manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
Unified Entrainment and Detrainment Closures for
1
Extended Eddy-Diffusivity Mass-Flux Schemes
2
Yair Cohen
1
, Ignacio Lopez-Gomez
1
, Anna Jaruga
1
,
2
, Jia He
1
, Colleen Kaul
3
,
3
Tapio Schneider
1
,
2
4
1
California Institute of Technology, Pasadena, California, USA.
5
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA.
6
3
Pacific Northwest National Laboratory, Washington, USA
7
Key Points:
8
•
An extended eddy-diffusivity mass-flux (EDMF) scheme successfully captures di-
9
verse regimes of convective motions.
10
•
Unified closures are presented for entrainment and detrainment across the differ-
11
ent convective regimes.
12
•
With the unified closures, the EDMF scheme can simulate dry convection, shal-
13
low cumulus, and deep cumulus.
14
Corresponding author: Tapio Schneider,
tapio@caltech.edu
–1–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
Abstract
15
We demonstrate that an extended eddy-diffusivity mass-flux (EDMF) scheme can be used
16
as a unified parameterization of subgrid-scale turbulence and convection across a range
17
of dynamical regimes, from dry convective boundary layers, over shallow convection, to
18
deep convection. Central to achieving this unified representation of subgrid-scale mo-
19
tions are entrainment and detrainment closures. We model entrainment and detrainment
20
rates as a combination of turbulent and dynamical processes. Turbulent entrainment/detrainment
21
is represented as downgradient diffusion between plumes and their environment. Dynam-
22
ical entrainment/detrainment are proportional to a ratio of buoyancy difference and ver-
23
tical velocity scale, partitioned based on buoyancy sorting approaches and modulated
24
by a function of relative humidity difference in cloud layer to represent buoyancy loss
25
owing to evaporation in mixing. We first evaluate the closures offline against entrain-
26
ment and detrainment rates diagnosed from large-eddy simulations (LES) in which trac-
27
ers are used to identify plumes, their turbulent environment, and mass and tracer ex-
28
changes between them. The LES are of canonical test cases of a dry convective bound-
29
ary layer, shallow convection, and deep convection, thus spanning a broad range of regimes.
30
We then compare the LES with the full EDMF scheme, including the new closures, in
31
a single column model (SCM). The results show good agreement between the SCM and
32
LES in quantities that are key for climate models, including thermodynamic profiles, cloud
33
liquid water profiles, and profiles of higher moments of turbulent statistics. The SCM
34
also captures well the diurnal cycle of convection and the onset of precipitation.
35
Plain Language Summary
36
The dynamics of clouds and their underlying turbulence are too small in scale to
37
be resolved in global models of the atmosphere, yet they play a crucial role controlling
38
weather and climate. Climate and weather forecasting models rely on parameterizations
39
to represent the dynamics of clouds and turbulence. Inadequacies in these parameter-
40
izations have hampered especially climate models for decades; they are the largest source
41
of physical uncertainties in climate predictions. It has proven challenging to represent
42
the wide range of cloud and turbulence regimes encountered in nature in a parameter-
43
ization that can capture them in a unified physical framework. Here we present a pa-
44
rameterization that does capture a wide range of cloud and turbulence regimes within
45
a single, unified physical framework, with relatively few parameters that can be adjusted
46
to fit data. The framework relies on a decomposition of turbulent flows into coherent up-
47
and downdrafts (i.e. plumes) and random turbulence in their environment. A key con-
48
tribution of this paper is to show how the interaction between the plumes and their tur-
49
bulent environment—the so-called entrainment and detrainment of air into and out of
50
plumes—can be modeled. We show that the resulting parameterization represents well
51
the most important features of dry convective boundary layers, shallow cumulus convec-
52
tion, and deep cumulonimbus convection.
53
1 Introduction
54
Turbulence and convection play an important role in the climate system. They trans-
55
port energy, moisture, and momentum vertically, thereby controlling the formation of
56
clouds and, especially in the tropics, the thermal stratification of the atmosphere. They
57
occur on a wide range of scales, from motions on scales of meters to tens of meters in
58
stable boundary layers and near the trade inversion, to motions on scales of kilometers
59
in deep convection. General Circulation Models (GCMs), with horizontal resolutions ap-
60
proaching tens of kilometers, are unable to resolve this spectrum of motions. Turbulence
61
and convection will remain unresolvable in GCMs for the foreseeable future (Schneider
62
et al., 2017), although some deep-convective motions, on scales of kilometers to tens of
63
–2–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
kilometers, are beginning to be resolved in short-term global simulations (Kajikawa et
64
al., 2016; Stevens et al., 2019).
65
Unable to resolve turbulence and convection explicitly, GCMs rely on parameter-
66
ization schemes to represent subgrid-scale (SGS) motions. Typically, GCMs have sev-
67
eral distinct parameterization schemes for representing, for example, boundary layer tur-
68
bulence, stratocumulus clouds, shallow convection, and deep convection. The different
69
parameterization schemes interact via trigger functions with discontinuous behavior in
70
parameter space, even though in reality the flow regimes they represent lie on a contin-
71
uous spectrum, (Xie et al., 2019). This fragmentary representation of SGS motion by
72
multiple schemes leads to a proliferation of adjustable parameters, including paramet-
73
ric triggering functions that switch between schemes. Moreover, most existing param-
74
eterizations rely on statistical equilibrium assumptions between the SGS motions and
75
the resolved scales. These assumptions become invalid as model resolution increases and,
76
for example, some aspects of deep convection begin to be explicitly resolved (Dirmeyer
77
et al., 2012; Gao et al., 2017). It is widely recognized that these issues make model cal-
78
ibration challenging and compromise our ability to make reliable climate predictions (Hourdin
79
et al., 2017; Schmidt et al., 2017; Schneider et al., 2017).
80
Many known biases in climate models and uncertainties in climate predictions are
81
attributed to difficulties in representing SGS turbulence and convection. For example,
82
biases in the diurnal cycle and the continental near-surface temperature, especially in
83
polar regions, have been traced to inadequacies in turbulence parameterizations for sta-
84
ble boundary layers (Holtslag et al., 2013). Across climate models, biases in how trop-
85
ical cloud cover co-varies with temperature and other environmental factors on seasonal
86
and interannual timescales are correlated with the equilibrium climate sensitivity, thus
87
revealing the important role the representation of tropical low clouds plays in uncertain-
88
ties in climate predictions (Bony & Dufresne, 2005; Teixeira et al., 2011; Nam et al., 2012;
89
Lin et al., 2014; Brient et al., 2016; Brient & Schneider, 2016; Ceppi et al., 2017; Cesana
90
et al., n.d.; Caldwell et al., 2018; Dong et al., 2019; Schneider et al., 2019). Differences
91
in moisture export from the mixed layer to the free troposphere by cumulus convection
92
lead to differences in the width and strength of the ascending branch of the Hadley cir-
93
culation (R. A. Neggers et al., 2007). And biases in the structure of the South Pacific
94
Convergence Zone have been traced to biases in the intensity of deep-convective updrafts
95
(Hirota et al., 2014). It is evident from these few examples that progress in the repre-
96
sentation of SGS turbulence and convection is crucial for progress in climate modeling
97
and prediction. At the same time, it is desirable to unify the representation of SGS mo-
98
tions in one continuous parameterization scheme, to reduce the number of adjustable pa-
99
rameters and obtain a scheme that more faithfully represents the underlying continuum
100
of physical processes.
101
Different approaches for a systematic coarse graining of the equations of motion,
102
leading to a unified parameterization, have been proposed (Lappen & Randall, 2001a;
103
de Rooy & Siebesma, 2010; Yano, 2014; Park, 2014a, 2014b; Thuburn et al., 2018; Tan
104
et al., 2018; Han & Bretherton, 2019; Rio et al., 2019; Suselj et al., 2019b). They typ-
105
ically entail a conditional averaging (or filtering) of the governing equations over several
106
subdomains (Weller & McIntyre, 2019), or an assumed probability density function (PDF)
107
ansatz for dynamical variables and generation of moment equations from the ansatz (Lappen
108
& Randall, 2001a; Golaz et al., 2002; Larson & Golaz, 2005; Larson et al., 2012). For
109
example, conditional averaging can lead to a partitioning of a GCM grid box into sub-
110
domains representing coherent ascending and descending plumes, or drafts, and a more
111
isotropically turbulent environment. Unclosed terms arise that, for example, represent
112
interactions among subdomains through entrainment and detrainment. Such unclosed
113
terms need to be specified through closure assumptions (de Rooy et al., 2013). Or, if mo-
114
ment equations are generated through an assumed PDF ansatz for dynamical and ther-
115
modynamic variabels, unclosed interactions among moments and dissipation terms need
116
–3–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
to be specified through closure assumptions (Lappen & Randall, 2001b; Golaz et al., 2002).
117
Our goal in this paper is to develop a unified set of closures that work across the range
118
of turbulent and convective motions, within one specific type of parameterization scheme
119
known as an eddy diffusivity/mass flux (EDMF) scheme (A. P. Siebesma & Teixeira, 2000;
120
A. P. Siebesma et al., 2007; Wu et al., 2020).
121
We build on the extended EDMF scheme of Tan et al. (2018), which extends the
122
original EDMF parameterization A. P. Siebesma and Teixeira (2000) by retaining ex-
123
plicit time dependence (SGS memory) and treating subdomain second-moment equations
124
consistently, so that, for example, energy exchange between plumes and their environ-
125
ment obeys conservation requirements. The explicit SGS memory avoids any statisti-
126
cal equilibrium assumption and allows the scheme to operate in the convective gray zone,
127
where deep convective motions begin to become resolved.
128
In this and a companion paper Lopez-Gomez et al. (2020), along with a revised pres-
129
sure closure (Jia He, personal communication), we present a set of unified closures that
130
allow the extended EDMF parameterization to simulate stable boundary layers, dry con-
131
vective boundary layers, stratocumulus-topped boundary layers, shallow convection, and
132
deep convection, all within a scheme with unified closures and a single set of parame-
133
ters. This paper focuses on unified entrainment and detrainment closures that are es-
134
sential for convective regime, and Lopez-Gomez et al. (2020) presents a closure for tur-
135
bulent mixing. To demonstrate the viability of our approach, we compare the resulting
136
parameterization scheme against large-eddy simulations (LES) of several canonical test
137
cases for different dynamical regimes.
138
This paper is organized as follows. In section 2, we present the general structure
139
of the extended EDMF scheme, including the subdomain decomposition and the prog-
140
nostic equations for subdomain moments. Section 3 introduces the entrainment and de-
141
trainment closures that are key for the scheme to work across different dynamical regimes.
142
Section 4 describes the numerical implementation of this scheme in a single column model
143
(SCM). In section 5, we describe the LES used in this study and how we compare terms
144
in the EDMF scheme against statistics derived from the LES. Section 6 compares results
145
from the EDMF scheme against LES of canonical test cases of dry convective boundary
146
layers, shallow and deep convection. Section 7 summarizes and discusses the main find-
147
ings.
148
2 Extended EDMF Scheme
149
2.1 Equations of Motion
150
The extended EDMF scheme is derived from the compressible equations of motion
151
of the host model. As thermodynamic variables, we choose the liquid-ice potential tem-
152
perature
θ
l
and the total water specific humidity
q
t
, but these choices can easily be mod-
153
ified and harmonized with the thermodynamic variables of the host model in which the
154
scheme is implemented. The unfiltered governing equations are:
155
∂ρ
∂t
+
∇
h
·
(
ρ
u
h
) +
∂
(
ρw
)
∂z
= 0
,
(1)
∂
(
ρ
u
h
)
∂t
+
∇
h
·
(
ρ
u
h
⊗
u
h
) +
∂
(
ρw
u
h
)
∂z
=
−∇
h
p
†
+
ρS
u
h
,
(2)
∂
(
ρw
)
∂t
+
∇
h
·
(
ρ
u
h
w
) +
∂
(
ρww
)
∂z
=
ρb
−
∂p
†
∂z
+
ρS
w
,
(3)
∂
(
ρθ
l
)
∂t
+
∇
h
·
(
ρ
u
h
θ
l
) +
∂
(
ρwθ
l
)
∂z
=
ρS
θ
l
,
(4)
∂
(
ρq
t
)
∂t
+
∇
h
·
(
ρ
u
h
q
t
) +
∂
(
ρwq
t
)
∂z
=
ρS
q
t
,
(5)
p
=
ρR
d
T
v
.
(6)
–4–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
In the momentum equation, to improve numerical stability, we have removed a reference
pressure profile
p
h
(
z
) in hydrostatic balance with a density
ρ
h
(
z
):
∂p
h
∂z
=
−
ρ
h
g,
where
g
is the gravitational acceleration. Therefore, the perturbation pressure
p
†
=
p
−
p
h
and the buoyancy
b
=
−
g
ρ
−
ρ
h
ρ
appear in the momentum equations in place of the full pressure
p
and gravitational ac-
156
celeration
g
. Otherwise, the notation is standard:
ρ
is density,
q
t
is the total water spe-
157
cific humidity,
T
v
is the virtual temperature,
R
d
is the gas constant for dry air, and
158
θ
l
=
T
(
p
p
s
)
R
d
/c
p
exp
(
L
v
(
q
l
+
q
i
)
c
p
T
)
(7)
is the liquid-ice potential temperature, with liquid and ice specific humidities
q
l
and
q
i
159
and reference surface pressure
p
s
= 10
5
Pa. In a common approximation that can eas-
160
ily be relaxed, we take the isobaric specific heat capacity of moist air
c
p
to be constant
161
and, consistent with Kirchhoff’s law, the latent heat of vaporization
L
v
to be a linear
162
function of temperature (Romps, 2008). The temperature
T
is obtained from the ther-
163
modynamic variables
θ
l
,
ρ
, and
q
t
by a saturation adjustment procedure, and the vir-
164
tual temperature
T
v
is computed from the temperature
T
and the specific humidities (Pressel
165
et al., 2015). The horizontal velocity vector is
u
h
, and
w
is the vertical velocity compo-
166
nent;
∇
h
is the horizontal nabla operator. The symbol
S
stands for sources and sinks.
167
For the velocities, the sources
S
u
h
and
S
w
include the molecular viscous stress and Cori-
168
olis forces, and for thermodynamic variables, the sources
S
θ
l
and
S
q
t
represent sources
169
from molecular diffusivity, microphysics, and radiation.
170
2.2 Domain Decomposition and Subdomain Moments
171
The extended EDMF scheme is derived from the equations of motion by decom-
172
posing the host model grid box into subdomains and averaging the equations over each
173
subdomain volume. We denote by
〈
φ
〉
the average of a scalar
φ
over the host model grid
174
box, with
φ
∗
=
φ
−〈
φ
〉
denoting fluctuations about the grid mean. Similarly,
̄
φ
i
is the
175
average of
φ
over the
i
-th subdomain, and
φ
′
i
=
φ
−
̄
φ
i
is the fluctuation about the mean
176
of subdomain
i
. The difference between the subdomain mean and grid mean then be-
177
comes
̄
φ
∗
i
=
̄
φ
i
− 〈
φ
〉
. Common terminology assigns an area fraction
a
i
=
A
i
/A
T
to
178
each subdomain, where
A
i
is the horizontal area of the
i
-th subdomain and
A
T
is the
179
horizontal area of the grid box. This
a
i
is more precisely a volume fraction, since
A
i
is
180
the vertically averaged horizontal area of the
i
-th subdomain within the grid box. We
181
retain here the terminology using subdomain area fractions, which reflect the subdomain
182
volume fractions, consistent with previous works (A. P. Siebesma et al., 2007).
183
With this decomposition, the subdomain zeroth moment (area fraction), first mo-
184
ment (mean), centered second moment (covariance), and centered third moment obey:
185
–5–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
∑
i
≥
0
a
i
= 1
,
(8)
〈
φ
〉
=
∑
i
≥
0
a
i
̄
φ
i
,
(9)
〈
φ
∗
ψ
∗
〉
=
∑
i
≥
0
a
i
[
̄
φ
∗
i
̄
ψ
∗
i
+
φ
′
i
ψ
′
i
]
,
=
∑
i
≥
0
[
a
i
φ
′
i
ψ
′
i
+
1
2
∑
j
≥
0
a
i
a
j
(
̄
φ
i
−
̄
φ
j
)(
̄
ψ
i
−
̄
ψ
j
)
]
,
(10)
〈
φ
∗
ψ
∗
w
∗
〉
=
∑
i
≥
0
[
a
i
(
ψ
′
i
φ
′
i
w
′
i
+
̄
φ
i
̄
ψ
i
̄
w
i
+
̄
ψ
i
w
′
i
φ
′
i
+
̄
φ
i
w
′
i
ψ
′
i
+ ̄
w
i
ψ
′
i
φ
′
i
)
]
−
[
〈
φ
〉〈
ψ
〉〈
w
〉
+
〈
φ
〉〈
ψ
∗
w
∗
〉
+
〈
ψ
〉〈
φ
∗
w
∗
〉
+
〈
w
〉〈
ψ
∗
φ
∗
〉
]
.
(11)
Equations (8) and (9) are self-evident; the derivation of (10) and (11) from (8) and (9)
186
is given in Appendix A. Equation (10) with
φ
=
w
is the vertical SGS flux of a scalar
187
ψ
, which is one of the key predictands of any parameterization scheme: the divergence
188
of this flux appears as a source in the equations for the resolved scales of the host model.
189
The decomposition in (9)–(11) only applies in general if
(
·
) is a Favre average—an av-
190
erage weighted by the density that appears in the continuity equation. However, in the
191
EDMF scheme we describe in what follows, we make the approximation of ignoring den-
192
sity variations across subdomains (except in buoyancy terms), so that Favre and volume
193
averages coincide within a grid box.
194
The central assumption in EDMF schemes is that within-subdomain covariances
195
such as
φ
′
i
ψ
′
i
and higher moments are neglected in all subdomains except one distinguished
196
subdomain, the environment, denoted by index
i
= 0. In the environment, covariances
197
φ
′
0
ψ
′
0
are retained, and third moments such as
w
′
0
φ
′
0
ψ
′
0
, which appear in second-moment
198
equations, are modeled with closures. The intuition underlying this assumption is that
199
the flow domain is subdivided into an isotropically turbulent environment (
i
= 0) and
200
into coherent structures, identified with plumes (
i
≥
1). The environment can have sub-
201
stantial within-environment covariances, whereas the plumes are taken to have compar-
202
atively little variance within them. Variance within plumes can be represented by hav-
203
ing an ensemble of plumes with different mean values (R. A. J. Neggers et al., 2002; R. Neg-
204
gers, 2012; Suˇselj et al., 2012). For the case of only two subdomains, an updraft (
i
=
205
1) and its environment (
i
= 0), the second-moment equation (10) then simplifies to
206
〈
φ
∗
ψ
∗
〉
=
a
1
φ
′
1
ψ
′
1
+ (1
−
a
1
)
φ
′
0
ψ
′
0
+
a
1
(1
−
a
1
)(
̄
φ
1
−
̄
φ
0
)(
̄
ψ
1
−
ψ
0
)
≈
(1
−
a
1
)
φ
′
0
ψ
′
0
︸
︷︷
︸
ED
+
a
1
(1
−
a
1
)(
̄
φ
1
−
̄
φ
0
)(
̄
ψ
1
−
ψ
0
)
︸
︷︷
︸
MF
,
(12)
where the approximation in the second line reflects the EDMF assumption of neglect-
207
ing within-plume covariances. The first equation states that the covariance on the grid
208
scale can be decomposed into the sum of the covariances within subdomains and the co-
209
variance among subdomain means, as in the analysis of variance (ANOVA) from statis-
210
tics (Mardia et al., 1979). In the second line, the first term is closed by a down-gradient
211
eddy diffusion (ED) closure; the second term is represented by a mass flux (MF) closure,
212
whence EDMF derives its name (A. P. Siebesma & Teixeira, 2000). Similarly, under the
213
EDMF assumption, the third-moment equation (11) for two subdomains, written for a
214
single scalar, simplifies to
215
〈
φ
∗
φ
∗
φ
∗
〉≈−
a
1
(1
−
a
1
)(
̄
φ
1
−
̄
φ
0
)
φ
′
0
φ
′
0
+ 3
a
1
(1
−
a
1
)(1
−
2
a
1
)(
̄
φ
1
−
̄
φ
0
)
3
.
(13)
That is, third moments (i.e., skewness) on the grid scale are represented through covari-
216
ances within the environment and through variations among means across subdomains
217
with differing area fractions.
218
–6–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
2.3 EDMF Assumptions
219
The extended EDMF scheme is obtained by applying this decomposition of grid-
220
scale variations to the equations of motion (1)–(6), making the following additional as-
221
sumptions:
222
1. We make the boundary layer approximation for subgrid scales, meaning that we
223
assume vertical derivatives to be much larger than horizontal derivatives. This in
224
particular means that the diffusive closure for fluxes in the environment only in-
225
volves vertical gradients,
226
w
′
i
φ
′
i
≈−
K
φ,i
∂
̄
φ
i
∂z
,
(14)
where
K
φ,i
is the eddy diffusivity (to be specified) for scalar
φ
in subdomain
i
. Con-
227
sistent with the EDMF assumptions, we assume
K
φ,i
= 0 for
i
6
= 0.
228
2. We use the same, grid-mean density
〈
ρ
〉
in all subdomains except in the buoyancy
229
term. This amounts to making an anelastic approximation on the subgrid scale,
230
to suppress additional acoustic modes that would otherwise arise through the do-
231
main decomposition. For notational simplicity, we use
ρ
rather than
〈
ρ
〉
for the
232
grid-mean density in what follows, and ̄
ρ
i
for the subdomain density that appears
233
only in the buoyancy term:
234
̄
b
i
=
−
g
̄
ρ
i
−
ρ
h
ρ
.
(15)
The grid-mean density
ρ
appears in the denominator, playing the role of the ref-
235
erence density in the anelastic approximation. The area fraction-weighted sum of
236
the subdomain buoyancies is the grid-mean buoyancy, ensuring consistency of this
237
decomposition:
238
〈
b
〉
=
∑
i
a
i
̄
b
i
=
−
g
ρ
−
ρ
h
ρ
.
(16)
3. We take the subdomain horizontal velocities to be equal to their grid-mean val-
239
ues,
240
̄
u
h,i
=
〈
u
h
〉
.
(17)
This simplification is commonly made in parameterizations for climate models (Larson
241
et al., 2019). It eliminates mass-flux contributions to the SGS vertical flux of hor-
242
izontal momentum.
243
2.4 EDMF Equations
244
The full derivation of the subdomain-mean and covariance equations from (1)–(6)
245
is given in Appendix B. The derivation largely follows Tan et al. (2018), except for a dis-
246
tinction between dynamical and turbulent entrainment and detrainment following de Rooy
247
and Siebesma (2010). The resulting extended EDMF equation for the subdomain area
248
fraction is
249
∂
(
ρa
i
)
∂t
+
∇
h
·
(
ρa
i
〈
u
h
〉
) +
∂
(
ρa
i
̄
w
i
)
∂z
=
∑
j
6
=
i
(
E
ij
−
∆
ij
)
;
(18)
–7–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.
manuscript submitted to
Journal of Advances in Modeling Earth Systems (JAMES)
the equation for the subdomain-mean vertical momentum is
∂
(
ρa
i
̄
w
i
)
∂t
+
∇
h
·
(
ρa
i
〈
u
h
〉
̄
w
i
) +
∂
(
ρa
i
̄
w
i
̄
w
i
)
∂z
=
∂
∂z
(
ρa
i
K
w,i
∂
̄
w
i
∂z
)
+
∑
j
6
=
i
[
(
E
ij
+
ˆ
E
ij
) ̄
w
j
−
(∆
ij
+
ˆ
E
ij
) ̄
w
i
]
+
ρa
i
(
̄
b
∗
i
+
〈
b
〉
)
−
ρa
i
∂
∂z
(
̄
p
∗
i
+
〈
p
†
〉
ρ
)
+
̄
S
w,i
; (19)
and the equation for the subdomain-mean of scalar
φ
is
∂
(
ρa
i
̄
φ
i
)
∂t
+
∇
h
·
(
ρa
i
〈
u
h
〉
̄
φ
i
) +
∂
(
ρa
i
̄
w
i
̄
φ
i
)
∂z
=
∂
∂z
(
ρa
i
K
φ,i
∂
̄
φ
i
∂z
)
+
∑
j
6
=
i
[
(
E
ij
+
ˆ
E
ij
)
̄
φ
j
−
(∆
ij
+
ˆ
E
ij
)
̄
φ
i
]
+
ρa
i
̄
S
φ,i
.
(20)
The dynamical entrainment rate from subdomain
j
into subdomain
i
is
E
ij
, and the de-
250
trainment rate from subdomain
i
into subdomain
j
is ∆
ij
. In addition to dynamical en-
251
trainment, there is turbulent entrainment from subdomain
j
into
i
, with rate
ˆ
E
ij
. Tur-
252
bulent entrainment differentially entrains tracers but not mass (see Appendix B).
253
The pressure and buoyancy terms in the vertical momentum equation (19) are writ-
ten as the sum of their grid-mean value and perturbations from their grid-mean value.
These perturbations vanish when summed over all subdomains because
∑
i
a
i
̄
φ
∗
i
= 0;
hence, the grid-mean values of the pressure and buoyancy terms are recovered upon sum-
ming over subdomains. Following Pauluis (2008), the pressure gradient term in (19) is
written with 1
/ρ
inside the gradient to ensure energy conservation in our SGS anelas-
tic approximation; see Appendix C for details. The subdomain density ̄
ρ
i
that is essen-
tial for the subdomain buoyancy is computed from the subdomain virtual temperature
̄
T
v,i
using the ideal gas law with the grid-mean pressure
〈
p
〉
:
̄
ρ
i
=
〈
p
〉
R
d
̄
T
v,i
.
(21)
In analogy with the anelastic approximation Pauluis (2008), this formulation of the ideal
254
gas law ensures that
∑
i
a
i
̄
ρ
i
̄
T
v,i
=
ρ
〈
T
v
〉
, while accounting for subdomain virtual tem-
255
perature effects, which play a key role in the buoyancy of updrafts in shallow convection.
256
The scalar equation (20) is applied to any thermodynamic variable, with its cor-
responding subdomain-averaged source
̄
S
φ,i
on the right-hand side. The terms on the
left-hand side represent the explicit time tendencies and fluxes of the subdomain-means,
which can be viewed as forming part of the dynamical core of the host model. The terms
on the right-hand side are sources and sinks that require closure. The covariance equa-
–8–
ESSOAr | https://doi.org/10.1002/essoar.10502905.1 | Non-exclusive | First posted online: Sat, 2 May 2020 02:40:35 | This content has not been peer reviewed.