of 28
Uni
fi
ed Entrainment and Detrainment Closures for
Extended Eddy
Diffusivity Mass
Flux Schemes
Yair Cohen
1
, Ignacio Lopez
Gomez
1
, Anna Jaruga
1,2
, Jia He
1
, Colleen M. Kaul
3
, and
Tapio Schneider
1,2
1
Department of Environmental Science and Engineering, California Institute of Technology, Pasadena, CA, USA,
2
Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA,
3
Paci
fi
c Northwest National Laboratory,
Richland, WA, USA
Abstract
We demonstrate that an extended eddy
diffusivity mass
fl
ux (EDMF) scheme can be used as a
uni
fi
ed parameterization of subgrid
scale turbulence and convection across a range of dynamical regimes,
from dry convective boundary layers, through shallow convection, to deep convection. Central to achieving
this uni
fi
ed representation of subgrid
scale motions are entrainment and detrainment closures. We model
entrainment and detrainment rates as a combination of turbulent and dynamical processes. Turbulent
entrainment/detrainment is represented as downgradient diffusion between plumes and their environment.
Dynamical entrainment/detrainment is proportional to a ratio of a relative buoyancy of a plume and a
vertical velocity scale, that is modulated by heuristic nondimensional functions which represent their
relative magnitudes and the enhanced detrainment due to evaporation from clouds in drier environment.
We
fi
rst evaluate the closures off
line against entrainment and detrainment rates diagnosed from large eddy
simulations (LESs) in which tracers are used to identify plumes, their turbulent environment, and mass and
tracer exchanges between them. The LES are of canonical test cases of a dry convective boundary layer,
shallow convection, and deep convection, thus spanning a broad rangeof regimes. We then compare the LES
with the full EDMF scheme, including the new closures, in a single
column model (SCM). The results show
good agreement between the SCM and LES in quantities that are key for climate models, including
thermodynamic pro
fi
les, cloud liquid water pro
fi
les, and pro
fi
les of higher moments of turbulent statistics.
The SCM also captures well the diurnal cycle of convection and the onset of precipitation.
Plain Language Summary
The dynamics of clouds and turbulence are too small in scale to be
resolved in global models of the atmosphere, yet they play a crucial role in controlling weather and
climate. These models rely on parameterizations for representing clouds and turbulence. Inadequacies in
these parameterizations have hampered especially climate models for decades; they are the largest source of
physical uncertainties in climate predictions. It has proven challenging to represent the wide rangeof cloud
and turbulence regimes encountered in nature in a single parameterization. Here we present such a
parameterization that does capture a wide range of cloud and turbulence regimes within a single, uni
fi
ed
physical framework, with relatively few parameters that can be adjusted to
fi
t data. The framework relies on
a decomposition of turbulent
fl
ows into coherent updraft and downdraft (i.e., plumes) and random
turbulence in their environment. A key contribution of this paper is to show how the exchange of mass and
properties between the plumes and their turbulent environment
the so
called entrainment and
detrainment of air into and out of plumes
can be modeled. We show that the resulting parameterization
represents well the most important features of dry convective boundary layers, shallow cumulus convection,
and deep cumulonimbus convection.
1. Introduction
Turbulence and convection play an important role in the climate system. They transport energy, moisture,
and momentum vertically, thereby controlling the formation of clouds and, especially in the tropics, the
thermal strati
fi
cation of the atmosphere. They occur on a wide range of scales, from motions on scales of
meters to tens of meters in stable boundary layers and near the trade inversion, to motions on scales of kilo-
meters in deep convection. General circulation models (GCMs), with horizontal resolutions approaching
tens of kilometers, are unable to resolve this spectrum of motions. Turbulence and convection will remain
unresolvable in GCMs for the foreseeable future (Schneider et al., 2017), although some deep convective
©2020. The Authors.
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.
RESEARCH ARTICLE
10.1029/2020MS002162
Key Points:
An extended eddy
diffusivity
mass
fl
ux (EDMF) scheme
successfully captures diverse
regimes of convective motions
Uni
fi
ed closures are presented for
entrainment and detrainment across
the different convective regimes
With the uni
fi
ed closures, the EDMF
scheme can simulate dry convection,
shallow cumulus, and deep cumulus
Correspondence to:
T. Schneider,
tapio@caltech.edu
Citation:
Cohen, Y., Lopez
Gomez, I., Jaruga, A.,
He, J., Kaul, C. M., & Schneider, T.
(2020). Uni
fi
ed entrainment and
detrainment closures for extended
eddy
diffusivity mass
fl
ux schemes.
Journal of Advances in Modeling Earth
Systems
,
12
, e2020MS002162. https://
doi.org/10.1029/2020MS002162
Received 1 MAY 2020
Accepted 29 JUL 2020
Accepted article online 4 AUG 2020
COHEN ET AL.
1of28
motions, on scales of kilometers to tens of kilometers, are beginning to be resolved in short
term global simu-
lations (Kajikawa et al., 2016; Stevens et al., 2019).
Unable to resolve turbulence and convection explicitly, GCMs rely on parameterization schemes to repre-
sent subgrid
scale (SGS) motions. Typically, GCMs have several distinct parameterization schemes for repre-
senting, for example, boundary layer turbulence, stratocumulus clouds, shallow convection, and deep
convection. The different parameterization schemes interact via trigger functions with discontinuous beha-
vior in parameter space, even though in reality the
fl
ow regimes they represent lie on a continuous spectrum
(Xie et al., 2019). This fragmentary representation of SGS motion by multiple schemes leads to a proliferation
of adjustable parameters, including parametric triggering functions that switch between schemes. Moreover,
most existing parameterizations rely on statistical equilibrium assumptions between the SGS motions and
the resolved scales. These assumptions become invalid as model resolution increases and, for example, some
aspects of deep convection begin to be explicitly resolved (Dirmeyer et al., 2012; Gao et al., 2017). It is widely
recognized that these issues make model calibration challenging and compromise our ability to make reli-
able climate predictions (Hourdin et al., 2017; Schmidt et al., 2017; Schneider et al., 2017).
Many known biases in climate models and uncertainties in climate predictions are attributed to dif
fi
culties
in representing SGS turbulence and convection. For example, biases in the diurnal cycle and the continental
near
surface temperature, especially in polar regions, have been traced to inadequacies in turbulence para-
meterizations for stable boundary layers (Holtslag et al., 2013). Across climate models, biases in how tropical
cloud cover covaries with temperature and other environmental factors on seasonal and interannual time
scales are correlated with the equilibrium climate sensitivity, thus revealing the important role the represen-
tation of tropical low clouds plays in uncertainties in climate predictions (Bony & Dufresne, 2005; Brient &
Schneider, 2016; Brient et al., 2016; Caldwell et al., 2018; Ceppi et al., 2017; Cesana et al., 2018; Dong et al.,
2019; Lin et al., 2014; Nam et al., 2012; Schneider et al., 2019; Teixeira et al., 2011). Differences in moisture
export from the mixed layer to the free troposphere by cumulus convection lead to differences in the width
and strength of the ascending branch of the Hadley circulation (Neggers et al., 2007). And biases in the struc-
ture of the South Paci
fi
c Convergence Zone have been traced to biases in the intensity of deep convective
updrafts (Hirota et al., 2014). It is evident from these few examples that progress in the representation of
SGS turbulence and convection is crucial for progress in climate modeling and prediction. At the same time,
it is desirable to unify the representation of SGS motions in one continuous parameterization scheme, to
reduce the number of adjustable parameters and obtain a scheme that more faithfully represents the under-
lying continuum of physical processes.
Different approaches for a systematic coarse graining of the equations of motion, leading to a uni
fi
ed para-
meterization, have been proposed (de Rooy & Siebesma, 2010; Han & Bretherton, 2019; Lappen & Randall,
2001a; Park, 2014a, 2014b; Rio et al., 2019; Suselj et al., 2019b; Tan et al., 2018; Thuburn et al., 2018; Yano,
2014). They typically entail a conditional averaging (or
fi
ltering) of the governing equations over several-
subdomains (Weller & McIntyre, 2019), or an assumed probability density function (PDF) ansatz for dyna-
mical variables and generation of moment equations from the ansatz (Golaz et al., 2002; Lappen & Randall,
2001a; Larson & Golaz, 2005; Larson et al., 2012). For example, conditional averaging can lead to a parti-
tioning of a GCM grid box into subdomains representing coherent ascending and descending plumes, or
drafts, and a more isotropically turbulent environment. Unclosed terms arise that, for example, to represen-
tinteractions among subdomains through entrainment and detrainment. Such unclosed terms need to be
speci
fi
ed through closure assumptions (de Rooy et al., 2013). Or, if moment equations are generated
through an assumed PDF ansatz for dynamical and thermodynamic variables, unclosed interactions among
moments and dissipation terms need to be speci
fi
ed through closure assumptions (Golaz et al., 2002;
Lappen & Randall, 2001b). Our goal in this paper is to develop a uni
fi
ed set of closures that work across
the range of turbulent and convective motions, within one speci
fi
c type of parameterization scheme known
as an eddy
diffusivity mass
fl
ux (EDMF) scheme (Siebesma & Teixeira, 2000; Siebesma et al., 2007;
Wu et al., 2020).
We build on the extended EDMF scheme of Tan et al. (2018), which extends the original EDMF scheme of
Siebesma and Teixeira (2000) by retaining explicit time dependence (SGS memory) and treating subdomain
second
moment equations consistently, so that, for example, energy exchange between plumes and their
environment obeys conservation requirements. The explicit SGS memory avoids any statistical
10.1029/2020MS002162
Journal of Advances in Modeling Earth Systems
COHEN ET AL.
2of28
equilibrium assumption. This is a necessary ingredient for the scheme to become scale aware and be able to
operate in the convective gray zone, where deep convective motions begin to become resolved.
In this and the companion paper Lopez
Gomez et al. (2020) we present a set of uni
fi
ed closures that allow the
extended EDMF parameterization to simulate stable boundary layers, dry convective boundary layers,
stratocumulus
topped boundary layers, shallow convection, and deep convection, all within a scheme with
uni
fi
ed closures and a single set of parameters. This paper focuses on uni
fi
ed entrainment and detrainment
closures that are essential for convective regime, and Lopez
Gomez et al. (2020) present a closure for turbu-
lent mixing. To demonstrate the viability of our approach, we compare the resulting parameterization
scheme against large
eddy simulations (LESs) of several canonical test cases for different dynamical regimes.
This paper is organized as follows. In section 2, we present the general structure of the extended EDMF
scheme, including the subdomain decomposition and the prognostic equations for subdomain moments.
Section 3 introduces the entrainment and detrainment closures that are key for the scheme to work across
different dynamical regimes. Section 4 describes the numerical implementation of this scheme in a
single
column model (SCM). In section 5, we describe the LES used in this study and how we compare terms
in the EDMF scheme against statistics derived from the LES. Section 6 compares results from the EDMF
scheme against LES of canonical test cases of dry convective boundary layers, shallow, and deep convection.
Section 7 summarizes and discusses the main
fi
ndings.
2. Extended EDMF Scheme
2.1. Equations of Motion
The extended EDMF scheme is derived from the compressible equations of motion of the host model. As
thermodynamic variables, we choose the liquid
ice potential temperature
θ
l
and the total water speci
fi
c
humidity
q
t
, but these choices can easily be modi
fi
ed and harmonized with the thermodynamic variables
of the host model in which the scheme is implemented. The un
fi
ltered governing equations are as follows:
ρ
t
þ
h
·
ð
ρ
u
h
Þþ
ð
ρ
w
Þ
z
¼
0
;
(1)
ð
ρ
u
h
Þ
t
þ
h
·
ð
ρ
u
h
u
h
Þþ
ð
ρ
w
u
h
Þ
z
¼
h
p
þ
ρ
S
u
h
;
(2)
ð
ρ
w
Þ
t
þ
h
·
ð
ρ
u
h
w
Þþ
ð
ρ
ww
Þ
z
¼
ρ
b
p
z
þ
ρ
S
w
;
(3)
ð
ρθ
l
Þ
t
þ
h
·
ð
ρ
u
h
θ
l
Þþ
ð
ρ
w
θ
l
Þ
z
¼
ρ
S
θ
l
;
(4)
ð
ρ
q
t
Þ
t
þ
h
·
ð
ρ
u
h
q
t
Þþ
ð
ρ
wq
t
Þ
z
¼
ρ
S
q
t
;
(5)
p
¼
ρ
R
d
T
v
:
(6)
In the momentum equation, to improve numerical stability, we have removed a reference pressure pro
fi
le
p
h
(
z
) in hydrostatic balance with a density
ρ
h
(
z
):
p
h
z
¼
ρ
h
g
;
where
g
is the gravitational acceleration. Therefore, the perturbation pressure
p
¼
p
p
h
and the buoyancy
b
¼
g
ρ
ρ
h
ρ
10.1029/2020MS002162
Journal of Advances in Modeling Earth Systems
COHEN ET AL.
3of28
appear in the momentum equations in place of the full pressure
p
and gravitational acceleration
g
.
Otherwise, the notation is standard:
ρ
is density,
q
t
is the total water speci
fi
c humidity,
T
v
is the virtual
temperature,
R
d
is the gas constant for dry air, and
θ
l
¼
T
p
s
p

R
d
=
c
p
exp
L
v
ð
q
l
þ
q
i
Þ
c
p
T

(7)
is the liquid
ice potential temperature, with liquid and ice speci
fi
c humidities
q
l
and
q
i
and reference sur-
face pressure
p
s
¼
10
5
Pa. In a common approximation that can easily be relaxed, we take the isobaric spe-
ci
fi
c heat capacity of moist air
c
p
to be constant and, consistent with Kirchhoff's law, the latent heat of
vaporization
L
v
to be a linear function of temperature (Romps, 2008). The temperature
T
is obtained from
the thermodynamic variables
θ
l
,
ρ
, and
q
t
by a saturation adjustment procedure, and the virtual tempera-
ture
T
v
is computed from the temperature
T
and the speci
fi
c humidities (Pressel et al., 2015). The horizon-
tal velocity vector is
u
h
, and
w
is the vertical velocity component;
h
is the horizontal nabla operator. The
symbol
S
stands for sources and sinks. For the velocities, the sources
S
u
h
and
S
w
include the molecular
viscous stress and Coriolis forces, and for thermodynamic variables, the sources
S
θ
l
and
S
q
t
represent
sources from molecular diffusivity, microphysics, and radiation.
When implemented in a GCM, the host model solves for the grid
averaged form of Equations 1
(6). In the
averaged equations, SGS
fl
uxes arise from the application of Reynolds averaging to quadratic and
higher
order terms. As is common, we make the boundary layer approximation and focus on the vertical
SGS
fl
uxes, neglecting horizontal SGS
fl
uxes. The role of the parameterization in the host model is to predict
these vertical SGS
fl
uxes, in addition to cloud properties that are used by radiation and microphysics
schemes. In the next section, a decomposition of grid boxes into subdomains expresses the vertical SGS
fl
uxes as a sum of turbulent
fl
uxes in the environment (ED) and convective mass
fl
uxes in plumes (MF).
To compute the MF component of the
fl
uxes, the EDMF scheme solves for
fi
rst moments of the host model's
prognostic variables (
w
,
θ
l
,
q
t
) in each of its subdomains, as well as for the area fraction of the subdomains.
To compute the ED component, the EDMF scheme solves additionally for the turbulence kinetic energy in
the environment. Finally, to compute cloud properties by sampling from implied SGS distributions of ther-
modynamic variables, the EDMF scheme also solves for variances and covariance of
θ
l
and
q
t
in the envir-
onment. A summary of the prognostic and diagnostic variables in the scheme is given in Table 1.
2.2. Domain Decomposition and Subdomain Moments
The extended EDMF scheme is derived from the equations of motion by decomposing the host model grid
box into subdomains and averaging the equations over each subdomain volume. We denote by
φ
the aver-
age of a scalar
φ
over the host model grid box, with
φ
¼
φ
φ
denoting
fl
uctuations about the grid mean.
Table 1
EDMF Scheme Variables
Symbol
Description
Unit
Prognostic
Diagnostic
ρ
;
ρ
i
Density
kg
m
3
upd, env, gm
p
i
;
p
Pressure
Pa
upd, env, gm
a
i
Subdomain area fraction
upd
env
θ
l
;
i
;
θ
l
Liquid
ice potential temperature
K
upd, gm
env
q
t
;
i
;
q
t
Total water speci
fi
c humidity
kg kg
1
upd, gm
env
w
i
;
w
Vertical velocity
m
s
1
upd, gm
env
u
h
;
i
¼
u
h
Horizontal velocity
m
s
1
gm
upd, env
b
i
;
b
Buoyancy
m s
2
env, upd, gm
̅
θ
2
l
;
0
;
θ
2
l
θ
l
variance
K
2
env
gm
̅
q
2
t
;
0
;
q
2
t
q
t
variance
kg
2
kg
2
env
gm
̅
θ
l
;
0
q
t
;
0
;
θ
l
q
t
Covariance of
θ
l
and
q
t
Kkgkg
1
env
gm
ē
0
;
e
Turbulence kinetic energy
m
2
s
2
env
gm
Note
. In the right two columns,
upd,
”“
env,
and
gm
stand for updrafts, environment, and grid mean, respectively, and these indicate whether a variable is
prognostic or diagnostic in that model subdomain.
10.1029/2020MS002162
Journal of Advances in Modeling Earth Systems
COHEN ET AL.
4of28