A Generalized Mixing Length Closure for Eddy
‐
Diffusivity
Mass
‐
Flux Schemes of Turbulence and Convection
Ignacio Lopez
‐
Gomez
1
, Yair Cohen
1
, Jia He
1
, Anna Jaruga
1,2
, 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
Abstract
Because of their limited spatial resolution, numerical weather prediction and climate models
have to rely on parameterizations to represent atmospheric turbulence and convection. Historically,
largely independent approaches have been used to represent boundary layer turbulence and convection,
neglecting important interactions at the subgrid scale. Here we build on an eddy
‐
diffusivity mass
‐
fl
ux
(EDMF) scheme that represents all subgrid
‐
scale mixing in a uni
fi
ed manner, partitioning subgrid
‐
scale
fl
uctuations into contributions from local diffusive mixing and coherent advective structures and allowing
them to interact within a single framework. The EDMF scheme requires closures for the interaction
between the turbulent environment and the plumes and for local mixing. A second
‐
order equation for
turbulence kinetic energy (TKE) provides one ingredient for the diffusive local mixing closure, leaving a
mixing length to be parameterized. Here, we propose a new mixing length formulation, based on constraints
derived from the TKE balance. It expresses local mixing in terms of the same physical processes in all
regimes of boundary layer
fl
ow. The formulation is tested at a range of resolutions and across a wide range of
boundary layer regimes, including a stably strati
fi
ed boundary layer, a stratocumulus
‐
topped marine
boundary layer, and dry convection. Comparison with large eddy simulations (LES) shows that the EDMF
scheme with this diffusive mixing parameterization accurately captures the structure of the boundary layer
and clouds in all cases considered.
Plain Language Summary
Turbulence and convection transport heat and moisture in the
atmosphere and are ultimately responsible for the formation of clouds. However, they act on scales far
too small to be resolved in current global atmosphere models. Instead, parameterizations have to be used to
approximate their average effect on the
fi
nite volumes that are resolved in a global model. These
parameterizations are often tailored to speci
fi
c atmospheric conditions and fail when those conditions
are not met. Here we propose a parameterization that aims to reproduce the average effect of turbulent heat
and moisture transport under all atmospheric conditions. Numerical simulations demonstrate the
accuracy of the parameterization in simulating turbulence in atmospheric boundary layers under stable
and convective conditions, including the simulation of stratocumulus clouds.
1. Introduction
Turbulence is ubiquitous in the planetary boundary layer. Small
‐
scale chaotic air motions enhance mixing of
energy and moisture in the lower troposphere. Under statically unstable conditions, convective updrafts and
downdrafts further increase the vertical transport of energy and moisture between the surface and the air
aloft. Together, turbulence and convection shape the vertical distribution of temperature and water vapor
that sustains clouds. However, these processes act on scales far too small to be resolved in global climate
models (GCMs), with resolutions constrained by current computational power (Schneider et al., 2017).
Although the unabated increase in processing power will make globally resolving deep convective processes
routine in the coming years (Kajikawa et al., 2016), resolving turbulent mixing and shallow convection will
remain an intractable problem for decades. Instead, parameterizations have to be used to approximate the
average effect of these subgrid
‐
scale processes on the grid scale.
Conventional parameterizations consider atmospheric turbulence and convection as independent processes,
neglecting interactions that alter their combined effect on the large scale. These parameterizations are often
regime dependent, leading to models that arti
fi
cially split the spectrum of atmospheric conditions into a dis-
crete number of cases. Examples of such case
‐
dependent approaches include parameterizations of cumulus
©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
10.1029/2020MS002161
Key Points:
•
EDMF schemes represent boundary
layer (BL) turbulence and
convection separately yet
consistently
•
A mixing length model based on
kinetic energy constraints represents
turbulent
fl
uxes in EDMF schemes
well
•
The resulting EDMF scheme
captures dynamic regimes ranging
from stable BLs to
stratocumulus
‐
topped BLs
Correspondence to:
T. Schneider,
tapio@caltech.edu
Citation:
Lopez
‐
Gomez, I., Cohen, Y., He, J.,
Jaruga, A., & Schneider, T. (2020). A
generalized mixing length closure for
eddy
‐
diffusivity mass
‐
fl
ux schemes of
turbulence and convection.
Journal of
Advances in Modeling Earth Systems
,
12
, e2020MS002161. https://doi.org/
10.1029/2020MS002161
Received 1 MAY 2020
Accepted 7 OCT 2020
Accepted article online 15 JUN 2020
LOPEZ
‐
GOMEZ ET AL.
1of28
(Arakawa, 2004) and stratocumulus clouds (Lilly, 1968; Schubert, 1976). However accurate, the use of dispa-
rate schemes for different conditions complicates a seamless representation of subgrid
‐
scale processes in the
lower troposphere.
Several approaches to obtain a uni
fi
ed model of turbulence and convection have been proposed (Lappen &
Randall, 2001; Park, 2014; Thuburn et al., 2018). Here we focus on the extended formulation of an
eddy
‐
diffusivity mass
‐
fl
ux (EDMF) scheme developed in Tan et al. (2018), which in turn built on work by
Siebesma and Teixeira (2000), Soares et al. (2004), Siebesma et al. (2007), and Angevine et al. (2010), among
others. In the EDMF framework, the
fl
ow within each grid cell is decomposed into several distinct subdo-
mains, representing coherent convective structures and their relatively isotropic turbulent environment.
Convective transport is captured by mass
fl
ux terms that depend on differences between subdomain mean
properties; more isotropic turbulent transport, associated with small
‐
scale
fl
uctuations within each subdo-
main, is captured by eddy diffusion closures.
The extended EDMF framework uses additional prognostic equations for subdomain variables, such as the
environmental turbulence kinetic energy (TKE), and it requires closures for local turbulent
fl
uxes and for
the mass exchange between subdomains (Tan et al., 2018). Even though the EDMF framework arises from
the need for a uni
fi
ed model of turbulence and convection, the parameterizations used for entrainment and
turbulent mixing are usually de
fi
ned differently for each regime (Suselj et al., 2013; Witek et al., 2011). The
development of regime
‐
independent parameterizations for the required closures is the last step in the con-
struction of a uni
fi
ed model of atmospheric turbulence and convection.
Here, a regime
‐
independent closure for turbulent mixing within the EDMF framework is proposed. Section
2 reviews the decomposition of subgrid
‐
scale
fl
uxes in the extended EDMF scheme. Section 3 introduces the
formulation of the closure. Section 4 illustrates the performance of the EDMF scheme with the turbulent
mixing closure in boundary layer regimes where vertical transport is strongly dependent on the turbulence
closure used: the stable boundary layer (SBL), the stratocumulus
‐
topped boundary layer (STBL), and dry
convection. The performance of the extended EDMF scheme with this closure in moist convective cases is
demonstrated in a companion paper (Cohen et al., 2020). Finally, section 5 summarizes the results and
conclusions.
2. EDMF Framework
In the EDMF framework, each grid cell volume is decomposed into
n
updrafts or downdrafts (labeled by
index
i
¼
1
;
...
;
n
) and an environment (labeled by index
i
¼
0) in which they are embedded. Following this
decomposition, the grid
‐
mean value of variable
ψ
may be written as
⟨
ψ
⟩
¼
∑
n
i
≥
0
a
i
ψ
i
:
(1)
Here, angle brackets
⟨
·
⟩
denote the grid mean,
ψ
i
denotes the Favre average of
ψ
over subdomain
i
, and
a
i
is
the mean horizontal cross
‐
sectional area covered by subdomain
i
within the grid cell. This partition is moti-
vated by the anisotropy of turbulent convective
fl
ows, in which isotropic turbulent eddies coexist with coher-
ent columnar structures that induce a strong vertical transport (Bjerknes, 1938). The subdomain
decomposition is simpli
fi
ed for the horizontal velocity vector
u
h
, which is taken to have the same mean value
for all subdomains,
u
h
;
i
¼
⟨
u
h
⟩
. Applying the subdomain decomposition to higher
‐
order moments intro-
duces additional terms associated with the difference between grid and subdomain means. For the vertical
subgrid
‐
scale
fl
ux of
ψ
, this leads to
⟨
w
∗
ψ
∗
⟩
¼
∑
n
i
≥
0
a
i
w
′
i
ψ
′
i
þ
w
∗
i
ψ
∗
i
:
(2)
Here,
w
is the vertical velocity,
ψ
∗
¼
ψ
−
⟨
ψ
⟩
,
ψ
′
i
¼
ψ
−
ψ
i
, and
ψ
∗
i
¼
ψ
i
−
⟨
ψ
⟩
. The decomposition (2) parti-
tions the subgrid
‐
scale
fl
ux into contributions from small
‐
scale
fl
uctuations, associated with turbulence, and
subdomain
‐
mean terms, representative of convection. In the following, we will refer to these contributions
as turbulent and convective
fl
uxes, respectively.
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The subdomain
‐
mean terms can be explicitly solved for by introducing
n
prognostic subdomain equations
for each variable and an additional equation for each plume area fraction
a
i
, which may be diagnostic or
prognostic. Cohen et al. (2020) derive the subdomain equations used in the EDMF framework, starting from
the Navier
‐
Stokes equations. The use of prognostic subdomain equations means that convective
fl
uxes such
as
w
∗
i
ψ
∗
i
in (2) are explicitly solved for, while turbulent
fl
uxes like
w
′
i
ψ
′
i
must be modeled. Turbulent
fl
uxes
within each subdomain are modeled as downgradient and proportional to an eddy diffusivity
K
ψ
,
i
, where
ψ
is the property being transported. For the vertical turbulent
fl
ux in (2), this gives
w
′
i
ψ
′
i
¼
−
K
ψ
;
i
∂
ψ
i
∂
z
:
(3)
The eddy diffusivity
K
ψ
,
i
is proportional to a characteristic velocity scale and the length scale of the eddies
driving the transport, both of which must be parameterized.
Proposed closures for the eddy diffusivity vary from simple diagnostic expressions to second
‐
order models
that introduce prognostic equations for both scales (Umlauf & Burchard, 2003). The 1.5
‐
order TKE model
is a particularly popular choice due to its balance between accuracy and computational ef
fi
ciency (Mellor
& Yamada, 1982). The 1.5
‐
order model, also referred to as the Level 2.5 model in the Mellor
‐
Yamada hier-
archy, makes use of a prognostic equation for TKE and a diagnostic expression for the mixing length. In
the EDMF framework, the grid
‐
mean TKE
⟨
e
⟩
can be decomposed following Expression (2) for
second
‐
order moments as
⟨
e
⟩
¼
∑
n
i
≥
0
a
i
ē
i
þ
w
∗
i
w
∗
i
2
;
(4)
where
ē
i
is the TKE of subdomain
i
and the second term represents the corresponding convective kinetic
energy. This expression can be simpli
fi
ed by assuming that for the updrafts and downdrafts (
i
> 0), the
contribution to the grid
‐
mean TKE from small
‐
scale turbulence is negligible compared to the convective
term, an assumption commonly made in EDMF schemes:
⟨
e
⟩
¼
a
0
ē
0
þ
∑
n
i
≥
0
a
i
w
∗
i
w
∗
i
2
:
(5)
Thus, grid
‐
mean TKE is given by the sum of environmental TKE and convective TKE. The TKE decomposi-
tion (5) can also be obtained by assuming a small updraft and downdraft area fraction and similar turbulence
intensity in all subdomains (Siebesma et al., 2007). However, the equations derived for the subdomain
second
‐
order moments with these two approaches differ in the source terms that appear due to entrainment
processes between subdomains. The former approximation is favored here to allow for the use of this frame-
work in high
‐
resolution models, where the assumption of slender updrafts may become inadequate
(Randall, 2013).
Given an updraft area fraction
a
i
, which may be diagnostic or prognostic (Tan et al., 2018), the grid
‐
mean
TKE is determined by the environmental TKE
ē
0
and the subdomain
‐
mean vertical velocities
w
i
. The
subdomain
‐
mean vertical velocity equation for subdomain
i
is
∂
ð
ρ
a
i
w
i
Þ
∂
t
þ
∂
ð
ρ
a
i
w
2
i
Þ
∂
z
þ
∇
h
·
ð
ρ
a
i
u
h
;
i
w
i
Þ¼
−
∂
ð
ρ
a
i
w
′
i
w
′
i
Þ
∂
z
−
∇
h
·
ð
ρ
a
i
u
′
h
;
i
w
′
i
Þ
þ
∑
j
≠
i
E
ij
w
j
−
Δ
ij
w
i
þ
^
E
ij
ð
w
j
−
w
i
Þ
þ
ρ
a
i
b
i
−
ρ
a
i
∂
Ψ
†
i
∂
z
;
(6)
where
∇
h
is the horizontal gradient operator,
Ψ
¼
p
=
ρ
is the pressure potential, and the turbulent trans-
port terms on the right
‐
hand side are negligible for all subdomains except the environment (
i
¼
0).
Subgrid density changes are only considered in the buoyancy term, such that
ρ
¼
⟨
ρ
⟩
in the previous equa-
tion, in order to avoid creation of spurious acoustic modes through the subdomain decomposition (Cohen
et al., 2020). The buoyancy
b
i
and the pressure potential anomaly
Ψ
†
i
are de
fi
ned with respect to a refer-
ence hydrostatic pressure pro
fi
le
p
h
(
z
) and density
ρ
h
(
z
), related by
∂
z
p
h
¼
−
ρ
h
g
:
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Journal of Advances in Modeling Earth Systems
LOPEZ
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b
i
¼
−
g
ρ
i
−
ρ
h
ρ
;
∂
Ψ
†
i
∂
z
¼
∂
∂
z
p
i
ρ
þ
g
ρ
h
ρ
:
(7)
Here,
p
i
is the subdomain
‐
mean pressure. Density appears inside the pressure gradients in (6) and (7) to
ensure thermodynamic consistency of the subgrid
‐
scale anelastic approximation (Cohen et al., 2020).
Interactions between subdomains are captured by entrainment and detrainment
fl
uxes. In the vertical velo-
city Equation 6,
Δ
ij
is the dynamical detrainment of air mass from subdomain
i
into subdomain
j
, and
E
ij
and
^
E
ij
are the dynamical and turbulent entrainment from subdomain
j
into subdomain
i
, respectively. It is
assumed that entrainment events occur over timescales much shorter than the eddy turnover rate
K
ψ
;
i
=
ē
i
,
so that entrained air carries the properties of the subdomain it detrains from. In addition, for now we assume
that entrainment occurs only between convective plumes and the environment, not among plumes.
The prognostic equation for environmental TKE can be written in non
‐
conservative form as (Cohen
et al., 2020)
∂
ē
0
∂
t
þ
w
0
∂
ē
0
∂
z
þ
⟨
u
h
⟩
·
∇
h
ē
0
¼
−
w
′
0
u
′
0
∂
⟨
u
⟩
∂
z
−
w
′
0
v
′
0
∂
⟨
v
⟩
∂
z
−
w
′
2
0
∂
w
0
∂
z
þ
w
′
0
b
′
0
−
P
−
1
ρ
a
0
∂
∂
z
ρ
a
0
w
′
0
e
′
0
þ
∑
i
>0
Δ
i
0
ρ
a
0
ð
w
i
−
w
0
Þ
2
2
−
ē
0
!
−
^
E
i
0
ρ
a
0
w
∗
0
ð
w
i
−
w
0
Þþ
ē
0
"#
−
D
−
1
ρ
a
0
∇
h
·
ρ
a
0
u
′
h
;
0
e
′
0
−
u
′
h
;
0
u
′
0
·
∇
h
⟨
u
⟩
−
u
′
h
;
0
v
′
0
·
∇
h
⟨
v
⟩
−
u
′
h
;
0
w
′
0
·
∇
h
w
0
:
(8)
Here,
⟨
u
⟩
and
⟨
v
⟩
are the components of
⟨
u
h
⟩
,
P
is the velocity
‐
pressure gradient correlation, and
D
is the
turbulent dissipation. All sources and sinks of
ē
0
account for unresolved processes on the grid scale, so they
must be parameterized. Subdomain covariances in (8) are modeled diffusively, with the environmental eddy
diffusivity
K
ψ
de
fi
ned as
K
ψ
¼
c
ψ
l
ē
0
ðÞ
1
=
2
;
(9)
where
l
is the mixing length and
c
ψ
is a
fi
tting parameter. The subscript 0 in the eddy diffusivity is dropped
to simplify notation. The coef
fi
cient
c
ψ
is taken to be equal to
c
h
for the diffusion of all
fi
elds except for
momentum, for which
c
ψ
=
c
m
. The eddy viscosity
K
m
is related to
K
h
through the turbulent Prandtl num-
ber Pr
t
, such that
K
m
¼
Pr
t
K
h
Under the assumption that subgrid
‐
scale pressure work on the grid mean is negligible,
P
is taken as opposite
to the pressure work on the plumes (Tan et al., 2018),
P
¼
w
′
0
∂
Ψ
∂
z
′
0
þ
u
′
0
∂
Ψ
∂
x
′
0
þ
v
′
0
∂
Ψ
∂
y
′
0
"#
¼
−
∑
i
>0
a
i
a
0
ð
w
∗
i
−
w
∗
0
Þ
∂
Ψ
∗
i
∂
z
:
(10)
The last term in (10) appears as a sink term in the convective TKE balance, which is derived in Appendix B.
Hence,
P
acts as a return
‐
to
‐
isotropy term on the full grid, transferring momentum from the strongly aniso-
tropic coherent structures into the relatively isotropic eddies in the environment. The pressure work on the
plumes is formulated in terms of contributions from a virtual mass term (Gregory, 2001), an advective term,
and a drag term (Romps & Charn, 2015), yielding the following expression for the velocity
‐
pressure gradient
correlation:
P
¼
−
∑
i
>0
a
i
a
0
ð
w
∗
i
−
w
∗
0
Þ
α
b
b
∗
i
−
α
a
w
∗
i
∂
w
∗
i
∂
z
þ
α
d
ð
w
∗
i
−
w
∗
0
Þj
w
∗
i
−
w
∗
0
j
H
i
;
(11)
where
α
a
and
α
d
are constant parameters,
H
i
is the plume height, and
α
b
is a function of the aspect ratio of
the plume. Finally, assuming statistical equilibrium at scales
l
(Vassilicos, 2015), turbulent dissipation can
be estimated from the spectral transport relation that follows from Kolmogorov's theory of inertial turbu-
lence, giving Taylor's dissipation surrogate
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Journal of Advances in Modeling Earth Systems
LOPEZ
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D
¼
c
d
ē
3
=
2
0
l
:
(12)
Here,
c
d
is an empirical coef
fi
cient and
l
is the dissipation length, taken to be equal to the mixing length in our
model. Expressions (3) and (5)
–
(12) provide closure to a 1.5
‐
order model of turbulence within the EDMF fra-
mework, given diagnostic expressions for the mixing length
l
and for entrainment and detrainment.
3. Mixing Length Formulation
We seek to obtain a regime
‐
independent eddy diffusivity closure that provides an accurate representation of
turbulent subgrid
‐
scale
fl
uxes, over a wide range of host model resolutions. Thus, the eddy diffusivity should
reduce to a large eddy simulation (LES)
‐
type closure at high resolution, while being able to account for the
processes that modify turbulent
fl
uxes at larger scales. The formulation of the closure is organized following
this logic.
In section 3.1, we
fi
rst adapt a minimum TKE dissipation closure proposed for LES subgrid models
(Abkar & Moin, 2017) to the EDMF framework. Given the diffusive closure (3) and the eddy diffusivity (9),
the minimum dissipation assumption can be used to construct a mixing length closure. This mixing
length closure is shown to be equivalent to other proposed closures (e.g., Grisogono, 2010) for stable stra-
ti
fi
cation, but additional entrainment terms appear in the general case. Section 3.2 highlights the limita-
tions of this closure for climate modeling and weather prediction purposes when a prognostic TKE
equation is used. Section 3.3 then introduces a modi
fi
ed mixing length closure, which builds on the mini-
mum dissipation model and corrects its shortcomings by introducing additional mechanisms of net TKE
dissipation.
3.1. Minimum Dissipation of Environmental TKE
As in Verstappen (2011) and Abkar and Moin (2017), we assume that at the small scales represented by the
environment in the EDMF scheme, TKE is dissipated at least at the rate at which it is produced. This condition
translates into an inequality for the production and dissipation terms in the environmental TKE budget (8):
w
′
0
b
′
0
−
w
′
0
u
′
0
∂
⟨
u
⟩
∂
z
−
w
′
0
v
′
0
∂
⟨
v
⟩
∂
z
−
w
′
2
0
∂
w
0
∂
z
−
u
′
h
;
0
u
′
0
·
∇
h
⟨
u
⟩
−
u
′
h
;
0
v
′
0
·
∇
h
⟨
v
⟩
−
u
′
h
;
0
w
′
0
·
∇
h
w
0
þ
∑
i
>0
Δ
i
0
ρ
a
0
ð
w
i
−
w
0
Þ
2
2
−
ē
0
!
−
^
E
i
0
ρ
a
0
w
∗
0
ð
w
i
−
w
0
Þþ
ē
0
"#
≤
D
:
(13)
Here, the terms involving TKE injection from entrained air are also taken to be locally balanced by dissipa-
tion, consistent with the assumption that entrainment events occur over timescales much shorter than the
eddy turnover time
K
ψ
;
i
=
ē
i
. Note that the net dissipation condition (13) does not include redistribution
terms, such as the turbulent transport or the velocity
‐
pressure gradient correlation
P
. Moreover, the inequal-
ity (13) represents a local condition for the environment, and it does not preclude net subgrid
‐
scale energy
production due to processes such as convection, represented by plumes. Denoting the difference between
the right
‐
hand side and the left
‐
hand side of (13) as the net environmental TKE dissipation
γ
0
, the prognostic
environmental TKE Equation 8 reduces to
∂
ē
0
∂
t
þ
w
0
∂
ē
0
∂
z
þ
⟨
u
h
⟩
·
∇
h
ē
0
¼
−
1
ρ
a
0
∂
∂
z
ρ
a
0
w
′
0
e
′
0
þ
∇
h
·
ρ
a
0
u
′
h
;
0
e
′
0
−
P
−
γ
0
:
(14)
Here,
P
captures the effect of plumes on the environmental TKE. The evolution of the grid mean TKE that
follows from decomposition (5) and the simpli
fi
ed prognostic Equation 14 is
∂
⟨
e
⟩
∂
t
þ
⟨
u
h
⟩
·
∇
h
⟨
e
⟩
þ
⟨
w
⟩
∂
⟨
e
⟩
∂
z
þ
1
ρ
∇
h
·
ð
ρ
⟨
u
∗
h
e
∗
⟩
Þþ
∂
ð
ρ
⟨
w
∗
e
∗
⟩
Þ
∂
z
¼
∑
i
a
i
w
∗
i
b
∗
i
−
w
∗
2
i
∂
⟨
w
⟩
∂
z
−
a
0
γ
0
þ
a
0
w
′
2
0
∂
w
∗
0
∂
z
þ
u
′
h
;
0
w
′
0
·
∇
h
w
∗
0
−
∑
i
>0
Δ
i
0
ρ
ð
w
i
−
w
0
Þ
2
2
−
^
E
i
0
ρ
w
∗
0
ð
w
i
−
w
0
Þ
"#
:
(15)
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A detailed derivation of this equation and the subgrid
‐
scale kinetic energy pathways in the extended EDMF
scheme is described in Appendix B. Under the net dissipation closure (13), grid mean TKE production occurs
through the
fi
rst two terms on the right
‐
hand side of (15): the convective buoyancy
fl
ux and the
subdomain
‐
scale shear production.
The net dissipation condition (13) can be written in terms of the mixing length by introducing the closures
described in section 2. Using Taylor's dissipation surrogate (12) and downgradient closures for the shear and
buoyancy terms of the form
∑
3
j
¼
1
∑
3
k
¼
1
−
u
′
k
;
0
u
′
j
;
0
∂
ū
j
;
0
∂
x
k
¼
∑
3
j
¼
1
∑
3
k
¼
1
K
m
∂
ū
j
;
0
∂
x
k
2
;
w
′
0
b
′
0
¼
−
K
h
∂
b
0
∂
z
;
(16)
the inequality (13) leads to a condition for the maximum value of the mixing length
l
at which the net dis-
sipation
γ
0
is still positive semide
fi
nite:
∑
3
k
¼
1
∂
⟨
u
⟩
∂
x
k
2
þ
∂
⟨
v
⟩
∂
x
k
2
þ
∂
w
0
∂
x
k
2
"#
−
1
Pr
t
∂
b
0
∂
z
()
l
2
þ
∑
i
>0
Δ
i
0
ρ
a
0
ð
w
i
−
w
0
Þ
2
2
−
ē
0
!
−
^
E
i
0
ρ
a
0
w
∗
0
ð
w
i
−
w
0
Þþ
ē
0
"#
l
≤
c
d
c
m
ē
0
:
(17)
Here, the environmental buoyancy gradient is computed following Tan et al. (2018), taking into account pos-
sible phase change effects. In (16) and (17),
x
k
and
u
k
,0
represent the
k
‐
th coordinate and
k
‐
th velocity com-
ponent in the environment, respectively. From the inequality (17), an expression for the mixing length that
minimizes turbulent dissipation can be obtained by solving for
l
. This is equivalent to setting
γ
0
¼
0 in (14)
and (15). For the resulting value of the mixing length, production and dissipation of TKE are locally
balanced:
l
tke
¼
ffiffiffiffi
Δ
p
−
I
2
ð
S
l
þ
B
l
Þ
¼
−
I
2
ð
S
l
þ
B
l
Þ
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I
2
þ
4
ð
S
þ
B
Þ
D
p
2
ð
S
l
þ
B
l
Þ
:
(18)
Here,
Δ
is the discriminant and the different terms are given by
S
l
þ
B
l
¼
c
m
ē
1
=
2
0
∑
3
k
¼
1
∂
⟨
u
⟩
∂
x
k
2
þ
∂
⟨
v
⟩
∂
x
k
2
þ
∂
w
0
∂
x
k
2
"#
−
1
Pr
t
∂
b
0
∂
z
()
;
I
¼
∑
i
>0
Δ
i
0
ρ
a
0
ð
w
i
−
w
0
Þ
2
2
−
ē
0
!
−
^
E
i
0
ρ
a
0
w
∗
0
ð
w
i
−
w
0
Þþ
ē
0
"#
;
S
þ
B
¼ð
S
l
þ
B
l
Þ
l
:
(19)
In (18), the product
ð
S
þ
B
Þ
D
is independent of the mixing length, so
l
tke
can be readily evaluated. Although
the term
ð
S
þ
B
Þ
is sign inde
fi
nite, the discriminant
Δ
¼
I
2
þ
4
ð
S
þ
B
Þ
D
in (18) can be shown to remain positive semide
fi
nite even when the shear and buoyancy terms result in
TKE destruction, provided that the inequality (13) holds. This is because the minimum dissipation balance
requires
I
¼
D
−
ð
S
þ
B
Þ
;
(20)
so that the expression for the discriminant
Δ
is of the form
Δ
¼½
D
−
ð
S
þ
B
Þ
2
þ
4
ð
S
þ
B
Þ
D
¼½
D
þð
S
þ
B
Þ
2
≥
0
:
(21)
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The mixing length
l
tke
depends on local characteristics of the environment and on the vertical velocity dif-
ference between subdomains, which enters the injection term
I
in (19). Hence, convection modi
fi
es the
environmental diffusive transport directly through entrainment processes. In addition, convection also reg-
ulates the time evolution of turbulent
fl
uxes through its effect on the prognostic environmental TKE
Equation 14, captured by
P
.
This approach can also be applied to turbulence models that retain covariance terms
w
′
i
ψ
′
i
for other subdo-
mains and not only for the environment. In this case, the minimum dissipation condition may be used to
obtain a characteristic mixing length
l
tke,
i
for each subdomain. However, variance within plumes can also
be accounted for by variance among plumes when the number of subdomains is increased.
In stably strati
fi
ed boundary layers, where convection is inhibited, pressure work and entrainment
fl
uxes
in (6) act to homogenize the different subdomains, such that
ψ
∗
i
→
0 for any variable
ψ
and
a
0
→
1 (i.e., there
are no convective plumes). Under these conditions, the minimum dissipation mixing length (18) reduces to
the expression proposed by Grisogono (2010) for steady
‐
state SBL
fl
ow:
l
tke
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
S
þ
B
Þ
D
p
ð
S
l
þ
B
l
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffi
c
d
c
m
⟨
e
⟩
r
∑
3
k
¼
1
∂
⟨
u
⟩
∂
x
k
2
þ
∂
⟨
v
⟩
∂
x
k
2
þ
∂
⟨
w
⟩
∂
x
k
2
"#
−
1
Pr
t
∂
⟨
b
⟩
∂
z
()
−
1
=
2
:
(22)
The balance between shear production, destruction due to strati
fi
cation, and dissipation, which arises when
using this mixing length, is a well
‐
known leading
‐
order state in neutral (Spalart, 1988) and moderately stable
boundary layer
fl
ows (Li et al., 2016).
3.2. Limitations of the Minimum
‐
Dissipation Closure
Expression (18) for the mixing length
l
tke
captures the leading
‐
order balance in the environmental TKE bud-
get at small scales. However, a model with a diffusive closure based on
l
tke
cannot fully describe the
dynamics of the boundary layer at the coarse resolutions typical of GCMs, on the order of 10
4
m in the hor-
izontal and 10
–
100 m in the vertical. At these scales, the resolved horizontal gradients are weak, and the
environmental TKE Equation 14 that results from using
l
tke
can be simpli
fi
ed using the boundary layer
approximation (neglecting horizontal relative to vertical derivatives):
∂
ē
0
∂
t
þ
w
0
∂
ē
0
∂
z
¼
−
1
ρ
a
0
∂
∂
z
ρ
a
0
w
′
0
e
′
0
−
P
:
(23)
Note that we set
γ
0
¼
0 to obtain (23), since we are considering the case where
l
=
l
tke
and production locally
balances dissipation. In stable conditions (
P
¼
0), integrating the conservative form of (23) from the surface
layer (
z
s
) to the free troposphere above (
z
i
) yields the evolution equation for the vertically integrated envir-
onmental TKE:
Z
z
i
z
s
∂
ð
ρ
a
0
ē
0
Þ
∂
t
d
z
¼
−
ρ
a
0
w
0
e
0
½
z
i
z
s
≈
−
ρ
a
0
K
m
∂
ē
0
∂
z
z
s
:
(24)
In stable conditions,
a
0
≈
1 and
ψ
∗
i
≈
0 for any variable
ψ
. In addition, the absence of plumes implies that
detrainment and entrainment processes are negligible. From (24), it follows that the evolution of the verti-
cally integrated TKE under the minimum dissipation closure only depends on the
fl
ux from the unresolved
surface layer in stable conditions. But unbalanced TKE dissipation has been observed to become increas-
ingly important as strati
fi
cation develops in
fi
eld studies of the atmospheric boundary layer
(Li et al., 2016), and it can be expected to play a role in conditions of strong surface cooling. The budget (24)
cannot capture unbalanced TKE destruction within the boundary layer due to strati
fi
cation. Furthermore,
the minimum dissipation mixing length
l
tke
leads to enhanced eddy diffusion with increasing strati
fi
cation,
as deduced from (22). This is contrary to the evidence of turbulent mixing being inhibited in strong strati
fi
-
cation, such as near strong inversions.
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