Impact of Entrainment Mixing and Turbulent Fluctuations on Droplet Size Distributions in a Cumulus Cloud: An Investigation Using Lagrangian Microphysics with a Subgrid-Scale Model

Kamal Kant Chandrakar aNational Center for Atmospheric Research, Boulder, Colorado

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Wojciech W. Grabowski aNational Center for Atmospheric Research, Boulder, Colorado

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Hugh Morrison aNational Center for Atmospheric Research, Boulder, Colorado

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George H. Bryan aNational Center for Atmospheric Research, Boulder, Colorado

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Abstract

Entrainment mixing and turbulent fluctuations critically impact cloud droplet size distributions (DSDs) in cumulus clouds. This problem is investigated via a new sophisticated modeling framework using the Cloud Model 1 (CM1) LES model and a Lagrangian cloud microphysics scheme—the “superdroplet method” (SDM)—coupled with subgrid-scale (SGS) schemes for particle transport and supersaturation fluctuations. This modeling framework is used to simulate a cumulus congestus cloud. Average DSDs in different cloud regions show broadening from entrainment and secondary cloud droplet activation (activation above the cloud base). DSD width increases with increasing entrainment-induced dilution as expected from past work, except in the most diluted cloud regions. The new modeling framework with SGS transport and supersaturation fluctuations allows a more sophisticated treatment of secondary activation compared to previous studies. In these simulations, it contributes about 25% of the cloud droplet population and impacts DSDs in two contrasting ways: narrowing in extremely diluted regions and broadening in relatively less diluted. SGS supersaturation fluctuations contribute significantly to an increase in DSD width via condensation growth and evaporation. Mixing of superdroplets from SGS velocity fluctuations also broadens DSDs. The relative dispersion (ratio of DSD dispersion and mean radius) negatively correlates with gridscale vertical velocity in updrafts but is positively correlated in downdrafts. The latter is from droplet activation driven by positive SGS supersaturation fluctuations in grid-mean subsaturated conditions. Finally, the sensitivity to model grid length is evaluated. The SGS schemes have greater influence as the grid length is increased, and they partially compensate for the reduced model resolution.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kamal Kant Chandrakar, kkchandr@ucar.edu

Abstract

Entrainment mixing and turbulent fluctuations critically impact cloud droplet size distributions (DSDs) in cumulus clouds. This problem is investigated via a new sophisticated modeling framework using the Cloud Model 1 (CM1) LES model and a Lagrangian cloud microphysics scheme—the “superdroplet method” (SDM)—coupled with subgrid-scale (SGS) schemes for particle transport and supersaturation fluctuations. This modeling framework is used to simulate a cumulus congestus cloud. Average DSDs in different cloud regions show broadening from entrainment and secondary cloud droplet activation (activation above the cloud base). DSD width increases with increasing entrainment-induced dilution as expected from past work, except in the most diluted cloud regions. The new modeling framework with SGS transport and supersaturation fluctuations allows a more sophisticated treatment of secondary activation compared to previous studies. In these simulations, it contributes about 25% of the cloud droplet population and impacts DSDs in two contrasting ways: narrowing in extremely diluted regions and broadening in relatively less diluted. SGS supersaturation fluctuations contribute significantly to an increase in DSD width via condensation growth and evaporation. Mixing of superdroplets from SGS velocity fluctuations also broadens DSDs. The relative dispersion (ratio of DSD dispersion and mean radius) negatively correlates with gridscale vertical velocity in updrafts but is positively correlated in downdrafts. The latter is from droplet activation driven by positive SGS supersaturation fluctuations in grid-mean subsaturated conditions. Finally, the sensitivity to model grid length is evaluated. The SGS schemes have greater influence as the grid length is increased, and they partially compensate for the reduced model resolution.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kamal Kant Chandrakar, kkchandr@ucar.edu

1. Introduction

The spectral width of cloud droplet size distributions (DSDs) is a critical aspect for processes like precipitation formation and interaction with solar radiation. Several past studies have confirmed that DSDs in clouds are much broader than what is expected from adiabatic condensation growth alone (e.g., Warner 1969; Lasher-Trapp et al. 2005; Brenguier et al. 2011; Schmeissner et al. 2015). The missing ingredients for broad distributions are the processes associated with turbulent flow (Manton 1979; Cooper 1989; Lasher-Trapp et al. 2005; Chandrakar et al. 2016; Grabowski and Abade 2017; Hoffmann et al. 2017), aerosol effects (giant CCN and Ostwald ripening) (Korolev 1995; Wood et al. 2002; Jensen and Nugent 2017; Yang et al. 2018), and the turbulence-enhanced collision–coalescence of droplets (Shaw 2003; Grabowski and Wang 2013; Devenish et al. 2012; Hoffmann et al. 2017). This article focuses on the processes governing DSD broadening before the onset of collision–coalescence.

Since Warner (1969) observed DSD broadening with altitude and the presence of bimodal distributions in cumulus clouds, different theories have evolved to explain the observed broadening. A central theme that emerges from these studies is the influence of turbulent entrainment and mixing on droplet sizes (e.g., Latham and Reed 1977; Baker et al. 1980; Blyth 1993; Brenguier and Grabowski 1993; Lasher-Trapp et al. 2005; Burnet and Brenguier 2007; Lehmann et al. 2009; Hoffmann et al. 2019). Turbulent flow influences cloud DSDs through interactions at various scales. At larger scales, turbulent eddies entrain surrounding clear air into clouds and cause dilution and evaporation of cloud droplets. Subsequent turbulent mixing of entrained air intrusions produces smaller filaments and scalar fluctuations at various scales. Cloud droplets interact with these fluctuations and grow or evaporate depending on the supersaturation/subsaturation they experience locally. Several other studies have shown that small-scale heterogeneity, including variability in thermodynamic and flow properties and mixing of droplets from different trajectories, promotes the formation of broad DSDs (e.g., Manton 1979; Telford et al. 1984; Cooper 1989; Lasher-Trapp et al. 2005; Cooper et al. 2013).

Several past studies used simplistic modeling frameworks to explore the problem of entrainment mixing in cumulus clouds, for example, the explicit mixing parcel model (EMPM) (Su et al. 1998; Tölle and Krueger 2014), a two-dimensional cloud-resolving model with bulk microphysics (Brenguier and Grabowski 1993), or a three-dimensional model with Lagrangian microphysics that neglected feedback between Lagrangian microphysics and the flow (Lasher-Trapp et al. 2005; Cooper et al. 2013). While these studies led to important advances in understanding DSD evolution from entrainment mixing, their frameworks were limited in representing microphysical interactions and feedback with the flow. Due to this limitation, they had to prescribe mixing type1 (homogeneous, inhomogeneous, or their combination) a priori in models instead of driven by the dynamics.

The recent development of Lagrangian particle-based microphysics schemes with detailed coupling and feedback between the dynamics and microphysics has made it feasible to study microphysical interactions in a more complete framework (e.g., Andrejczuk et al. 2008; Shima et al. 2009; Abade et al. 2018; Hoffmann et al. 2019). A Lagrangian microphysics model called the “superdroplet method” (SDM; Shima et al. 2009, 2020) is used in the current study. In the SDM, statistically representative particles (superdroplets) are transported and feed back with the model-predicted flow along their Lagrangian trajectories. Compared to its Eulerian counterpart that also resolves the DSD—spectral or bin microphysics—SDM has several advantages (Grabowski et al. 2019). In particular, it is free from the numerical diffusion that is present in spectral schemes due to finite size (or mass) discretizations (Morrison et al. 2018). It also allows fairly straightforward approaches for coupling microphysics with subgrid-scale (SGS) dynamics (Abade et al. 2018; Grabowski and Abade 2017; Hoffmann et al. 2019). Studies have indicated such interactions are critical for an accurate representation of cloud microphysical processes. For example, small-scale turbulent fluctuations have been shown to impact droplet activation and DSD broadening based on theory (e.g., Cooper 1989; Paoli and Shariff 2009; Chandrakar et al. 2016), laboratory experiments (e.g., Chandrakar et al. 2016; Prabhakaran et al. 2020), direct numerical simulations (e.g., Paoli and Shariff 2009; Sardina et al. 2015), and cloud models with SGS parameterizations (e.g., Grabowski and Abade 2017; Abade et al. 2018; Hoffmann et al. 2019). Small-scale fluctuations of scalar quantities (water vapor and temperature) driven by turbulent fluxes (e.g., entrainment mixing and vertical velocity fluctuations) produce SGS fluctuations in supersaturation (Politovich and Cooper 1988; Siebert and Shaw 2017; Chandrakar et al. 2020). These fluctuations in supersaturation in turn affect droplet activation and condensation growth and can be important sources of DSD broadening. Additionally, small-scale velocity fluctuations influence transport and mixing of particles in turbulent flow (Thomson 1987; Weil et al. 2004; Cooper 1989) that can also affect DSD width.

Typical LES grid sizes of O(10) m are insufficient to resolve cloud variability at meter and smaller scales. However, interactions of turbulent fluctuations with cloud droplets at these scales can be modeled using SGS Lagrangian (Grabowski and Abade 2017; Sardina et al. 2018; Chandrakar et al. 2018) or Eulerian (only mixing between the particles) (Hoffmann et al. 2019) stochastic approaches. For example, Hoffmann et al. (2019) used the linear-eddy model (Kerstein 1988) with Lagrangian microphysics to parameterize the effects of localized mixing on supersaturation fluctuations and droplet condensation. They also investigated the role of inhomogeneous mixing in the formation of precipitation embryos. In their approach, resolution of the linear-eddy model depends on the number of superdroplets per grid box. Moreover, a fixed cloud droplet concentration was used without explicitly solving for the droplet activation process. In the current study, we build on the work of Hoffmann et al. (2019) using a different SGS scheme that solves a Lagrangian stochastic differential equation similar to Abade et al. (2018) and Chandrakar et al. (2018). In this approach, the magnitude of supersaturation fluctuations is determined from SGS vertical velocity fluctuations and droplet activation and deactivation are explicitly calculated and coupled to these SGS fluctuations.

Broadly, the problem of entrainment mixing and its influence on cloud microphysical processes in a cumulus-congestus cloud is the focus of this study. A few previous studies have investigated this problem using Lagrangian microphysics. Lasher-Trapp et al. (2005) showed that variations in droplet trajectories and growth histories contribute to broadening of DSDs using Lagrangian parcel calculations that excluded feedback to the dynamics and neglected SGS supersaturation fluctuations. Does our model with fully coupled Lagrangian microphysics and dynamics as well as SGS schemes show a similar effect, and what can be learned using this more detailed approach? Naumann and Seifert (2016) and Hoffmann et al. (2017) also used a coupled Lagrangian microphysics–dynamics approach, but without SGS schemes to track droplet growth histories and with a focus on rain drop formation. Hoffmann et al. (2015) used Lagrangian microphysics to study secondary droplet activation, but without explicit coupling to a SGS scheme. Besides differences in the modeling framework, particularly the SGS schemes in our study, compared to Hoffmann et al. (2015, 2017) we also use a different thermodynamic sounding and more realistic surface forcing to initiate convection rather than warm bubbles. Similar to observational studies (e.g., Martin et al. 1994; Brenguier et al. 2011; Lu et al. 2012), our simulations can serve as a benchmark for parameterizing spectral width and other DSD moments relevant to cloud optical properties and the autoconversion process in simpler microphysics schemes. Relationships between some of these moments with vertical velocity and cloud dilution are also investigated here.

Here a setup motivated from the Small Cumulus Microphysics Study (SCMS) field experiment, and based on Lasher-Trapp et al. (2005) and the International Cloud Modeling Workshop (2021) (Shima and Grabowski 2020), is used to model an isolated cumulus congestus cloud. The superdroplet method is implemented in the Cloud Model 1 (CM1; Bryan and Fritsch 2002) to investigate this problem. As mentioned earlier, SGS interactions of superdroplets with the turbulent flow fields are modeled using a Lagrangian stochastic approach. The primary purpose of this study is to investigate the influence of turbulent fluctuations and the entrainment-mixing process on DSDs and their moments using a sophisticated modeling framework with a more complete coupling of microphysics–dynamics across scales (particularly subgrid scales) compared to previous studies.

Specifically, we address the following science questions:

  • Where does secondary activation occur, and in what locations does it most impact the DSD width?

  • What is the relation between relative dispersion and vertical velocity, and how does the k parameter (a function of the second and third DSD moments) used in effective radius parameterization vary with cloud dilution through entrainment?

  • Does an improved representation of the SGS supersaturation and velocity fluctuations produce notable impacts on the microphysical properties of the cloud, specifically droplet number concentration and DSD width?

  • What is the sensitivity of results to the dynamical model grid resolution coupled with SDM, and can the SGS supersaturation and transport schemes used in the current framework compensate for a lower grid resolution?

2. Modeling framework and simulation configuration

a. Model setup

The CM1 dynamical core developed by Bryan and Fritsch (2002) is used for the simulations presented here. It is a three-dimensional nonhydrostatic model solving a nonlinear conservation equation set to simulate idealized atmospheric phenomena. In our study, a large-eddy-simulation (LES) configuration of CM1 with a prognostic turbulent kinetic energy (TKE) SGS mixing scheme is employed. A set of compressible governing equations is solved with Klemp–Wilhelmson time splitting, and a three-step Runge–Kutta time integration with a fifth-order advection scheme is used. Additionally, Rayleigh damping is applied at the topmost 1 km of the domain.

A Lagrangian microphysics scheme (SDM) that explicitly solves for the transport and growth of statistically representative particles (superdroplets) is used here to investigate the role of entrainment mixing on microphysical processes. The superdroplet method developed by Shima et al. (2009) is implemented in CM1 with some modifications. The number of superdroplets (SDs) at each grid location is initially distributed based on the local air density in each grid, making the aerosol (and superdroplet) mixing ratio uniform. This allows a more uniform distribution of mixing ratio of SDs at later stages of transport compared to, for example, applying initially uniform or random numbers of SDs in grid cells. Additionally, the initial position of each SD within a grid box is random. Velocities averaged over the acoustic substeps from CM1 are interpolated linearly to the location of each SD and used to drive the transport of SDs. An interpolation scheme similar to Arabas et al. (2015) and Grabowski et al. (2018) that preserves divergence at subgrid scales is used for this purpose. The superdroplets’ equations of motion are solved using a predictor–corrector method (Grabowski et al. 2018) with horizontally periodic and vertically reflective boundary conditions. The transport equations neglect the drag and inertial terms (i.e., following the fluid flow as a passive tracer). The Stokes number of droplets in the simulations presented here is less than 0.1. Thus, the inertial and drag contributions (except gravitational settling) can be neglected. Since droplet sizes are small, we also neglected the gravitational settling for simplicity. Additional simulations with sedimentation turned on (but neglecting the effect of sedimentation on SGS fluctuations) give similar results, within the spread of different flow realizations (not shown). As in the host CM1 dynamical core, a distributed memory parallelization scheme is used for the SDM code via Message Passing Interface (MPI). Superdroplets and their location, multiplicity,2 and attributes3 are communicated horizontally to their neighboring computational subdomains after each transport step if they cross the subdomain (MPI tile) boundaries.

Activation and condensation of SDs are driven by the thermodynamic variables (water vapor mixing ratio, potential temperature, and air density) from CM1 at scalar grid points. For simplicity the scalar variables are not interpolated to the location of SDs. These thermodynamic variables are updated after each condensation substep. An implicit Euler scheme with the Newton–Raphson method is used to solve the nonlinear condensation growth equation that includes the solute and curvature terms. Additionally, Köhler theory is considered for droplet activation and the growth/evaporation of hydrated aerosols. After performing the microphysical substeps, the liquid water mixing ratio, net latent heating/cooling, and water vapor tendencies are calculated based on SDs at each grid location to provide feedback to the dynamics. The droplet collision–coalescence process was turned off for these simulations.

Lagrangian subgrid-scale models for particle transport and condensation

The Lagrangian frame of reference in the SDM facilitates implementation of SGS schemes to model interactions between superdroplets and the resolved-scale flow and thermodynamic fields. A Lagrangian stochastic model from Weil et al. [2004, Eq. (15)] is implemented for the SGS transport of superdroplets. The Lagrangian stochastic differential equation in this approach can be expressed as (Thomson 1987; Weil et al. 2004)
dui(t)=ui(t)τtdt+12[1σu2dσu2dtui(t)+σu2xi]dt+(2σu2τtdt)1/2ηi(t),
where ui is the ith SGS velocity component of a SD, τt=2σu2/(Coϵ) is the Lagrangian time scale of turbulent fluctuations, σu2=2/3 eu, eu is the SGS turbulent kinetic energy, Co = 3 is a constant coefficient (Kolmogorov constant), ϵ is the kinetic energy dissipation rate, and ηi is a delta correlated Gaussian random noise. The second and third terms in the above equation are approximated as dσu2/dt(2/3)(deu/dt) and σu2/xi(2/3)(eu/xi). The subgrid-scale components of the inertial and drag terms are neglected in the current stochastic model. These components would have minimal influence on the spatial distribution of cloud droplets for the current case (Stokes number < 0.1).
Similarly, a Lagrangian stochastic model for SGS supersaturation fluctuations based on the Langevin approach to represent scalar fluctuations in a turbulent flow field is implemented for the SGS supersaturation (Gardiner et al. 1985; Pope 2001). In this approach, the magnitude of SGS supersaturation fluctuations along Lagrangian trajectories is obtained from SGS vertical velocity fluctuations. It neglects any contributions from the variance of water vapor mixing ratio and the covariance between temperature and water vapor (Chandrakar et al. 2020). Additionally, the covariance between integral radius (the first moment of a DSD) and supersaturation fluctuations is neglected (i.e., cloud droplets are assumed to be uniformly distributed within a grid box), which could also influence the fluctuation magnitude (Cooper 1989; Desai et al. 2018). Gravitational settling can also slightly decorrelate the SGS supersaturation field and cloud droplets, but this is not incorporated in the SGS supersaturation scheme since the effect is small and we neglect sedimentation anyway. It is our future interest to also account for these terms in the SGS supersaturation fluctuations model. With these assumptions and approximations, the Lagrangian differential equation of supersaturation fluctuations along the SD’s trajectory can be expressed as (Paoli and Shariff 2009; Sardina et al. 2015; Chandrakar et al. 2016; Grabowski and Abade 2017; Abade et al. 2018; Chandrakar et al. 2018)
ds(t)=[s(t)τt+s(t)τc]dt+α(T)w(t)dt+(2σs2dtτt)1/2ξ(t),
where s′ is the SGS supersaturation fluctuation experienced by an SD, τt is the Lagrangian time scale of turbulent fluctuations, τc=(4πDr¯n¯)1 is the phase relaxation time scale of droplets within a grid box during condensation substep, D is a thermodynamic factor [modified water vapor diffusivity; see Eq. (A1) in the appendix] that includes additional terms associated with the phase change except the molecular kinetic, curvature, and solute effects, r¯ is the mean droplet radius, n¯ is the droplet number concentration within a grid box, σs2=α(T)2σw2τt2/2, σw (=σu) is the magnitude of SGS vertical velocity fluctuations, α(T) is a constant coefficient that depends on temperature(T), w' (u3) is the SGS vertical velocity fluctuation of an SD, and ξ is a delta correlated Gaussian random noise. In some past modeling studies (e.g., Abade et al. 2018; Grabowski and Abade 2017) the Wiener diffusion term [the last term in Eq. (2)] was excluded, but in others it was included (e.g., Sardina et al. 2015). Without this term, the variance of SGS supersaturation fluctuations experienced by superdroplets will be lower than the expected value. Physically, including the Weiner term means that supersaturation fluctuation is not merely a response to vertical velocity fluctuation. That is, although vertical velocity fluctuation provides a drift term to drive supersaturation fluctuation at time scale 1/α(T), the presence of another drift term associated with turbulent eddies (−s′/τt) implies supersaturation fluctuation is a separate stochastic process. Thus, a complementary Wiener term is needed consistent with any drift-diffusion model (Langevin model) of a scalar field in turbulent flow (Pope 2001).

b. Simulation configurations and outputs

The current simulation setup follows the framework presented in Lasher-Trapp et al. (2005). A similar setup was used recently in Grabowski (2020) to compare Lagrangian and bin microphysics schemes. Here, horizontally uniform surface fluxes of heat (0.1 K m s−1) and moisture (4 × 10−5 kg kg−1 m s−1) are applied for an hour in the beginning to develop a convective boundary layer. Additionally, random perturbations in the potential temperature (amplitude: 0.01 K) and water vapor (amplitude: 2.5 × 10−5 kg kg−1) fields are applied in the lowest 1 km of the domain to initiate turbulent convection. After an hour when a turbulent boundary layer is fully developed, Gaussian distributions of surface fluxes of heat (0.3 K m s−1) and moisture (1.2 × 10−4 kg kg−1 m s−1) with a characteristic length scale of 1500 m are applied. The momentum flux is driven by a constant friction velocity (0.28 m s−1) and horizontal velocities at the lowest model grid level. The initial sounding data are from the SCMS (Lasher-Trapp et al. 2005).

The domain size is 10.4 km × 10.4 km (horizontally) × 8 km (vertically) with a grid spacing of 50 m in each direction. We have also repeated the same case with 25 and 100 m grid spacings to investigate the sensitivity to grid resolution. The integration time step for dynamics is 0.5 s. A smaller value of microphysical time step (Δtm = 0.1 s) expected to satisfy Δtm < τc and τt, is used to fulfill the requirements both for the SGS schemes and to appropriately represent the condensation process. A bimodal lognormal distribution of ammonium bisulfate aerosols with mode radii of 30 and 140 nm, geometric standard deviation of 1.28 and 1.75, and concentrations of 990 and 165 cm−3 is used to determine multiplicity and aerosol proprieties of superdroplets.

An ensemble of five different realizations for each case presented here (described below) is used to improve robustness of the analysis, for a total of 40 runs. Different realizations are generated using different random number seeds for the initial superdroplet positions and low-level theta perturbations. The bulk scalar quantities (water vapor mixing ratio, liquid water, potential temperature, etc.) and velocity fields are saved every minute. Detailed superdroplet data are saved at desired times for analysis.

c. Simulation cases

As summarized in Table 1, we have performed eight different sets of simulations. In all cases, the total number of superdroplets is the same to have a fair comparison. The simulations with both SGS supersaturation and transport schemes at 50-m grid spacing (CTL) are considered as a reference case to compare other simulation conditions. The results from this case are also used to demonstrate the general microphysical features of the simulated cumulus cell. Additionally, this case is used to study the effects of entrainment on DSDs and their moments. The impact of secondary activation above cloud base on DSDs is evaluated by preventing droplet activation above 1600-m altitude (NOINCLD), and other conditions are the same as CTL. To prevent activation above 1600 m, the condensation growth of all particles (not activated as cloud droplets) below their critical radius is turned off. The significance of SGS interactions of cloud particles with thermodynamic and dynamic fields are studied by running simulations without SGS supersaturation but including SGS transport (SGST) and without either of the SGS schemes (NOSGS). Comparison of SGST and NOSGS is expected to reveal the importance of improved mixing and dispersion of particles in cloud properties using the SGS transport scheme. In addition, comparing CTL and SGST shows how improved representation of SGS supersaturation fluctuations affects cloud properties. Simulations with two additional grid spacings (CTL25/NOSGS25: 25 m; and CTL100/NOSGS100: 100 m) are also performed to test the sensitivity of results to grid resolution. These simulations also help to assess the effectiveness of the SGS schemes to compensate for lower grid resolutions. Another important aspect of this analysis is to evaluate the extent to which the unresolved inhomogeneous mixing associated with entrainment influences results.

Table 1.

Summary of all simulation cases presented in this article.

Table 1.

3. Results

a. Cloud properties in the control case CTL

Figure 1 shows the cloud-base and -top heights as a function of time from multiple simulation realizations. The figure implies that the cloud base is consistent, but the cloud top varies for different flow realizations. These different flow realizations cause some variation in cloud properties, making comparisons between different cases difficult if only a single simulation is used for each experiment. This is addressed by analyzing an ensemble of simulations for each case, providing a statistically accurate way to compare these results. Variation among ensemble members causes some uncertainties in the ensemble average. For example, the statistical variability in the ensemble average vertical profile of droplet number concentration and radius standard deviation is approximately 5%–10% for most heights but does not mask the significant differences among the simulation ensembles presented here. Figure 2 shows results from one of the simulation cases with the SGS transport and condensation schemes (CTL). It displays a snapshot of the vertical profile of liquid water mixing ratio and droplet number concentration at 110 min of simulation time. These profiles are similar to what we would expect in an actively growing cumulus cloud cell. Qualitatively, the results are consistent with those presented in Grabowski (2020), despite the different dynamical model used here. However, there are quantitative differences due to the use of the SGS schemes and explicit treatment of droplet activation in the current simulations. The mean liquid water mixing ratio increases with height above the cloud base, and most of the grid points are within the adiabatic value. However, there are some fluctuations above the adiabatic value as well. A limited superdroplet concentration per grid box may contribute to these fluctuations as indicated by sensitivity tests with different initial numbers (80 and 240) of SDs per grid box. The adiabatic cloud water mixing ratio qad is calculated from dqad/dz=(dqs/dT)(dT/dz), where qs is the saturation vapor mixing ratio and dT/dz is the moist adiabatic lapse rate, by integrating qad and T upward from cloud base (lowest cloudy grid cell) using a simple Euler forward method with a 50-m grid spacing. The pressure (needed to obtain qs) is the horizontally averaged pressure profile from the model. The mean droplet concentration also increases above the cloud base, reaches a peak, and then slightly decreases and remains relatively steady further aloft. A peak in droplet concentration near the cloud base is expected since the mean saturation ratio has a peak near the cloud base. The region with fairly uniform droplet concentration aloft is most likely the result of a balance between cloud dilution and secondary droplet activation, which will be discussed in more detail later.

Fig. 1.
Fig. 1.

Cloud base (dotted lines) and top (solid lines) as a function of the simulation time for different realizations (CTL). The cloud base and top are defined based on the lowest and highest cloudy grid cell’s vertical location, and a threshold of 0.000 01 kg kg−1 is used for determining cloudy grid cells.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Fig. 2.
Fig. 2.

2D histogram of (a) cloud water mixing ratio, (b) cloud droplet number concentration, (c) cloud droplet mean radius, and (d) radius dispersion (standard deviation) along the vertical axis (from CTL). The data are from all ensemble members. Here, square symbols are the horizontally averaged value, bars represent 10th and 90th percentiles of the data, the solid black curve shows adiabatic cloud water mixing ratio, the vertical line at 500 cm−3 is a reference line for comparison, and the color scale shows normalized counts per bin. The data presented here are from t = 110 min of simulation time, and a threshold of 0.000 01 kg kg−1 is used for determining cloudy grid cells.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

The arithmetic mean cloud droplet size (Fig. 2c) increases above cloud base. The mean droplet size dispersion (standard deviation of droplet radius)4 within a grid box (Fig. 2d) is almost constant up to around 1600 m, except for a small increase right at the cloud base, and then increases higher up. The trend of increasing spectral dispersion with height was also reported in recent high-frequency observations of shallow cumulus clouds by Schmeissner et al. (2015). The region of nearly constant mean dispersion near cloud base is hypothesized to result from two competing effects: narrowing during condensation associated with the mean supersaturation and broadening due to supersaturation fluctuations (SGS plus gridscale horizontal variability). Significant activation of small size droplets from the cloud base CCNs likely also contributes to a constant DSD width near the cloud base. The horizontally averaged value of the quasi-steady SGS supersaturation standard deviation including the relaxation from SGS condensation (not shown) is higher at the cloud base (0.25%), which decreases to around 0.1% at 1500 m. It increases again with height, reaches approximately 0.2%, and then drops to 0.1% near the cloud top. The horizontal variability of the SGS supersaturation standard deviation also increases with altitude, likely because of the cloud dilution. These SGS supersaturation fluctuations are likely affecting the droplet activation and DSD broadening at different altitudes. The increase in droplet spectral dispersion above cloud base is hypothesized to result from cloud dilution/evaporation and new droplet activation through entrainment. Both of these hypotheses are tested by analyzing the other simulation cases listed in Table 1, which is described later in this section.

Figure 3 shows average DSDs at six different altitude ranges, from which we infer the role of specific microphysical processes occurring in the developing cumulus cloud. In each altitude range, distributions are classified based on the adiabatic fraction (AF) value, equal to the ratio of the actual liquid water content and the adiabatic value. Therefore, the DSDs in Fig. 3 represent an average distribution of cloud droplets within specific AF and altitude range from all ensemble members. As evident from the figure, the mode radius of the distributions increases with altitude. Moreover, near the cloud base, there are notable variations in the mode radius (from about 2 to 5 μm) with increasing adiabatic fraction. This variability could be due to gridscale spatiotemporal variability of updraft velocity and saturation ratio near the cloud base or due to entrainment from the subsiding shell around the cloud. However, at higher altitudes, the primary mode radius is the same across all of the adiabatic fraction ranges within a given altitude range. Interestingly, a secondary mode at smaller sizes develops above cloud base at lower adiabatic fractions. This result hints at contributions from variations in the growth history and activation of new droplets in diluted regions as CCN are entrained. Overall, these DSD features are consistent with previous observational and modeling studies (Warner 1969; Baker and Latham 1979; Brenguier and Grabowski 1993; Lasher-Trapp et al. 2005; Schmeissner et al. 2015).

Fig. 3.
Fig. 3.

Cloud DSDs at different altitudes from the case with both SGS supersaturation and transport (CTL). These average DSDs at each altitude range are calculated using the SD data from cloudy grid cells within different adiabatic fraction ranges in all ensemble members. Here, the y axis shows probability density.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

1) Trajectory analysis

Droplet trajectories are a natural output of the Lagrangian approach in contrast to Eulerian microphysics, where one needs to back-calculate trajectories based on output from the dynamics model at different times (e.g., Lasher-Trapp et al. 2005; Cooper et al. 2013). The current study utilizes this feature of the SDM to investigate SD trajectories and their contribution to microphysical variability, similar to Naumann and Seifert (2016) and Hoffmann et al. (2017) but with a focus on microphysical processes before the onset of precipitation formation. The main purpose of this analysis is to demonstrate that the model is working correctly and producing expected results. Figure 4 shows a few examples of SD back trajectories reaching different target grid cells after 110 min of simulation time. The target cells are located inside the cloud at four different altitudes. In the figure, cells 1 and 5 correspond to 1750 m, cells 2 and 6 to 2250 m, cells 3 and 7 to 2700 m, and cells 4 and 8 to 3200 m. All trajectories are sampled for 10 min at a 20-s interval, and only the parts of trajectories above z = 1000 m (approximate height of cloud base) are recorded for analysis. The superdroplet trajectories demonstrate some expected features of the flow. For example, trajectories in the top four panels (target grid cells 1–4) show SDs carried mostly by the core updraft flow. The bottom four panels (cells 5–8) show entrainment features associated with the toroidal circulations of cloud thermals. Overall, these results suggest that SD transport in CM1 is appropriately represented.

Fig. 4.
Fig. 4.

Sample trajectories of superdroplets reaching a few target grid cells at different altitudes. These trajectories are overlaid on isosurfaces of cloud water mixing ratio (qc = 0.000 01 kg kg−1) at 110 min of simulation time. Figure labels show the identity of a target grid.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Distributions of the origin height of these sample trajectories are presented in Fig. 5. In the top four panels (cells 1–4), a significant proportion of trajectories travel through the cloud base and reach a grid cell location near the convective cell’s updraft core. The destination grid cells at progressively higher altitudes (cells 2–4) have relatively more contributions from trajectories originating above cloud base. Trajectories reaching destination grid cells 5–8 (bottom panels in Fig. 5) have much more variability in origin location. DSDs at the target grid cells are shown in Fig. 6. Results indicate some trend with the variability in back trajectory origin altitude. Specifically, DSDs at lower altitudes (cells 1, 2, 5, and 6) are substantially different between the cases with (cells 5 and 6) and without (cells 1 and 2) laterally entraining SD trajectories at the same target altitude. In cell 4, 5, and 6, the entraining trajectories lead to a pronounced bimodal radius distribution. In the other entraining cases, particularly for cells 7 and 8, there is an indication of a secondary small droplet mode around 3–5 μm. Greater variability in droplet trajectories is clearly associated with broader DSDs qualitatively. This supports the role of “eddy hopping” (Cooper 1989; Lasher-Trapp et al. 2005; Grabowski and Abade 2017) in producing relatively broad DSDs in these simulations.

Fig. 5.
Fig. 5.

Distributions of origin altitude of SDs reaching the target grid cells in Fig. 4. A fixed bin width (200 m) is used here, and all origin points below 1000 m (close to the cloud base) are placed in the lowest bin.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Fig. 6.
Fig. 6.

DSDs at the target grid cells in Fig. 4. The y axis shows concentration density per bin.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

2) Entrainment effects on droplet spectral dispersion

The influence of entrainment on spectral dispersion is investigated further by plotting size-dispersion (standard deviation of droplet radius within a grid box) against AF at different altitudes. As seen in Fig. 7, the spectral dispersion increases with a decrease in AF for AF values > 0.25 but decreases at smaller AF. This trend is most evident well above cloud base (altitudes above 1750 m). The increase in dispersion with a decrease in AF for AF > 0.25 is due to entrainment-induced evaporation and subsequent mixing of droplets with different growth histories. A part of this dispersion change could also be due to homogeneous evaporation; homogeneous evaporation causes broadening since the rate of change of droplet radius is inversely proportional to the droplet radius. However, evaporation is not the only contributor. Turbulent mixing of cloud droplets from regions with different growth/evaporation histories and the secondary activation of new cloud droplets also contribute to this dispersion change. Moreover, supersaturation fluctuations allow secondary activation to occur even in regions away from cloud edges. This is examined later by comparing the spectral dispersion for the current case with results from simulations without SGS supersaturation. Burnet and Brenguier (2007) similarly reported increasing spectral width with decreasing AF in cumulus clouds from observations. That increase was more than the estimated spectral width change from the homogeneous evaporation assumption.

Fig. 7.
Fig. 7.

2D histogram of droplet radius dispersion (standard deviation) vs cloud adiabatic fraction at different altitudes (from CTL). Data from all ensemble members are included. In these plots, square symbols show the mean value of radius dispersion at each adiabatic fraction bin, and the color scale shows count density per bin.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

We hypothesize the decreasing trend of σr for small AF values (AF < 0.25) is due to significant secondary activation of cloud droplets, which is further analyzed in the next subsection. In cloudy regions with substantial entrainment-induced dilution, the condensation sink of supersaturation is low owing to the small droplet concentrations and sizes. If these diluted parcels subsequently experience ascent, significant secondary activation can occur from the entrained CCN. Consequently, the droplet dispersion becomes narrower because activation of these CCN produces a large concentration of small droplets within a narrow size range. Thus, secondary activation can have two distinct effects on DSD width depending on the adiabatic fraction range. In highly diluted regions (AF < 0.25), secondary activation decreases DSD width, while it increases the width in less diluted cloud regions (AF > 0.25). The decreasing trend of σr for small AF values was not observed by Burnet and Brenguier (2007), probably due to limitations of the measuring instrument (FSSP/Fast-FSSP) to sample small cloud droplets.

b. Role of secondary droplet activation in cloud properties

Observational and modeling studies (e.g., Pinsky and Khain 2002; Segal et al. 2003; Slawinska et al. 2012; Schmeissner et al. 2015; Hoffmann et al. 2015; Yang et al. 2018; Fan et al. 2018; Hernández Pardo et al. 2020) demonstrated a significant impact of secondary activation on microphysical properties like number concentration, effective radius, and spectral width. However, many of the previous modeling studies have been limited by using bulk schemes or bins schemes without explicit aerosol tracking or one-dimensional parcel models without entrainment. For example, bin and Lagrangian schemes with Twomey activation or without explicit aerosol tracking cannot correctly represent the activation of cloud droplets from entrained CCNs (Grabowski et al. 2018). Additionally, these schemes produce spurious activation in instantaneously high supersaturation regions (Grabowski et al. 2018; Hoffmann 2016). Similar to Hoffmann et al. (2015), the current SDM scheme allows this problem to be studied with the advantage of explicit aerosol activation and tracking in a three-dimensional model. Unlike the previous studies, the current framework also includes the effect of SGS supersaturation fluctuations on droplet activation.

Secondary activation can occur through different pathways (Hoffmann et al. 2015). Aerosols that enter the cloud through its base but do not activate there can still activate aloft owing to increasing supersaturation with height. Such an increase in supersaturation is possible by either a significantly accelerated updraft or strong collision–coalescence, reducing the droplet number concentration (e.g., Pinsky and Khain 2002). Note the latter effect does not occur here since collision–coalescence is excluded from the current case. Aerosols can also be entrained and activated above the cloud base. To distinguish between the secondary activation of entrained aerosols and activation aloft of aerosols that entered from the cloud base, we have tracked their entrainment and activation height. A similar analysis was done in Hoffmann et al. (2015) (Fig. 6) but for a case without SGS supersaturation fluctuations and with warm bubbles to initiate convection rather than surface flux forcing. The activation height of the CCN was recorded when its size crossed the critical radius. The activation height was reset when the size was reduced to lower than the critical radius or when the particle moved outside the cloudy volume. The entrainment height is defined as the level where the CCN enters the cloudy volume (cloud mixing ratio ≥ 0.01 g kg−1). Figure 8 shows the entrainment versus activation height and vertical profiles of cumulative aerosol entrainment and droplet activation of currently activated droplets. In the left panel of Fig. 8, the horizontal-aligned mode at 1100–1350 m represents CCN activated at the cloud base. The vertically aligned mode represents CCN that went through the cloud base and were activated at higher levels. The mode along the diagonal represents CCN entrained above the cloud base and activated close to their entrainment level. There are many points above the diagonal. These are entrained CCN that are activated above their entrainment level. However, there are also points below the diagonal—CCN that are entrained and then activated below their entrainment level. Points significantly below the diagonal (<1% of total points) are due to the classification of cloudy points based on a threshold (QC > 0.01 g kg−1) that does not incorporate all grid boxes with cloud droplets. Most activation (~75%) occurs near the cloud base, clearly seen by the sharp increase in cumulative percentage of the activation profile at and just above cloud base (the right panel of Fig. 8). Secondary activation increases the cumulative activation percentage nearly linearly inside the cloud, except near the top. The entrainment height has a similar profile. Overall, the secondary activation amounts to about a quarter of all CCN activation. Secondary activation that results from entrainment of aerosols above cloud base is 20% of the total activation (80% of the total secondary activation). The remaining secondary activation, 5% of the total activation (20% of the total secondary activation), occurs from aerosols entering the cloud through its base. Hoffmann et al. (2015) reported slightly larger (approximately 30% of the total activation) secondary activation from entrainment but without considering SGS supersaturation fluctuations.

Fig. 8.
Fig. 8.

(a) Activation (Zact) vs entrainment (Zent) heights of aerosols that are activated as cloud droplets. The color scale shows normalized count per bin. (b) Vertical profiles of the cumulative percentage of droplets activated or entrained at or below that level. These plots are obtained from the data at 110 min of the simulation time and a single flow realization.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

To investigate the influence of secondary droplet activation on DSD evolution we have performed a set of simulations where cloud droplet activation above 1600 m (the height of peak droplet concentration) is turned off (the NOINCLD case). The mean droplet concentration without secondary activation (not shown) has a decreasing profile with altitude (after the peak) owing to dilution, as expected based on Slawinska et al. (2012). The mean concentration has a slight increase near cloud top because parcels that reach the top are relatively less diluted as argued in Slawinska et al. (2012). In the less diluted regions, droplet number concentration is expected to be larger since the entertainment-induced evaporation and reduction in concentration are relatively small. Consistent with the lower mean droplet concentration in these simulations compared to the control (CTL), a larger mean cloud droplet radius is also evident.

Vertical profiles of the spectral dispersion with and without secondary activation are compared in Fig. 9. Values of the mean spectral dispersion near cloud base are similar (1–1.5 μm) for CTL and NOINCLD. However, they differ significantly at higher altitudes, increasing with height with large spatial variations in CTL compared to constant with height (around 1 μm) and smaller spatial variations in NOINCLD. The maximum values at midlevels are also different (≈5 vs 4 μm). Activation of new cloud droplets at various levels in CTL could allow the cloud to reach a higher altitude, but this could also be a coincidence due to a limited number of flow realizations. The average DSDs (Fig. 10) follow a similar trend as the case with secondary activation. The distributions from the altitude ranges 1000 < Z < 1425 m and 1425 < Z < 1750 m are very close to the case with secondary activation as the droplet activation is turned off only above 1600 m. However, at higher altitudes, the left shoulder of the distributions (radius < 5 μm) has a smaller contribution than in the case with secondary activation. Contrasting with the CTL case, the secondary mode is absent in NOINCLD. This result suggests the role of secondary activation in producing bimodality of DSDs in diluted cloud regions. This is consistent with our trajectory analysis described above that showed, for many cloudy grid cells, how superdroplets laterally entrained into the cloud mix with those ascending from the cloud base in the updraft core to broaden DSDs.

Fig. 9.
Fig. 9.

2D histogram of radius dispersion of cloud droplets along the vertical axis [(a) with secondary activation (CTL), and (b) without secondary activation (NOINCLD)]. Data from all ensemble members are included. Here, square symbols are the mean value, bars represent 10th and 90th percentiles of the data, and the color scale shows normalized counts per bin. A vertical line at 2 μm is added for comparison.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Fig. 10.
Fig. 10.

As in Fig. 3, but for NOINCLD ensemble.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Figure 11 shows variation of spectral dispersion with AF for the case without secondary activation, similar to Fig. 7. In the previous subsection, we hypothesized that the decreasing trend of spectral dispersion with AF (for AF < 0.25) in the control case is due to the secondary activation of new droplets. Figure 11 corroborates this hypothesis. Without secondary activation, the decreasing trend in σR versus AF is much less significant and limited to only a couple of smallest AF bins. Compared to CTL, the data points are slightly more concentrated around the mean value at a given AF. Moreover, the mean spectral dispersion at higher altitudes is smaller compared to the case with secondary activation. These results confirm our previous assertion that secondary activation has two distinct effects on DSD width: narrowing in highly diluted regions (AF < 0.25) and broadening in less diluted regions (AF > 0.25).

Fig. 11.
Fig. 11.

As in Fig. 7, but for NOINCLD ensemble.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

c. Importance of subgrid-scale transport and condensation of superdroplets

SGS fluctuations in particle velocities have been shown to be critical for accurately modeling particle dispersion in a turbulent flow field (Thomson 1987; Weil et al. 2004; Fede and Simonin 2006). A Lagrangian stochastic SGS model, similar to Weil et al. (2004), was adapted here to model the SGS particle transport. Figure 12 (first and second columns) compares cloud droplet concentration and spectral dispersion for the cases without (NOSGS) and with (SGST) SGS transport. The peak horizontally averaged droplet concentration is approximately similar for both the cases. However, the mean concentration decreases with altitude at a greater rate for SGST, and this decreasing trend is present in all ensemble members. This suggests more turbulent mixing of SDs across the cloud boundary and evaporation of drops outside of the cloud leads to a decrease in droplet concentration. Moreover, droplet size dispersion for SGST is slightly larger than NOSGS. This could be due to better transport and improved mixing of SDs of different growth histories when the SGS transport model is used. However, one should also be cautious about the impact of different flow realizations and small ensembles while comparing these cases.

Fig. 12.
Fig. 12.

Effects of the SGS supersaturation and transport schemes on the vertical profile of (a)–(c) cloud droplet number concentration and (d)–(f) spectral dispersion: it compares results from (left) NOSGS, (center) SGST, and (right) CTL. Here, each plot includes data from all ensemble members. All other properties of this figure are as in Figs. 2b and 2d. Figures 2b and 2d are repeated here for the convenience of the reader.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Like SGS transport, several studies have indicated that SGS supersaturation is an essential aspect of the droplet condensation growth in a turbulent cloud condition (Cooper 1989; Paoli and Shariff 2009; Sardina et al. 2015; Chandrakar et al. 2016; Abade et al. 2018). In the current model setup, we partially accounted for that growth by considering fluctuations in supersaturation via SGS vertical velocity fluctuations. This SGS supersaturation scheme neglects any additional contributions that arise due to fluctuations in integral radius and water vapor and covariance between water vapor and temperature fluctuations (Cooper 1989; Chandrakar et al. 2020; Siebert and Shaw 2017). Those terms could also be important to include, specifically in regions most affected by entrainment. Nevertheless, the current scheme is expected to capture aspects of SGS supersaturation, as suggested by Abade et al. (2018) and Sardina et al. (2018). Figure 12 (top row) compares vertical profiles of droplet number concentration for the cases with (CTL) and without (SGST) SGS supersaturation. SGS fluctuations allow activation of droplets experiencing supersaturation above their critical value, even if the grid-mean supersaturation is zero or less. If these droplets grow due to condensation, they deplete mean supersaturation faster. This reduction in mean supersaturation delays activation of other CCN that could have been activated in the absence of SGS supersaturation fluctuations (Abade et al. 2018). The activation of those potential CCN at higher altitudes therefore increases secondary activation. This secondary activation compensates for the reduction in droplet concentration from entrainment and dilution and produces a relatively constant mean vertical profile above midcloud levels in CTL in contrast to the decrease of droplet concentration with height in SGST. The cloud top height is also slightly higher for CTL compared to NOSGS and SGST. But, as can be seen in Fig. 1, it could be just a coincidence due simply to different flow realizations.

Another significant contribution of SGS supersaturation fluctuations is on broadening of the DSDs. In an adiabatically rising cloud parcel, condensation growth associated with the grid-mean supersaturation narrows the droplet size spectrum with altitude. The presence of supersaturation fluctuations acts against this narrowing effect (see Figs. 2 and 3 in Grabowski and Abade 2017). Figure 12 (bottom row) demonstrates this broadening effect; spectral dispersion has a larger mean value for the case with SGS supersaturation (CTL) than without (SGST). This spectral broadening is caused by both more secondary activation of entrained CCN and growth of existing cloud droplets. However, there is a negative feedback between these two contributions. An increase in droplet number concentration due to more secondary activation decreases the phase relaxation time of cloud droplets, and thus limits the increase in dispersion from SGS supersaturation of existing droplets (Chandrakar et al. 2016). Additionally, increased activation and condensation due to positive SGS supersaturation fluctuations (even in grid boxes with a small mean subsaturation) produce a smaller magnitude of the horizontally mean saturation ratio inside the cloud than the case without SGS microphysics.

d. Influence of entrainment on relative dispersion and effective radius parameterization

1) Relation between relative dispersion and vertical velocity

A relation between the relative dispersion of cloud droplets (ϵ=σr/r¯) and vertical velocity (W) could be significant for parameterizing the dispersion effect in models with bulk microphysics (Lu et al. 2012). However, the presence of turbulent fluctuations and entrainment breaks the connection between supersaturation and grid-mean vertical velocity (Manton 1979; Cooper 1989). Using a theoretical framework, Cooper (1989) showed that the microphysical variability due to turbulent mixing of parcels originating from different cloud regions and fluctuations in vertical velocity could lead to broad DSDs and large relative dispersions even in undiluted cloud regions. However, they did not consider entrainment-induced evaporation and activation in their derivation. In the next subsection we show that SGS fluctuations in supersaturation also influence ε. Therefore, a question worth considering is, How do these factors impact the relation between ε and W?

Figure 13 sheds some light on this question. It shows examples of ε versus W from a horizontal slice at 2250 < z < 2750 m. Here, the vertical velocity (W) is the instantaneous gridscale vertical velocity. Although instantaneous values might not be a perfect quantity for this analysis, they can still provide some valuable insight. In this figure, the peak value of mean ε is highest for the cases with both SGS supersaturation and transport (CTL). Additionally, the mean ε negatively correlates with updraft velocity for W > 1 to 2 m s−1 in this and all other cases. This inverse relation with updraft velocity is also expected in a turbulent nonentraining parcel (Cooper 1989; Chandrakar et al. 2018). In updraft regions, a larger updraft velocity most likely occurs near the relatively undilute cloud core. Vertical velocity decreases toward boundaries affected by entrainment, and the spectral width is also higher near the cloud boundaries due to entrainment. Moreover, the mean size of cloud droplets increases with the updraft velocity. Both of these factors explain the decrease in mean ε with increasing updraft velocity (for W > ~1 m s−1) in all cases as seen in Fig. 13.

Fig. 13.
Fig. 13.

2D histogram of relative dispersion vs vertical velocity (w) from a horizontal section at z = 2250–2750 m. It compares the σr vs w relation from simulations (including all ensemble members) with (a) SGS supersaturation and transport (CTL), (b) SGS supersaturation and transport but no secondary activation (NOINCLD), (c) only SGS transport(SGST), and (d) no SGS supersaturation and transport (NOSGS). Here, square symbols are the mean relative dispersion at each vertical velocity bin, and the color scale shows count density per bin.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Interestingly, there is a negative correlation of ε and W (i.e., larger W implies smaller ε) in updrafts and a positive correlation in downdraft regions in case CTL, contrasting with the other cases (NOINCLD, SGST, and NOSGS). The peak mean ε occurs at small (but nonzero) positive vertical velocity of ~1 m s−1. Comparison of CTL with NOINCLD suggests that the positive correlation in downdrafts results from secondary activation of cloud droplets; the positive correlation for downdrafts in CTL is not seen in NOINCLD. The spectral width for the NOINCLD case is also smaller (and the mean radius is larger) than the CTL case. Thus, ε is much smaller than in the CTL case. Additionally, there is much less spatial variability of ε (maximum value less than 0.7), and data points are concentrated near 0.05–0.2 in the NOINCLD case. Without secondary activation, subsaturation in downdrafts decreases the cloud droplet mean size, while the DSD width changes relatively little. Without secondary activation, the change in spectral width with downdraft velocity is mainly due to evaporation-induced broadening (opposite to condensation narrowing) and SGS supersaturation fluctuations. However, these changes depend on growth/evaporation history, not just instantaneous conditions (i.e., downdraft velocity). Consequently, ε increases with downdraft velocity in NOINCLD down to about −3 m s−1, below which it slightly decreases with further decreases in W. In contrast, when secondary activation is allowed the activation of new cloud droplets increases the spectral width. In downdraft regions, new droplet activation generally occurs only when SGS fluctuations produce positive supersaturations locally. With an increase in downdraft velocity, the probability of this occurring decreases. Thus, the spectral width of DSDs and mean ε decrease with increasing downdraft velocity (i.e., decreasing W when W < 0) in CTL. Although the evaporation of cloud droplets reduces the mean size, the overall change in ε is dominated by a decreased probability of cloud droplet activation with increasing downdraft velocity. Another factor that could cause the narrowing of DSDs in downdrafts is the complete evaporation of small cloud droplets. In the CTL case, the concentration of newly activated droplets could be more significant. This would explain why the narrowing effect is larger for that case than NOINCLD.

For SGST, the peak value of mean ε versus W is slightly lower than in CTL. Also, the spatial variability is lower than in the CTL case. Similar to CTL, the peak occurs at W of 1–2 m s−1. From the peak, mean ε decreases to W = 0 m s−1 but then remains nearly constant with further decreases in W. This result indicates that both mean radius and spectral dispersion decrease at a similar rate with increasing downdraft velocity. In contrast, mean ε increases with downdraft velocity for NOSGS, and this case also has a lower peak mean ε and reduced spatial variability than CTL and SGST.

2) Influence of entrainment on effective radius parameterization

The ratio of effective and mean volume radius cubed (k=re3/r3¯) is a parameter that accounts for the shape of the DSD relevant to parameterization of cloud optical properties (Pontikis and Hicks 1992; Martin et al. 1994). Here, the effective radius is the ratio of the third and second moments of DSD (rer3¯/r2¯). The parameter k depends on the relative dispersion and skewness of the DSD. Martin et al. (1994) reported measurements of average values of k in continental and maritime cloud conditions. However, they did not explicitly consider the effect of entrainment and spatial variation of k. Based on the SCMS field experiment, Brenguier et al. (2011) showed that k decreases with increasing cloud dilution due to entrainment in cumulus clouds. Measurements also suggest vertical variation of k. However, Brenguier et al. (2011) suggested that more detailed investigation of the spatial variation of k is needed, which we discuss here. The horizontal mean k value increases with altitude just above cloud base and decreases aloft up to the cloud top (not shown). There is also significant horizontal variability in the magnitude of k. We have also analyzed the influence of entrainment on k by comparing its value at different adiabatic fraction (AF) values (Fig. 14). Similar to the measurements in Brenguier et al. (2011, Fig. 3b), mean k decreases with a decrease in AF for relatively undiluted grid cells. The decrease in mean k is greatest for the case with both SGS supersaturation and transport (CTL), indicating the important role of SGS fluctuations. However, in highly diluted regions, mean k increases slightly with a further decrease in AF. This is caused by secondary droplet activation, as indicated by results from the case without secondary activation (NOINCLD) in Fig. 14b.

Fig. 14.
Fig. 14.

2D histogram of k parameter vs adiabatic fraction from a horizontal section at z = 2250–2750 m. It compares the k vs AF relation from simulations (including all ensemble members) with (a) SGS supersaturation and transport (CTL), (b) SGS supersaturation and transport but no secondary activation (NOINCLD), (c) only SGS transport (SGST), and (d) no SGS supersaturation and transport (NOSGS). Square symbols are the mean k parameter at each adiabatic fraction bin, and the color scale shows count density per bin.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

e. Impact of grid resolution on representation of microphysical processes

Model grid resolution can influence the representation of microphysical processes and transport of droplets in any numerical framework, especially when the thermodynamics and flow properties are not sufficiently resolved or modeled to the scale of droplet interactions. To a certain extent, the influence of grid resolution on bulk cloud properties simulated using the SDM microphysics in a LES framework has been investigated by Sato et al. (2018). However, a more detailed investigation of the microphysical impacts is needed. To investigate the impact of grid resolution, we have performed a set of simulations with 25-, 50-, and 100-m grid resolutions with and without the SGS transport and condensation schemes. These simulations also serve as a test for the SGS schemes to evaluate whether they can compensate for a coarser grid resolution. Simulations with 25- and 100-m grid lengths are processed similarly as the 50 m cases (same spatial binning) for a fair comparison. Figures 15 and 16 show results from these sensitivity tests.

Fig. 15.
Fig. 15.

Comparison of vertical profiles of droplet concentration from simulations (including all ensemble members) with (a),(b) 25- and (c),(d) 100-m grid sizes. Columns show results (a),(c) without any SGS microphysics schemes, and (b),(d) with SGS supersaturation and transport schemes. Other properties of the plots are as in Fig. 2b.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

Fig. 16.
Fig. 16.

Comparison of vertical profiles of size dispersion from the simulations (including all ensemble members) with (a),(b) 25- and (c),(d) 100-m grid sizes. Columns show results (a),(c) without any SGS microphysics schemes, and (b),(d) with SGS supersaturation and transport schemes. Other properties of the plots are as in Fig. 2d.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0281.1

The mean profile of cloud droplet number concentration (Fig. 15) is qualitatively similar for all the cases, but the magnitude differs noticeably. The peak number concentration near the cloud base decreases with increasing grid length. A lower peak concentration is expected if grid resolution is not small enough to resolve peak supersaturation near the cloud base (Clark 1974; Morrison and Grabowski 2008). Above the peak, cloud droplet concentration decreases sharply with altitude up to midcloud level for NOSGS25 and CTL25. In NOSGS (Fig. 12, first column), the decrease is less and continues up to the top, whereas the decrease with height is not evident in the 100-m grid-length case. Additionally, horizontal variability in droplet concentration systematically increases with grid resolution. These results indicate that the turbulence structures and thermodynamic variability are underresolved in the 100- and 50-m cases.

Vertical profiles of droplet size dispersion (left columns in Figs. 16 and 12) for NOSGS and NOSGS100 are similar but have a few differences. NOSGS100 has an additional decreasing trend near cloud top, probably due to poor resolution of vertical transport of SDs. For the ensemble of simulations with a 25-m grid size (NOSGS25), the mean spectral dispersion is slightly greater, and horizontal variability is smaller compared to the lower resolution cases. The small ensemble and flow variability could also affect this comparison to some extent. These results suggest that 50- and 100-m grid sizes are not small enough to resolve inhomogeneous mixing during entertainment sufficiently. Additional simulations at higher resolutions are needed to check the convergence of results below 25-m grid size.

A comparison of results using different model grid lengths indicates that the SGS microphysics schemes improve droplet number concentration and spectral dispersion profiles in the lower resolution cases (relative to the 25-m case). As expected, the SGS schemes have a much greater impact for the lower resolution cases, with little impact at 25-m grid length, which validates the fidelity of the SGS schemes implemented in this study. Profiles of spectral dispersion for the simulations with the SGS microphysics schemes are similar for the 25- and 50-m cases, suggesting convergence. Moreover, the liquid water mixing ratio is similar using 100-, 50-, and 25-m grid lengths with the SGS schemes. However, the mean concentration profile does not appear converged at 25-m grid length, and similarly the mean droplet size changes with grid length down to 25 m. Overall, these results suggest that the SGS schemes can partially compensate for changes in grid length, especially for the DSD dispersion, but they cannot fully compensate for lower model resolution. In particular, inhomogeneity in the water vapor field and covariance between SGS temperature and water vapor fluctuations during the entrainment-mixing process could be important. Including these other components in the SGS supersaturation scheme might further improve results at lower resolution.

4. Discussion and conclusions

The problem of entrainment mixing and its influence on DSDs has been studied extensively in the past, mostly using relatively simple modeling frameworks in which many details of cloud–turbulence interactions were not included. This motivated a thorough investigation of the topic using a microphysics scheme that realistically simulates cloud droplets’ natural interactions with turbulent flow dynamics. Our study examined the entrainment-mixing problem in detail using a sophisticated modeling framework that includes several factors neglected in previous studies. A Lagrangian microphysics model (“superdroplet method”) was implemented in CM1 and used to investigate microphysical properties in an isolated cumulus-congestus cloud. The droplet activation process was solved explicitly, and subgrid-scale fluctuations of supersaturation and superdroplet velocities in a turbulent field were modeled using Lagrangian stochastic approaches. However, only the component of supersaturation inhomogeneity arising from SGS vertical velocity fluctuations was considered for SGS supersaturation and droplet activation. Other contributions from SGS scalar fluctuations (e.g., water vapor) will be a focus of future investigations.

Compared to previous studies with Lagrangian microphysics (e.g., Hoffmann et al. 2019, 2017; Abade et al. 2018), the current study provides new insights about the spatial variability of DSDs and their variation with the amount of entrainment-driven dilution. This analysis also demonstrated the transport scheme’s effectiveness in capturing basic flow features of entraining cumulus convection. DSDs from the current simulations had features typically observed in cumulus clouds. For example, the model produced bimodal DSDs in entrainment affected regions similar to the observations of Warner (1969) and Schmeissner et al. (2015). Analysis of superdroplet trajectories at different locations revealed contributions from the mixing of droplets having different growth histories in spectral broadening and production of bimodal distributions. With an increasing number of trajectories originating above the cloud base, the DSD width became broader. Lasher-Trapp et al. (2005) also showed entrainment-driven DSD broadening, but their Lagrangian parcel calculations did not feedback to the model dynamics. The current study did not have this limitation and hence represented the impact of different broadening mechanisms in a more complete way.

In this study, we also examined the role of secondary activation in highly diluted and less diluted cloud regions. Our findings in this regard build on the studies of Hernández Pardo et al. (2020) and Lasher-Trapp et al. (2005), who used Eulerian bin microphysics and offline Lagrangian schemes, respectively. Those studies showed the contribution of secondary activation toward DSD broadening but used modeling frameworks that did not include explicit aerosol tracking and SGS interactions. The current study overcomes these limitations, thus provides further understanding of the role of secondary activation in different cloud regions. In less diluted cloud regions (AF > 0.25), we showed that secondary activation enhanced the broadening of DSD width. Droplet spectral width had an increasing trend with decreasing adiabatic fraction for AF > 0.25, similar to measurements during the SCMS field campaign (e.g., Burnet and Brenguier 2007). This result is also consistent with Lasher-Trapp et al. (2005) and Hernández Pardo et al. (2020). However, in contrast to those modeling studies, we showed that in highly diluted regions (AF < 0.25) this trend reversed, and spectral width decreased with decreasing AF. This behavior was also caused by secondary droplet activation. Secondary activation also significantly modified the vertical profile of DSD width.

This study also provides novel insights into the roles of SGS velocity fluctuations (affecting particle trajectories) and supersaturation fluctuations on DSD evolution. Apart from homogeneous evaporation due to entrainment, SGS supersaturation fluctuations and turbulent mixing of droplets with different growth histories influenced the droplet size dispersion. Moreover, the SGS supersaturation and transport schemes drove activation/deactivation and mixing of cloud droplets. Abade et al. (2018) also showed that turbulent fluctuations affect the activation of cloud droplets and DSD width during entrainment events, but they used a parcel approach with fixed turbulence intensity and entrainment rate. They found that SGS supersaturation fluctuations delay the activation of cloud droplets inside the cloud but did not examine the impact of these fluctuations on secondary activation. Our study showed significant increases in both cloud base and secondary activation owing to SGS supersaturation fluctuations. We also showed that secondary droplet activation induced by SGS supersaturation fluctuations increased DSD width noticeably and led to changes in the droplet concentration profile. Hoffmann et al. (2019) found that SGS mixing leads to broad DSDs for cloud cases with low turbulent intensity and predominantly inhomogeneous mixing conditions, but not for a cumulus case with more homogeneous SGS mixing and evaporation. Moreover, they neglected the effects of SGS fluctuations on droplet activation, which were found to be important here.

A variable crucial for the parameterization of autoconversion rate and optical properties using bulk microphysics schemes—the relative size dispersion of cloud droplets—was also analyzed here. Specifically, the relationship with grid-mean vertical velocity W was investigated. Similar to observations and theory (Cooper 1989; Lu et al. 2012), we showed that mean relative dispersion was negatively correlated with gridscale updraft velocity and peaked at low vertical velocity (~1 m s−1). However, in downdrafts, the mean relative dispersion decreased with increasing downdraft velocity. A decrease in mean relative dispersion with decreasing W in downdraft regions and increasing W in updrafts was linked to secondary droplet activation and SGS supersaturation fluctuations. Moreover, the mean relative dispersion’s peak value was highest when the model included both SGS activation/condensation and transport schemes. A higher magnitude of relative dispersion for this case highlights the importance of SGS interactions for microphysical processes. Enhancement of secondary activation due to SGS supersaturation fluctuations significantly increased the mean relative dispersion and its spatial variability. The parameter k, which accounts for the DSD shape in parameterizing the effective radius, had a vertical structure and magnitude close to observations in cumulus clouds (Figs. 3b and 5a; Brenguier et al. 2011). Additionally, the effects of entrainment on k were analyzed in the context of the observed trend in k versus AF. The parameter k decreased with decreasing AF in less diluted regions, whereas in highly diluted regions it increased slightly. This trend reversal was also due to the secondary activation of cloud droplets. Secondary activation and SGS supersaturation fluctuations also decreased k and affected its spatial variation.

An analysis of simulations using various grid lengths revealed that some microphysical interactions are not sufficiently resolved above 25-m grid size without any SGS microphysics models. Thus, more simulations with higher grid resolutions are needed to check the convergence of microphysical properties at or below 25-m grid size. Significant differences between different grid resolution cases were evident for the cloud droplet number concentration and spectral dispersion profiles. It appears that peak supersaturation near the cloud base and turbulent entrainment-mixing features remain unresolved above 25-m grid length in the absence of SGS microphysics models. Including the SGS microphysics schemes improves results by decreasing the sensitivity to model grid length. For example, the spectral dispersion and liquid water mixing ratio profiles appear converged at 50-m grid length.

Overall, the combined CM1-SDM framework developed here successfully demonstrated the fundamental physical characteristics of an isolated cumulus congestus cloud and provided new insights into the role of secondary activation and SGS turbulent fluctuations. Additionally, implementing compatible SGS microphysics schemes utilizing a Lagrangian stochastic approach improved how the physics of cloud–turbulence interactions are represented. Further improving the model and comparing it with an Eulerian bin microphysics scheme, including rain formation by collision–coalescence, will be a focus of future studies.

Acknowledgments

This work was supported by U.S. Department of Energy Atmospheric System Research Grant DE-SC0020118. KKC was supported by the NCAR ASP postdoctoral fellowship program. We would like to acknowledge high-performance computing support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. We are thankful to Shin-ichiro Shima for providing the SCALE-SDM code. The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Data availability statement

The current study used the CM1 model (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.

APPENDIX

Modified Water Vapor Diffusivity

The droplet condensation growth equation used in SDM [Eq. (2), Shima et al. 2009]:
RidRidt=(S1)aRi+bRi3[ρl(LRυT1)LkT+ρlRυTDes(T)],
where Ri is the radius of a droplet, S is the saturation ratio of water vapor, a and b are the factors associated with the curvature and solute effects, ρi is the density of water, L is the latent heat of vaporization, T is the air temperature, k is the thermal conductivity of air, D is the molecular diffusivity of water vapor, and es(T) is the saturation vapor pressure as a function of the air temperature. The above equation can be converted to an expression for the rate of change mass of a cloud droplet due to condensation:
dMidt=4πRi(S1)aRi+bRi3[(LRυT1)LkT+RυTDes(T)],
dMidt=4πRiρυs(T)D(S1)aRi+bRi3[1+(LRυT1)DLρυs(T)kT]=4πRiDρυs(T)[(S1)aRi+bRi3].
Here, ρυs(T) is the saturation vapor density as a function of the air temperature. The modified diffusivity from the above expression is (Kumar et al. 2013)
D=D[1+(LRυT1)DLρυs(T)kT]1.

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1

When the turbulent mixing time scale is smaller than the response time of cloud droplets, all droplets experience a well-mixed subsaturated environment and evaporate uniformly (“homogeneous mixing”). In contrast, if the mixing time scale is longer than the response time of cloud droplets, droplets in the vicinity of the cloudy–clear-air interface evaporate substantially or entirely, and others do not change significantly (“inhomogeneous mixing”).

2

The multiplicity of a superdroplet is the number of physical particles each superdroplet represents (Shima et al. 2009).

3

The attributes of a superdroplet are the state variables that characterize it (Shima et al. 2009). In the current case, they are radius, solute mass, SGS velocity components, and SGS supersaturation.

4

The standard deviation of the droplet radius is calculated based on the weighted sample variance of superdroplets within a sample volume of a gridbox size (1.25 × 105 m3).

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