A Mass-Flux Scheme View of a High-Resolution Simulation of a Transition from
Shallow to Deep Cumulus Convection
ZHIMING KUANG*
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California
CHRISTOPHER S. BRETHERTON
Department of Atmospheric Sciences, University of Washington, Seattle, Washington
(Manuscript received 11 April 2005, in final form 11 November 2005)
ABSTRACT
In this paper, an idealized, high-resolution simulation of a gradually forced transition from shallow,
nonprecipitating to deep, precipitating cumulus convection is described; how the cloud and transport
statistics evolve as the convection deepens is explored; and the collected statistics are used to evaluate
assumptions in current cumulus schemes. The statistical analysis methodologies that are used do not require
tracing the history of individual clouds or air parcels; instead they rely on probing the ensemble charac-
teristics of cumulus convection in the large model dataset. They appear to be an attractive way for analyzing
outputs from cloud-resolving numerical experiments. Throughout the simulation, it is found that 1) the
initial thermodynamic properties of the updrafts at the cloud base have rather tight distributions; 2) contrary
to the assumption made in many cumulus schemes, nearly undiluted air parcels are too infrequent to be
relevant to any stage of the simulated convection; and 3) a simple model with a spectrum of entraining
plumes appears to reproduce most features of the cloudy updrafts, but significantly overpredicts the mass
flux as the updrafts approach their levels of zero buoyancy. A buoyancy-sorting model was suggested as a
potential remedy. The organized circulations of cold pools seem to create clouds with larger-sized bases and
may correspondingly contribute to their smaller lateral entrainment rates. Our results do not support a
mass-flux closure based solely on convective available potential energy (CAPE), and are in general agree-
ment with a convective inhibition (CIN)-based closure. The general similarity in the ensemble character-
istics of shallow and deep convection and the continuous evolution of the thermodynamic structure during
the transition provide justification for developing a single unified cumulus parameterization that encom-
passes both shallow and deep convection.
1. Introduction
Parameterizations of cumulus convection in large-
scale models currently employ a wide variety of as-
sumptions about how the cumulus cloud ensemble and
associated fluxes of heat, moisture, and momentum re-
late to large-scale variables. Many of these assumptions
have not yet been adequately evaluated. Such evalua-
tions are difficult to make because they require data
with a level of detail and accuracy that is very difficult
to obtain from observations. In recent years, cloud-
resolving models [CRMs; here defined to include large-
eddy-resolving two-dimensional (2D) as well as three-
dimensional (3D) models] have emerged as an alterna-
tive source of information.
For shallow (by which we mean almost nonprecipi-
tating) cumulus convection, studies by Siebesma and
Cuijpers (1995), Siebesma et al. (2003), Zhao and Aus-
tin (2005a,b), and others compared CRM results with
mass-flux schemes, the most widely used type of cumu-
lus parameterizations. The former two papers com-
pared vertical moisture and heat-flux profiles from a
CRM with predictions of a single bulk entraining–
detraining plume model, and diagnosed optimal frac-
tional entrainment and detrainment rates for such a
* Current affiliation: Department of Earth and Planetary Sci-
ences, and Division of Engineering and Applied Sciences, Har-
vard University, Cambridge, Massachusetts.
Corresponding author address: Zhiming Kuang, Dept. of Earth
and Planetary Sciences, Harvard University, 20 Oxford St., Cam-
bridge, MA 02138.
E-mail: kuang@fas.harvard.edu
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© 2006 American Meteorological Society
JAS3723
model. The latter two papers examined turbulent trans-
port and mixing processes in six simulated shallow cu-
muli of different sizes, demonstrating the importance of
buoyancy-sorting processes (Raymond and Blyth 1986)
and of all phases of the cumulus life cycle.
CRM studies of individual cumulus congestus clouds
and deep (significantly precipitating) cumulus clouds by
Carpenter et al. (1998) and Cohen (2000) reached simi-
lar conclusions to Zhao and Austin. There have also
been CRM studies aimed at evaluating parameteriza-
tion schemes for deep cumulus convection (Lin 1999:
Lin and Arakawa 1997), though these studies were lim-
ited to a 2D geometry and a coarse numerical resolu-
tion. Lin and Arakawa (1997) showed that if clouds
were categorized by their maximum cloud-top height as
determined by trajectory analysis, an entraining plume
model similar to that used in the parameterization of
Arakawa and Schubert (1974) could predict many of
the cloud properties. Lin (1999) used back trajectories
to examine the properties of air entering the base of the
cumuli.
Several key issues for mass-flux cumulus parameter-
ization have not been adequately addressed by prior
CRM studies. These include mass-flux closure (what
regulates the overall cloud-base mass flux) and parti-
tioning (how the cumulus mass flux at each level is
partitioned between different mixtures of cloud base
and environmental air). Furthermore, many large-scale
models use separate parameterizations for shallow and
deep cumulus convection. This is most justifiable if
transitions between the two are discrete, as might occur
in continental deep cumulus convection in highly con-
ditionally unstable environments. But is such a transi-
tion still discrete when the cloud ensemble is produced
by a more gradually evolving forcing, such as might
occur over a warm ocean?
In this study, we describe an idealized, high-
resolution CRM simulation of a gradually forced tran-
sition from shallow, nonprecipitating to deep, precipi-
tating cumulus convection; explore how the cloud and
transport statistics evolve as the convection deepens;
and use the collected statistics to evaluate assumptions
in current cumulus schemes, including cloud-base prop-
erties, the utility of an entraining plume perspective,
and the mass-flux closure and partitioning problems.
We compare the statistical character of shallow and
deep convection in order to gain new insights into
whether and how unified schemes should be developed
to handle both shallow and deep convection. Our study
differs from previous ones in that we use a significantly
higher resolution and a 3D geometry so that we can
better resolve turbulent mixing. We have also devised
convenient new ways to analyze the simulated cloud
ensemble, which lead to more direct evaluation of cu-
mulus schemes and do not require tracing the history of
individual clouds or air parcels.
One of our goals is to examine the utility of an en-
training plume model of mixing in cumulus clouds as an
organizing principle for understanding the statistical
distribution of buoyancy, vertical velocity, and liquid
water content as functions of height. In framing this
goal, we do not imply that individual cumuli act as en-
training plumes. The above-referenced CRM studies
show that individual clouds are made up of a complex
and time-varying spectrum of mixtures of cloud base
and environmental air from many levels. Each cloud
during each phase of its life cycle may therefore be
regarded as contributing to many plumes of many en-
trainment rates in a cumulus parameterization using an
entraining plume ensemble formulation, for example,
see Arakawa and Schubert (1974). One should under-
stand the analogy more in terms of fluid parcels within
cumulus clouds, each undergoing a mixing history that
can be roughly idealized in an ensemble sense as be-
having somewhat like an entraining plume.
Section 2 contains a brief description of the model
used in this study and an overview of the simulation.
Analyses of the model output and implications for cu-
mulus schemes are presented in section 3, followed by
a brief summary (section 4).
2. Model and an overview of the simulation
We use the System for Atmospheric Modeling
(SAM), which is an updated version of the Colorado
State University Large-Eddy Simulation/Cloud-
Resolving Model (Khairoutdinov and Randall 2003).
The model uses the anelastic equations of motion with
bulk microphysics. Its prognostic thermodynamic vari-
ables are the liquid–ice static energy s
li
⫽ c
p
T ⫹ gz ⫺
L(q
n
⫹ q
p
) ⫺ L
f
(q
i
⫹ q
s
⫹ q
g
), the total (nonprecipi-
tating) water specific humidity
1
q
t
⫽ q
⫹ q
n
and the
total precipitating water specific humidity q
p
. Here T is
temperature, z is height, g is the gravitational accelera-
tion, c
p
is isobaric specific heat, L and L
f
are the latent
heats of vaporization and freezing, and q
n
⫽ q
l
⫹ q
i
is
the nonprecipitating condensate specific humidity,
which is the sum of the specific humidities of cloud
water q
l
and cloud ice q
i
. In addition, q
is water vapor
specific humidity, and q
s
and q
g
are the specific humid-
ity of the ice phase precipitating water, snow and grau-
1
While mixing ratios were used in the original description of
SAM (Khairoutdinov and Randall, 2003), the actual model uses
specific humidity (M. Khairoutdinov 2005, personal communica-
tion).
1896 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 63
pel, respectively. The readers are referred to Khairout-
dinov and Randall (2003) for further details about the
model. For this study, we use a simple Smagorinsky-
type scheme to represent the effect of subgrid-scale tur-
bulence.
We use a doubly periodic domain with the model top
placed at 19 km. There are 256 grid points in the ver-
tical, and the vertical grid size is uniformly 50 m below
12 km and gradually increases above that. A wave-
absorbing layer is placed in the upper third of the do-
main. We use a relatively small horizontal grid size of
100 m in order to resolve the mixing processes between
cumulus clouds of all sizes and their environment, a key
aspect of the present study. This resolution is similar to
that of a recent CRM intercomparison study of shallow
cumulus convection (Siebesma et al. 2003). Computa-
tional constraints mandate the use of a fairly small do-
main size (19.2 km ⫻ 19.2 km). While this domain size
is insufficient to correctly simulate the long-term be-
havior of fully developed deep convective systems
(such as in radiative–convective equilibrium simula-
tions), it appears sufficient to accommodate the transi-
tion from shallow to deep cumulus convection, and is
adequate for examining the evolution of cloud and
transport statistics during this transition.
Our simulation starts from a classic and well-studied
oceanic trade cumulus case derived from observations
taken during the Barbados Oceanography and Meteo-
rology Experiment (BOMEX; Holland and Rasmusson
1973; Nitta and Esbensen 1974). We use the initial
sounding, surface heat and moisture fluxes, horizontal-
mean advective forcings, subsidence velocity, and con-
stant, cloud-independent radiative forcings specified in
the recent CRM intercomparison study of this case by
Siebesma et al. (2003). Unlike Siebesma et al. (2003),
we start with no horizontal winds, apply no Coriolis
force, and do not nudge the mean horizontal winds at
later times. These forcings allow a nearly statistically
steady trade cumulus cloud ensemble to develop within
2–3 h. The forcings all decrease linearly with height
above 1500 m and are zero at all heights above 2.5 km.
This is artificial, but is a nice simplification for our pur-
poses since above 2.5 km, we can assume that moist
convection will be the only source of moisture and heat.
After 12 h of simulation, we slowly ramp up the surface
latent and sensible heat fluxes (Fig. 1a), with the Bowen
ratio kept constant. Note that the dashed line in Fig. 1a
is the sensible heat flux scaled up by a factor of 10.
An overview of the domain mean precipitation and
profiles of potential temperature, total water content,
FIG. 1. (a) Surface latent heat (solid) and 10 times the sensible heat (dashed) fluxes applied in the
experiment, and the evolution of domain-mean (b) precipitation, and profiles of (c) temperature (con-
tours) and water vapor specific humidity (shading), and (d) cloud fraction.
J
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and cloud fraction is provided in Fig. 1b–d. During the
first 12 h, a nearly statistically steady shallow convec-
tive regime is established. As the surface fluxes in-
crease, convection deepens. The inversion originally lo-
cated at 2 km gets eroded and increases in height. At
the end of day 2, the inversion is no longer discernable.
Convection develops into a cumulus congestus regime.
At this time there is substantial convectively available
potential energy (CAPE) for an undilute near-surface
air parcel to rise to the upper troposphere, as shown in
Fig. 2, but deep convection does not occur because the
midtroposphere is still relatively dry. A dry midtropo-
sphere is unfavorable to deep convection because lat-
eral entrainment of drier environmental air by the ris-
ing plumes leads to more evaporative cooling, and
hence negative buoyancy (Derbyshire et al. 2004).
Deep convection develops after the middle troposphere
is moistened by the cumulus congestus, as indicated in
Fig. 1 by the appearance of substantial upper tropo-
spheric anvil clouds around the beginning of day 5. Fig-
ure 1 clearly shows that the thermodynamic structure
evolves continuously during the transition, without a
discrete eruption into deep convection, providing justi-
fication for developing unified shallow and deep cumu-
lus parameterizations.
In the following discussion, we shall refer to the time
period from 9 to 12 h as the shallow cumulus regime,
day 2.5 to day 3.5 as the cumulus congestus regime, and
the last day as the deep cumulus regime. Instantaneous
three-dimensional (3D) fields were saved after every 20
min of simulated time, and horizontally averaged quan-
tities were sampled every 30 s and saved as 30-min
averages.
To further examine the effect of midtroposphere
moisture, we have conducted an additional experiment,
where the simulation was restarted at the end of day 2.5
of the control experiment. Horizontally uniform values
were added to the q
field above 3 km so that the do-
main averaged q
profile above 3 km matches that at
the end of day 5 of the nominal run (Fig. 3a). In this
simulation, deep convection develops rapidly (Fig. 3b),
confirming the role of midtroposphere dryness in de-
laying the development of deep convection in the con-
trol case (Fig. 1d).
FIG. 2. Domain-mean profiles for a 30-min period during day 2
of environmental virtual potential temperature (thick) and the
density potential temperature of a cloudy cloud-base particle
lifted adiabatically without mixing (all condensates are retained in
the parcel).
FIG. 3. (a) Domain-averaged q
profile at the end of day 2.5 (solid) and day 5 (dashed) of the control
run. The difference between the dotted line and the solid line was added to the new run, which restarted
at the end of day 2.5 of the nominal run. (b) The evolution of cloud fraction profile in the restarted run.
The first 2.5 days are the same as the nominal run shown in Fig. 1d.
1898 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 63
While one could choose to simulate the transition
from shallow to deep cumulus convection in the more
realistic setting of the continental diurnal cycle
(Grabowski et al. 2006), the present idealized experi-
mental setup was chosen so that the transition occurs
slowly (over 2–3 days instead of 1–2 h). This affords us
adequate sampling with a relatively small domain.
Equivalent sampling in a diurnal cycle study requires a
domain that is too large for the memory of our present
computer cluster to accommodate.
3. Analyses and results
In our analysis we will extensively use thermody-
namic variables that are (within the numerical formu-
lation of the simulation) conserved in adiabatic fluid
motions including phase changes of water, and which
are also approximately linearly mixing. Two such vari-
ables introduced in section 2 were SAM’s prognostic
thermodynamic variables, the total water specific hu-
midity q
t
and the liquid-ice static energy s
li
. We shall
exclude the effect of precipitating water from s
li
; that is,
redefine s
li
⫽ c
p
T ⫹ gz ⫺ Lq
n
⫺ L
f
q
i
, so that it is better
conserved in the presence of precipitation. It is useful in
addition to define two further variables. The first is the
frozen moist-static energy h ⫽ s
li
⫹ Lq
t
⫽ c
p
T ⫹ gz ⫹
Lq
⫺ L
f
q
i
, hereafter referred to as MSE. This variable
is conserved by the model even for air parcels in which
there is production or evaporation of liquid precipita-
tion. The second is the liquid water virtual potential
temperature
l
⫽
l
(1 ⫹ 0.61q
), where
l
⫽ (T ⫺
Lq
l
/c
p
)(1000 hPa/p)
R/cp
, and R is the specific gas con-
stant. In the numerical model, this variable is approxi-
mately but not exactly conserved and linearly mixing. It
is useful for comparing the densities of air parcels from
different levels, after they are adiabatically brought to a
common level at which they are unsaturated.
a. Cloud-base mass flux
An important component of a mass-flux scheme is to
determine the cloud-base mass flux. While it has been
popular to build the closure assumption solely on
CAPE (Arakawa and Schubert 1974; Bechtold et al.
2001; Fritsch and Chappell 1980), it is clear that during
the transition from shallow to deep cumulus, such a
formulation is not valid. Figure 4 shows the time evo-
lution of 1) the cloud-base mass flux and 2) the CAPE.
CAPE is computed by reversibly displacing an air par-
cel with the mean thermodynamic properties of cloudy
cloud-base air parcels. At ⬃28 h, the displaced undi-
luted parcel overcomes the trade inversion and gains
access to the conditional instability above the inversion,
leading to the sudden increase in CAPE. After that,
CAPE continues to increase and gains another factor of
3 by the end of the simulation. In contrast, the cloud-
base mass flux is slightly smaller during the latter peri-
ods of the simulation. The cloud base here is identified
as the level of maximum cloud fraction. Variations in
the cloud-base mass flux were also found to be small in
a study of the transition of shallow to deep convection
transition over land (Grabowski et al. 2006). A recent
CRM study of the diurnal cycle of shallow cumulus
convection (Neggers et al. 2004) also reached negative
conclusions about the CAPE closure.
A more recent closure suggestion is that the cloud-
base mass flux is regulated through its interaction with
the weak stable layer atop the subcloud layer. This
weak stable layer provides convective inhibition (CIN)
to subcloud layer air parcels. Borrowing a concept from
statistical mechanics, Mapes (2000) considered the role
of subcloud layer fluctuations in overcoming CIN and
triggering convection, and proposed that the cloud base
mass flux be parameterized in the form W exp(⫺kCIN/
W
2
), where W is some measure of typical updraft ver-
tical velocity at cloud base and k is a constant. When
CIN is too small, cloud mass flux increases and causes
stronger compensating subsidence, which heats the cu-
mulus layer and increases CIN until a balance is estab-
lished. Bretherton et al. (2004) demonstrated the viabil-
ity of this approach as a combined mass-flux closure
and trigger implemented in a parameterization of shal-
low cumulus convection, taking W
2
equal to the sub-
FIG. 4. Time evolution of (a) the cloud-base mass flux and (b)
CAPE. CAPE is computed by reversibly displacing an air parcel
with the mean thermodynamic properties of cloudy cloud-base air
parcels.
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