1. Introduction
Supersaturation is a key variable for cloud droplet activation and growth and depends linearly on water vapor mixing ratio and nonlinearly on temperature approximately through the Clausius–Clapeyron equation. Here, supersaturation is defined as the ratio of water vapor mixing ratio to its equilibrium value over a flat surface minus one. Supersaturation fluctuations can be generated through fluctuations in water vapor and temperature fields in a turbulent environment. For example, temperature fluctuations generated from vertical velocity fluctuations in a stratified environment drive supersaturation fluctuations (Politovich and Cooper 1988; Korolev and Mazin 2003). Similarly, temperature and water vapor fluctuations during the isobaric mixing process (Korolev and Isaac 2000; Gerber 1991) can generate a fluctuating field of supersaturation. In general, multiscalar transport in turbulent flow is relevant to a variety of processes that depend on multiple scalars—chemical reactions and transport of reactive species (Gao and Wesely 1994), engineering applications and fundamental fluid problems (Warhaft 1981), wave propagation in a turbulent medium (Wyngaard et al. 1978), etc. For example, the refractive index of air depends on both water vapor and temperature, and covariance of water vapor and temperature fluctuations affect the electromagnetic and acoustic wave propagation in turbulent flows (Wyngaard et al. 1978). The focus of the current study is on the supersaturation fluctuations driven by the process of scalar mixing in turbulence.
In the Michigan Tech Pi Convection Cloud Chamber (Chang et al. 2016; Chandrakar et al. 2016), supersaturation fluctuations are generated via scalar mixing in Rayleigh–Bénard convection. The moist Rayleigh number,
In actual cloud turbulence, there is an enormous range of scales from the O(1 mm) Kolmogorov microscale to O(1–10)km integral scales. Interactions of cloud and aerosol particles with thermodynamic and dynamic fields occur even below the Kolmogorov microscale. Thus, an accurate representation of clouds in atmospheric models across scales poses a significant challenge, and smallscale cloud processes are a significant source of uncertainty in weather prediction and climate models (Morrison et al. 2020). The turbulent flow field affects clouds at macroscales, such as entrainment of noncloudy air and largescale organization (Mellado 2017; De Rooy et al. 2013; Feingold et al. 2010; Pauluis and Schumacher 2011). Microscale interactions of turbulence and clouds affect droplet activation (Chandrakar et al. 2017; Abade et al. 2018; Prabhakaran et al. 2020), condensation growth (Cooper 1989; Vaillancourt et al. 2002; Lanotte et al. 2009; Paoli and Shariff 2009; Field et al. 2014; Sardina et al. 2015; Chandrakar et al. 2016, 2018, 2020b; Grabowski and Abade 2017; Desai et al. 2018; Saito et al. 2019; Hoffmann et al. 2019; Thomas et al. 2020; Chandrakar et al. 2021), and collisional growth of droplets (Shaw 2003; Grabowski and Wang 2013; Devenish et al. 2012). How can we account for the impact of SGS supersaturation fluctuations on droplet activation and growth in models? A few different approaches based on Gaussian stochastic models have gained recent popularity (e.g., Paoli and Shariff 2009; Sardina et al. 2015; Chandrakar et al. 2016; Grabowski and Abade 2017; Hoffmann et al. 2019). Most of these SGS schemes are suitable for parcel models or Lagrangian microphysics schemes in a threedimensional dynamical model. Some of these schemes require estimating the strength of the supersaturation fluctuation forcing. A few of them (e.g., Sardina et al. 2015; Grabowski and Abade 2017) represent supersaturation fluctuations driven by SGS vertical velocity fluctuations and neglect contributions from the scalar mixing process. Hoffmann et al. (2019) and Hoffmann and Feingold (2019) account for SGS entrainment mixing using the onedimensional lineareddy model (Kerstein 1988) but without separately treating water vapor and temperature fluctuations. Thus, the contribution of SGS water vapor and temperature covariance to supersaturation fluctuations was not accounted for explicitly. The current paper introduces a new SGS model of supersaturation fluctuations that separately treats water vapor and temperature fluctuations and accounts for SGS supersaturation driven by scalar mixing process in addition to SGS vertical velocity fluctuations.
Supersaturation variance depends on water vapor and temperature variances and their covariance (Kulmala et al. 1997). In turbulent convection, scalar variance is directly related to scalar fluxes, e.g., surface fluxes or turbulent fluxes at different levels of turbulent Rayleigh–Bénard and atmospheric boundary layer convection, entrainment flux in cloud parcels. The covariance of different scalars (in this case, water vapor and temperature) depends on their relative fluxes. In the atmospheric boundary layer or clouds, entrainment of air from the free troposphere, condensation/evaporation, and surface inhomogeneities introduce relative fluxes of water vapor and heat. This can make the normalized covariance close to +1 near the surface but completely anticorrelated (−1) near the top of the boundary layer (Wyngaard et al. 1978; Wyngaard and LeMone 1980; Williams et al. 1997). The topdown mixing of entrained air in the boundary layer is thought of as a significant source of decrease in the correlation of the two scalars in a wellmixed layer (Wyngaard et al. 1978; Wyngaard and LeMone 1980; Williams et al. 1997; Katul et al. 2008). Via highresolution measurements, Siebert and Shaw (2017) reported normalized covariance between water vapor and temperature significantly less than one in shallow cumulus clouds. Similarly, the presence of sidewalls in the Pi Cloud Chamber introduces relative fluxes of heat and moisture (Thomas et al. 2019). To understand how the relative fluxes of heat and water vapor affect their covariance in turbulent convection, we compare DNS of the Rayleigh–Bénard convection with adiabatic sidewalls to DNS with different sidewall conditions that drive different degrees of relative fluxes of both scalars.
This study introduces a new SGS model for supersaturation fluctuations that accounts for the generation of supersaturation fluctuations from scalar mixing apart from the vertical velocity fluctuations. For that purpose, the model explicitly predicts water vapor and temperature variances and their covariance in an LES model. The approach is similar to that of Deardorff (1974a). A DNS of Rayleigh–Bénard convection with realistic sidewall boundaries (motivated by Michigan Tech Pi Cloud Chamber) is used as a benchmark to test the SGS scheme in LES at varying grid resolutions. We also investigate the roles of scalar mixing and relative fluxes of water vapor and sensible heat on supersaturation fluctuations using DNS with different sidewall conditions. The current study builds on past studies of Rayleigh–Bénard convection in the Pi Chamber (e.g., Chandrakar et al. 2020a; Thomas et al. 2021; MacMillan et al. 2022) using more detailed and realistic simulations (e.g., similar Rayleigh number as the real Pi Chamber and with realistic sidewall boundaries).
We focus on the following science questions:

How well does the new SGS supersaturation model (based on scalar variance and covariance prognosis) perform compared to a benchmark DNS of turbulent Rayleigh–Bénard convection?

How do the sidewall fluxes of heat and moisture affect turbulence statistics for Rayleigh–Bénard convection?

How do the relative fluxes of heat and moisture during turbulent mixing influence supersaturation fluctuations?
2. Subgridscale model of scalar fluctuations
a. Subgrid supersaturation variance
As discussed in the introduction, some previous approaches to represent SGS supersaturation fluctuations (e.g., Field et al. 2014; Sardina et al. 2015; Grabowski and Abade 2017; Chandrakar et al. 2021) only considered fluctuations driven from vertical velocity and therefore had to assume zero external water vapor and heat fluxes in the cloud system. However, supersaturation is a function of temperature and water vapor, and in a turbulent environment both fluctuate if there is a flux convergence or divergence of either locally. Other approaches account for SGS scalar mixing in supersaturation fluctuations (e.g., Hoffmann et al. 2019) but do not treat water vapor and temperature fluctuations separately. Our new SGS modeling approach includes supersaturation fluctuations generated from fluctuations in temperature and water vapor during turbulent mixing apart from supersaturation fluctuations driven by vertical velocity. Thus, this model can also predict SGS supersaturation fluctuations driven by turbulent entrainment mixing in clouds. Moreover, a separate treatment of water vapor and temperature fluctuations is a more rigorous way to determine the SGS supersaturation fluctuations since the relative fluxes of water vapor and temperature affect the supersaturation statistics significantly (as shown in section 4a).
b. Subgridscale model for water vapor and potential temperature variance and covariance
The dissipation terms in the above equations for variances and covariance are modeled in terms of SGS TKE and TKE dissipation rate by assuming proportionality between the turbulent mixing time scale for momentum and scalars (Deardorff 1974b; Jiménez et al. 2001; Shi et al. 2019). Therefore, the dissipation rate for a scalar (ϕ) is
3. Model description and simulation setup
The CM1 dynamical model (Bryan and Fritsch 2002) is used in DNS and LES configurations for the simulations discussed in this article. The compressible Boussinesq equation set (Boussinesq approximation for buoyancy in the momentum equation with a prognostic pressure equation) is solved using a threestep Runge–Kutta time integration with a fifthorder advection scheme. Scalars are advected using a weighted essentially nonoscillatory (WENO) scheme. The compressible solver employed here uses Klemp–Wilhelmson timesplit steps for acoustic terms. The above compressible Boussinesq solver was successfully used in the past for an idealized simulation of a channel flow with shear [see Bryan and Rotunno (2014) for details]. Here, a compressible solver is used instead of an incompressible one because it has been optimized for distributedmemory supercomputers, which allows us to use large domains with
The simulations presented here are motivated by the Pi Cloud Chamber that generates supersaturation variability through turbulent Rayleigh–Bénard convection (Chang et al. 2016; Chandrakar et al. 2016; Niedermeier et al. 2018). The present simulations are similar to the model setup of Thomas et al. (2019). The vertical dimension of the domain is the same (1 m), but the aspect ratio (width to height ratio) is Γ = 1 compared to Γ = 2 for the Pi Chamber. The aspect ratio of the chamber primarily influences the largescale circulation, and influences the turbulence statistics slightly. However, the main parameter governing Rayleigh–Bénard convection, the Rayleigh number (1.52 × 10^{9}), is the same in our DNS as the Pi Chamber. The top and bottom boundaries of the chamber are maintained at water saturation at different temperatures to create a steady moist environment. The temperature and water vapor differences between the top and bottom boundaries drive turbulent Rayleigh–Bénard convection with humidity fluctuations. Ideally, the isobaric mixing of air from the upper and lower boundaries should create a supersaturated condition inside the chamber, even for a minimal temperature gradient. However, the presence of sidewalls depletes water vapor substantially through condensation and allows domain mean supersaturation to be maintained only for certain conditions (Thomas et al. 2019). The sidewalls are generally held at the average temperature and are not saturated with water since condensation forms droplets on the sidewalls (not a uniform film of water) if they do not consist of porous material.
Lateral entrainment of dry environmental air in a rising cumulus thermal or cloudtop entrainment in boundary layer clouds brings additional flux of scalars (water vapor and heat) inside clouds. Subsequent mixing of dry environmental or free tropospheric and cloudy air can drive fluctuations in individual scalars as well as dependent variables, including supersaturation (Siebert and Shaw 2017). The effects of these fluxes do not just influence the turbulent fluctuations at the interface but also propagate deeper into the boundary layer or inside clouds (Mellado et al. 2017; Deardorff 1974b). To investigate the effects of scalar mixing on turbulence statistics, especially supersaturation, we performed DNS of turbulent Rayleigh–Bénard convection with different sidewall conditions to mimic the effects of entraining dry air in our simplified setup. Comparing simulations using nonadiabatic sidewalls with the simulation using adiabatic sidewalls allows investigating the role of heat and moisture relative fluxes on covariance and supersaturation statistics. This simple convection configuration allows investigating this problem in detail without interference from other feedbacks (such as atmospheric radiation) present in the atmospheric boundary layer and clouds. We have modified CM1 to treat sidewall conditions as described below.
A summary of the simulation cases is presented in Table 1. The top and bottom boundaries are kept saturated with water and at 282 and 294 K, respectively. Sidewalls are at the mean temperature of the top and bottom boundaries (288 K) (except for DNSAD). For DNSAD, a no flux boundary condition for scalars is applied at the sides. Different water vapor boundary conditions are applied at all sides in the DNS and LES to investigate the effects of sidewall fluxes on scalar fluctuations, assuming 90%, 95%, and 100% of the saturation vapor mixing ratio. For momentum, the noslip boundary condition is applied to all boundaries for the DNS runs, and for the LES runs, the flux boundary condition is used along all surfaces. The scalar and momentum fluxes for the LES runs are modeled based on a bulk aerodynamic method assuming constant exchange coefficients for momentum and scalars (water vapor and temperature) and depend on the gridscale velocity magnitude near each surface. The simpler formulation of fluxes above is used instead of the Monin–Obukhov similarity to avoid complications at vertical boundaries. The value of these coefficients (same for all surfaces) is tuned to match the bottom surface heat flux (Nusselt number Nu) and turbulent kinetic energy from DNSS95. LES of the DNSS95 case are done to determine the SGS water vapor and temperature variances and their covariance and supersaturation variance using the SGS model introduced in section 2. Apart from the control case at 3.125cm grid spacing (LESCTL), two additional simulations at a higher (LESHI) and a lower grid resolution (LESLO) are performed to evaluate the scaling of SGS fluctuations with the grid spacing. The SGS model is expected to produce higher fluctuations relative to the resolved scale with an increase in the grid spacing. The turbulent statistics are calculated from the DNS and LES runs by spatiotemporal averaging of data over several hundred freefall buoyancy time scale (τ_{f} = 1.5 s). The freefall time scale is a characteristic time scale associated with buoyancy in turbulent convection. It is the ratio of the vertical length scale of the system and the buoyancy velocity (
Summary of all simulation cases presented in this article. Here, 〈η〉 is the average Kolmogorov length scale in the convection core from DNS runs, and
4. DNS of turbulent Rayleigh–Bénard convection with water vapor
DNS results for Rayleigh–Bénard convection with water vapor (the Pi Chamber simulations) are presented in this section. The statistics presented here exclude grid cells near the sidewall boundaries (6 cm on each side) to limit direct wall effects. However, this does not exclude the grid cells within the largescale circulation (LSC). A coherent flow structure, the LSC is an inherent feature of Rayleigh–Bénard convection and forms from the organization of plumes between the top and bottom boundaries (Krishnamurti and Howard 1981). Because of the LSC, scalar fluctuations and turbulent flow properties vary spatially, even away from the viscous layer near walls. Anderson et al. (2021) showed how Eulerian point measurements of water vapor and temperature fluctuations vary depending on the location relative to the LSC. The analysis presented here includes both updraft and downdraft branches of the circulation to avoid biases in statistics. Figure 1 shows a snapshot of isosurfaces of supersaturated (S = 1.001) regions for one of the cases (DNSS95). In this figure, different scales of supersaturated volumes are evident. Although the domain average saturation ratio is less than one for this case, supersaturated regions are present in the domain. For this particular case, coherent structures of supersaturation seem to align with the LSC, which in this case features a convection roll along a diagonal plane.
Pi Chamber measurements of the temperature and water vapor standard deviation (without cloud droplets) at the same ΔT = 12 K as the current simulations reported in Anderson et al. (2021) are 0.18 K (updraft: 0.12 K; mid: 0.19 K; and downdraft: 0.22 K) and 0.07 g kg^{−1} (updraft: 0.05 g kg^{−1}; mid: 0.08 g kg^{−1}; and downdraft: 0.09 g kg^{−1}), respectively. Additionally, the measured value of supersaturation standard deviation is 0.9% (updraft: 0.69%; mid: 0.82%; and downdraft: 1.19%). In our DNS, the magnitudes of the temperature and water vapor standard deviation away from boundaries are 0.21–0.22 K and 0.16–0.18 g kg^{−1}. These values are in good agreement with the measurements in the Pi Chamber, considering significant spatial filtering of fluctuations in the measurements due to a large sample volume of instruments. The supersaturation fluctuations in our DNS vary with the sidewall condition. However, for a realistic sidewall condition (DNSS90), the standard deviation of supersaturation fluctuations is 0.68% (close to measurements). Also, the turbulent kinetic energy at the midheight of the domain is approximately 4 × 10^{−3} m^{2} s^{−2}, nearly matching the particle image velocimetry measurements at the center of the Pi Chamber (Thomas et al. 2019).
Effects of relative fluxes of heat and moisture on turbulence statistics
The sidewall boundary condition is known to influence heat transfer in Rayleigh–Bénard convection, even with the isothermal boundary condition at the mean of the top and bottom temperatures (Ahlers 2000; Stevens et al. 2014). In past DNS and experimental studies, the effects of sidewalls were explored in dry convection where the temperature difference was the only driver. Changes in the sidewall temperature not only affect the heat transfer but also affect the flow. The focus here is on the sidewall flux of water vapor. Although both temperature and water vapor differences drive buoyant convection, in our case the impact of water vapor on the buoyancy is very small (i.e., the buoyant force owing to gradients in water vapor is significantly smaller than that from temperature gradients). Thus, changes in the sidewall saturation have little impact on the flow and provide a different context for examining the effects of sidewall fluxes on scalar statistics in Rayleigh–Bénard convection compared to past studies. Figure 2 shows nondimensional heat (Nusselt number: Nu) and moisture (Sherwood number: Sh) fluxes at the top and bottom boundaries for different sidewall boundary conditions as listed in Table 1. For adiabatic sidewalls, Nu is higher than Sh. This result is also predicted by the scaling relation presented in Chandrakar et al. (2020a). As expected, the top and bottom fluxes of heat and moisture are nearly identical for DNSAD. For other nonadiabatic sidewall conditions, the heat flux at the top boundary is slightly higher than the bottom boundary consistent with a small net heat flux from the side. However, the top and bottom Nu difference is within one standard deviation. At high Ra, as in the current case, the effect of sidewall heat transfer at the mean temperature is expected to be negligible (Stevens et al. 2014) consistent with the current case. However, the difference between the top and bottom moisture fluxes is significant and increases with a decrease in the sidewall saturation level. The presence of saturated or subsaturated sidewalls “short circuits” the moisture fluxes at the top and bottom boundaries. The deviation of sidewall water vapor concentration from the mean of the top and bottom boundaries enhances the local water vapor gradient. It drives more sidewall flux near the bottom boundary and subsequently reduces the water vapor gradient at the top boundary, a “short circuit” of water vapor flux. Thus, the amount of moisture flux at the bottom boundary increases with decreasing the sidewall saturation level, but the top flux decreases nearly identically compared to DNSAD.
Nusselt (squares) and Sherwood (circles) numbers at the top (black) and bottom (red) boundaries for different DNS cases as listed in Table 1. The error bars show the standard deviation of the flux data (the standard errors are approximately the size of the symbols). The sign of the top fluxes is negative but they are displayed as positive nondimensional values.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Nusselt (squares) and Sherwood (circles) numbers at the top (black) and bottom (red) boundaries for different DNS cases as listed in Table 1. The error bars show the standard deviation of the flux data (the standard errors are approximately the size of the symbols). The sign of the top fluxes is negative but they are displayed as positive nondimensional values.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Nusselt (squares) and Sherwood (circles) numbers at the top (black) and bottom (red) boundaries for different DNS cases as listed in Table 1. The error bars show the standard deviation of the flux data (the standard errors are approximately the size of the symbols). The sign of the top fluxes is negative but they are displayed as positive nondimensional values.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Figure 3 shows the mean temperature and water vapor mixing ratio profiles for the various sidewall conditions. The shape of the scalar profiles is very similar to past studies of moist Rayleigh–Bénard convection and Rayleigh–Bénard convection generally (e.g., Chandrakar et al. 2020a; Ahlers et al. 2012, and references therein). The temperature profile is nearly identical for all cases consistent with the heat flux data shown in Fig. 2. The water vapor mixing ratio profiles have different mean values in the core region that decrease with a reduction of the sidewall saturation, similar to the LES study of Thomas et al. (2019). In the viscous layer near the boundaries, the slope of mean water vapor profiles also changes with the sidewall saturation. A steeper slope near the bottom boundary and shallower slope near the top for the lowest sidewall saturation is also consistent with the flux data presented in Fig. 2.
Vertical profiles of mean temperature and water vapor mixing ratio for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of mean temperature and water vapor mixing ratio for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of mean temperature and water vapor mixing ratio for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Figure 4 shows vertical profiles of water vapor and potential temperature fluctuations, their covariance, and supersaturation fluctuations. The temperature and water vapor fluctuations peak near the top and bottom boundaries and decrease toward the center, similar to past studies of dry Rayleigh–Bénard convection and Rayleigh–Bénard convection with water vapor (e.g., Ahlers et al. 2012; Chandrakar et al. 2020a, and references therein). They are also consistent with the scaling presented in Chandrakar et al. (2020a) to explain the profile of scalar fluctuations outside the viscous layer. The temperature fluctuation profiles are nearly vertically symmetric and overlap for all cases. The water vapor fluctuations are also symmetric for DNSAD. However, the magnitude of water vapor fluctuations is higher near the bottom boundary for other sidewall conditions. This vertical asymmetry also increases with a decrease in the sidewall saturation level; fluctuations near the bottom boundary increase and near the top boundary decrease with a reduction in the sidewall saturation. In turbulent convection, scalar variance is directly proportional to net scalar flux. The variance of a scalar field obeys a scaling,
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for the various DNS cases.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
In convection with saturated top and bottom boundaries, as is the case here, the normalized covariance is expected to be close to unity. However, the different diffusivities of water vapor and heat decorrelate fluctuations in the water vapor and temperature fields near boundaries. Still, the mean normalized covariance increases away from the boundaries due to greater mixing from turbulence (Chandrakar et al. 2020a). A similar profile of the normalized covariance is observed in the current study for DNSAD (it is not clearly visible in Fig. 4c since its value is close to unity). As mentioned earlier, natural convection, whether in the atmospheric boundary layer/clouds or the Pi Chamber, is not like this idealized scenario since relative fluxes vary for different scalars. As can be seen in Fig. 4c, the normalized covariance value decreases with a decrease in the sidewall saturation level, even in the regions away from all boundaries. For all the cases with sidewall fluxes, the covariance decreases away from the top and boundaries. It is relatively flat in the center region, 20 cm away from both boundaries. The decrease in the sidewall saturation causes an increase in water vapor flux relative to the heat flux. This relative flux from the side decreases the covariance away from the top and bottom boundaries. At the top and bottom boundaries, heat and water vapor fluxes are highly correlated, producing a normalized covariance close to unity (slight deviation due to relative diffusivities of both scalars).
Supersaturation fluctuations strongly depend on the water vapor and temperature covariance (Kulmala et al. 1997; Siebert and Shaw 2017; Chandrakar et al. 2020a; Thomas et al. 2021), which is seen here in Fig. 4d. Supersaturation fluctuations increase with a decrease in the sidewall saturation level consistent with the covariance profiles and Eq. (2). Contributions from differences in the water vapor and temperature variances are relatively small since they do not change significantly for the various sidewall conditions. Nonetheless, the increase in the asymmetry of water vapor fluctuations with a decrease in the sidewall saturation also affects the asymmetry of the supersaturation fluctuation profiles.
Effects of sidewall fluxes are further investigated by analyzing the scalar fluctuation statistics similar to Fig. 4 but in a horizontal direction (Fig. 5). For DNSAD, the horizontal profiles of temperature and water vapor standard deviation are the largest near side boundaries and decrease toward the center because the LSC carries plumes from the top and bottom boundaries that are not well mixed along sidewalls. Moreover, the noslip boundary condition for velocity causes variation in turbulent mixing normal to sidewalls. The presence of nonadiabatic sidewalls causes nonzero heat and water vapor fluxes depending on the updraft and downdraft branches of the LSC. On the updraft side, a higher temperature and water vapor induce negative fluxes of both scalars. However, colder and less moist fluid from the top boundary drives positive fluxes on the downdraft side. These fluxes cause an immediate decrease of the fluctuations right at the sidewall boundaries with a sharp increase nearby and a peak within 5 cm of the boundaries. The fluctuation magnitudes then decrease from these peak values near the boundaries toward the center due to greater mixing at the center. Since the sidewall temperature is at the mean of the top and bottom boundaries and water vapor is not, the magnitude of the heat flux is the same on both sides but not the water vapor flux. This asymmetry produces larger fluctuations of water vapor on one side than the other. The peak water vapor fluctuations on the right side also increase with a decrease in the sidewall saturation.
Profiles of turbulence statistics as in Fig. 4 but in the Y direction. The statistics presented here exclude grid cells near the boundaries (6 cm on each side) to limit direct wall effects.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Profiles of turbulence statistics as in Fig. 4 but in the Y direction. The statistics presented here exclude grid cells near the boundaries (6 cm on each side) to limit direct wall effects.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Profiles of turbulence statistics as in Fig. 4 but in the Y direction. The statistics presented here exclude grid cells near the boundaries (6 cm on each side) to limit direct wall effects.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Apart from the difference in sidewall heat and water vapor fluxes due to the LSC, the sidewall induces a water vapor flux but not heat flux for a wellmixed fluid since the sidewalls are at the mean temperature. This flux of water vapor relative to heat decreases the correlation of water vapor and temperature along the sidewalls. The correlation increases toward the center due to better mixing in the convection core (away from the boundaries). As explained above, the covariance profile is also asymmetric, more for DNSS90, due to the asymmetry in the relative flux. In the convection core, covariance decreases with sidewall saturation due to the mixing of fluid from the sidewall boundaries similar to entrainment and mixing of dry environmental air inside atmospheric clouds. The standard deviation of supersaturation fluctuations has a relatively flat horizontal profile for DNSAD even though water vapor and temperature fluctuations peak near the side boundaries. This is due to a higher value of the covariance and a lower value of the saturation ratio near the side boundaries (not shown). For other sidewall cases, supersaturation fluctuations peak near the sidewalls similar to water vapor and temperature fluctuations. In the cases with nonadiabatic sidewalls, a sharp decrease in the covariance near the boundaries causes supersaturation fluctuations to increase. Similar to the vertical profiles, the magnitude of supersaturation fluctuations in the convection core increases with a decrease in the sidewall saturation. The effects of sidewall fluxes on scalar fluctuations and covariance are analogous to the impact one would expect in the presence of entrainment and mixing of dry environmental air into clouds.
5. Evaluation of the subgridscale model: LES of Rayleigh–Bénard convection
The SGS model introduced in section 2 was implemented in CM1 for the LES configuration and evaluated here by comparing the scalar fluctuation statistics with the reference DNS case (DNSS95). To evaluate the performance of an SGS scheme, traditionally, there are two methods: “a posteriori” and “a priori” tests (Meneveau and Katz 2000). In a posteriori test, the benchmark DNS or measurement data are compared with the model output. We use the a posteriori method to evaluate the performance of the LESSGS scheme proposed here by comparing net scalar statistics with corresponding reference DNS output. The a priori test compares subfiltered components of DNS data with modeled SGS components at the length scale of LES. Although the a priori test is left here for future investigation, it may shed some light on regimes where the proposed SGS model may not perform well. In the spirit of the a priori method, we compared filtered and subfiltered parts^{2} of the scalar statistics with the LES resolved and SGS output statistics at different grid spacing. It is not precisely an a priori test since the model output is compared instead of the modeled SGS terms before running the model. However, it can further elucidate the SGS model performance compared to the a posteriori test. For this comparison, the DNS output is filtered with a spatial box filter using different filter widths of the size of LES grid spacings (10, 16, and 25 DNS grid points).
As mentioned in section 3, only the average heat flux at the bottom surface and turbulent kinetic energy are matched with the DNS data to determine the exchange coefficient for scalars (heat and moisture) and momentum. Therefore, the water vapor flux at the bottom boundary in LES is slightly (less than 5%) different than DNSS95. We choose the bottom heat flux instead of the water vapor flux to tune the scalar exchange coefficient since it is the main driver of the convection in the current case. The mean profiles of water vapor and temperature from LESCTL matches well with the DNSS95 mean profiles (see Fig. 6). A minor difference in the mean water vapor profile slope is related to the differences in the water vapor flux. Figures 7 and 8 show the water vapor and temperature standard deviations, their covariance, and supersaturation standard deviation profiles for the resolved and SGS scales and their total magnitude. The total magnitudes of the water vapor and temperature fluctuations follow the DNS data very well, even near the boundaries. Here, the total magnitude of fluctuations is defined as the square root of the sum of SGS and resolvedscale variance (or as the sum of the covariance contributions for the covariance profile). Although the numerical dissipation due to the discretization on the coarse LES grid is expected to reduce scalar fluctuations, the LES fluctuation magnitudes of individual scalars are slightly higher than DNS. Additionally, the normalized covariance is slightly smaller in LES. These results are most likely due to a somewhat higher value of the bottom water vapor flux in the LES. The scalar and momentum exchange coefficients are tuned based on the bottom boundary. Thus, sidewall fluxes could be slightly different from DNS, which could also cause deviation of the LES result from DNS. Moreover, the numerical Prandtl and Schmidt numbers are the same in LES. This could also affect the scalar covariance and individual scalar fluctuations.
Vertical profiles of (a) mean temperature and (b) water vapor mixing ratio for LESCTL compared with DNSS95 results.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) mean temperature and (b) water vapor mixing ratio for LESCTL compared with DNSS95 results.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) mean temperature and (b) water vapor mixing ratio for LESCTL compared with DNSS95 results.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for LESCTL. Profiles from DNSS95 are also plotted as dashed lines for comparison. The net fluctuation magnitude is calculated by taking the square root of the sum of resolved and SGS variances
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for LESCTL. Profiles from DNSS95 are also plotted as dashed lines for comparison. The net fluctuation magnitude is calculated by taking the square root of the sum of resolved and SGS variances
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Vertical profiles of (a) standard deviation of water vapor mixing ratio fluctuations, (b) standard deviation of potential temperature fluctuations, (c) covariance between potential temperature and water vapor mixing ratio fluctuations, and (d) standard deviation of supersaturation fluctuations for LESCTL. Profiles from DNSS95 are also plotted as dashed lines for comparison. The net fluctuation magnitude is calculated by taking the square root of the sum of resolved and SGS variances
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
Figure 8 compares the resolved and SGS scalar statistics from LES to filtered and subfiltered DNS statistics. The LESSGS water vapor and temperature statistics very well capture the subfilteredscale DNS statistics. They are slightly smaller than DNS output near the boundaries, and the covariance value is somewhat larger in the convection core. The SGS supersaturation standard deviation is also moderately larger for LESSGS, especially near the boundaries. Similarly, the resolvedscale LES statistics of water vapor and temperature are close to the DNS filteredscale statistics (slight deviation near the convection core).
As in Fig. 7, but showing the resolved (solid blue) and SGS (solid red) LES scalar statistics and the filtered (blue dashed) and subfiltered (red dashed) DNS statistics. The DNS data are filtered with a spatial box filter using a filter width of 16 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but showing the resolved (solid blue) and SGS (solid red) LES scalar statistics and the filtered (blue dashed) and subfiltered (red dashed) DNS statistics. The DNS data are filtered with a spatial box filter using a filter width of 16 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but showing the resolved (solid blue) and SGS (solid red) LES scalar statistics and the filtered (blue dashed) and subfiltered (red dashed) DNS statistics. The DNS data are filtered with a spatial box filter using a filter width of 16 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
The water vapor standard deviation from the LES is also able to capture the vertical asymmetry in the DNS profile as explained in the previous section. SGS scalar fluctuations near the boundaries are larger than the resolved fluctuations. However, SGS fluctuations are significantly smaller in the convection core than the resolved fluctuations, especially for temperature and water vapor. The total normalized covariance from the SGS model also captures the DNS profile reasonably well quantitatively. The SGS covariance profile decreases away from the boundaries and is nearly flat in the convection core, similar to the DNS data. Thus, it can simulate the decrease in covariance due to relative sidewall fluxes of scalars. The supersaturation standard deviation calculated from the scalar variances and covariance at resolved and SGS scales [Eq. (2)] is also able to quantitatively well match the supersaturation standard deviation from the DNS. However, the LES results do not fully capture the asymmetry in the supersaturation standard deviation profile. The most likely reason for this dissimilarity is the use of the same diffusivity and exchange coefficient for both water vapor and temperature in the LES. A slight deviation of the mean water vapor profile compared to the DNS run is also evident in Fig. 6 because of the use of the same exchange coefficient for water vapor and temperature. Similar to the water vapor and temperature standard deviation profiles, the magnitude of SGS supersaturation is also larger than the resolvedscale fluctuations near the boundaries. In the convection core, SGS fluctuations in supersaturation are smaller than the resolvedscale fluctuations, but the difference is less than the differences in the temperature and water vapor standard deviations. This smaller difference is a result of a smaller SGS covariance magnitude [see Eq. (2)].
The sensitivity of the LESSGS results to the model grid spacing is also tested here by performing two additional simulations with a smaller (2 cm in LESHI) and larger (5 cm in LESLO) grid spacing. LESHI and LESLO both show a good agreement with the DNS results similar to LESCTL (Figs. 9–12). The resolved and SGS LES scalar statistics consistently match filtered and subfiltered DNS statistics across a wide range of filter scales. However, the resolved supersaturation standard deviation profile varies more from the filtered DNS magnitude at a lower resolution, likely due to deviation in sidewall fluxes. SGS fluctuations in water vapor, temperature, and supersaturation relative to resolvedscale fluctuations for LESLO are larger than in the LESHI and LESCTL cases as expected. The change in relative SGS fluctuations with a change in grid spacing is more apparent for supersaturation than the individual scalars. For supersaturation fluctuations, the smallscale mixing of scalars is important. Thus, it scales more with the grid spacing. The normalized covariance of SGS temperature and water vapor fluctuations in the convection core is also smaller for LESLO than LESHI. In the LES here, SGS water vapor and temperature fluctuations are small compared to those at the resolved scale in the current Pi Chamber simulations owing to a small inertial range that is limited by the small length scale of the domain. The situation would be different in cloudscale simulations with typical LES grid spacing due to a greater range of length scales involved. Overall, these results suggest that the SGS supersaturation scheme is appropriately scaling across different LES grid spacings. These grid sensitivity test results build confidence in the SGS supersaturation model introduced here.
As in Fig. 7, but for LESHI.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but for LESHI.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but for LESHI.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESHI. The DNS data are filtered using a spatial box filter with a filter width of 10 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESHI. The DNS data are filtered using a spatial box filter with a filter width of 10 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESHI. The DNS data are filtered using a spatial box filter with a filter width of 10 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but for LESLO.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but for LESLO.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 7, but for LESLO.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESLO. The DNS data are filtered with a spatial box filter using a filter width of 25 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESLO. The DNS data are filtered with a spatial box filter using a filter width of 25 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
As in Fig. 8, but for LESLO. The DNS data are filtered with a spatial box filter using a filter width of 25 DNS grid cells.
Citation: Journal of the Atmospheric Sciences 79, 4; 10.1175/JASD210250.1
6. Implications for applying the SGS supersaturation model to bulk and Lagrangian microphysics schemes
The SGS supersaturation variance obtained from the model described in section 2 can be applied to parameterize the activation of cloud droplets driven by turbulent fluctuations (e.g., Prabhakaran et al. 2020; Shawon et al. 2021) with an assumption about the shape of the joint PDF of supersaturation fluctuations and aerosols. Additionally, bulk microphysics schemes commonly assume a constant shape parameter (or spectral width) of the droplet spectrum. The current SGS scheme can be used in bulk schemes to solve the prognostic equation of droplet spectral width [see Eq. (16) in Chandrakar et al. 2018] that in turn can be used to calculate the effective radius and microphysical process rates (e.g., autoconversion). Another application of the new SGS model is in bin microphysics schemes. It can be applied for modeling spectral broadening due to turbulence as a diffusion term (Saito et al. 2019). In this case, the diffusivity
It is straightforward to apply the SGS supersaturation scheme in Lagrangian particlebased models using Lagrangian stochastic differential equations. There are two different approaches to do that. The first is to use separate equations for SGS water vapor and temperature along individual particle trajectories with the assumption of Gaussian Wiener forcing and neglecting any deviation from the Lagrangian fluid trajectories (Paoli and Shariff 2009). In that case, a part of the Wiener forcing terms for both the scalars would be correlated depending on the correlation coefficient between the two
7. Discussion and conclusions
Processes involving multiscalar interactions in turbulent convection are interesting from the standpoint of fundamental fluid dynamics and have several atmospheric science applications (e.g., transport of reactive species, wave propagation, cloud formation and growth). The focus here is on multiscalar interactions governing the supersaturation—a nonlinear function of water vapor and temperature. Supersaturation fluctuations play a critical role in cloud particle activation and growth in turbulent moist convection. This article introduces a new SGS model for supersaturation fluctuations based on scalar variance and covariance. Conservation equations for prognostic water vapor and temperature variances and their covariance based on Deardorff (1974a) were implemented in CM1 for this purpose. The new SGS model was tested in LES and compared to benchmark DNS of turbulent Rayleigh–Bénard convection with water vapor (simulations of the Pi Cloud Chamber). This setup, without cloud droplets, provided an ideal framework for testing the SGS model without complications from external feedbacks (such as atmospheric radiation). DNS results were found to compare favorably with the Pi Chamber measurements reported in Anderson et al. (2021) and Thomas et al. (2019) for similar conditions. DNS was also conducted with different sidewall conditions to investigate the influence of external fluxes on turbulent statistics for the twoscalar Rayleigh–Bénard convection. A followup study will investigate the role of cloud droplet growth.
Comparison of the DNS results with measurements in the Pi Chamber at similar conditions (e.g., Anderson et al. 2021; Thomas et al. 2019) showed a good agreement. There are some limitations of the scalar and turbulence statistics measurements in the Pi Chamber, e.g., a large sample volume of instruments, Eulerian point measurements. There are also challenges in obtaining the supersaturation mean and fluctuations from measurements because collocated temperature and water vapor measurements are required. The current DNS overcomes those limitations and provides more detailed information about scalar and turbulence statistics to complement the measurements.
Comparison of the DNS of Pi Chamber with adiabatic sidewalls to DNS with other sidewalls conditions showed that the net sidewall heat flux is not significant if the sidewalls are at the mean temperature of the top and bottom boundaries. The water vapor flux increased at the bottom and decreased at the top boundary as the sidewall saturation was decreased, influencing the profiles of the mean and standard deviation of water vapor. The presence of relative scalar fluxes from the side boundaries decorrelated scalars, and this was not just limited to the layer near the boundaries but also affected the convection core. The decrease in covariance between water vapor and temperature fluctuations increased the supersaturation fluctuations. An analogy can be made between the production of supersaturation fluctuations in the current simplified setting and entrainment mixing in atmospheric clouds. In both cases, external water vapor and thermal fluxes are present, and will affect supersaturation fluctuations in a similar way. However, in atmospheric clouds, condensation/evaporation and dynamical feedbacks make this process more complicated.
The new SGS model is a significant advancement over existing schemes since it captures supersaturation fluctuations generated from scalar mixing apart from vertical velocity fluctuations. By treating water vapor and temperature fluctuations separately, it accounts for SGS supersaturation variability during turbulent mixing. It can be used with bulk, bin, or Lagrangian microphysics schemes to represent cloud–turbulence interactions at subgrid scales. Additionally, the model is compatible with higherorder turbulence closure schemes since it solves prognostic equations for scalar variances and covariance. Comparison of the Pi Chamber LES incorporating the new SGS model with the benchmark DNS showed a close match of profiles of different scalar statistics. The SGS model well reproduced profiles of water vapor and temperature variance and their covariance. It also captured asymmetry in the vertical profile of
The SGS supersaturation variability in the Pi Chamber case is driven by scalar mixing. However, vertical velocity fluctuations are also an important contributor to supersaturation variability for atmospheric convection in a stratified environment, and the new SGS model also accounts for this effect. The diabatic contributions of condensation/evaporation to the scalar variance and covariance are also zero in the current simulations. These terms will reduce the magnitude of supersaturation or subsaturation fluctuations in simulations with cloud. In our SGS model, the effect of condensation can be introduced as an additional sink term (−s′/τ_{c}), and it would reduce the supersaturation variance at the rate of
Overall, the SGS model presented here showed promising results compared to benchmark DNS in the current idealized setting of turbulent Rayleigh–Bénard convection. Further investigation of the SGS model and terms associated with condensation in cloudy conditions is a subject of future work. Application of the model to realistic atmospheric cloud conditions will also be a part of the subsequent studies.
Acknowledgments.
This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977. This work was also supported in part by U.S. Department of Energy Atmospheric System Research Grant DESC0020118. KKC was supported by the NCAR ASP postdoctoral fellowship program. We would like to acknowledge highperformance computing support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. RAS acknowledges support from NSF Grant AGS2133229. We also thank three anonymous reviewers for their constructive comments and suggestions that improved the paper.
Data availability statement.
The current study used CM1 (https://www2.mmm.ucar.edu/people/bryan/cm1/) for the simulations presented here. The model outputs used in this study are too large to archive. Instead, the input and namelist files for the simulation runs will be made available upon request to the first author.
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The gridscale magnitude is represented as
If 〈f〉 represents a filtered scalar data, the subfiltered scalar field is defined as f − 〈f〉.