Augmenting the Double-Gaussian Representation of Atmospheric Turbulence and Convection via a Coupled Stochastic Multi-Plume Mass-Flux Scheme

Mikael K. Witte; Adam Herrington; Joao Teixeira; Marcin J. Kurowski; Maria J. Chinita; Rachel L. Storer; Kay Suselj; Georgios Matheou; Julio Bacmeister

doi:10.1175/MWR-D-21-0215.1

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Abstract
- Significance Statement
1. Introduction
2. Model description
3. Results
- a. Trade-wind shallow cumulus
- b. Continental shallow cumulus
4. Discussion
5. Conclusions
Acknowledgments.
Data availability statement.
APPENDIX
- Table of Non-Default SCAM Parameters
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Fig. 1.

Mean profiles over simulated hours 4–6 from the BOMEX case. (top) Mean liquid-water potential temperature $\bar{θ_{l}}$ , mean total water mixing ratio $\bar{q_{t}}$ , mean cloud water mixing ratio $\bar{q_{c}}$ , and cloud fraction f_c. Insets of $\bar{θ_{l}}$ and $\bar{q_{t}}$ panels show the deviation of SCAM from LES profiles (interpolated to SCAM vertical grid) between 1100- and 1700-m altitude. (bottom) Temperature flux $\bar{w^{'} θ_{l}^{'}}$ , moisture flux $\bar{w^{'} q_{t}^{'}}$ , buoyancy flux $\bar{w^{'} θ_{υ}^{'}}$ , and total plume area fraction $\sum a_{i}$ . The dashed and dotted curves in the flux panels denote the individual contributions of CLUBB and MF, respectively. In the plume area fraction panel (bottom rightmost), the solid line shows dry plume area fraction and the dashed line moist plume area fraction.

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Fig. 2.

Profiles of scalar variances of (left) vertical velocity, (center) temperature, and (right) moisture for the BOMEX case averaged over simulation hours 4–6. LES total variance includes the contribution from the subgrid-scale.

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Fig. 3.

Joint probabilities of perturbation $q_{t}^{'}$ and $θ_{l}^{'}$ relative to level mean $\bar{q_{t}}$ and $\bar{θ_{l}}$ at t = 6 h for three levels: (bottom) z = 250 m, (middle) z = 730 m, and (top) z = 1000 m from the BOMEX simulations for LES (gray contours), CLUBB (black contours), and MF plumes (black dots). Red crosses denote mean surface properties relative to the mean environmental properties aloft. Probabilities are expressed as 1 minus a unitless cumulative probability, C(θ_l, q_t), such that the mean of the joint distributions has a value of 1, decreasing radially outward. The contours intervals are logarithmic and identical for both CLUBB and LES. The outermost contour represents a cumulative probability of 3.5% and the innermost contour represents a cumulative probability of 70%. The CLUBB contours refer to the (left) CLUBB-only and (right) CLUBB+MF simulations, with the black dots denoting the $q_{t}^{'}$ and $θ_{l}^{'}$ values of individual members of the MF plume ensemble.

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Fig. 4.

Time–height curtain plots of (top),(middle) $\bar{θ_{l}}$ difference between SCAM and LES and (bottom) mean liquid-water potential temperature $\bar{θ_{l}}$ from LES from the ARM shallow cumulus case. For the difference panels, SCAM output was interpolated to the LES altitude grid and LES output was interpolated to the SCAM time grid.

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Fig. 5.

Time–height curtain plots of (top),(middle) $\bar{q_{t}}$ difference between SCAM and LES and (bottom) mean liquid-water potential temperature $\bar{q_{t}}$ from LES for the ARM shallow cumulus case. For the difference panels, SCAM output was linearly interpolated to the LES altitude grid and LES output was interpolated to the SCAM time grid.

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Fig. 6.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of cloud fraction for the ARM shallow cumulus case.

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Fig. 7.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of mean cloud liquid-water mixing ratio $\bar{q_{c}}$ for the ARM shallow cumulus case.

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Fig. 8.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of liquid-water potential temperature flux $\bar{w^{'} θ_{l}^{'}}$ for the ARM shallow cumulus case.

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Fig. 9.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of total water mixing ratio flux $\bar{w^{'} q_{t}^{'}}$ for the ARM shallow cumulus case.

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Fig. 10.

Time–height curtain of the fraction of $\bar{w^{'} q_{t}^{'}}$ due to the MF component from the SCAM CLUBB+MF simulation of the ARM shallow cumulus case.

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Editorial Type:: Article

Article Type:: Research Article

Augmenting the Double-Gaussian Representation of Atmospheric Turbulence and Convection via a Coupled Stochastic Multi-Plume Mass-Flux Scheme

Mikael K. Witte

Mikael K. Witte ^aNaval Postgraduate School, Monterey, California
^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
^cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Adam Herrington

Adam Herrington ^dNational Center for Atmospheric Research, Boulder, Colorado

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Joao Teixeira

Joao Teixeira ^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
^cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Marcin J. Kurowski

Marcin J. Kurowski ^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Maria J. Chinita

Maria J. Chinita ^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
^cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Rachel L. Storer

Rachel L. Storer ^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
^cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Kay Suselj

Kay Suselj ^bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Georgios Matheou

Georgios Matheou ^eUniversity of Connecticut, Storrs, Connecticut

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Julio Bacmeister

Julio Bacmeister ^dNational Center for Atmospheric Research, Boulder, Colorado

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Online Publication:: 07 Sep 2022

Print Publication:: 01 Sep 2022

DOI:: https://doi.org/10.1175/MWR-D-21-0215.1

Page(s):: 2339–2355

Received:: 03 Aug 2021

Accepted:: 20 May 2022

Published Online:: 07 Sep 2022

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Modern general circulation models continue to require parameterizations of subgrid transport due to planetary boundary layer (PBL) turbulence and convection. Some schemes that unify these processes rely on assumed joint probability distributions of vertical velocity and moist conserved thermodynamic variables to predict the subgrid-scale contribution to the mean state of the atmosphere. The multivariate double-Gaussian mixture has been proposed as an appropriate model for PBL turbulence and shallow convection, but it is unable to reproduce important features of shallow cumulus convection. In this study, a novel unified PBL turbulence–convection–cloud macrophysics scheme is presented based on the eddy-diffusivity/mass-flux framework. The new scheme augments the double-Gaussian representation of subgrid variability with multiple stochastic mass-flux plumes at minimal added computational cost. Improved results for steady-state maritime and transient continental shallow convection from a single-column model implementation of the new scheme are shown with respect to reference large-eddy simulations. Improvements are seen in the cloud layer due to mass-flux plumes occupying the extreme moist, low liquid-water potential temperature tail of the joint temperature–moisture distribution.

Significance Statement

Computer models of the atmosphere used to predict future climate are unable to directly represent air motion at small spatial scales because it would take too long to run the model over the entire planet. Instead, models typically use coarse model grid spacing and a simplified statistical representation of the physical processes that cause small-scale motions. This paper improves a particular simplified representation by adding a mechanism to represent statistically rare events of strong small-scale air motion that coherently transport air from near the surface to higher in the atmosphere. This increased transport also improves the representation of clouds, a particularly difficult phenomenon to simulate in models.

Suselj’s current affiliation: Running Tide, Portland, Maine.

Corresponding author: Mikael K. Witte, mikael.witte@nps.edu

Abstract

Significance Statement

Suselj’s current affiliation: Running Tide, Portland, Maine.

Corresponding author: Mikael K. Witte, mikael.witte@nps.edu

Keywords: Clouds; Convective-scale processes; Small scale processes; Turbulence; Boundary layer; Cloud cover; Cumulus clouds; Climate models; Cloud parameterizations; Convective parameterization; Parameterization; Single column models; Stochastic models; Subgrid-scale processes

1. Introduction

As a computational necessity, numerical models of the atmosphere used for operational prediction of weather and climate resolve spatial scales much larger than the characteristic scale(s) of many of the physical processes the models must represent. This is especially true of climate projections performed with general circulation models (GCMs), for which simulations span decades to centuries and horizontal grid spacing is minimally on the of order tens of kilometers. To maximize computational efficiency and account for the disconnect in resolved versus physically relevant spatial scales, parameterizations are required to encapsulate the effects of unresolved physics.

GCM parameterizations fall in two broad categories: those describing subgrid-scale fluid dynamics (e.g., convection, gravity waves, small-scale turbulence) and other physical processes that represent unresolved thermodynamic source terms in the primitive equations (e.g., radiation, cloud macro- and microphysics, chemistry). Some of these parameterizations explicitly account for subgrid-scale variability by the use of assumed probability distributions. Early examples were often specific to predicting cloud macrophysical quantities (e.g., partial cloudiness) from thermodynamic variables (Sommeria and Deardorff 1977; Mellor 1977; Bougeault 1981, 1982). More recent efforts also unify subgrid dynamical processes such as turbulence and convection with cloud macrophysics (e.g., Lappen and Randall 2001; Golaz et al. 2002; Tompkins 2002; Bogenschutz and Krueger 2013; Suselj et al. 2013, 2019b; Park 2014; Tan et al. 2018; Cohen et al. 2020) since it is critical to represent physics-dynamics feedbacks. A hallmark of these “unified” schemes is the attempt to use an internally consistent multivariate assumed probability distribution (AD) to describe the suite of processes represented, rather than a disjoint set of siloed ADs that may be mutually inconsistent. Unified schemes are also designed to seamlessly simulate transitions among cloud and convective regimes, e.g., from stratocumulus to shallow cumulus, without the use of artificial “trigger functions” that can create nonphysical discontinuities in space and time (e.g., Suselj et al. 2019b).

A particular subclass of unified schemes combines an AD with a higher-order closure (AD-HOC) model for turbulence and macrophysics (e.g., Lappen and Randall 2001; Golaz et al. 2002; Bogenschutz and Krueger 2013). The ADs for such schemes are simple and analytically straightforward (e.g., double delta or double Gaussian) such that the choice of distribution shape constrains the closure problem without creating an impractical number of additional prognostic variables.

AD-HOC schemes assume relationships among the unclosed higher-order moments of the ADs, which can lead to biases (Firl and Randall 2015; Fitch 2019; Huang et al. 2020). Firl and Randall (2015) diagnostically evaluated the ability of a double-Gaussian mixture to capture resolved variability from large-eddy simulations (LES), showing that while the double-Gaussian AD performed well for a stratocumulus case and, perhaps surprisingly, in deep convection as well, the scheme significantly underdiagnosed higher-order moments such as $\bar{{w^{'}}^{3}}$ and $\bar{w^{'} θ_{υ}^{'}}$ in shallow convection. Fitch (2019) also found that the double-Gaussian AD has major challenges in representing shallow convection, particularly with respect to the long updraft tail and therefore higher-order moments (e.g., skewness and kurtosis) of the vertical velocity distribution. Huang et al. (2020) directly evaluated the simplifying assumptions of the double-Gaussian AD used in a specific AD-HOC parameterization, the Cloud Layers Unified By Binormals (CLUBB) scheme (Golaz et al. 2002; Larson and Golaz 2005). Using LES of the transition from shallow to deep convection over land, they found that CLUBB typically overpredicts cloud macrophysical properties (i.e., cloud cover and liquid-water amount) and liquid-water flux near cloud base; it simultaneously underpredicts those quantities near cloud top. These biases were particularly pronounced during the simulated period of shallow convection and could be improved by introducing empirical scalar skewness values derived from LES, although the general applicability of such an approach to other convective regimes is not clear.

Another category of unified parameterizations, known as eddy-diffusivity/mass-flux (EDMF) schemes (Siebesma et al. 2007), was originally formulated to unify local turbulence in the non-convective environment, which is traditionally parameterized by downgradient diffusion using an eddy-diffusivity (ED) parameterization, and nonlocal transport via strong convective plumes that are represented using a mass-flux (MF) model (for a comprehensive overview of MF models, see Arakawa 2004). Most modern EDMF schemes now diagnose macrophysics as well. In the multi-plume EDMF framework, transport due to the ED and MF components is combined in a “unified” solver, i.e., for a generic thermodynamic scalar φ, the tendency due to turbulent transport

\bar{w^{'} φ^{'}}

can be given by

\frac{\partial \bar{φ}}{\partial t} = - \frac{\partial \bar{w^{'} φ^{'}}}{\partial z},

(1)

and the total turbulent transport is given by

\bar{w^{'} φ^{'}} = - K \frac{\partial \bar{φ}}{\partial z} + \sum_{i = 0}^{I} a_{i} (w_{i} - \bar{w}) (φ_{i} - \bar{φ}),

(2)

where K is eddy diffusivity, a_i is the fractional area of the ith updraft plume, w_i is the vertical velocity of the ith plume, and the overbar denotes the grid mean. Note that Eq. (2) follows the approximate

\bar{w^{'} φ^{'}}

definition of Siebesma et al. [2007, their Eq. (4)] and we omit a multiplicative factor, the non-convective area fraction

a_{e} = 1 - \sum a_{i}

, applied to the ED term since we assume a_e ≈ 1.

There are many examples of EDMF implemented in operational weather forecast models (e.g., Köhler et al. 2011; Suselj et al. 2014, 2021; Han et al. 2016; Olson et al. 2019), and each of these implementations is fairly unique because EDMF is a flexible framework that is not necessarily tied to a particular scheme for either the ED or MF component. In general, though, the cloud macrophysical state of the non-convective environment is diagnosed from an AD-type scheme (often assuming a normal distribution of thermodynamic variables) while the MF plumes are “top-hats” (i.e., a horizontal slice through a plume is characterized by uniform thermodynamic and dynamic properties) whose fractional area and condensate mixing ratio are summed to describe the convective macrophysics. Microphysics in the environment is typically handled by an external “stratiform” microphysics scheme, many of which also use ADs to predict precipitation growth rates (e.g., Morrison and Gettelman 2008), although unified schemes that include microphysics have recently been proposed (Larson et al. 2005; Larson and Schanen 2013; Storer et al. 2015; Thayer-Calder et al. 2015; Suselj et al. 2022).

As currently formulated, nearly all state-of-the-art GCMs utilize some combination of AD, HOC, ED, and MF schemes to parameterize planetary boundary layer (PBL) turbulence, shallow convection, deep convection, and cloud macrophysics. The prevailing trend in GCM development is toward increased unification of these interrelated processes, and this study represents another step in that direction. Given the adoption of AD-HOC schemes in recent years and the shortcomings of the double-Gaussian AD in representing shallow convection, we propose a new class of unified scheme combining an AD-HOC scheme with a MF model to create what we term an AD-HOC-MF unified parameterization. We show that the addition of MF to a double-Gaussian-based AD-HOC scheme in the Community Atmosphere Model (CAM) increases turbulent transport to the top of the convective layer, thus improving the representation of shallow convection. Furthermore, AD-HOC-MF is computationally economical because it introduces no new prognostic variables to the host model beyond the requirements of the AD-HOC scheme. In the future, this parameterization will be extended to represent deep convection as well, given the promising results of Suselj et al. (2019b) in representing deep convection with EDMF.

The remainder of the paper is organized as follows. Section 2 describes the single-column CAM (SCAM) configuration, AD-HOC-MF implementation and LES model used as a reference. Results from the SCM and LES are presented in section 3, followed by discussion in section 4. Finally, concluding remarks are given in section 5.

2. Model description

The AD-HOC-MF parameterization implemented in this study is a combination of the CLUBB scheme (Golaz et al. 2002; Larson et al. 2005; Bogenschutz et al. 2012, 2013) with the stochastic multi-plume mass-flux scheme described in Sušelj et al. (2013), Suselj et al. (2019a) and Kurowski et al. (2019). This implementation differs significantly from Kurowski et al. (2019), who implemented a traditional EDMF scheme in version 5 of CAM (CAM5) by combining the Holtslag–Boville–Rasch PBL parameterization with a diagnostic multi-plume MF scheme. Model output in this study uses the single-column model configuration of CAM6.3 (SCAM; Gettelman et al. 2019) with 256 vertical levels, corresponding to approximately 20-m spacing in the planetary boundary layer. The model time step is set to 50 s, which, with the leapfrog time-integration used by SCAM, equates to a physics and CLUBB+MF time step of 100 s. Both the vertical and temporal resolutions are significantly higher than the standard settings for three-dimensional CAM6.3 simulations, but the purpose of this study is to demonstrate the fidelity of SCAM output with respect to external reference solutions from large-eddy simulations that operate at even higher resolution in both space and time.

To simplify the comparison with reference LES output, the SCAM setup deviates from the standard physics configuration (see the appendix) to allow modeling of idealized case studies of non-precipitating shallow convection. That is, the deep convection, gravity wave and radiation schemes are deactivated; cloud drop number concentration is set to a fixed value of 10¹⁰ m⁻³ to minimize precipitation; and surface fluxes and temperature are prescribed following the idealized case setup. Large-scale microphysics and aerosol processes are active, although their effect is minimal because cloud drop number concentration is fixed. Finally, we nudge the model temperature field following the default SCAM configuration (Gettelman et al. 2019) to a time-varying, case-specific profile with a time scale of 10 days at the surface, linearly decreasing to 2 days above 200 hPa; moisture freely evolves. The long nudging time scale at the surface results in minimal impact on the evolution of the PBL and low altitude convective cloud layer for the cases studied while the upper atmosphere does not drift from the specified input sounding. Compare this to the approach taken in LES, where a specified upper air sounding is typically imposed above a model domain top of a few km for simulations of shallow convection.

a. Brief description of CLUBB

CLUBB is an AD-HOC scheme that uses a mixture of two Gaussians to represent planetary boundary layer turbulence, shallow convection, and cloud macrophysics (Golaz et al. 2002; Larson and Golaz 2005). The CLUBB and MF components share a set of prognostic moist conserved thermodynamic variables: liquid-water potential temperature θ_l and total water mixing ratio q_t. They are defined as follows:

θ_{l} = \frac{1}{Π} (T - \frac{L_{υ}}{c_{p}} q_{c}),

(3)

q_{t} = q_{υ} + q_{c},

(4)

where T is temperature, L_υ is the latent heat of vaporization, c_p is the specific heat capacity of air, q_c is the mixing ratio of cloud liquid water, and q_υ is the mixing ratio of water vapor. The Exner function is defined as

Π = {(p / p_{0})}^{R_{d} / c_{p}}

where p is pressure, p₀ is a reference pressure (in CAM, sea level pressure; in LES, p₀ = 1 × 10⁵ Pa), and R_d is the gas constant of dry air. For plotting purposes, CAM θ_l is recomputed with reference to the LES p₀.

Bogenschutz et al. (2012) first implemented CLUBB in CAM5 and showed significant improvement in low cloud properties in case study simulations with SCAM as well as in global simulations of present-day climate (Bogenschutz et al. 2013). The scheme is now the default parameterization for PBL turbulence, shallow convection, and cloud macrophysics in CAM6 (Danabasoglu et al. 2020). CLUBB predicts nine prognostic higher-order moments that contribute to the time tendencies of CAM’s prognostic thermodynamic variables and are used to diagnose cloud macrophysical properties. In the rest of the paper, an overline above a variable denotes the mean over a gridbox or a subdomain thereof. The CLUBB prognostic variables are as follows: variances of horizontal momentum components $\bar{{u^{'}}^{2}}$ and $\bar{{υ^{'}}^{2}}$ ; vertical velocity variance $\bar{{w^{'}}^{2}}$ ; the third moment of vertical velocity $\bar{{w^{'}}^{3}}$ ; the variance of liquid-water potential temperature $\bar{{θ^{'}}_{l}^{2}}$ ; the variance of total water mixing ratio $\bar{{q^{'}}_{t}^{2}}$ ; the covariance of θ_l and q_t, $\bar{θ_{l}^{'} q_{t}^{'}}$ ; and the turbulent fluxes of thermodynamic variables $\bar{w^{'} θ_{l}^{'}}$ and $\bar{w^{'} q_{t}^{'}}$ . The turbulent momentum fluxes $\bar{u^{'} w^{'}}$ and $\bar{υ^{'} w^{'}}$ are diagnosed (Golaz et al. 2002; although these may soon be prognostic as well, Larson et al. 2019) and all third-order and higher moments besides $\bar{{w^{'}}^{3}}$ are obtained from the AD closure. No changes are made to the internal structure of CLUBB in this study and we use the default CLUBB parameter tuning for CAM6.3 in the SCAM simulations shown. We intentionally choose not to perturb CLUBB tunable parameters to demonstrate how the naïve coupling approach described in section 2c results in improved SCAM simulations of shallow cumulus.

Of greatest interest in this work are the budget equations for $\bar{{w^{'}}^{2}}$ and the variances and turbulent fluxes of thermodynamic quantities. We therefore reproduce and annotate these equations here because we will refer to them later in a discussion of how a deeper coupling of CLUBB and MF may be pursued than that presented in this study. We omit the budget of the thermodynamic covariance $\bar{θ_{l}^{'} q_{t}^{'}}$ for brevity since it is not discussed.

The budget of vertical velocity variance

\bar{{w^{'}}^{2}}

is given by

\frac{\partial \bar{{w^{'}}^{2}}}{\partial t} = - \bar{w} \frac{\partial \bar{{w^{'}}^{2}}}{\partial z} - \frac{1}{ρ_{0}} \frac{\partial ρ_{0} \bar{{w^{'}}^{3}}}{\partial z} - 2 \bar{{w^{'}}^{2}} \frac{\partial \bar{w}}{\partial z} + \frac{2 g}{θ_{υ, 0}} \bar{w^{'} θ_{υ}^{'}} - \frac{2}{ρ_{0}} \bar{w^{'} \frac{\partial p^{'}}{\partial z}} - ϵ_{w w},

(5)

where

\bar{w}

is the mean vertical velocity (from the host model), ρ₀ is reference air density, g is the acceleration due to gravity, θ_υ_,0 is a reference virtual potential temperature,

\bar{w^{'} θ_{υ}^{'}}

is the diagnostic turbulent buoyancy flux,

\bar{w^{'} (\partial p^{'} / \partial z)}

is the pressure correlation term and ϵ_ww is dissipation. The right-hand side (rhs) terms represent (in order): mean advection, turbulent advection, accumulation, buoyancy production, pressure correlation and dissipation. The closed equation for the pressure correlation term for

\bar{{w^{'}}^{2}}

includes contributions from shear production and horizontal momentum variances and is expressed as

\begin{array}{l} - \frac{2}{ρ_{0}} \bar{w^{'} \frac{\partial p^{'}}{\partial z}} = - \frac{C_{4}}{τ_{1}} (\bar{{w^{'}}^{2}} - \frac{2}{3} \bar{e}) - C_{5} (- 2 \bar{{w^{'}}^{2}} \frac{\partial \bar{w}}{\partial z} + \frac{2 g}{θ_{υ, 0}} \bar{w^{'} θ_{υ}^{'}}) \\ + \frac{2}{3} C_{5} (\frac{g}{θ_{υ, 0}} \bar{w^{'} θ_{υ}^{'}} - \bar{u^{'} w^{'}} \frac{\partial \bar{u}}{\partial z} - \bar{υ^{'} w^{'}} \frac{\partial \bar{υ}}{\partial z}), \end{array}

(6)

where C₄ and C₅ are tunable constants, τ₁ is a dissipation time scale,

\bar{e} = (1 / 2) (\bar{{u^{'}}^{2}} + \bar{{υ^{'}}^{2}} + \bar{{w^{'}}^{2}})

is turbulent kinetic energy (TKE), and

- \bar{u^{'} w^{'}} (\partial \bar{u} / \partial z) - \bar{υ^{'} w^{'}} (\partial \bar{υ} / \partial z)

is shear production of turbulence. We note that in the original formulation of CLUBB, TKE was only dependent on

\bar{{w^{'}}^{2}}

because the horizontal momentum variances were not considered. The full three-dimensional expression for TKE was first introduced in Larson et al. (2012).

The variance budget of a generic scalar φ is given by

\frac{\partial \bar{{φ^{'}}^{2}}}{\partial t} = - \bar{w} \frac{\partial \bar{{φ^{'}}^{2}}}{\partial z} - \frac{1}{ρ_{0}} \frac{\partial ρ_{0} \bar{w^{'} {φ^{'}}^{2}}}{\partial z} - 2 \bar{w^{'} φ^{'}} \frac{\partial \bar{φ}}{\partial z} - ϵ_{φ φ},

(7)

where

\bar{w^{'} {φ^{'}}^{2}}

is a diagnostic third-order moment and ϵ_φφ is a dissipation term. The rhs terms represent (in order): mean advection, turbulent advection, turbulent production, and dissipation. And finally, the turbulent flux budget of φ is given by

\frac{\partial \bar{w^{'} φ^{'}}}{\partial t} = - \bar{w} \frac{\partial \bar{w^{'} φ^{'}}}{\partial z} - \frac{1}{ρ_{0}} \frac{\partial ρ_{0} \bar{{w^{'}}^{2} φ^{'}}}{\partial z} - \bar{{w^{'}}^{2}} \frac{\partial \bar{φ}}{\partial z} - \bar{w^{'} φ^{'}} \frac{\partial \bar{w}}{\partial z} + \frac{g}{θ_{υ, 0}} \bar{φ^{'} θ_{υ}^{'}} - \frac{1}{ρ_{0}} \bar{φ^{'} \frac{\partial p^{'}}{\partial z}} - ϵ_{w φ},

(8)

where

\bar{{w^{'}}^{2} φ^{'}}

is a diagnostic third moment,

\bar{φ^{'} θ_{υ}^{'}}

is the diagnostic covariance of φ and θ_υ,

\bar{φ^{'} (\partial p^{'} / \partial z)}

is the pressure correlation term, and ϵ_wφ is dissipation. Similar to the

\bar{{w^{'}}^{2}}

budget, rhs terms represent (in order): mean advection, turbulent advection, turbulent production, accumulation, buoyancy production, pressure correlation, and dissipation.

b. Brief description of the mass-flux scheme

The multi-plume mass-flux (MF) scheme closely follows that described by Suselj et al. (2013), Suselj et al. (2019a), and Kurowski et al. (2019). Assuming steady state, multiple diagnostic plumes are initialized at the surface and integrated upward to the level of zero vertical velocity with the following equation:

\frac{1}{2} \frac{\partial w_{i}^{2}}{\partial z} = a_{w} B_{i} - b_{w} ε_{i} w_{i}^{2},

(9)

where a_w = 1 and b_w = 1.5 are constants,

B_{i} = g [(θ_{υ, i} / \bar{θ_{υ}}) - 1]

is the buoyancy of the ith plume, and ε_i is the entrainment rate.

Initialization of plumes follows the method of Suselj et al. (2019a). Specifically, plumes are deterministically initialized when grid-mean surface buoyancy flux ${\bar{w^{'} θ_{υ}^{'}}}_{sfc} > 0$ with values of (w, θ_l, q_t) drawn from the right tail of an assumed single Gaussian distribution. Surface vertical velocity spans the range $1.5 σ_{w} < w_{i} - \bar{w} < 3 σ_{w}$ in N evenly spaced intervals. Integration over this range results in a total fractional area of 6.5% at the surface (and therefore individual plume fractional area a_i = 0.065% with N = 100), lower than the 15% used in Kurowski et al. (2019) who extended the minimum updraft intensity to 1σ_w. This choice is justified by Suselj et al. (2019a), who explored the sensitivity of an EDMF scheme to MF surface plume area and found that results were relatively insensitive to fractional area > 5%. A more consistent approach would be to sample the right tail of CLUBB’s “updraft” Gaussian, but CLUBB PDF parameters are diagnosed after the MF plumes are integrated thus this is left for future development. In any case, the approach taken follows Suselj et al. (2019a), who found little sensitivity of plume evolution to surface updraft properties in their results.

Plume initial thermodynamic conditions are then generated as

θ_{υ, i, sfc} = {\bar{θ_{υ}}}_{, sfc} + c (w, θ_{υ}) w_{i, sfc} \frac{σ_{θ_{υ}}}{σ_{w}},

(10)

q_{t, i, sfc} = {\bar{q_{t}}}_{, sfc} + c (w, q_{t}) w_{i, sfc} \frac{σ_{q_{t}}}{σ_{w}},

(11)

where the covariance c(w, θ_υ) = c(w, q_t) = 0.58 and the surface standard deviations (σ_w, σ_θυ, σ_qt) are given by the following, assuming a velocity scale

w_{*}^{3} = (g / θ_{υ}) {\bar{w^{'} θ_{υ}^{'}}}_{sfc} z_{top}

, where z_top is the CAM-diagnosed PBL height:

σ_{w} = 0.57 w_{*},

(12)

σ_{q_{t}} = 2.89 \frac{{\bar{w^{'} q_{t}^{'}}}_{sfc}}{w_{*}},

(13)

σ_{θ_{υ}} = 2.89 \frac{{\bar{w^{'} θ_{υ}^{'}}}_{sfc}}{w_{*}} .

(14)

We note that the surface fluxes (i.e., ${\bar{w^{'} θ_{υ}^{'}}}_{sfc}$ and ${\bar{w^{'} q_{t}^{'}}}_{sfc}$ ) are not taken from CLUBB but rather are externally imposed; the MF scheme has no direct dependence on CLUBB covariances.

Lateral entrainment is represented as a stochastic process and is expressed as

ε_{i} (Δ z) = \frac{ε_{0}}{Δ z} P_{i} (\frac{Δ z}{L_{ε}}),

(15)

where ε_i is the entrainment rate in the ith plume, Δz is vertical grid spacing, ε₀ the fractional entrainment rate per event, L_ε the average distance between entrainment events and

P_{i} (λ)

a random number drawn from a Poisson distribution with parameter λ. The entrainment parameters ε₀ and L_ε represent one of the greatest sources of model uncertainty (Suselj et al. 2020; Smalley et al. 2022). The fractional entrainment rate ε₀ is set to a constant value of 0.2 and L_ε is tuned on a case-by-case basis for the simulations presented in this study. Work is currently underway to implement a diagnostic L_ε formulation that depends on the convective mixed layer depth following a previous EDMF implementation in CAM5 (Kurowski et al. 2019). Condensation occurs via saturation adjustment when plume total water mixing ratio q_ti exceeds the saturation vapor mixing ratio q_si(T_i, p) and the MF cloud fraction is simply the sum of the fractional area of plumes with nonzero condensate. Cases simulated for this study contain only warm clouds, thus the ice phase is neglected.

c. Coupling of CLUBB and MF

CLUBB and MF are coupled via the mean thermodynamic fields ( $\bar{θ_{l}}$ , $\bar{q_{t}}$ ) in the CLUBB diffusion solver. The underlying concept is similar to the eddy diffusivity/mass-flux (EDMF) approach (Siebesma et al. 2007; Sušelj et al. 2013; Tan et al. 2018), but the application is specific to this implementation. The “minimalist” coupling approach is typical of EDMF parameterizations and in this study is taken to demonstrate the feasibility of the AD-HOC-MF framework. Given the breadth of higher-order moments prognosed by AD-HOC schemes, it is possible to introduce MF terms directly into the tendencies of turbulent fluxes, scalar variances, or any other prognosed higher-order moments of the AD (i.e., $\bar{{w^{'}}^{2}}$ , $\bar{{w^{'}}^{3}}$ , $\bar{θ_{l}^{'} q_{t}^{'}}$ ). Future work will evaluate the theoretical basis for a broader coupling and seek to quantify the contribution of coherent dynamical structures on grid-mean higher-order moments, e.g., by extending the flux partitioning method of Chinita et al. (2018) to a joint double Gaussian.

Briefly, we adopt an EDMF-like decomposition of the total turbulent flux of a generic scalar φ:

\bar{w^{'} φ^{'}} = {\bar{w^{'} φ^{'}}}_{CLUBB} + \sum_{i = 1}^{I} a_{i} (w_{i} - \bar{w}) (φ_{i} - \bar{φ}),

(16)

where

{\bar{w^{'} φ^{'}}}_{CLUBB}

is the prognostic flux from CLUBB and the rightmost term is the mass-flux transport due to i = 1, …, I subgrid-scale plumes. A large number of plumes per time step are used in this study (I = 100) since CAM is run in single-column mode. In an operational three-dimensional configuration for which a smaller number of plumes would be used [i.e.,

I \sim O (10)

], intermittency arising from a small plume number (Suselj et al. 2019a) is partially smoothed by the CAM physics sub-stepping procedure: CLUBB and large-scale cloud microphysics are advanced on a 5-min (300 s) time step while the full physics time step is 20–30 min (1200–1800 s) such that CLUBB+MF is integrated 4–6 times per full time step, respectively. The stochastic entrainment process for MF plumes is uncorrelated across time steps, which likely contributes to intermittency.

By default, CLUBB’s prognostic equations for a thermodynamic mean field and its associated turbulent flux are advanced simultaneously using a five-diagonal banded matrix and a semi-implicit time stepping method (Larson 2020). The left-hand-side matrix contains the implicit terms while the right-hand-side vector contains explicit terms. Thermodynamic forcing due to the MF scheme is incorporated as an explicit term on the mean field bands of the five-diagonal banded matrix; no MF terms appear in the turbulent flux bands. This maintains separation of the AD-HOC and MF contributions to total flux; while it is expected that the AD-HOC contribution to total turbulent fluxes will adjust due to the mean field coupling, there is no explicit contribution from MF. The MF forcing term applied to the mean field tendency is the divergence of the convective transport in Eq. (16), omitting terms at the surface since surface fluxes are applied externally. The implementation discussed herein assumes that plume area is much smaller than a grid cell and is therefore neglected except in computing turbulent fluxes and grid-mean cloud fraction. For the latter, the MF contribution is simply taken as the sum of moist plume area.

The current formulation allows CLUBB and MF to be coupled at the interface between CLUBB and CAM rather than within CLUBB. A major advantage of this approach is that it maintains the modularity of the CLUBB and MF codebases; i.e., minor changes to the CLUBB software do not impact the coupling strategy for CLUBB+MF. Finally, diagnostics presented as “combined” quantities are summed in postprocessing, e.g., total turbulent fluxes of θ_l, θ_υ, and q_t; total cloud fraction and condensate mixing ratio.

d. Cases simulated

For the initial implementation of CLUBB+MF, it is most important to evaluate changes in turbulent transport and cloud macrophysics without the complication of microphysical interactions from precipitation or condensate phase changes (e.g., freezing of liquid hydrometeors). Accordingly, we focus on two cases of non-precipitating warm shallow convection: the BOMEX case (Siebesma et al. 2003) is a well-known example of near-steady-state non-precipitating shallow maritime convection and the ARM shallow cumulus case (Brown et al. 2002) typifies non-precipitating shallow convection over land. The BOMEX case is preconfigured for use in SCAM (Gettelman et al. 2019) and the ARM case setup was taken from the E3SM single-column model library (Bogenschutz et al. 2020), although it has previously been simulated in SCAM (Kurowski et al. 2019). The MF entrainment parameter L_ε is set to 75 m for the BOMEX case and 150 m for the ARM case. Other shallow convection cases are available in SCAM (cf. Table 1 of Gettelman et al. 2019) such as ATEX (marine cumulus rising into stratocumulus) or CGILS S6 (another idealized trade-cumulus case), but these cases are not as easily interpretable because cloud field evolution is rather sensitive to other model parameterizations such as microphysics and radiation. We do not assess stratocumulus as our focus is on shallow cumulus convection.

e. Large-eddy simulation model

High-resolution LES model output is used as reference data for parameterization development. The LES model of Matheou and Chung (2014) is used to simulate the BOMEX and ARM cases. The LES model forcing follows Siebesma et al. (2003) and Brown et al. (2002), respectively. In both cases the domain is 10.24 km in the horizontal. The domain height is 3 km in the LES of BOMEX and 4.5 km in the LES of ARM. The grid spacing is isotropic Δx = Δy = Δz = 20 m. The LES domain is doubly periodic in the horizontal directions. Surface fluxes are prescribed and a Rayleigh damping layer is used in the top 500 m of the domain to limit undesirable gravity wave reflections. The sixth-order centered fully conservative finite-difference scheme of Morinishi et al. (1998) adapted for the anelastic approximation (Matheou et al. 2016) is used for momentum and scalar advection. The buoyancy adjusted stretched vortex subgrid scale turbulence closure is used to account for the effects of unresolved turbulence. Following the case descriptions (Brown et al. 2002; Siebesma et al. 2003) precipitation is not included in the LES model and all water condensate is assumed suspended using an “all or nothing” saturation adjustment scheme based on the local mean state in each grid cell. The simulations are carried out in the frame of reference of the domain-mean wind to reduce numerical errors (Lamaakel and Matheou 2021). The LES model was successfully used in several previous studies spanning a diverse set of meteorological conditions (Chung et al. 2012; Matheou and Chung 2014; Matheou 2018; Matheou and Teixeira 2019; Couvreux et al. 2020; Chinita et al. 2022).

For future reference, the LES and SCM use identical surface fluxes, initial conditions and large-scale forcing from the reference case descriptions. Thus, the LES results (represented in solid black) are used as the benchmark to which the single-column model is compared in all the figures that follow.

3. Results

a. Trade-wind shallow cumulus

Results for the BOMEX case are shown in Fig. 1 and demonstrate improvements in the profiles of turbulent fluxes and macrophysical quantities. Beginning with the thermodynamic profiles, SCAM profiles of $\bar{θ_{l}}$ and $\bar{q_{t}}$ closely match the LES for both CLUBB-only and CLUBB+MF configurations. The CLUBB+MF simulation mixes slightly too vigorously in the cloud layer, as evidenced by the slightly moister region from 1200 to 1500 m (see insets of top-left panels, Fig. 1); this is confirmed by the substantially stronger fluxes of θ_l and q_t relative to CLUBB alone (bottom-left panels, Fig. 1). Despite this, the shoulder in the $\bar{θ_{l}}$ and $\bar{q_{t}}$ curves produced by CLUBB+MF just below 1500 m is consistent with the LES profile. This slight recalibration of PBL thermodynamic profiles from the addition of MF, though it appears minor in magnitude, leads to noticeable improvements in the macrophysical fields (top-right panels, Fig. 1), for which mean cloud water mixing ratio $\bar{q_{c}}$ and cloud fraction f_c are about 50% larger for CLUBB+MF than CLUBB alone and are in much better agreement with LES.

Fig. 1.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

It is clear from the turbulent flux and cloud macrophysics profiles in Fig. 1 that there is a major adjustment to the CLUBB profiles with MF active. The partitioning of total flux between CLUBB and MF in the CLUBB+MF simulation is broadly consistent with what is seen in the EDMF decomposition (e.g., Suselj et al. 2019a; Kurowski et al. 2019; Cohen et al. 2020): CLUBB dominates transport in the subcloud layer and has a decreasing contribution with height in the conditionally unstable cumulus cloud layer. In contrast with a traditional ED scheme, CLUBB produces nonzero counter-gradient transport in this region.

Curiously, there is relatively little change in buoyancy flux for CLUBB+MF versus CLUBB alone, with the main difference being a slightly stronger cloud layer peak in the CLUBB+MF simulation. This minor change in CLUBB+MF buoyancy flux is a consequence of adjustments in CLUBB prognostic fluxes and macrophysics, the latter of which are diagnosed from the double-Gaussian AD. The reductions in CLUBB $\bar{{θ^{'}}_{l}^{2}}$ and $\bar{{q^{'}}_{t}^{2}}$ shown in Fig. 2 arise from a complicated feedback in the CLUBB budgets for scalar variances and turbulent fluxes. The turbulent production term of the scalar variance budget is the product of the turbulent flux and the mean field gradient [Eq. (7)]. The flux budget turbulent production term [Eq. (8)] also depends on the mean field gradient (multiplied by the vertical velocity variance). Thus, a reduction in mean field gradient magnitude due to MF mixing results in lower CLUBB fluxes and scalar variances. This also explains the reduction in CLUBB’s contribution to cloud fraction and $\bar{q_{c}}$ , for which the CLUBB AD closure has a strong dependence on $\bar{{θ^{'}}_{l}^{2}}$ and $\bar{{q^{'}}_{t}^{2}}$ . It is possible that retuning of some subset of the many CLUBB parameters [i.e., those relating to the dissipation term in Eq. (7)] could reduce the adverse impact of MF on CLUBB variances. We have not done this because the purpose of this work is to investigate whether there is a benefit of combining CLUBB+MF; thus a “naïve,” untuned coupling is a better demonstration of this potential.

Fig. 2.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

While the CLUBB scalar variances are of smaller magnitude for the CLUBB+MF simulation, the variances from the CLUBB-only simulations do not satisfactorily reproduce the LES variances of temperature and moisture, either. To take a broader view of the joint distribution of q_t and θ_l, Fig. 3 shows a snapshot of the joint probabilities of $θ_{l}^{'}$ and $q_{t}^{'}$ (for a generic variable φ, the perturbation $φ^{'} = φ - \bar{φ}$ ) in the cloud layer at three altitudes (subcloud, near cloud base, and midcloud) at t = 6 h. It can be seen that the number of plumes decreases with height. This is an expected consequence of the formulation of the vertical velocity equation: plumes terminate when their vertical velocity reaches 0 and the “survival” of a plume (especially above the PBL) primarily depends on the number of entrainment events it experiences (e.g., Suselj et al. 2019a).

Fig. 3.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

Figure 3 also demonstrates the extent to which the LES distribution is not adequately simulated by CLUBB for either the CLUBB+MF or CLUBB-only configurations. Conversely, the MF plumes (black dots) are located at the extremes of the LES joint distribution, demonstrating that they successfully capture the more skewed region of the LES distribution, consistent with other EDMF implementations (e.g., Suselj et al. 2013). The plumes naturally occupy the $θ_{l}^{'}$ < 0, $q_{t}^{'}$ > 0 quadrant of the joint $θ_{l}^{'} - q_{t}^{'}$ PDF because they bring their entrainment-diluted initial properties (the mean surface properties are shown as red crosses in Fig. 3) to the cloud layer. The only region in which the plume ensemble has uniformly greater θ_l and q_t than the mean surface properties is in the subcloud layer, where most plumes have yet to experience any entrainment and the initial surface perturbations are still apparent in a well-mixed environment. It is notable that the LES distribution has such a $θ_{l}^{'}$ > 0, $q_{t}^{'}$ > 0 lobe as well, while CLUBB does not reproduce this feature. The overlap of LES and plume distributions in this quadrant is evidence that the plume distributions are properly initialized.

In the cloud layer, the updrafts essentially fall on a mixing line between the mean surface properties and the mean environmental properties at a given level. The plume-environment contrast initially increases as the plumes rise (especially in the cloud layer, which is characterized by large vertical gradients in θ_l and q_t), and the role of entrainment is to decrease this contrast. As the MF points occupy the moistest, lowest θ_l portion of the LES distribution in the cloud layer, the plumes provide an outsize contribution to turbulent fluxes, cloud cover and q_c despite their small fractional area. Note also the similar orientations of the LES, CLUBB and MF distributions in the cloud layer, indicating that the MF scheme satisfactorily captures the LES-generated covariance of q_t and θ_l.

In summary, CLUBB+MF is in better agreement with LES and produces stronger turbulent fluxes, greater cloud cover and more condensate than CLUBB alone. Changes in CLUBB’s contribution to the macrophysical fields can be traced to a combination of increased mixing efficiency from MF in the PBL (i.e., reduced gradient magnitude), decreases in CLUBB turbulent fluxes via the prognostic scalar variance budgets and the ability of MF plumes to populate the extreme tail of the LES joint (θ_l, q_t) distribution by coherently transporting surface/PBL properties to the conditionally unstable cloud layer.

b. Continental shallow cumulus

While much can be learned analyzing the steady-state behavior of SCAM, recent studies such as Huang et al. (2020) demonstrated the shortcomings of double-Gaussian-based AD-HOC schemes in representing transient episodes of shallow convection. The ARM case is representative of such events. Thus, results for this case are presented as time–height curtain plots.

Beginning again with the mean thermodynamic fields in Figs. 4 and 5, difference plots of SCAM minus LES show broadly similar trends for the two SCAM configurations, consistent with the steady-state results from BOMEX. The “error” relative to LES is smaller for the CLUBB+MF simulation for both θ_l and q_t, with the greatest decrease in error magnitude for CLUBB+MF in the region near 2 km during the last 4 h of the simulations (i.e., during the period of peak cumulus convection). Additionally, SCAM captures the general features of the LES PBL evolution, though there are some inconsistencies. Specifically, the SCM simulations accumulate heat and moisture more quickly than LES in the PBL and the SCAM simulations have considerably higher θ_l above 2500 m. The latter is due to the stretched SCAM grid, but the former is ostensibly due to physics. In addition, the SCAM simulations do not adequately ventilate the PBL later in the run while the LES does, thus leaving a rather moist PBL in the SCAM runs after 10 h.

Fig. 4.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

Fig. 5.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

We next turn to the cloud macrophysical variables, cloud fraction (Fig. 6) and $\bar{q_{c}}$ (Fig. 7). Neither SCAM configuration matches the maximum magnitude of cloud fraction from LES. Nevertheless, CLUBB+MF produces a deeper cloud field overall (by ≈300 m during the period of deepest cloud from 11 to 12 h) than CLUBB alone with a significantly deeper layer of f_c > 0.03. The evolution of $\bar{q_{c}}$ broadly mimics that of f_c, with maximum $\bar{q_{c}}$ from CLUBB+MF at cloud base similar to CLUBB but lower than LES. Above cloud base, CLUBB+MF shows much improved agreement with LES $\bar{q_{c}}$ . Notably, CLUBB+MF improves the timing of the onset of cloud cover (just before 5 h for LES and CLUBB+MF, about 6 h for the CLUBB-only SCAM configuration).

Fig. 6.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of cloud fraction for the ARM shallow cumulus case.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

Fig. 7.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of mean cloud liquid-water mixing ratio $\bar{q_{c}}$ for the ARM shallow cumulus case.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

As with the BOMEX case, improvements in the cloud macrophysical state of the SCAM simulations relative to LES can be attributed to increased turbulent fluxes in the conditionally unstable layer. This is shown in the curtain plots of $\bar{w^{'} θ_{l}^{'}}$ (Fig. 8) and $\bar{w^{'} q_{t}^{'}}$ (Fig. 9). The curtain plots of $\bar{w^{'} θ_{υ}^{'}}$ are of similar magnitude for the same reasons as discussed for BOMEX (see section 3a). As seen in Fig. 8, the most important difference between CLUBB and CLUBB+MF is the more vigorous transport of PBL thermodynamic quantities (i.e., of low θ_l, high q_t air; expressed as more strongly negative $\bar{w^{'} θ_{l}^{'}}$ for CLUBB+MF) to the convective cloud layer, for which CLUBB+MF more closely follows the LES output while CLUBB is unable to reproduce the minimum at PBL top. Returning to the cloud macrophysical fields (Figs. 6 and 7), the lack of negative θ_l transport in the cloud layer in the CLUBB simulation (along with weaker q_t transport) is likely the cause of the lower cloud top height from CLUBB alone compared to CLUBB+MF and LES. A decomposition of $\bar{w^{'} θ_{l}^{'}}$ similar to that in Fig. 1 (not shown) shows that the increase in transport through the cloud layer is solely due to MF.

Fig. 8.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of liquid-water potential temperature flux $\bar{w^{'} θ_{l}^{'}}$ for the ARM shallow cumulus case.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

Fig. 9.

Time–height curtain plots of (top),(middle) SCAM and (bottom) LES output of total water mixing ratio flux $\bar{w^{'} q_{t}^{'}}$ for the ARM shallow cumulus case.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

The curtains of moisture flux $\bar{w^{'} q_{t}^{'}}$ (Fig. 9) exhibit analogous behavior to $\bar{w^{'} θ_{l}^{'}}$ : increased transport of cooler, moister PBL air into the cloud layer leads to a marked increase of total $\bar{w^{'} q_{t}^{'}}$ in the cloud layer during the latter half of the simulation. This occurs throughout the depth of the cloud layer as a result of simulating a representative population of convective updrafts that terminate at different heights due to stochastic entrainment. The increased moistening of the conditionally unstable cumulus layer is a particularly promising result for future work in which the transition to deep convection is to be simulated.

4. Discussion

The results shown demonstrate that CLUBB+MF improves the representation of shallow cumulus convection relative to the standard CLUBB implementation in SCAM. We attribute this improvement to four aspects of the MF scheme:

Counter-gradient transport is more readily accomplished by the MF plumes than CLUBB. This can be seen in Fig. 10, which shows the fraction of $\bar{w^{'} q_{t}^{'}}$ due to MF plumes as a function of time and height. MF dominates the total moisture flux above cloud base (in the cumulus layer) throughout the simulation while the region with nonzero CLUBB contribution (i.e., with fraction due to MF < 1) approximately corresponds to the well-mixed subcloud layer.
As a result, MF plumes produce more vigorous vertical mixing within and at the top of the convective cloud layer (Figs. 8 and 9), thus deepening the PBL and cumulus layer (Figs. 6 and 7) more effectively than CLUBB alone.
In the cloud layer, the MF plumes populate the long tail of the joint PDF of moisture and temperature with the correct covariance (Fig. 3). The source of this anomalously cool, moist air is the PBL.
The vertical coherence of MF plumes is critical for PBL depth evolution. This point is discussed in greater detail in the following paragraph.

Fig. 10.

Time–height curtain of the fraction of $\bar{w^{'} q_{t}^{'}}$ due to the MF component from the SCAM CLUBB+MF simulation of the ARM shallow cumulus case.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0215.1

Vertical coherence of MF plumes with respect to surface conditions and dynamical structure is a key feature differentiating AD-HOC-MF from purely AD-HOC schemes. Even when a method for representing sub-column heterogeneity is applied to CLUBB (e.g., Larson and Schanen 2013), CLUBB PDF parameters are computed separately at each level. Furthermore, sub-column samples are still drawn from a double-Gaussian distribution, effectively limiting selection of the extreme perturbation temperature and moisture values that characterize areas of active shallow convection. Thus, in CLUBB, the extremes of the joint PDF are dictated by the local higher-order moments (e.g., $\bar{{θ^{'}}_{l}^{2}}$ and $\bar{{q^{'}}_{t}^{2}}$ ). In contrast, MF properties aloft are intrinsically related to the surface at every time step. This explains why MF populates the long tail of the joint thermodynamic PDF: surface properties are modulated by lateral entrainment mixing with the surrounding environment as a plume rises such that, even in the cloud layer, the plume has “memory” of its initial conditions.

Profiles of CLUBB prognostic variables (Figs. 1 and 2) show that CLUBB responds to MF in a consistent fashion. Where MF mixes efficiently, gradients decrease. Decreasing gradient magnitude impacts the turbulent production terms of both turbulent flux and scalar variance budgets. The scalar variances are particularly impacted because the turbulent production term of the budget of a thermodynamic scalar φ is the product $\bar{w^{'} φ^{'}} (\partial \bar{φ} / \partial z)$ [Eq. (7)]. Finally, this leads to reduced cloud fraction and condensate mixing ratio from CLUBB, although for the cases examined, MF compensates, leading to greater total f_c and $\bar{q_{c}}$ than the CLUBB-only solution (Figs. 1, 6, and 7).

We must note that we do not see evidence of the overprediction of cloud macrophysical properties near cloud base observed by, e.g., Huang et al. (2020). In addition, SCAM with CLUBB alone rather significantly underpredicts turbulent fluxes for both the BOMEX and ARM cases. We attribute these differences to the choice of modeling framework: Huang et al. (2020) applied the CLUBB PDF closure directly to LES output with different tunable parameter values than are used in the CAM implementation. The sheer number of parameters and model options available to a CLUBB user makes it all the more important to utilize a consistent model framework when comparing different configurations of CLUBB.

Compared with other recent studies that presented SCM results for the BOMEX and ARM cases using more traditional EDMF parameterizations (Suselj et al. 2019a, BOMEX only; Cohen et al. 2020, both cases), we find improved SCM-LES agreement using CLUBB+MF in terms of the mean thermodynamic fields (especially in the cloud layer), comparable agreement with cloud macrophysical properties, and slightly worse agreement with the profiles of turbulent fluxes and scalar variances. We stress that any comparison of parameterizations across different host models is purely contextual; without implementing schemes in the same host model, results cannot be directly compared. In the results of Suselj et al. (2019a; cf. their results using an updraft surface area of 5%) and Cohen et al. (2020), θ_l and q_t are underpredicted at cloud base and overpredicted at cloud top compared to CLUBB+MF and LES—in other words, both fields are too well-mixed. Despite these biases, Cohen et al. (2020) better match LES variance and turbulent flux profiles than CLUBB+MF. While this is clearly an area for improvement, we note that CLUBB+MF’s shortcomings in simulating higher-order moments have a limited impact on the quantities returned to the host model (viz., mean field tendencies and cloud macrophysical properties). In the cumulus regime the cloud macrophysical properties are strongly dependent on the thermodynamic properties of the plumes, which do not depend on the total scalar variances (Fig. 2), but rather contribute to them.

Looking forward, there are a number of opportunities for future development that are anticipated to extend the applicability of this AD-HOC-MF implementation. First, a dynamic L_ε is desirable, in particular for smoothly simulating the growth/decay of the boundary layer as part of the diurnal cycle (Fig. 8) or the transition to deep convection (Huang et al. 2020). Other recent EDMF formulations solely intended to represent shallow convection have utilized constant L_ε (Suselj et al. 2021) but the results shown here indicate that is not appropriate for accurately representing, e.g., continental shallow cumulus. Another important feature is precipitation in the MF plumes, which is crucial for representing any cloud regime dominated by convective precipitation. Precipitation is also of fundamental importance for parameterizing deep convection, for which the thermodynamic effects (i.e., latent heating in cloud and cold pool formation below cloud) are vital for organizing convection and stabilizing the atmosphere.

Finally, there is further potential for coupling MF to the CLUBB AD. We give one example for future development that involves more tightly coupling MF to CLUBB’s flux and variance prognostics and one regarding plume initialization. MF plumes may be directly coupled to CLUBB’s prognostic turbulent flux budget equations [Eqs. (5) and (8)] for vertical velocity and thermodynamic scalars with the MF contribution as an added explicit turbulent production term. This additional source term is intended to mitigate the reduction of CLUBB fluxes and scalar variances in the current configuration, consequently providing maximal impact from MF plumes with minimal intervention in the internal structure of CLUBB.

To initialize plumes, the current MF implementation uses a single Gaussian with prescribed correlations among the velocity, temperature and moisture marginal distributions based on observational evidence from Mahrt and Paumier (1984). Suselj et al. (2019a) showed relatively weak sensitivity to the details of surface conditions using a similar MF model to that described here, but as land surface models move toward a more heterogeneous representation of surface properties and therefore fluxes (e.g., Simon et al. 2021), this aspect may become more important, especially for continental convection. Using consistent subgrid assumptions at the surface in both CLUBB and MF is important to represent such heterogeneity.

5. Conclusions

A new class of unified convection, turbulence and cloud macrophysics parameterization based on a combination of assumed probability distribution function, higher-order closure, and mass-flux schemes (AD-HOC-MF) is implemented in the Community Atmosphere Model version 6.3. This specific implementation combines CLUBB and a stochastic multi-plume MF model, termed CLUBB+MF. MF is straightforwardly coupled to CLUBB via the mean thermodynamic fields in CLUBB’s diffusion solver. The performance of the new CLUBB+MF scheme is evaluated in single-column mode against LES output for two canonical cases of non-precipitating shallow cumulus convection: quasi-steady-state maritime trade-wind cumulus and the diurnal cycle of continental shallow cumulus. For both cases, CLUBB+MF shows improved agreement with LES in terms of turbulent fluxes and cloud macrophysical characteristics. CLUBB+MF also produces a deeper cloud layer and stronger vertical mixing at the top of the PBL and within the cumulus layer than CLUBB alone. Improved simulation outcomes with CLUBB+MF can be attributed to four aspects of the MF framework: increased counter-gradient transport, more vigorous vertical mixing at PBL top, the ability to capture extremes of the joint moisture–temperature distribution, and vertical coherence of plumes.

While several recent studies have identified shortfalls of the AD-HOC unified parameterization approach (Firl and Randall 2015; Fitch 2019; Huang et al. 2020), to our knowledge this study is the first to propose an economical solution to those issues. With no added prognostic variables, an AD-HOC-MF scheme can populate the updraft tail of the joint (w, θ_l, q_t) distribution in a manner consistent with LES (Fig. 3), thus addressing the main weakness of double-Gaussian-based AD-HOC models in representing shallow convection. Based on recent developments in the EDMF framework (Suselj et al. 2019b, 2021) and a clear path to consistent coupling of the AD-HOC and MF components, AD-HOC-MF schemes show great promise in serving as “fully” unified parameterizations that seamlessly simulate stable and dry convective boundary layers, stratiform and convective clouds, precipitation, and transitions between various cloud regimes.

Acknowledgments.

This research was supported by the National Science Foundation Climate Process Team (CPT) program (AGS 1916619). Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We thank the three anonymous reviewers for their insightful comments.

Data availability statement.

The CAM code with CLUBB+MF is available via github on the “cam_development” branch (https://github.com/ESCOMP/CAM/tree/cam_development). The ARM shallow cumulus case can be obtained at https://github.com/E3SM-Project/scmlib. Reference LES output is available on request from Georgios Matheou (georgios.matheou@uconn.edu).

APPENDIX

Table of Non-Default SCAM Parameters

Table A1 gives a list of numerical parameters used to produce the SCAM simulations shown in the main text that were modified from their default values. All parameters except L_ε are identical for both BOMEX and ARM simulations.

Table A1

SCAM parameters modified for simulations shown in the main text.

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Monthly Weather Review

Abstract
- Significance Statement
1. Introduction
2. Model description
3. Results
- a. Trade-wind shallow cumulus
- b. Continental shallow cumulus
4. Discussion
5. Conclusions
Acknowledgments.
Data availability statement.
APPENDIX
- Table of Non-Default SCAM Parameters
REFERENCES

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Abstract

Significance Statement

Abstract

Significance Statement

1. Introduction

2. Model description

a. Brief description of CLUBB

b. Brief description of the mass-flux scheme

c. Coupling of CLUBB and MF

d. Cases simulated

e. Large-eddy simulation model

3. Results

a. Trade-wind shallow cumulus

b. Continental shallow cumulus

4. Discussion

5. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX

Table of Non-Default SCAM Parameters

REFERENCES

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