1. Introduction
The motivation for numerical simulations reported in this paper, the first of its kind to the author’s knowledge, is to use the turbulent fluid flow environment of a laboratory cloud chamber to compare microphysical properties of a simulated cloud between two drastically different simulation methodologies for the droplet spectral evolution. The first approach, referred to as the Eulerian bin microphysics, solves the evolution equation for the spectral density function. The droplet spectrum is represented by a finite number of droplet size (or mass) spectral density bins. Each bin has to be advected in the physical space, and all bins are combined in each model grid volume to represent droplet growth through the transport (advection) of the spectral density from one bin to another. This is the traditional approach to model spectral evolution of cloud and precipitation particles [see Khain et al. (2015) and Grabowski et al. (2019), and references therein]. The second approach is to represent the evolution of the droplet spectrum by a judiciously selected ensemble of Lagrangian point particles called superdroplets. Superdroplets are traced in physical space using the model-predicted flow field, and they grow or evaporate as they move with the flow. Each superdroplet represents a multitude of natural cloud particles, and an additional parameter, the multiplicity, is used to describe the total number of real particles each superdroplet represents. The Lagrangian particle-based approach is often referred to as the superdroplet method (Shima et al. 2009). The motivation for the comparison comes from the numerical broadening of simulated droplet spectra in bin microphysics as discussed in Morrison et al. (2018) and Grabowski et al. (2019). The Lagrangian particle-based approach is void of numerical broadening, but faces limitations because of the finite (and usually relatively small) number of superdroplets that can be used in a numerical simulation. These aspects are illustrated in simulations discussed in the current paper.
There is a long history of comparing warm-rain and ice microphysics schemes in simulations of cloud dynamics. These include, among many others, McCumber et al. (1991), Seifert et al. (2006), Morrison et al. (2009), Molthan and Colle (2012), Varble et al. (2011), Shipway and Hill (2012), and Xue et al. (2017); see also Khain et al. (2015). However, all of those past studies report comparisons between various Eulerian schemes, typically bulk (either single moment or multimoment) or bin. Since the Lagrangian particle-based approach is a relatively novel way to model cloud microphysics, there are no comparisons between simulations featuring Eulerian and Lagrangian schemes aiming at detail comparisons of simulated droplet spectra. Sato et al. (2018) compares results from shallow cumulus cloud field simulation applying either a Lagrangian scheme (the superdroplet method) or a bulk double-moment scheme focusing on the convergence of numerical solutions with the increasing spatial resolution. Current study compares simulations of idealized cloud dynamics and microphysics applying either the Lagrangian particle-based approach or the Eulerian bin scheme.
The setup of numerical simulations follows laboratory experiments with a cloud chamber, referred to as the Pi Chamber, an apparatus at the Michigan Technological University built for experimental studies of CCN activation and droplet growth as well as ice processes in a turbulent environment (Chang et al. 2016; Chandrakar et al. 2016, C16 hereinafter; see the list of relevant publications at http://phy.sites.mtu.edu/cloudchamber/publications/). The name comes from the fact that in its cylindrical configuration the chamber volume is π cubic meters (1 m height and 1 m radius). However, the typical configuration in which laboratory studies are conducted is a 2 m × 2 m square in the horizontal. The water-saturated lower and upper uniform boundaries are kept at prescribed temperatures, with the higher (lower) temperature at the lower (upper) boundary. The temperature difference (between a few and a couple tens of degrees) drives turbulent Rayleigh–Bénard convection. If no CCN are present inside the chamber, no formation of cloud droplets is observed despite significant supersaturations. The supersaturations come from the isobaric turbulent mixing of warm and humid air from near the lower boundary with colder and less humid air from near the upper boundary. The reason is the convex shape of the saturated water vapor pressure as a function of temperature, the Clausius–Clapeyron relationship. When CCN are present, formation of cloud droplets is observed. CCN characteristics (size and chemical composition) can be selected in specific experiments, and they can be introduced into the chamber at a controlled rate that affects the resulting mean droplet concentration. Either with or without the droplets, the chamber relatively quickly (within minutes) reaches a dynamic quasi equilibrium. Microphysical and to some extent macrophysical characteristics of the conditions inside the chamber depend on the rate at which CCN are inserted. The chamber sidewalls play an important role in the chamber mean thermodynamic conditions; see section 3.1 in Thomas et al. (2019; T19 hereinafter) for a discussion.
T19 discusses results of Pi Chamber large-eddy simulations applying bin microphysics. The chamber is modeled applying a small-scale fluid flow model featuring top, bottom, and sidewalls with prescribed temperatures. The temperature, moisture, and momentum exchanges between the walls and the air inside the chamber is modeled applying Monin–Obukhov similarity theory. A subgrid-scale scheme is applied to represent unresolved turbulent transport. Simulation results compare favorably with laboratory experiments, but specific differences between simulations and observations are also noted. The simulations reported in this paper apply a similar setup to T19, but include significant differences in specific details (e.g., the way CCN activation and droplet growth are represented). These differences are irrelevant for the comparison between results obtained applying the two microphysical schemes, but they make the comparison with experimental results more difficult.
The next section presents the model and model setup together with the details of the two approaches to model spectral characteristics of the Pi Chamber cloud. Section 3 presents simulation results, starting with simulations without condensation and followed by simulations predicting droplet spectral evolution with the two microphysics schemes. Results of additional sensitivity simulations are presented in section 4. Discussion in section 5 that includes computational cost comparison concludes the paper.
2. The model, methodology, and model setup
a. Dynamics
The dynamic model is a simplified serial version of the 3D nonhydrostatic anelastic Eulerian–semi-Lagrangian (EULAG) model (http://www.mmm.ucar.edu/eulag/) referred to as babyEULAG. It features Eulerian dynamics and has been applied previously to simulations of shallow and deep convection (e.g., Grabowski 2014, 2015; Grabowski and Morrison 2016, 2017). Here, we apply babyEULAG to simulate Rayleigh–Bénard convection inside the Pi Chamber. The model has no subgrid-scale transport scheme and relies on the monotone multidimensional positive-definite advection transport algorithm (MPDATA) advection scheme to provide small-scale kinetic energy dissipation. This is in the spirit of the so-called implicit large-eddy simulation (ILES; Margolin and Rider 2002; Andrejczuk et al. 2004; Margolin et al. 2006; Grinstein et al. 2007). The computational domain has spatial dimensions exactly as the Pi Chamber in its cuboid configuration, 2 m × 2 m in the horizontal and 1 m in the vertical. The domain is covered with a 513 uniform grid, implying horizontal and vertical grid lengths of 0.04 and 0.02 m, respectively. The time step applied in all simulations (dictated by the CFL stability criterion of the fluid flow) is 0.025 s.
The standard babyEULAG setup features periodic lateral boundary conditions and rigid-lid lower and upper boundaries. The latter is suitable for the Pi Chamber simulations provided that heat, moisture, and momentum exchanges with the boundaries are properly represented. A suite of dry simulations (i.e., solving transports of the temperature and water vapor mixing ratio with no condensation) was performed to select the simplest model setup before proceeding to simulations with condensation. The turbulent kinetic energy (TKE) simulated in the central part of the chamber was compared to that reported in T19 and used as a guide for the setup selection. The approach eventually selected uses an extremely simple method where lower- and upper-boundary temperature and moisture variables are reset at each time step to the prescribed values, and free-slip conditions for the horizontal velocity components are used. More complicated approaches considered calculation of the temperature, moisture, and horizontal momentum fluxes from the lower and upper boundaries and then distributing those fluxes over a few grid points close to the horizontal boundaries as typically done in LES studies of atmospheric boundary layer flows. The fluxes were calculated applying either a simple bulk aerodynamic flux formulation (with different drag coefficients in different tests) or applying molecular exchange coefficients in simulations with a stretched vertical grid that featured a vertical grid length of about 1 mm near the boundaries. T19 applied Monin–Obukhov similarity theory to calculate the fluxes, a cumbersome technique that we prefer to avoid considering the microphysical emphasis of our investigation. Overall, we found a significant sensitivity of the simulated TKE not only to the flux formulation but also to the depth of the layer near the surface over which the fluxes were distributed. As a result, we decided to follow the simple approach of enforcing temperature and moisture values on lower and upper boundaries that provided TKE values similar to those reported in T19 (see Fig. 6 therein and Fig. 1 here to be discussed in the results section).
Standard babyEULAG’s periodic lateral boundary conditions are not suitable for Pi Chamber simulations. This is because of the small horizontal extent of the Pi Chamber where the presence of lateral boundaries has a significant impact on turbulence and cloud processes within the chamber (see discussion in T19). Rather than replacing periodic boundary conditions with vertical walls as in T19, we decided to use a simpler approach that mimics the presence of vertical walls in otherwise horizontally periodic domain. The idea is to relax model fields (3D velocity components, temperature, and water vapor mixing ratio) within narrow three-gridpoint zones in two x–z and y–z vertical planes. These zones will be referred to as “walls” throughout the paper. Velocity components are relaxed toward zero within each “wall,” whereas thermodynamic fields are relaxed toward the prescribed values selected following the specific laboratory setup. To mimic the solid wall, a short relaxation time scale is selected as 0.1 s (i.e., four model time steps). Inclusion of “walls” helps the development of chamber-scale overturning circulation observed in the laboratory and mimics the impact of chamber walls on temperature and moisture budgets as discussed in T19.
The specific Pi Chamber experimental setup selected has a lower-boundary temperature of 299 K and an upper-boundary temperature of 280 K, that is, a 19 K difference between the lower and upper boundaries. This is one of the setups applied in T19. We decided to limit the corresponding water vapor mixing ratio from the 100% relative humidity (RH) as assumed in the laboratory experiments to 95% on both boundaries. The “wall” temperature was assumed to be the arithmetic mean of the lower- and upper-boundary temperatures (i.e., 288.5 K as in simulations discussed in T19), and the water vapor was taken at 95% RH at the “wall” temperature. Limiting RH at all boundaries results in the mean supersaturation in simulations without condensation more in line of the laboratory measurements (see section 3a). Pressure inside the simulated chamber is assumed as 1000 hPa. Initial conditions inside the chamber are prescribed as no motion and uniform temperature and moisture corresponding to the “wall” conditions. To initiate convective motions, a random noise of 0.1 K amplitude is added to the initial temperature field. Simulations are for either 60 or 30 min (see details below), which is long enough to reach quasi-steady-state conditions as documented by model results.
b. Microphysics
Two different methods are used to model microphysical processes within the simulated chamber, an Eulerian bin microphysics scheme and a particle-based Lagrangian scheme. The two schemes consider only CCN activation and condensational growth of cloud droplets. When a droplet falls through the lower boundary due to droplet sedimentation or evaporates completely near the subsaturated boundaries, it is removed from the simulation in both schemes. Specific details of the schemes are selected in such a way that the resulting processes are simulated as similarly as possible.
The Eulerian bin microphysics scheme is the one applied in shallow nonprecipitating convection simulations described in Grabowski and Jarecka (2015). Note, however, that the model grid box in the current simulations is orders of magnitude smaller than in Grabowski and Jarecka (2015) and thus it typically features a relatively small number of droplets (e.g., 3200 per grid box for a droplet concentration of 100 cm−3). Bin microphysics has been successfully applied in simulations with even smaller grid boxes (e.g., 1 cm−3 as in Andrejczuk et al. 2004). The scheme setup considers 40 equally spaced bins with a bin size of 0.5 μm (thus spanning the range from 0 up to 20 μm radius). Note that the bin scheme in T19 features a significantly lower bin resolution (especially for the condensational growth simulations) as T19 apply only 33 mass-doubling bins for the entire cloud droplet to raindrop range. Grabowski et al. (2011) shows that a significantly larger number of bins is needed to approach numerical convergence for the combined condensational and collisional growth. The number of bins applied here is approximately similar to the number of bins in the 0–20 μm radius in the setup resulting in almost converged numerical solutions in Grabowski et al. (2011). Activated droplets are inserted into an appropriate bin based on the CCN activation radius. Each bin is independently advected in the physical space (including or excluding sedimentation) using the same MPDATA advection scheme as applied to the momentum, temperature, and water vapor mixing ratio. All bins are combined within each grid box to calculate the evolution of the spectrum due to the local supersaturation.
The particle-based Lagrangian scheme follows Grabowski et al. (2018). The specific implementation considers 40 Lagrangian particles (superdroplets) per grid box, each featuring the same multiplicity. The number was selected considering a simple “degrees of freedom” argument for comparison with the bin scheme (i.e., 40 bins and 40 superdroplets per grid box), and additional simulations with a different number of Lagrangian particles (10 and 80 per grid box) will also be presented. When comparing to the bin microphysics, one needs to keep in mind that the number of superdroplets per grid box is on the order of 1% of real droplets. This is much larger than in traditional LES simulations where the ratio between the number of superdroplets and the number of real droplets is several orders of magnitude smaller. The initial particle radius is taken as the CCN activation radius with a random perturbation of 0.2 μm. The perturbation is supposed to mimic the bin scheme finite bin width. Each activated superdroplet is placed in a random position within the grid box and it is subsequently advected applying the model flow field interpolated to the droplet position using an interpolation scheme that preserves incompressibility of the interpolated velocity field; see discussion in section 2.4 in Grabowski et al. (2018). As in the bin scheme, all droplets within a given grid box grow or evaporate in response to the mean gridbox supersaturation, and droplet sedimentation can be included or excluded in the superdroplet transport.
We apply Twomey CCN activation to represent the initial formation of cloud droplets in the two schemes. As in Pi Chamber laboratory experiments, we select a CCN particle size and vary their concentration in different simulations. The CCN is taken as NaCl with a dry radius of 50 nm. This corresponds to an activation supersaturation of 0.1% and a cloud droplet initial (activation) radius of about 0.7 μm (W. Cantrell 2019, personal communication). In the numerical implementation, the concentration of cloud droplets within each grid box is calculated first in both schemes. If the concentration is smaller than the prescribed CCN concentration and the supersaturation is larger than the activation supersaturation (i.e., 0.1%), additional droplets are added to the grid box up to the prescribed CCN concentration. This is straightforward for the bin scheme, but requires additional considerations for the Lagrangian scheme because of the limited number of equal-multiplicity superdroplets, 40 in the most of current simulations. Assuming that the prescribed CCN concentration N provides the upper limit of the local droplet concentration, each superdroplet is assumed to represent N/40 real droplets. It follows that new superdroplets are created within a given grid box in installments of N/40. This approach is a variant of the Twomey activation scheme described in Grabowski et al. (2018, see section 2.3 and Fig. 1 therein). Note that the Twomey activation used here provides only a small computational advantage over the traditional superdroplet technique (i.e., when deliquescence and activation of individual CCN particles are explicitly calculated) because droplets fill almost the entire computational domain and the model time step is sufficiently small to explicitly calculate CCN deliquescence. This is in contrast to simulations discussed in Grabowski et al. (2018) where a cloud occupies only a small fraction of the computational domain and a time step of a few seconds is applied. It should be also stressed that the treatment of CCN applied here is different from that in the Pi Chamber experiments and T19 simulations: we assume that the entire domain is always filled with CCN rather than representing a point CCN source. This likely affects comparison between our simulations on one side and T19 and laboratory experiments on the other, but it is irrelevant for the comparison of the two microphysical schemes.
c. Model simulations
The following simulations are presented in this paper:
Simulation without CCN activation, referred to as NO.CCN. This simulation is run for 60 min from motionless initial conditions described above. A set of such simulations is used to select specific aspects of the model setup as discussed above.
Simulations with bin microphysics. Three different CCN concentrations are considered: 20, 100, and 500 cm−3. The simulations are referred to as BIN.20, BIN.100, and BIN.500, where the number depicts the CCN concentration. Bin simulations are run for 30 min from motionless initial conditions.
Simulations with superdroplets and CCN concentrations of 20, 100, and 500 cm−3. These are referred to as SDs.20, SDs.100, and SDs.500. These simulations are initiated from motionless initial conditions as bin simulations and are run for 30 min.
Sensitivity simulations with CCN concentrations of 100 cm−3 and with no droplet sedimentation. These are referred to as BIN.100.nosed and SDs.100.nosed and run in the same way as simulations 2 and 3. See additional comments below.
Additional sensitivity simulations with Lagrangian microphysics, similar to SDs.100 but applying either 10 (referred to as sd.10) or 80 (referred to as sd.80) superdroplets per grid volume instead of 40 in simulation set 3 above.
Simulation data are archived as snapshots of the fluid velocity, temperature, water vapor, and cloud water mixing ratios (the latter for simulations with condensation) every 20 s. Droplet data (i.e., spectral density function for bin simulations and superdroplet data for Lagrangian microphysics simulations) are saved every minute. These results are applied in the analysis presented below.
3. Results
a. General features of model results
Because of the heating and cooling of the motionless air in contact with the lower and upper boundary, respectively, vertical motions start within a minute of the simulations. Warm plumes rising from the lower boundary meet cold plumes descending from the upper boundary, and the turbulent mixing between the two plume families commences. The resulting small-scale turbulent flow within the chamber is superimposed on more coherent chamber-scale intermittent overturning circulations that feature vertical motions along vertical chamber “walls.” These large-scale flow features affect the mean TKE within the simulated chamber. The quasi-steady conditions are established within several minutes (see Figs. 1 and 4), with significantly fluctuating macrophysical and microphysical characteristics as shown below. As in simulations discussed in T19, horizontally averaged temperature and moisture profiles within the chamber are uniform in height except near the bottom and top boundaries (cf. Fig. 5 in T19). However, spatial variability of the temperature and moisture at any level is significant, with typical standard deviations of around 0.5 K and 0.5 g kg−1 for the temperature and moisture, respectively (not shown).
Evolutions of the mean (top) supersaturation and (bottom) TKE averaged over the chamber volume away from boundaries for NO.CCN simulation.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
b. Simulation NO.CCN without condensation
Figure 1 shows evolutions of the supersaturation and TKE averaged over part of the simulated chamber volume away from horizontal boundaries, 0.2–0.8 m in height, 0.3–1.7 m in both horizontal directions. The TKE fluctuates around 10−2 m2 s−2 with fluctuations dominated by chamber-scale overturning circulations. Inside the Pi Chamber, these oscillations have a period of about a minute (see Fig. 6 in T19). The mean supersaturation fluctuates between approximately 4% and 6%.
The supersaturation also fluctuates in space due to chaotic turbulent motions. This is documented in Fig. 2 that shows PDF of the supersaturation inside the chamber. The mean value is around 5% (as already shown in Fig. 1) and the standard deviation is about 0.9%. The latter is similar to about 1% standard deviation seen in the Pi Chamber observations (see Fig. 8 in T19).
PDF of the supersaturation averaged over the chamber volume away from boundaries for the second half of the NO.CCN simulation. The mean and the standard deviation are given at the top of the plot.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
c. Simulations with CCN activation
Allowing CCN activation and droplet growth brings down the supersaturation to values close to zero. This is documented in Fig. 3 that shows PDFs of the supersaturation inside the chamber away from the boundaries (0.2–0.8 m in height, 0.3–1.7 m in both horizontal directions as in Fig. 1) with data collected during the second half of the simulations. Both the mean value and the standard deviation are much smaller than in the NO.CCN simulation. The key result is that the PDFs are similar between BIN and SDs simulations, with shift toward larger mean and wider distributions with decreasing droplet concentrations. The decrease is consistent with the reduction of the phase relaxation time that depends on the droplet concentration to the power of −2/3 for a constant liquid water mixing ratio. Small differences between BIN and SDs for the same CCN concentrations are likely resulting from differences in the mean droplet concentration as discussed below.
PDFs of the supersaturation averaged over the chamber volume away from boundaries for the second half of the (left) BIN and (right) SDs simulations. The mean and the standard deviation (value at the left and the value in the parenthesis, respectively) are given in each panel.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Differences in the mean supersaturation for simulations with different CCN concentrations explain differences in the mean cloud water mixing ratio and TKE away from chamber boundaries shown in Fig. 4. The most noticeable difference in the cloud water evolution (similar between BIN and SDs simulations) is the lower mean cloud water mixing ratio for the simulations with CCN concentration of 20 cm−3. The difference is consistent with the mean supersaturation difference because the 0.1% supersaturation corresponds to about 0.01 g kg−1 of the liquid water mixing ratio for the averaged water vapor mixing ratio inside the chamber of about 12 g kg−1. Simulations with CCN concentrations of 100 and 500 cm−3 show similar mean values of the liquid water mixing ratio; this can be considered inconsistent with the mean supersaturation difference. TKE appears only weakly affected by the CCN differences, although differences in the TKE averaged for the last 20 min of the simulations are consistent with the finite supersaturation reducing cloud buoyancy as discussed in Grabowski and Jarecka (2015, Fig. 1 therein in particular). The time- and space-averaged TKE is 1.14, 1.09, and 1.02 m2 s−2 for BIN.500, BIN.100, and BIN.20 simulations, respectively, with almost the same values for the corresponding SDs simulations.
Evolutions of (top) the mean cloud water mixing ratio and (bottome) the TKE averaged over the chamber volume away from boundaries for the (left) BIN and (right) SDs simulations. Vertical lines in the top panels represent twice the standard deviation of the spatial variability in the second half of the simulations. The lines are drawn every 3 min and are separated in the horizontal for different simulations so they do not overlap.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Spatial variability of the cloud water mixing ratio shown by the vertical lines for the second half of the simulations in the top panels of Fig. 4 increases with the droplet concentration (roughly by a factor of 2 between 20 and 500 cm−3 for both BIN and SDs simulations). An explanation likely comes from a faster relaxation of the cloud water toward vanishing supersaturation for higher droplet concentration that increase spatial variability. The spatial variability of the mean cloud water mixing ratio is significantly larger in SDs simulations. This is because of the inherent feature of the Lagrangian microphysics related to the finite (and usually small) total number of superdroplets (about 5 million in the current simulations). With about 40 superdroplets per grid box, statistical fluctuations are expected to be large because the standard deviation of the superdroplet number due to transport from grid box to grid box is about 15% (i.e., one over the square root of the mean number per grid box).
Evolutions of microphysical properties such as the mean droplet concentration, mean droplet radius, and mean spectral width (i.e., the standard deviation of the droplet size distribution) for BIN and SDs simulations are compared in Fig. 5. Overall, the agreement between corresponding BIN and SDs simulations is good. SDs simulations feature higher droplet concentrations as well as larger standard deviations of the spatial variability when compared to BIN. This again comes from the inherent limitation of the Lagrangian microphysics with Twomey activation as already noted in Grabowski et al. (2018). The key point is that the CCN activation in each grid box is based on the local droplet concentration within the box, with the concentration calculated from the number of superdroplets within the box. As a result, statistical fluctuations of the superdroplet number impact CCN activation. In particular, if a grid box features a smaller number of superdroplets because of a statistical fluctuation, additional superdroplets are created to increase the droplet concentration toward the activation limit. At the same time, a grid box with a larger number of superdroplets (even above the activation limit) remains unchanged. It follows, Twomey activation always adjusts upward the number of superdroplets and this leads to a higher mean droplet concentration when compared to the bin microphysics. Despite this difference, the mean droplet radius is similar between corresponding BIN and SDs simulations, arguably because the small droplet concentration difference leads to even smaller difference in the droplet radius (the radius changes as the concentration to the power of −1/3 for the constant cloud water mixing ratio). The mean spectral widths in SDs are slightly smaller than in BIN.
Evolutions of (top) mean spectral width, (middle) mean radius, and (bottom) mean droplet concentration for (left) BIN and (right) SDs simulations. Data are averaged over the chamber volume away from boundaries. Vertical lines in the bottom panels (plotted every 2 min starting at minute 10) show twice the standard deviation of the spatial variability.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
d. Comparison of bin and Lagrangian microphysics
The results presented so far show a close similarity between Eulerian bin and Lagrangian particle-based microphysics. Here we further document the similarities including simulations without droplet sedimentation. Figure 6 shows time- and space-averaged means and standard deviations of the droplet concentration and the corresponding cloud water mixing ratio. The figure further documents BIN and SDs differences in the droplet concentration and cloud water mixing ratios, together with their spatial variability. The figure also shows that the simulations without droplet sedimentation, SDs.100.nosed and BIN.100.nosed, feature larger cloud water mixing ratios than the corresponding simulations with sedimentation. The reason for the increase cannot be explained by the mean supersaturation differences as the supersaturations are practically the same in simulations with and without sedimentation (not shown).
Cloud water mixing ratio vs droplet concentration for all BIN and SDs simulations averaged away from chamber boundaries. The length of lines represents twice the time-averaged standard deviation of the spatial variability. The mean values are at the intersection of the two lines. Simulations are marked with their names close to the lines.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Figure 7 shows the mean droplet radius and its time-averaged spatial variability as a function of the droplet concentration in the format similar to Fig. 6. As noted before, SDs simulations show higher mean droplet concentrations and thus smaller mean droplet radius. The simulations are aligned to approximately follow the radius–concentration relationship for a constant cloud water mixing ratio as one might expect. Excluding droplet sedimentation seems to have small impact on the results. Overall, the mean radii shown in Fig. 7 do not match Pi Chamber observations. For instance, the mean radii of about 8 and 4 μm for droplet concentration around 20 and 500 cm−3 was reported in C16; see Table 1 therein. Our values are close to 6 and 2 μm for similar droplet concentrations. Results reported in T19 (Fig. 9 therein) approximately agree with our 100 cm−3 simulations (i.e., mean radius around 4 μm). T19 mention possible reasons for the discrepancy between simulations and observations, concerning details of both. These also apply to our results, with an additional point that the simplified droplet growth equation [Eq. (1)] as well as the simplified method of introducing CCN into the chamber also affects the comparison between our results with T19 simulations and Pi Chamber observations.
Mean droplet radius vs droplet concentration for simulations with condensation. The length of lines represents twice the time-averaged standard deviation of the spatial variability. The mean values are at the intersection of the two lines. The dashed line represents −1/3 power-law scaling valid with constant liquid water mixing ratio.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Finally, Fig. 8 shows spectral width and relative dispersion (the ratio between the spectral width and the mean radius) as a function of the mean radius for BIN and SDs simulations. As in previous figures, data come from the simulated chamber away from boundaries and are averaged over the second half of the simulations. Overall, the spectral width increases with the increasing mean radius (i.e., decreasing droplet concentration). This agrees with results reported in T19 (Fig. 9 therein) and C16 (Table 1 therein). The width reported in T19 for 100 cm−3 is significantly larger than in our simulations (about 4 vs 2–3 μm in our simulations). This is likely because the bin microphysics applied in T19 features low spatial resolution (just a few bins below 10 μm radius). Some numerical broadening of the bin scheme (i.e., Morrison et al. 2018; Grabowski et al. 2019) can also be noticed in our simulations because the width is smaller in SDs when compared to the corresponding simulations in BIN. The BIN–SDs difference is the largest for the smallest mean radius, which is again consistent with the impact of numerical broadening in the bin scheme. The relative dispersion decreases with the increase of the mean radius (i.e., with the decrease of droplet concentration), which is the opposite trend than reported in C16 Pi Chamber observations (Table 1 therein) and in some natural cloud observations (e.g., Liu and Daum 2002, and references therein). Specific values of the spectral width and relative dispersion shown in Fig. 8 are much larger than those reported in stratocumulus observations by Pawlowska et al. (2006). Including or excluding droplet sedimentation has small impact on the results.
(top) Spectral width and (bottom) relative dispersion as a function of the mean droplet radius for simulations with condensation for the (left) BIN and (right) SDs simulations. Data come from averaging over the chamber volume away from the boundaries for the second half of the simulations. Results for simulations without sedimentation are marked as “nosed” in the bottom panels.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Despite all these differences, the overall conclusion is that the two approaches to simulate microphysical processes compare favorably for the time- and space-averaged droplet population characteristics. The next section looks at the details of the droplet spectra.
e. Droplet spectra in Eulerian and Lagrangian simulations
Figure 9 compares droplet spectra at the end (30 min) of BIN and SDs simulations. The spectra are calculated using grid boxes away from the boundaries as in other figures above. Linear and log scales are used to show the shape near the peak and in the tail, respectively. The spectra from both schemes agree well, especially for the tail that gets to the 10−3 level at about the same droplet size. The differences in the small-droplet end (i.e., for radii smaller than 3 μm) are statistically insignificant as this part of the spectrum changes in time depending on the flow dynamics within the chamber (i.e., the TKE level). This is because the dynamics affects activation and deactivation of CCN during turbulent mixing. This can be documented by considering minute-by-minute evolution of the mean spectra (not shown).
Droplet spectral density functions (cm−3 μm−1) averaged over the chamber volume away from boundaries and presented applying the (left) linear and (right) log scales for (top) BIN and (bottom) SDs simulations. Thin dashed lines in the right panels are included to better expose similarity between the spectra. The scale of the vertical axis in panels with the linear scale (marked as “max”) is 3, 15, and 150 cm−3 μm−1 for CCN concentration of 20, 100, and 500 cm−3, respectively.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Although the mean spectra agree well between bin and particle-based microphysics, a closer look exposes inherent differences between the two simulation methodologies. This is illustrated in Fig. 10 that compares spectra in nine neighboring grid boxes randomly selected from the middle of the Pi Chamber simulation volume for the 100 cm−3 CCN case. The differences, perhaps to be expected, are remarkable. The bin microphysics spectra are smooth and differ little between neighboring grid volumes. When plotted applying the log scale on the vertical axis to see the tail, the BIN.100 results are similar to the mean spectra shown in Fig. 9. In contrast, the SDs.100 spectra only vaguely represent the mean spectrum. This is of course because of the small average number of superdroplets per grid box (40 in the current simulations). With the droplet concentration of about 100 cm−3 and the simulation gridbox volume of 32 cm−3, the average grid box contains approximately 3200 droplets. It can thus be argued that one needs to average superdroplet spectra from about 80 grid boxes to compensate for the limiting number of superdroplets. Spectra averaged over 80 grid boxes are indeed much smoother and they resemble those shown in Fig. 9 (not shown).
Local droplet spectral density functions (cm−3 μm−1) over 3 × 3 grid boxes presented applying the linear scale for (top) BIN.100 and (bottom) SDs.100 simulations.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
4. Sensitivity simulations with Lagrangian microphysics
The Lagrangian microphysics simulations discussed above apply a relatively small number of superdroplets per grid box when compared to other studies. Grabowski et al. (2018) considered a range from 50 to 4000 in two-dimensional rising thermal simulations without droplet collisions. Arabas and Shima (2013) applied a range from 8 to 512 in three-dimensional simulations of a precipitating shallow convection cloud field. Hoffmann et al. (2019) applied between 100 and 500 superdroplets per grid box in a study of the impact of cloud–environment mixing on warm-rain initiation. Motivated by these (and other) studies, additional simulations were run with either 10 or 80 superdroplets per grid box, sd.10 and sd.80, respectively. Figures 11 and 12 document results of these simulations.
Evolutions of the (top two rows) cloud water mixing ratio and TKE as in Fig. 4 and (bottom three rows) the mean spectral width, mean radius, and mean droplet concentration (including twice the standard deviation of concentration spatial variability shown as vertical lines in the bottom panel) as in Fig. 5, but with a reduced range on the vertical axes.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
As in Fig. 10, but for (top) sd.10 and (bottom) sd.80.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0216.1
Figure 11 shows evolutions of the cloud water mixing ratio, TKE, mean spectral width, mean radius, and mean droplet concentrations in the format of Figs. 4 and 5. The figure shows that even as small number of superdroplets as in sd.10 is sufficient to provide sensible prediction of the mean properties of the Pi Chamber cloud. The differences between sd.10 and sd.80 simulations include larger spatial variability of the cloud water field, larger mean droplet concentration and thus smaller mean droplet radius in sd.10. The spatial variability difference comes from the impact of statistical fluctuations whose amplitude decreases with the increase of the superdroplet number. The increased droplet concentration in sd.10 comes from the specific implementation of the Twomey activation as discussed in section 3c. Spectral widths are similar between the two simulations. However, the differences in the local spectra illustrated in Fig. 10 remain. This is illustrated in Fig. 12 that compares local spectra from sd.10 and sd.80 simulations. Although local spectra in sd.80 are improved compared to SD.100 in Fig. 10, significant fluctuations between neighboring grid boxes do remain. This is because the mean number of superdroplets per grid box is still much smaller than the mean number of real droplets, 3200 for droplet concentration of 100 cm−3.
5. Discussion and conclusions
This paper presents results of numerical simulations of the Pi Cloud Chamber dynamics and microphysics applying either a traditional Eulerian bin scheme or a Lagrangian particle-based scheme (i.e., using superdroplets). The Pi Cloud Chamber, a laboratory apparatus at the Michigan Technological University (see http://phy.sites.mtu.edu/cloudchamber/), forms a turbulent cloud because of the temperature and humidity differences between lower and upper horizontal boundaries that drive turbulent Rayleigh–Bénard convection. As in laboratory experiments, simulations presented here reach dynamic steady states featuring rising warm plumes and descending cold plumes that mix within the chamber. Because both plume families are close to being saturated, the mixing leads to supersaturated volumes and results in formation of cloud droplets when CCN are present. We introduce simulation results with no condensation (i.e., similar to laboratory air void of CCN and thus no cloud formation) and then focus on the comparison between the Eulerian and Lagrangian methods for the droplet spectrum simulation. In agreement with laboratory experiments, mean droplet radii are relatively small (between 2 and 6 μm for droplet concentrations between 500 and 20 cm−3, respectively) and spectra are wide, with the mean spectral width between 1 and close to 4 μm. The relative dispersion increases when the droplet concentration decreases. The latter is in contrast to Pi Chamber observations (see Table 1 in C16), likely because of the simplifications of the CCN activation in the model simulations (i.e., we assume that the chamber is filled with CCN rather than predicting CCN spread in the chamber from a point source as in the laboratory apparatus) and because of the simplified droplet growth equation used in both schemes.
Overall, the two methods to simulate droplet spectra give very similar results when spectra averaged over the entire chamber are considered. Most surprising is the agreement within the tail of the distribution that describes droplet numbers in the hundreds for the entire chamber volume. This is surprising because bin microphysics is known to artificially broaden droplet spectra (e.g., Morrison et al. 2018). Apparently, this is not the case when the droplet spectra—and spectral broadening in particular—come from the isobaric turbulent mixing and stochastic condensation. Excluding droplet sedimentation has a small but noticeable impact on model results. The likely explanation is that the sedimentation velocity (a fraction of 1 cm s−1 for typically small droplets in the chamber) is much smaller than the mean flow velocity in the chamber that is on the order of the square root of the TKE (i.e., around 10 cm s−1).
The computational cost of the bin and superdroplet simulations presented in this paper are comparable, with the 40-per-gridbox superdroplet simulations requiring about 25% more computational time than bin simulations. Since transport of bins and superdroplets in the physical space constitutes most of the computational effort, doubling the number of bins or the number of superdroplets approximately doubles the simulation time. However, if a cloud occupies only a fraction of the computational domain volume (which is not the case in simulations presented here), then the superdroplets with Twomey activation require proportionally less time to complete the simulation because of a smaller total number of superdroplets. At the same time, the bin microphysics simulation time is independent of the cloud volume fraction because the cost of each bin advection does not depend on the number of cloudy grid volumes. It follows that computational effort of the Lagrangian microphysics with Twomey activation can be even smaller than the bin microphysics when the cloud covers a small fraction of the computational domain volume and the number of superdroplets per grid volume is comparable to the number of bins. Since the cloud in current simulations covers almost the entire domain and a small time step has to be used, the cost of the Twomey superdroplet simulations reported here should be comparable to simulations with traditional superdroplets (e.g., Shima et al. 2009; Arabas et al. 2015) without droplet collisions.
There are important differences between microphysical processes within the Pi Chamber (the real one and the simulated one) and in natural clouds that need to be kept in mind when considering the simulation results. First, natural clouds form because of the cooling of air, typically either because of the adiabatic expansion due to vertical motion or because of radiative cooling of the air. In contrast, the cloud inside the chamber forms because of the mixing of air parcels with different temperature and moisture characteristics (i.e., from near the upper and lower boundaries). Clouds can form this way in nature as well, with classical examples being contrails resulting from the mixing of jet engine exhaust with the environmental air or a “cloud” when we breathe in a cool weather. The vertical motion inside the Pi Chamber has an insignificant impact on droplet growth and it has no impact in the model as the pressure inside the simulated chamber is assumed constant. It follows that the CCN activation and subsequent droplet growth come only from the turbulent mixing with no impact of the vertical air motion. The latter can explain the minimal numerical spreading in the Eulerian bin scheme results. This is because the spreading was argued to come from the vertical advection of droplet bins in a stratified environment that leads to supersaturation and diffusional growth of cloud droplets (Morrison et al. 2018; Grabowski et al. 2019). In simulations presented here, droplets grow and evaporate only because of the isobaric mixing.
The insensitivity of the droplet growth to the vertical motion has a subtle but interesting effect on the argument concerning the impact of turbulence on droplet growth through the process called eddy hopping (see section 3.5 in Grabowski and Wang 2013). Eddy hopping refers to a spectral broadening resulting from droplets following different trajectories in a turbulent environment and experiencing different growth histories (Cooper 1989). The key point [suggested in Grabowski and Wang (2013) and shown to be valid in Grabowski and Abade (2017); see Fig. 4 therein] is that the largest impact on the adiabatic droplet growth in a stratified environment comes from the largest eddies. This is because the largest eddies feature the largest vertical velocities that affect local supersaturation the most. Moreover, these deviations from the mean supersaturation affect the droplet growth for the longest time. Such argument no longer applies to the Pi Chamber because—unlike in natural clouds—vertical motions do not lead to droplet growth by condensation. However, large-scale eddies are the key to the mixing between air parcels originating from lower and upper boundaries, and thus they affect droplet spectra regardless of the stratification. A series of simulations with increasing spatial resolution, down to DNS that provides a natural limit of the superdroplet method, would be needed to resolve the issue of the impact of large versus small eddies on the droplet growth in the Pi Chamber. We leave this aspect for future investigations.
Acknowledgments
The author acknowledges partial supported from the U.S. DOE ASR Grants DE-SC0016476 and DE-SC0020118 and from the National Center of Meteorology, Abu Dhabi, UAE, under the UAE Research Program for Rain Enhancement Science (UAE-NATURE project). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Center of Meteorology, Abu Dhabi, UAE. Prof. Will Cantrell from Michigan Tech provided Pi Chamber CCN data applied in the simulations. Comments on the early draft of this manuscript by Gustavo Abade, Hugh Morrison, Shin-ichiro Shima, and Lian-Ping Wang are acknowledged. Comments by Prof. Raymond Shaw and two anonymous reviewers resulted in the final version of the manuscript. NCAR is sponsored by the National Science Foundation.
REFERENCES
Andrejczuk, M., W. W. Grabowski, S. P. Malinowski, and P. K. Smolarkiewicz, 2004: Numerical simulation of cloud–clear air interfacial mixing. J. Atmos. Sci., 61, 1726–1739, https://doi.org/10.1175/1520-0469(2004)061<1726:NSOCAI>2.0.CO;2.
Arabas, S., and S. Shima, 2013: Large-eddy simulations of trade wind cumuli using particle-based microphysics with Monte Carlo coalescence. J. Atmos. Sci., 70, 2768–2777, https://doi.org/10.1175/JAS-D-12-0295.1.
Arabas, S., A. Jaruga, H. Pawlowska, and W. W. Grabowski, 2015: Libcloudph++ 0.2: Single-moment bulk, double-moment bulk, and particle-based warm-rain microphysics library in C++. Geosci. Model Dev., 8, 1677–1707, https://doi.org/10.5194/gmd-8-1677-2015.
Chandrakar, K. K., W. Cantrell, K. Chang, D. Ciochetto, D. Niedermeier, M. Ovchinnikov, R. A. Shawa, and F. Yang, 2016: Aerosol indirect effect from turbulence-induced broadening of cloud-droplet size distributions. Proc. Natl. Acad. Sci. USA, 113, 14 243–14 248, https://doi.org/10.1073/pnas.1612686113.
Chang, K., and Coauthors, 2016: A laboratory facility to study gas–aerosol–cloud interactions in a turbulent environment: The Π chamber. Bull. Amer. Meteor. Soc., 97, 2343–2358, https://doi.org/10.1175/BAMS-D-15-00203.1.
Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory. J. Atmos. Sci., 46, 1301–1311, https://doi.org/10.1175/1520-0469(1989)046%3C1301:EOVDGH%3E2.0.CO;2.
Grabowski, W. W., 2014: Extracting microphysical impacts in large-eddy simulations of shallow convection. J. Atmos. Sci., 71, 4493–4499, https://doi.org/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, https://doi.org/10.1175/JAS-D-14-0307.1.
Grabowski, W. W., and L.-P. Wang, 2013: Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid. Mech., 45, 293–324, https://doi.org/10.1146/annurev-fluid-011212-140750.
Grabowski, W. W., and D. Jarecka, 2015: Modeling condensation in shallow nonprecipitating convection. J. Atmos. Sci., 72, 4661–4679, https://doi.org/10.1175/JAS-D-15-0091.1.
Grabowski, W. W., and H. Morrison, 2016: Untangling microphysical impacts on deep convection applying a novel modeling methodology. Part II: Double-moment microphysics. J. Atmos. Sci., 73, 3749–3770, https://doi.org/10.1175/JAS-D-15-0367.1.
Grabowski, W. W., and G. C. Abade, 2017: Broadening of cloud droplet spectra through eddy hopping: Turbulent adiabatic parcel simulations. J. Atmos. Sci., 74, 1485–1493, https://doi.org/10.1175/JAS-D-17-0043.1.
Grabowski, W. W., and H. Morrison, 2017: Modeling condensation in deep convection. J. Atmos. Sci., 74, 2247–2267, https://doi.org/10.1175/JAS-D-16-0255.1.
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, https://doi.org/10.1016/j.atmosres.2010.10.020.
Grabowski, W. W., P. Dziekan, and H. Pawlowska, 2018: Lagrangian condensation microphysics with Twomey CCN activation. Geosci. Model Dev., 11, 103–120, https://doi.org/10.5194/gmd-11-103-2018.
Grabowski, W. W., H. Morrison, S. Shima, G. C. Abade, P. Dziekan, and H. Pawlowska, 2019: Modeling of cloud microphysics: Can we do better? Bull. Amer. Meteor. Soc., 100, 655–672, https://doi.org/10.1175/BAMS-D-18-0005.1.
Grinstein, F. F., L. G. Margolin, and W. J. Rider, 2007: Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press, 578 pp.
Hoffmann, F., T. Yamaguchi, and G. Feingold, 2019: Inhomogeneous mixing in Lagrangian cloud models: Effects on the production of precipitation embryos. J. Atmos. Sci., 76, 113–133, https://doi.org/10.1175/JAS-D-18-0087.1.
Khain, A. P., and Coauthors, 2015: Representation of microphysical processes in cloud-resolving models: Spectral (bin) microphysics versus bulk parameterization. Rev. Geophys., 53, 247–322, https://doi.org/10.1002/2014RG000468.
Liu, Y., and P. H. Daum, 2002: Indirect warming effect from dispersion forcing. Nature, 419, 580–581, https://doi.org/10.1038/419580a.
Margolin, L. G., and W. J. Rider, 2002: A rationale for implicit turbulence modelling. Int. J. Numer. Methods Fluids, 39, 821–841, https://doi.org/10.1002/fld.331.
Margolin, L. G., W. J. Rider, and F. F. Grinstein, 2006: Modeling turbulent flow with implicit LES. J. Turbul., 7, N15, https://doi.org/10.1080/14685240500331595.
McCumber, M., W.-K. Tao, J. Simpson, R. Penc, and S.-T. Soong, 1991: Comparison of ice-phase microphysical parameterization schemes using numerical simulations of tropical convection. J. Appl. Meteor., 30, 985–1004, https://doi.org/10.1175/1520-0450-30.7.985.
Molthan, A. L., and B. A. Colle, 2012: Comparisons of single- and double-moment microphysics schemes in the simulation of a synoptic-scale snowfall event. Mon. Wea. Rev., 140, 2982–3002, https://doi.org/10.1175/MWR-D-11-00292.1.
Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 991–1007, https://doi.org/10.1175/2008MWR2556.1.
Morrison, H., M. Witte, G. H. Bryan, J. Y. Harrington, and Z. J. Lebo, 2018: Broadening of modeled cloud droplet spectra using bin microphysics in an Eulerian spatial domain. J. Atmos. Sci., 75, 4005–4030, https://doi.org/10.1175/JAS-D-18-0055.1.
Pawlowska, H., W. W. Grabowski, and J.-L. Brenguier, 2006: Observations of the width of cloud droplet spectra in stratocumulus. Geophys. Res. Lett., 33, L19810, https://doi.org/10.1029/2006GL026841.
Sato, Y., S. Shima, and H. Tomita, 2018: Numerical convergence of shallow convection cloud field simulations: Comparison between double-moment Eulerian and particle-based Lagrangian microphysics coupled to the same dynamical core. J. Adv. Model. Earth Syst., 10, 1495–1512, https://doi.org/10.1029/2018MS001285.
Seifert, A., A. Khain, A. Pokrovsky, and K. D. Beheng, 2006: A comparison of spectral bin and two-moment bulk mixed-phase cloud microphysics. Atmos. Res., 80, 46–66, https://doi.org/10.1016/j.atmosres.2005.06.009.
Shima, S.-I., K. Kusano, A. Kawano, T. Sugiyama, and S. Kawahara, 2009: The superdroplet method for the numerical simulation of clouds and precipitation: A particle-based and probabilistic microphysics model coupled with a non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 135, 1307–1320, https://doi.org/10.1002/qj.441.
Shipway, B. J., and A. A. Hill, 2012: Diagnosis of systematic differences between multiple parameterizations of warm rain microphysics using a kinematic framework. Quart. J. Roy. Meteor. Soc., 138, 2196–2211, https://doi.org/10.1002/qj.1913.
Thomas, S., M. Ovchinnikov, F. Yang, D. van der Voort, W. Cantrell, S. K. Krueger, and R. A. Shaw, 2019: Scaling of an atmospheric model to simulate turbulence and cloud microphysics in the Pi Chamber. J. Adv. Model. Earth Syst., 11, 1981–1994, https://doi.org/10.1029/2019MS001670.
Varble, A., and Coauthors, 2011: Evaluation of cloud-resolving model intercomparison simulations using TWP-ICE observations: Precipitation and cloud structure. J. Geophys. Res., 116, D12206, https://doi.org/10.1029/2010JD015180.
Xue, L., and Coauthors, 2017: Idealized simulations of a squall line from the MC3E field campaign applying three bin microphysics schemes: Dynamic and thermodynamic structure. Mon. Wea. Rev., 145, 4789–4812, https://doi.org/10.1175/MWR-D-16-0385.1.
