin simulation. The decrease of the number of updrafts with the decrease of droplet concentration in most bins is consistent with the reduction of cloud buoyancy between bulk and bin schemes. This effect, however, seems to be overpowered by the reduction in the number of downdrafts with the decrease of droplet concentration (see the bottom-right panels in the two figures), arguably documenting key difference due to the cloud water evaporation between bulk and bin schemes.
Histograms of vertical velocity at 640 m for SPOL, POL, PRI, SPRI, and BLK schemes driving the simulations. BLK distribution is the same in all panels. The bin size is 0.5 m s−1 and the bin between −0.25 and 0.25 m s−1 is not shown.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
As in Fig. 9, but at height of 1.5 km.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
As in Grabowski (2015), microphysical piggybacking allows direct comparison, in the same grid boxes, of the instantaneous cloud buoyancy between two sets of thermodynamic variables driving and piggybacking the simulation. Figures 11 and 12 present results of such an analysis for volumes referred to as cloudy updrafts and cloud-edge downdrafts, respectively, at height of 1.5 km and for simulations driven by the bulk scheme. As before, the analysis is for hours 3–6 of the simulations. Cloudy updraft grid points are defined as in Grabowski et al. (2015)—that is, with the vertical velocity w larger than 1 m s−1 and the cloud water mixing ratio qc larger than 0.1 g kg−1. Cloud-edge downdrafts refer to grid points with w < −1 m s−1 and qc < 0.1 g kg−1—that is, significant downdrafts with trace of or no cloud water as in Grabowski et al. (2015). The motivation for such a separation is to distinguish effects associated with the growth of cloud droplets in cloudy updrafts from the impact of droplet evaporation near cloud edges.
Scatterplots of buoyancy in cloudy updrafts from piggybacking SPRI, PRI, POL, and SPOL simulations driven by BLK scheme. Horizontal axis shows buoyancy in the bulk scheme, and the vertical axis shows the difference between buoyancies calculated using bin and bulk sets of thermodynamic variables. Scales on the vertical axes are different in different panels. Thin dashed lines are zero lines.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
As in Fig. 11, but for cloud-edge downdrafts. The scales on vertical axes are larger than in Fig. 11.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-15-0091.1
For the cloudy updrafts (Fig. 11), positive buoyancies in the bulk scheme correspond to negative differences between bin and bulk buoyancies. This agrees with the expectation that the no-supersaturation case results in the largest buoyancy for positive supersaturations (see section 2). The magnitude of the difference increases with the reduction of the droplet concentration, from below 0.001 m s−2 (i.e., the density temperature difference on the order of 0.01 K) for SPOL to around 0.01 m s−2 (i.e., the density temperature difference on the order of 0.1 K) for SPRI. For cloud-edge downdrafts (Fig. 12), the differences between BLK and bin schemes are significantly larger (note different vertical scales in Figs. 11 and 12), with buoyancy difference typically having the opposite sign than the BLK buoyancy. This implies, for instance, that negative buoyancies in BLK scheme correspond to less negative buoyancies in the bin schemes in agreement with the impact of droplet evaporation rate. The differences between bulk and bin schemes are significantly smaller in SPOL than in SPRI as expected from differences in evaporation of small and large cloud droplets.
In summary, the differences between results from bulk and bin simulations are relatively small and they increase with the decreasing droplet concentrations. However, the differences come not from the condensation but rather from the evaporation of cloud water near cloud edges due to entrainment and mixing.
6. Discussion and conclusions
The issue of the finite supersaturation in the bin scheme versus no supersaturation in the bulk scheme is important. First, as mentioned in the introduction, there are suggestions that polluted warm clouds become invigorated when compared to pristine clouds. This arguably has implications for deep convection as well. Second, there have been studies applying cloud-scale flows from bulk microphysics simulations to drive more complex schemes—for instance, the b2 scheme in Brenguier and Grabowski (1993) or the hybrid bulk–bin scheme of Grabowski et al. (2010). Such studies implicitly rely on the assumption that supersaturations in natural clouds are small and their presence has a negligible impact on the dynamics. The current study addresses this very issue.
Theoretical analysis in section 2 suggests that realistic supersaturations have a small impact on the buoyancy field. For instance, 1% supersaturation corresponds to the density temperature that is about 0.1 K smaller than in the situation when the supersaturation is removed through condensation. The supersaturations in natural clouds away from the cloud base are typically smaller [e.g., Politovich and Cooper (1988) and references therein], and the difference in supersaturation between pristine and polluted clouds away from the cloud base is arguably even smaller, on the order of 0.1% (cf. Fig. 3). This corresponds to the density temperature difference of a few hundredths of 1 K at the most. Arguably, one should not expect such differences to play a significant role in the cloud dynamics, perhaps with the exception of the difference between extremely clean and extremely polluted clouds.
The simulations applying a 1D constant-updraft advection–condensation setup show that the density temperature differences agree with the theoretical analysis summarized above. Moreover, the results also document the high-vertical-resolution requirement of the bin scheme—a few meters, which is difficult to meet in typical simulations of shallow convection. The high resolution is needed because one has to resolve the vertical profile of the supersaturation and its maximum that determines the concentration of activated cloud droplets. With lower resolution, the concentration can differ by several tens of percent as already observed in similar simulations discussed in section 4a of Morrison and Grabowski (2008) and section 4.2 of GAW11. This issue is especially relevant for bin simulations of deep convection where vertical grid length is typically on the order of 100 m or coarser. For that very reason, a better approach is to parameterize cloud-base activation (rather than trying to resolve it) as a function of the updraft speed and aerosol characteristics by applying a Lagrangian adiabatic-parcel model and developing a lookup table to be used during model run (e.g., Saleeby and Cotton 2004). Another possibility is to use a bulk condensation model and prescribe droplet concentration based on the characteristics of the air mass in which the simulated cloud develops, which is an approach used in the model applied in Stevens and Seifert (2008).
Large-eddy simulations of the shallow convection cloud field applying microphysical piggybacking with bin and bulk schemes show unequivocally that finite supersaturations in the bin scheme have an opposite impact on macroscopic cloud field characteristics (cloud fraction profiles, domain-averaged LWP, cloud cover, etc.) to the one expected in the convective invigoration hypothesis. This is argued to come from the impact of the droplet size on the evaporation near cloud edges, suppressed in pristine environments, and consistent with the discussion in Xue and Feingold (2006) and in Grabowski et al. (2015). The comparison of buoyancies in bulk and bin schemes, separately in cloud updrafts and cloud-edge downdrafts, shows that bulk and bin buoyancies differ more significantly in the cloud-edge downdrafts than in cloud updrafts. The vertical velocity statistics show that there is a detectable signal consistent with the invigoration of polluted clouds, but it is overpowered by the impact of the cloud-edge evaporation. Overall, the differences in macroscopic characteristics are relatively minor, up to about 10% between extremely polluted (SPOL) and extremely pristine simulations (SPRI; see Table 1). Differences between polluted and pristine simulations (POL vs PRI) are a mere few percent, arguably impossible to extract with confidence without the piggybacking methodology.
The cloud water evaporation near cloud edges is driven by a combination of an uncertain subgrid-scale parameterization and numerical diffusion [e.g., Grabowski and Smolarkiewicz (1990); see also discussion in the appendix in Slawinska et al. (2012)]. Grabowski (2007) points out that instantaneous evaporation of cloud water near cloud edges in the bulk scheme is unrealistic not only because of the finite time scale of droplet evaporation but also because of the fundamental characteristics of the turbulent mixing between a cloud and its environment. For the latter, droplet evaporation is delayed because turbulent mixing between initially separated volumes of cloudy and cloud-free environmental air proceeds through a gradual filamentation of these volumes, with progressively increasing evaporation of cloud water during the approach to final homogenization; see also Jarecka et al. (2009). Jarecka et al. (2013) present an approach to model these processes in the context of a double-moment microphysics scheme. Overall, one might expect that the specific conclusions concerning the relative importance of condensation in cloud updrafts and evaporation at cloud edges from the cloud simulations discussed here depend on the grid resolution. Higher-spatial-resolution simulations should be conducted in the future to explore this issue, perhaps applying similar methodology to the one developed in Jarecka et al. (2013).
As far as deep convection is concerned, the buoyancy differences resulting from either including or neglecting in-cloud supersaturations are arguably less significant because of larger in-cloud buoyancies. At the same time, however, larger vertical velocities imply larger supersaturations and thus larger differences between polluted and pristine deep convective clouds. Such considerations warrant follow-up investigations targeting deep convection.
Acknowledgments
Comments on an early version of this manuscript by Dr. Yefim Kogan are acknowledged. WWG was partially supported by the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP; managed by Colorado State University under Cooperative Agreement ATM- 0425247) and by the DOE ASR Grant DE-SC0008648. DJ was supported by the Polish Ministry of Science and Higher Education (Project 1119/MOB/13/2014/0).
APPENDIX
Bulk and Bin Equations and Numerical Methods
Numerical implementation of the bin scheme is discussed in GAW11, except that collision–coalescence is excluded and thus smaller number of bins can be used. Because of computational limitations, different grids in the radius space and different number of bins are considered in 1D advection–condensation tests and in 3D large eddy simulations (LES). The grid setup in 1D advection–condensation tests is as for the 112 radius bins setup of GAW11 [i.e., α = 0.2 μm and s = 3.0; see section 3.3 and (8) therein]. The total number of bins is limited to 50 covering the range up to 54 μm. In LES simulations, 24 bins are considered covering the range up 48 μm; the parameters in (8) of GAW11 are α = 0.5 μm and s = 1.5.
In 1D cloud updraft simulations, the time step is 0.1 s. In the BOMEX simulations, Δt = 2 s and Δτ = 0.25 s are used.
REFERENCES
Árnason, G., and R. S. Greenfield, 1972: Micro- and macro-structures of numerically simulated convective clouds. J. Atmos. Sci., 29, 342–367, doi:10.1175/1520-0469(1972)029<0342:MAMSON>2.0.CO;2.
Brenguier, J.-L., and W. W. Grabowski, 1993: Cumulus entrainment and cloud droplet spectra: A numerical model within a two-dimensional dynamical framework. J. Atmos. Sci., 50, 120–136, doi:10.1175/1520-0469(1993)050<0120:CEACDS>2.0.CO;2.
Grabowski, W. W., 2007: Representation of turbulent mixing and buoyancy reversal in bulk cloud models. J. Atmos. Sci., 64, 3666–3680, doi:10.1175/JAS4047.1.
Grabowski, W. W., 2014: Extracting microphysical impacts in large-eddy simulations of shallow convection. J. Atmos. Sci., 71, 4493–4499, doi:10.1175/JAS-D-14-0231.1.
Grabowski, W. W., 2015: Untangling microphysical impacts on deep convection applying a novel modeling methodology. J. Atmos. Sci., 72, 2446–2464, doi:10.1175/JAS-D-14-0307.1.
Grabowski, W. W., and P. K. Smolarkiewicz, 1990: Monotone finite-difference approximations to the advection-condensation problem. Mon. Wea. Rev., 118, 2082–2098, doi:10.1175/1520-0493(1990)118<2082:MFDATT>2.0.CO;2.
Grabowski, W. W., and P. K. Smolarkiewicz, 2002: A multiscale anelastic model for meteorological research. Mon. Wea. Rev., 130, 939–955, doi:10.1175/1520-0493(2002)130<0939:AMAMFM>2.0.CO;2.
Grabowski, W. W., and L.-P. Wang, 2009: Diffusional and accretional growth of water drops in a rising adiabatic parcel: Effects of the turbulent collision kernel. Atmos. Chem. Phys., 9, 2335–2353, doi:10.5194/acp-9-2335-2009.
Grabowski, W. W., and L.-P. Wang, 2013: Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech., 45, 293–324, doi:10.1146/annurev-fluid-011212-140750.
Grabowski, W. W., O. Thouron, J.-P. Pinty, and J.-L. Brenguier, 2010: A hybrid bulk-bin approach to model warm-rain processes. J. Atmos. Sci., 67, 385–399, doi:10.1175/2009JAS3155.1.
Grabowski, W. W., M. Andrejczuk, and L.-P. Wang, 2011: Droplet growth in a bin warm-rain scheme with Twomey CCN activation. Atmos. Res., 99, 290–301, doi:10.1016/j.atmosres.2010.10.020.
Grabowski, W. W., L.-P. Wang, and T. V. Prabha, 2015: Macroscopic impacts of cloud and precipitation processes on maritime shallow convection as simulated by a large eddy simulation model with bin microphysics. Atmos. Chem. Phys., 15, 913–926, doi:10.5194/acp-15-913-2015.
Grinstein, F. F., L. G. Margolin, and W. J. Rider, 2007: Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press, 578 pp.
Holland, J. Z., and E. M. Rasmusson, 1973: Measurements of the atmospheric mass, energy, and momentum budgets over a 500-kilometer square of tropical ocean. Mon. Wea. Rev., 101, 44–55, doi:10.1175/1520-0493(1973)101<0044:MOTAME>2.3.CO;2.
Igel, A. L., M. R. Igel, and S. C. van den Heever, 2015: Make it a double? Sobering results from simulations using single-moment microphysics schemes. J. Atmos. Sci., 72, 910–925, doi:10.1175/JAS-D-14-0107.1.
Jarecka, D., W. W. Grabowski, and H. Pawlowska, 2009: Modeling of subgrid-scale mixing in large-eddy simulation of shallow convection. J. Atmos. Sci., 66, 2125–2133, doi:10.1175/2009JAS2929.1.
Jarecka, D., W. W. Grabowski, H. Morrison, and H. Pawlowska, 2013: Homogeneity of the subgrid-scale turbulent mixing in large-eddy simulation of shallow convection. J. Atmos. Sci., 70, 2751–2767, doi:10.1175/JAS-D-13-042.1.
Kogan, Y. L., 1991: The simulation of a convective cloud in a 3-D model with explicit microphysics. Part I: Model description and sensitivity experiments. J. Atmos. Sci., 48, 1160–1189, doi:10.1175/1520-0469(1991)048<1160:TSOACC>2.0.CO;2.
Kogan, Y. L., and W. J. Martin, 1994: Parameterization of bulk condensation in numerical cloud models. J. Atmos. Sci., 51, 1728–1739, doi:10.1175/1520-0469(1994)051<1728:POBCIN>2.0.CO;2.
Kohler, H., 1936: The nucleus in and the growth of hygroscopic droplets. Trans. Faraday Soc., 32, 1152–1161, doi:10.1039/TF9363201152.
Margolin, L. G., W. J. Rider, and F. F. Grinstein, 2006: Modeling turbulent flow with implicit les. J. Turbul., 7, N15, doi:10.1080/14685240500331595.
Morrison, H., and W. W. Grabowski, 2007: Comparison of bulk and bin warm-rain microphysics models using a kinematic framework. J. Atmos. Sci., 64, 2839–2861, doi:10.1175/JAS3980.
Morrison, H., and W. W. Grabowski, 2008: Modeling supersaturation and subgrid-scale mixing with two-moment bulk warm microphysics. J. Atmos. Sci., 65, 792–812, doi:10.1175/2007JAS2374.1.
Petters, M. D., and S. M. Kreidenweis, 2007: A single parameter representation of hygroscopic growth and cloud condensation nucleus activity. Atmos. Chem. Phys., 7, 1961–1971, doi:10.5194/acp-7-1961-2007.
Politovich, M. K., and W. A. Cooper, 1988: Variability of the supersaturation in cumulus clouds. J. Atmos. Sci., 45, 1651–1664, doi:10.1175/1520-0469(1988)045<1651:VOTSIC>2.0.CO;2.
Prabha, T. V., A. Khain, R. Maheshkumar, G. Pandithurai, J. Kulkarni, M. Konwar, and B. Goswami, 2011: Microphysics of premonsoon and monsoon clouds as seen from in situ measurements during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX). J. Atmos. Sci., 68, 1882–1901, doi:10.1175/2011JAS3707.1.
Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. Kluwer Academic Publishers, 954 pp.
Prusa, J. M., P. K. Smolarkiewicz, and A. A. Wyszogrodzki, 2008: EULAG, a computational model for multiscale flows. Comput. Fluids, 37, 1193–1207, doi:10.1016/j.compfluid.2007.12.001.
Rogers, R. R., and M. K. Yau, 1989: A Short Course in Cloud Physics. 3rd ed. Butterworth-Heinemann, 304 pp.
Rosenfeld, D., U. Lohmann, G. B. Raga, C. D. O’Dowd, M. Kulmala, S. Fuzzi, A. Reissell, and M. O. Andreae, 2008: Flood or drought: How do aerosols affect precipitation? Science, 321, 1309–1313, doi:10.1126/science.1160606.
Saleeby, S. M., and W. R. Cotton, 2004: A large-droplet mode and prognostic number concentration of cloud droplets in the Colorado State University Regional Atmospheric Modeling System (RAMS). Part I: Module descriptions and supercell test simulations. J. Appl. Meteor., 43, 182–195, doi:10.1175/1520-0450(2004)043<0182:ALMAPN>2.0.CO;2.
Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219, doi:10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.
Slawinska, J., W. W. Grabowski, H. Pawlowska, and H. Morrison, 2012: Droplet activation and mixing in large-eddy simulation of a shallow cumulus field. J. Atmos. Sci., 69, 444–462, doi:10.1175/JAS-D-11-054.1.
Smolarkiewicz, P. K., 2006: Multidimensional positive definite advection transport algorithm: An overview. Int. J. Numer. Methods Fluids, 50, 1123–1144, doi:10.1002/fld.1071.
Soong, S.-T., and Y. Ogura, 1973: A comparison between axisymmetric and slab-symmetric cumulus cloud models. J. Atmos. Sci., 30, 879–893, doi:10.1175/1520-0469(1973)030<0879:ACBAAS>2.0.CO;2.
Squires, P., 1952: The growth of cloud drops by condensation. I. General characteristics. Aust. J. Chem., 5, 59–86, doi:10.1071/CH9520059.
Stevens, B., and A. Seifert, 2008: Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection. J. Meteor. Soc. Japan, 86A, 143–162, doi:10.2151/jmsj.86A.143.
Twomey, S., 1959: The nuclei of natural cloud formation part II: The supersaturation in natural clouds and the variation of cloud droplet concentration. Geofis. Pura Appl., 43, 243–249, doi:10.1007/BF01993560.
Warner, J., 1968: The supersaturation in natural clouds. J. Rech. Atmos., 3, 233–237.
Wyszogrodzki, A. A., W. W. Grabowski, L.-P. Wang, and O. Ayala, 2013: Turbulent collision-coalescence in maritime shallow convection. Atmos. Chem. Phys., 13, 8471–8487, doi:10.5194/acp-13-8471-2013.
Xue, H., and G. Feingold, 2006: Large-eddy simulations of trade wind cumuli: Investigation of aerosol indirect effects. J. Atmos. Sci., 63, 1605–1622, doi:10.1175/JAS3706.1.
Another possibility is to apply a double-moment microphysics—that is, to predict the mass and number mixing ratios as in Morrison and Grabowski (2007, 2008)—an option we do not consider here.
