April 1988
K. Nakajimaand T. Matsuno
Numerical Experiments
309
Concerning the Origin
of Cloud Clusters in the Tropical Atmosphere
By Kensuke Nakajima and Taroh Matsuno
Geophysical
Institute,Universityof Tokyo,
Bunkyo-ku,Tokyo,113,Japan
(Manuscriptreceived27 April1987,in revisedform 15 February1988)
Abstract
A large-domain,
two-dimensional
cloudconvectionmodelwasusedfor the purposeof examining
the naturalpropertiesof cloud convectionsunder idealizedconditions:the atmospherereceivedheat
and moisturefroman underlyinguniformlywarmwatersurface,and, at the sametime,the atmosphere
was cooled at a constantrate. Fiveexperimentswerecarriedout with differentsetsof microphysical
processes.
Quasi-steadystatesnaturallyattainedin the experimentsshoweddifferentspatialand temporal
structuresof convection.In the casewhererain was not generated,a cellularstructuresimilarto the
Benardconvectionappeared.In the case where rain was generatedbut did not evaporate,there
appearedonly a singledeep narrowcloudwhoselifetimewas unrealistically
long.In the casewiththe
full set of the standardmicrophysics,there appeareda 'double-scale'structure. That is, a numberof
deepconvectivecloudshavinga horizontalscaleof O(lkm)and a lifetimeof O(1hour)weregenerated,
and thesecloudswere spontaneouslyorganizedto form severalcloudsystemshavinga lifetimelonger
than 10 hours.Eachof the cloudsystemsinduceda rainfallovera regionhavinga widthof 30-100km
duringits lifecycle.
The principalmechanismfor the generationof the double-scalestructurein the last caseis the
formationof a cold air pool at the foot of eachdeepcloudby rain waterevaporation.The formation
of cold air limitsthe lifetimeof individualcloud and thus determinesthe characteristictime scaleof
the shorter-lived,smaller-scalestructure,i.e., individualcloud. At the same time, the cold air pool
spreadsout in the form of a densitycurrent and triggersnew cloudsat the edgesof the pool.In this
mannerit producesand maintainsthe longer-lived
larger-scale
structure,i.e., cloudsystem.
The double-scalestructure which was naturally obtained in the case with the full set of
microphysicsresemblesthe double-scalestructureof the convectionover the earth's tropicalocean;
i.e., short-lived,small-scalecumulonimbiare organizedto form longer-lived,larger-scaleclustersof
clouds. Furthermore,the role of the cold air pools in the maintenanceof the cloud systemsis
consistentwith that observedin the cloudclustersin the atmosphere.Thesestronglysuggestthat the
double-scalestructurein the earth's tropical atmosphereis the naturalform of precipitatingcloud
convectiondrivenby verticaldifferentialheating.The resultsof the experimentsalso showthat the
originof the double-scalestructureof the tropicalconvectionis the existenceof cloudmicrophysical
processes.
1.
Introduction
In the
tropics
C1988, Meteorological
a number
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of deep convective
of Japan
clouds are observed. The horizontal dimension of
an individual convective cloud is 1-10km, and its
lifetime is O(1 hour). Each convective cloud is
characterized by strong vertical motions of air
310
Journal
of the Meteorological
Society of Japan
Vol. 66, No. 2
through a considerable depth of the troposphere,
and by the occurrence of active phase changes of
water substances associated with that motion. It
is widely known that the vertical structure of the
earth's atmosphere is maintained through a
balance of radiative and convective processes.
The former cause an energy surplus at the ground
and a deficit in the troposphere, whereas the
Fig. 1. A part of the IR imagerytaken from the (;MS
latter transport the energy from the ground to
on 17 April 1985 (from Monthly Report of
the troposphere (e.g., Manabe and Strickler,
Meteorological
SatelliteCenter,April,1985)
1964). The 'convective' processes are, at least in
the tropics, the activity of convective clouds.
The central concern of this study is to
Indeed the vertical mass transport in deep clouds,
determine the cause of the double-scale structure
i.e., cumulonimbi, contributes largely to the
of the tropical cloud convection. The generation
upward transport of energy in the Intertropical
of cloud clusters has often been explained as a
Convergence Zone (Riehl and Simpson, 1979).
result of large-scaleconvergent motion in the low
Therefore the occurrence of convective clouds is
level atmosphere. If this is the case, then the
indispensable for the maintenance of the earth's
double-scale structure of cloud convection is
present climate.
merely a reflection of the existence of large-scale
A remarkable feature of deep convection in
disturbances in the flow field; only the smaller
the tropical atmosphere is that individual
scale is an intrinsic property of convection due to
cumulonimbi are not separated from one
vertical differential heating. The larger scale is
another; they usually appear in the form of
due to external2) dynamics. We propose,
groups which are much larger than a single cloud.
however, that this is not correct. Both the
This property of tropical cumulonimbi was first
smaller and larger scale structures are intrinsic
revealed when satellite observations became
properties of cloud convection in the tropical
available in the 1960's. We call such groups of
atmosphere caused by vertical differential
cumulonimbi 'cloud clusters'. Examples of cloud
heating. If the above speculation is correct, the
clusters are seen in Fig. 1, which shows an IR
double-scale structure can be realized without
imagery of cloud activity in the western tropical
any large-scale external dynamics. In particular,
Pacific. In the picture we can see that almost all
cloud clusters will emerge in an atmosphere
of the deep convective clouds occur in clusters, under the idealized conditions of horizontally
which are identified as white patches with
uniform heating below and cooling from above.
various shapes. Their horizontal dimensions are
The grounds for our above speculation are as
100-500km; their lifetimes are estimated at
follows:
(10 hours) from sequences of IR imageries.
O
(1) Active cloud convection almost always
From the facts above, we can state that cloud takes the form of cloud clusters whenever it
convection over the tropical oceans has a
"double -scale" structure: small-scale, short-lived occurs over a warm sea surface, regardless of the
differences of geographical locations. In several
individual clouds are organized into larger-scale, observational programs conducted in various
longer-lived clusters 1). [For more details on
parts of the tropics since the 1960's (i.e., Line
cloud clusters, see e.g., a review article by Houze Island Experiment, BOMEX, GATE and
and Betts, 1981.]
WMONEX),cloud clusters were always observed.
1) Hayashi and Sumi (1986) pointed out that cloud
(2) The internal structures of cloud clusters in
clusters
in the equatorial
latitude
are sometimes
organized
further
into the so-called "super cluster",
whose horizontal
scale is around 3000km. In the present
paper the term "cloud cluster" or "cluster"
refer to a
group of individual
clouds; it does not refer to "super
cluster".
2) He
re
`external'
dynamics
refer
to
dynamics
having a characteristic
scale larger than that of an
individual
cloud;
one example
is a geostrophicallybalanced
vortex
whose characteristic
spatial
scale
corresponds to the radius of deformation.
April 1988
K. Nakajima and T. Matsuno
various areas in the tropics are largely the same,
despite differences in large scale conditions or
initiation mechanisms. For example, Houze et al.
(1981) reported that there are structural
similarities between the cloud clusters in the
vicinity of north Borneo initiated by land-sea
breeze circulation and cloud clusters in other
parts of the tropics which are not related to such
local circulations.
(3) Even if a large-scale flow field exists, the
influence of the large-scale flow on cloud clusters
does not seem to be definitive. For example, by
examining the GATE data, Payne and McGarry
(1977) found that the location of cloud
cluster-genesis was related to the phase of
African waves only in a statistical sense, and that
the movements of individual clusters did not
closely follow the propagation of the waves.
(4) A possible key process in the maintenance
of a cloud cluster is downdrafts driven by the
evaporation of raindrops. This triggers the
generation of new convective clouds immediately
adjacent to the older convective clouds (e.g.,
Leary and Houze, 1979). This process is an
intrinsic property of the cloud microphysics, and
does not depend on external forcing.
(5) In numerical experiments using a cumulusresolving axisymmetric tropical cyclone model,
Yamasaki (1983) found that simulated clouds
occurred neither randomly nor uniformly within
the simulated disturbance of size O(100)km.
Rather, they occurred in the form of a very
long-livedcloud system whose size was O(1Okm).
This means that the spatial structure of cloud
activity may not be directly controlled by the
large-scalemotions.
However, if we take the standpoint that the
double-scale structure is an intrinsic and
therefore natural morphology of cloud convection, we still have an unanswered fundamental
question: why is the natural morphology of
cloud convection so different from that of
Benard-Rayleigh convection? The BenardRayleigh convection is the most standard form of
fluid motion which is observed in a fluid layer
which is heated from below and cooled from
above. Over a fairly wide range of Rayleigh
numbers, the flow observed in Benard-Rayleigh
convection is characterized by more or less
311
steady and regular cellular patterns. Such cells
are commonly observed in centimeter-scale
laboratory experiments as well as in 103km-scale
convection in the solar photosphere (i.e., solar
granulations). The cells have O(1) aspect ratios
and are distributed almost uniformly over the
fluid (e.g, Krishnamurti, 1970a,b, 1973). At
very high Rayleigh numbers, the fluid motion is
dominated by the intermittent ascent of
thermals', which have the size of the heat
'
conduction layer (e.g., Sparrow et al., 1970), and
becomes almost turbulent. Even in such cases,
however, the thermals are aligned in persistent
cellular patterns having horizontal dimensions
several times the depth of the fluid (e.g.,
Fitzjarrald, 1976; Willis and Deardroff, 1979).
Compared to the Benard-Rayleigh convection,
the morphology of tropical cloud convection is
far more complex. Cloud clusters have a much
greater diversity of shapes, sizes and lifetimes in
contrast to the persistent organization of
thermals in Benard convection. The great
irregularity of the configuration of cloud clusters
is also fundamentally different from the
relatively regular spatial structure of ordinary
Benard convection. Therefore, cloud clusters do
not seem to be the atmospheric counterpart to
the spatial organization of thermals which appear
in turbulent Benard-Rayleigh convection.
From the discussion above it has become clear
that we need to examine what the natural form
of cloud convection caused by vertical
differential heating really is. However, we are
unable to answer the above question by
observing clouds in the real atmosphere. This is
because there exist a number of factors such as a
large-scale flow field and the inhomogeneity of
sea surface temperature whose interaction with
clouds are inevitable. The most straightforward
and practical way to clarify such elementary
properties of cloud convection is to conduct
numerical experiments under simplified and
idealized conditions.
For the purpose of examining the natural
properties of cloud convection over the tropical
ocean, the numerical model should be designed
as follows:
(1) The numerical model should have a high
enough resolution and should include cloud
312
Journal
of the Meteorological
microphysics, so that cloud dynamics can be
simulated explicitly.
(2) The size of the computational domain of
the model should be large, so that the
development of clouds in the model is not
seriously affected by lateral boundary conditions.
Furthermore, the experiments should be
designed as follows:
(1) The vertical boundary conditions, e.g., the
temperature of the sea surface below, should be
horizontally uniform.
(2) The experiments should start from
horizontally uniform initial conditions.
(3) The period of the integration should be
long enough to ensure that not only the behavior
of clouds but also the temperature and moisture
structures of the "environment" settle down to a
mutually consistent and quasi-steady state. This
is because we are interested in behavior of cloud
convection which does not depend on the initial
conditions which are chosen more or less
arbitrarily. In this respect our experimental
design is similar to that used in the numerical
modeling of the general circulation, where the
model integration is carried out until the initial
conditions are "forgotten" and the internal
variables of the model reach a quasi-steady,
self-consistent state.
(4) Only horizontally uniform and temporally
steady thermodynamical forcings should be
incorporated as destabilization factors. This
situation is identical to that of laboratory
experiments of Benard-Rayleigh convection. No
externally specified dynamical forcing should be
imposed. In these respects the present experimental design is quite different from that of
Soong and Tao (1980) of Kreuger (1985). They
also used a large-domain cloud model. However,
they imposed a time-dependent "large-scale
lifting" which was taken from the observational
data of the GATE network. They chose this
forcing because their main purpose was to
simulate the response of convective clouds to the
specified forcing.
(5) In order to investigate the behavior of
cloud convection in its purest form, the effect of
the earth's rotation should be excluded. In this
respect, the present approach is different from
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of Japan
Vol.
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No.
2
the numerical studies by Yamasaki (1975, 1983,
1984 etc.), in which the interaction between the
cloud convection and large-scale geostrophic
motions was investigated.
Following the above considerations,
we
conducted a series of long-term integrations of a
512-km domain two-dimensional cloud convection model. In the numerical experiments, we
found that a number of clouds are indeed formed
successively, one next to another as observed in a
cloud cluster. This strongly supports our earlier
speculation that the double-scale structure is one
of the intrinsic and fundamental properties of
cloud convection. We now wish to address the
question: "why does the earth's cloud convection chooses the double-scale structure as its
natural morphology, rather than choosing the
more familiar cellular form of Benard Rayleigh
convection?" The most probable answer to the
above question is as follows: "Because cloud
microphysical processes exist in the earth's cloud
convection, whereas they do not exist in
Benard Rayleigh convection. "Therefore we have
to know how the cloud microphysical processes
affect the natural morphology of convection in
the atmosphere.
In this paper we present the results of
numerical experiments which were designed to
simulate the natural behavior of cloud convection under the different combinations of cloud
microphysics. By doing so, we can easily isolate
the main mechanism for the generation of the
double-scale structure and we can determine the
role of the individual microphysical processes. In
the next section, a brief description of the
numerical model is given. The design of the
experiments is explained in section 3. Results of
numerical experiments are presented and discussed in section 4, and conclusions are given in
the last section.
Because of the limitation
of computer
capacity, some of the integrations reported in the
present paper were terminated before reaching
equilibrium states with sufficient accuracy. After
the completion of the paper, we were able to do
extended integrations owing to an increase of
computer power. It eventuated that all conclusions in this paper were confirmed.
April 1988
2.
K. Nakajima and T. Matsuno
313
Model description
The model we used in this study is essentially
a two-dimensional cloud convection model. A
marked point of the model is its exceptionally
large computational domain in the horizontal
direction, 512km, which is almost 10 times as
large as the size of standard thunderstorm
simulators. This large size allows several clouds to
develop simultaneously. Constant body cooling
which simulates the effect of radiative processes
and surface fluxes from the lower boundary are
included. The effect of the earth's rotation is
excluded so that we can treat convective
processes free from the interaction with geostrophically balanced motions.
The basic equations and values of parameters
used in the present study were mostly taken
from Yamasaki (1975). The dynamical framework is based on the modified form of the
anelastic system developed by Ogura and Phillips
(1962).
Cloud microphysical
processes are
parameterized following Kessler (1969). A list of
symbols can be found in Appendix A.
As noted in the introduction, we assume a
different set of cloud microphysics for each of
the experiments. This is done by introducing
three "switching" parameters. They are *drag,
evap and *conv which control the *inclusion
(value=l) or the exclusion (value=0) of the drag
force, the rainwater evaporation and the change
from cloud water to rainwater, respectively.
Their specifications in each experiment will be
given in the next section.
The equations for the x- and z-components of
velocity are
D represents
defined
the
eddy
diffusion
term
which
is
as
where Y represents a variable which is subject to
diffusion. The drag force due to liquid water
loading is represented by the fifth term in the
right hand side of (2). *drag
is the switch
parameter for the drag force. The horizontal and
vertical eddy diffusion coefficients KII and KV
are assumed to depend
on the velocity
deformation and the static stability as follows:
The
equation
Using
the
stream
function *
above
of mass continuity
equation,
such
we
is
define
the
mass
that
In later sections, the flow field in the model
be displayed in terms of *.
The first law of thermodynamics
is
will
and
where
Here C is the rate of condensation
(or
evaporation of cloud water) which is calculated
using a saturation adjustment, and Er is the rate
of rainwater evaporation. *evap
is the switch
parameter for the evaporation of rainwater. The
314
Journal
of the Meteorological
effect of radiative processes is represented
crudely by the sum of a Newtonian cooling term
with the damping constant DR (specified to be
1/5 day) and a horizontally homogeneous body
cooling term QR. In order that no horizontal
inhomogeneity is introduced by the Newtonian
cooling, its rate is calculated using a horizontal
averageof the perturbation temperature
Society
processes
Fluxes
water
of Japan
are presented
in Appendix
of momentum,
sensible
vapor
calculated
formulae
Vol. 66, No. 2
from
by
the
underlying
employing
the
sea
B.
heat,
surface
standard
and
are
bulk
as follows:
where
where *(z)
is the horizontal
average of the
perturbation potential temperature.
profile of QR is specified as
The vertical
which is a simplified form of radiative cooling in
the tropical troposphere 3).
The continuity
equations
for water vapor q*,
cloud water qc, and rainwater qr are
The change from cloud to rain Prc consists of
autoconversion and collection. *conv is the
switching parameter for the change from cloud
to rain. The third term on the right hand side of
(12) represents the falling of rainwater with
volume median terminal velocity VT relative to
the air. Detailed treatments of microphysical
3)
The magnitude of QR specified here is, in fact,
about twice as large as that of the net radiative cooling
in the real tropical atmosphere (see e.g., Fig. 7 of
Freeman and Liou, 1979). Additional experiments with
a reduced QR revealed that the false choice of the
radiative cooling intensity did not affect the main
conclusions of this study.
and
Here Tsfc is a fixed value of the model sea
surface temperature, and is specified to be 302
K. The transverse velocity component *0, which
is specified to be 3m/s, is included in order to
keep the fluxes at a reasonable magnitude even
when the simulated wind in the X-Z plane is
weak. The surface drag coefficient CD is
specified to be 0.0015. The fluxes are given at
the grid points in the lowest level.
Cyclic boundary conditions are used for all
variables in the lateral direction to ensure
horizontal homogeneity. The bottom and the top
boundaries are assumed to be rigid, so that
where Ztop=22.6km.
All of the above equations are solved using the
following
finite
difference
method
on
a
rectangular
grid in space. The grid interval is
1000m
in the lateral direction,
whereas the
vertical grid spacing varies from 300m near the
surface to 1200m
at the top. We adopt a
staggered grid, in which points of u, w, and * are
placed at different positions. *,
q*, qc. and qr are
placed at the same points as w. The total number
of grid points is 512*34. Space derivatives in the
advection
terms
are
approximated
by the
first-order
upstream
scheme. Other space derivatives are approximated
by the second-order
centered
differences.
Time
integrations
are
performed
by using the leap-frog scheme except
for the friction and diffusion terms for which the
time-forward
scheme is used. The time-forward
April 1988
K. Nakajima and T. Matsuno
scheme is inserted at every 20 steps in order to
avoid splitting of prognostic variables at even and
odd time steps. The interval of the time steps is
typically 10sec.
3.
Designof the experiments.
Specifications of cloud physical processes
The cloud physical processes whose effects we
investigate in this study are: (1) the change from
cloud water to rainwater, (2) the evaporation of
rainwater into unsaturated air, and (3) the drag
force due to cloud and rainwater. Whether we
include a particular process in a given experiment or not is controlled by setting the value
of the corresponding switch parameter, i.e.,
315
due to liquid water is included in addition to the
set of cloud physics assumed in the case R. In
other words, the interaction
between rain water
and air is only
dynamical
in nature;
the
thermodynamical
interaction
through
the
evaporation
of the rain is excluded. This case and
the two cases below are conducted
to clarify
which of the roles played by the rain is primarily
responsible for the generation of the double-scale
structure.
(4) Case F (Full set; Rain and Drag and
Evaporation): This is the standard experiment in
which all of the realistic cloud physics are
included. The full set of the bulk cloud physical
parameterization of Kessler (1969) is used.
Rainwater produced from cloud water not only
acts as a drag force on the air but also cools the
air through evaporation.
(5) Case RE (Rain and Evporation): Here we
exclude the drag force from the set of cloud
physics assumed in the case F. Thus the rain
water interacts with the air thermodynamically
through evaporative cooling but does not interact
dynamically.
Initial conditions and integration time
We initialize the model in the following way.
First, all variables are set to be horizontally
uniform. The velocity components are zero
everywhere. The vertical distributions of the
temperature and the moisture are taken from
Table 1 of Yamasaki (1983), except for the
lower levels where the relative humidity is
increased slightly. This is basically the tropical
standard atmosphere. (These values are also used
to calculate the basic state variables). Note that
the initially specified thermodynamic structures
Table
1.
Specifications
of *conv, *evap
and *drag
for
each
case
of the experiments.
316
Journal
of the Meteorological
are rapidly forgotten in the first few hours of
each experiment. As a result, the 'environmental'
conditions for cloud development at the later
stages, upon which our interest is focused, may
be considerably different from the initial one.
This indeed occurs in case NR and will be
presented later. Therefore the details of the
initial conditions are not important.
Next, in order to seed convective motion, a
random potential temperature perturbation
whose value ranges from -0.3K to +0.3K is
specified for all grid points at Z=300m. This
random noise is applied only at the beginning of
each experiment. Thus our seeding technique is
different from those which are adopted in the
cloud ensemble models of Soong and Tao (1980)
and Tao and Simpson (1984). In those studies
random noise was introduced in order to
simulate the inhomogeneity in the surface
boundary layer due to turbulence. Thus they
added random perturbations repeatedly throughout the integration period. We do not adopt an
assumption like theirs. Nevertheless, in our
experiments, a number of clouds do appear
without such continuous randomization.
The numerical model was run typically for 50
hours in each experiment. This integration period
is long enough for the spatial structure and
temporal behavior of cloud convection to settle
down to a quasi-steady state. However, small
tendencies in the horizontally averaged
temperature and humidity were found in some of
the cases; much longer integrations would be
necessary to produce a perfectly equilibrated
situation. We terminated the experiments at this
integration period because we believe that the
basic characteristics of the behavior of the cloud
convection would not change greatly if the
integration were continued for a longer time."
The typical CPU time required for a 24 hour run
is about 30 minutes using a HITAC S-810/20 at
the Computer Center of the University of Tokyo.
4)
Extended
after
the
500
hours
numerical
completion
is required
environmental'
noticed
convections
presented
of
in
states.
the
in
in this
gross
the
experiments
this
to
carried
revealed
produce
However,
that
almost
little
characteristics
extended
paper.
paper
experiments
Society
of Japan
Vol . 66,
No.
4. Results of the experiments
4.1 Case NR
In this experiment, the effect of latent heat
associated with the phase change of water is
included, but the change from cloud water to
rainwater is not. The cloud physical process
assumed in this case is so different from the
reality that the final state of the atmosphere is
completely different from the real tropical
atmosphere. Because of this, a very long time,
longer than 500 hours, is required for the model
to reach an equilibrium state. In order to save
computational time, we set the horizontal
domain size to be 128km instead of the 512km
which is for the other cases. This is justifiable , a
posteriori, because the characteristic horizontal
scale of convection obtained in this case was
about 10km, which is an order of magnitude
smaller than the reduced domain size.
Fig. 2a-d show spatial distributions (in theX-Z
section) of the liquid water mixing ratio , the
water vapor mixing ratio , the potential
temperature (deviation from the basic state) , and
the mass stream function at the final state for
Fig. 2a. Mixing ratio of cloud water in theX-Zsection in
case NR. Contour interval is lg/kg. The maximum
value is 17.9g/kg.
out
around
equilibrated
differences
'
of
and
the
the
were
cloud
cases
2
Fig. 2b.
vapor.
Same
as
Fig.
2a
but
for
mixing
ratio
of water
April 1988
Fig. 2c. Same as Fig. 2a but for perturbation
temperature.
Contour interval is 0.5K.
K. Nakajima and T. Matsuno
potential
Fig.2d. Same as Fig. 2a but for massstreamfunction.
Contourintervalis 2000kg m-1s-1.
this case. Liquid water exists everywhere in the
domain; the whole atmosphere is completely
saturated. The mixing ratios of vapor and liquid
water are basically functions of height only:
mixing ratio of cloud water increases with height,
whereas that of the water vapor decreases with
height. Note that the sum of the two, i.e., the
total water content, is nearly constant in the
convection layer. This is quite understandable
because the total water must be conserved since
no water removal process exists. The mass stream
function field exhibits. a cellular pattern which
extends from the surface to a height of several
kilometers. The horizontal size of a cell is also
several kilometers. The intensities of updrafts
and downdrafts are of the same order. The
characteristic velocity is 5-10m/s. The horizontal
component of wind velocity is nearly equal to
the vertical component because the aspect ratio
of each cell is nearly unity. Both the potential
temperature and the water vapor mixing ratio are
positively correlated with the vertical velocity, so
that heat and moisture are transported upward.
The structure of the convective motion in this
case, i.e., a cellular pattern with O(1) aspect
317
ratio, is the same as that of ordinary
Benard-Rayleigh convection. This is true even
though a phase change of water is taking place
and one would expect the latent heat to strongly
affect the thermodynamics. It is clear that water
vapor is condensing in the updrafts whereas
liquid water is evaporating in the downdrafts.
Thus the latent heat of condensation is given to
the air in the updrafts while the latent heat of
vaporization is extracted from the air in the
downdrafts. In this sense, the updrafts and the
downdrafts are "symmetric"; both are moistadiabatic. Clearly there is no difference between
this convective motion and simple dry adiabatic
convection, if we replace the dry adiabatic
process by a moist adiabatic process. Therefore
the appearance of Benard-Rayleigh convection
and the realization of an almost moist-adiabatic
lapse rate are quite natural.
The origin of the up-down thermodynamic
symmetry is the abundance of liquid water
floating in the air. When precipitation processes
exist, the liquid water content in the atmosphere
will be drastically reduced. As a result of this
drying of the atmosphere, the thermodynamical
and the dynamical symmetries between the
upward and downward motions are lost.
4.2 Case R
In addition to the phase change of water
substances and the associated latent heating, in
this experiment we consider the conversion
process from cloud to rain. However, neither the
drag force due to liquid water nor the
evaporation of rainwater is included.
In the steady state of this experiment, only
one narrow cloud appears in the 512km domain.
Fig. 3 shows the time evolution of the horizontal
distribution (X-T section) of surface precipitation intensity. In the first few hours, a number of
clouds develop from the initially given random
seedings. Their lifetime is no more than a few
hours. As they decay, a cloud gradually develops
at around X=490km
and persists almost
steadily for more than 20 hours until the end of
the experiment. This cloud does not move. No
other clouds develop in the domain after this
cloud is established.
Fig. 4.a-e show the 12 hour-time average of
318
Journal
of the Meteorological
Society
of Japan
Vol . 66, No.
Fig. 4c. Same as Fig. 4a but for mixing ratio
rainwater. The maximum value is 11.2g/kg.
Fig. 3. Time evolution of the horizontal distribution of
precipitation intensity at the surface in case R.
Contour interval is 40mm/hr. The maximum value is
193mm/hr.
Fig.
4a.
12
X-Zsection
m-1s-1.
hour
averaged
in
case
mass
R.
stream
Contour
function
interval
in the
is 500kg
Hg. 4b. Same as Fig. 4a but for mixing ratio of cloud
water. Contour interval is 0.05g/kg. The maximum
value is 0.48g/kg.
2
of
Fig. 4d. Same as Fig. 4a but for mixing ratio of water
vapor. Contour interval is 1g/kg.
Fig. 4e. Same as Fig. 4a but for perturbation potential
temperature. Contour interval is 0.5K.
mass stream function, cloud water mixing ratio,
rainwater mixing ratio, water vapor mixing ratio,
and potential temperature deviation. The timeaveraging operation was performed in order to
filter out short-period gravity waves . In Fig. 4a, a
narrow, strong updraft is seen; the vertical
velocity is about 10m/s, and its width is only a
few kilometers.In the~rest of the domain there' is a
uniform weak downward motion. As seen in Fig.
4b and 4c, both cloud and rain water exist only
in the neighbourhood of the updraft. In Fig. 4d,
it is seen that there is a distinct contrast between
the water vapor mixing ratios on the inside and
April 1988
K. Nakajima and T. Matsuno
outside of the cloudy column. The width of the
moist column is only several kilometers. In the
subcloud layer, however, moisture is uniformly
distributed all over the domain including the area
just under the updraft. The potential
temperature is almost uniform in the x-direction
outside the cloud. The vertical stratification of
temperature is conditionally unstable. Note that
the cloudy column is cooler than the environment in the lower half of the atmosphere.
However, the buoyancy is larger there than in the
surrounding air because of the distinctly higher
moisture content in the cloudy column.
The results of this case and case NR differ in
the following respects: (a) In this case liquid
water is almost absent in the air, whereas it was
abundant everywhere in the case NR. (b) Except
in the narrow updraft, the atmosphere in this
case is unsaturated, whereas it was completely
saturated everywhere in case NR. (c) Here the
phase change of water substances occurs only in
the updrafts, whereas it occurred both in the
updrafts and in the downdrafts in case NR. (d)
The mean vertical temperature structure is
conditionally unstable here, whereas it was
almost moist-adiabatic in case NR. (e) Here the
upward and downward motions are asymmeteric
with respect to their intensities and width,
whereas the intensities of the updrafts and the
downdrafts in case NR was almost the same.
The differences above are clearly due to the
generation of rainwater. By the conversion to
rainwater and its precipitation, liquid condensate
is rapidly removed from the atmosphere. As a
result there is hardly any liquid water available
for evaporation in the downward motion.
Therefore the descending motion is dry
adiabatic, whereas the ascending motion is moist
adiabatic. Because of this, the mean vertical
temperature gradient is maintained at a magnitude between the temperature gradients
associated with the dry and moist adiabatic
processes. Thus, this combination of asymmetric
thermodynamics in the vertical motions keeps
the
atmosphere
conditionally
unstable.
According to linear theories of cloud convection
in a conditionally unstable atmosphere (e.g.,
Kuo, 1961) or the classical "slice method" of
Bjerknes (1938), the combination of a strong
319
narrow updraft and a weak widespread downward motion is preferred: this is the very same
spatial structure of connective motion realized in
the present case. Note that the result of the case
R strongly suggests the predictions of these linear
theories to be correct even in the finiteamplitude case, if cloud physical processes other
than those assumed here are absent.
Compared to the results of case NR, the
unsaturated
and
conditionally
unstable
atmosphere developed in this case agrees much
better with the real atmosphere. The narrow
strong updraft region is also similar to that of
real convective clouds which occupy very small
areas in the atmosphere.
There are, however, features which seriously
disagree with observations. In this experiment,
the updraft seems to persist for a very long time,
perhaps infinity, whereas the typical lifetime of
observed convective clouds is O(1
hour).
Moreover, in this case, only one isolated cloud
appeared in our very large domain, whereas there
is a much larger number of clouds in the real
atmosphere. Therefore the classical linear
instability theories of cloud convection, whose
realization in the large-amplitude case was
demonstrated here, cannot be accepted as
satisfactory understandings of cloud convection
in the real atmosphere.
4.3 Case RD
In addition to the set of cloud physical
processes in the previous case, the drag force due
to liquid water is considered here.
Fig. S shows the time change of the horizontal
distribution (X-T section) of the surface
precipitation intensity. In the first few hours,
just as in case R, a number of clouds which are
distributed randomly develop. However, they
soon decay. After that, rainfall occurs only at a
few places each of whose width is several
kilometers. Through the model integration, these
narrow precipitation spots are nearly fixed in
space, except for a small migration within a range
of several kilometers.
The following differences are notable when
we compare the X-T sections of precipitation
intensity of this case with that of case R: First,
even after several hours of running this
320
Journal
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Society of Japan
Vol. 66, No. 2
Fig. 6. The 12 hour averaged mass stream function in
the X-Z section in case RD. Contour interval is 100
kgm'ls'1g
Fig.5. Time evolution of horizontal distributionof
surface precipitation intensity in case RD. The
maximumvalueis 101mm/hr.
experiment, there are two or three precipitation
areas as opposed to only one in case R. However,
the fact that one of the precipitation areas dies
out at T=20h
suggests that they tend to
suppress each other. Therefore, we may expect
that one of the two updrafts which exist at the
end of the experiment would die after a
sufficiently long time. Second, in contrast to the
steadiness of the cloud in case R, the rainfall
intensity at each of the precipitation areas shows
significant oscillatory variations in time; the
period of the oscillation is several hours.
Fig. 6 shows the time-averaged mass stream
function in the time interval from T=24h to T
=36h . Corresponding to the two precipitation
areas seen in Fig. 5, two strong, narrow updrafts
exist. However, the downward motions are weak
and extend over the remaining part of the
domain. This feature of the motion field
resembles that realized in case R in the
asymmetry between upward and downward
motions. On the other hand, the time sequence
of the mass stream function (not presented here)
shows that the concentrated updrafts in this case
are highly time-dependent; this is in contrast to
the updraft in case R whose intensity was almost
constant in time.
The cause of the oscillations
of the clouds,
whose period is several hours, is believed to be
the drag force due to liquid water, in particular,
rain water. Unlike cloud water, rainwater
can
accumulate
in the updrafts
by falling down
relative
to the air flow. Thus its drag force
eventually
becomes
large enough to overcome
the buoyancy
due to high temperature
in the
cloud, thereby suppressing the updraft.
The persistence
of updraft
areas in spite of
their violent pulsation also requires explanation.
Fig. 7 shows the water vapor mixing ratio at T=
45h,
at which time the updraft spot at X=65
km is in a break. A zone of highly increased
moisture
content
is seen around X=64km
which corresponds
to the updraft area, although
upward motion does not exist at this moment. It
is the conservation
property
of the water vapor
mixing ratio that serves as a memory
for the
position of an updraft over a time interval of
longer than ten hours.
The inclusion of the liquid water drag effect
results in a double-scale
temporal structure
of
Fig. 7. The mixing ratio of water
Contour interval is 1g/kg.
vapor
at T=45h.
April 1988
cloud convection:
K. Nakajima and T. Matsuno
the long-lived updrafts which
pulsate with a short time scale appear. However,
no double-scale spatial structure is obtained.
Thus the inclusion of the drag effect is not
enough to explain the double-scale structure of
the observed cloud convection.
4.4 Case F
In this case the full set of warm rain cloud
physical processes is included following Kessler's
parameterization. When an equilibrium state is
realized in the experiment,
a number of
convective clouds are generated one after
another. Furthermore, almost all of the clouds
form groups whose lifetime is much longer than
that of individual clouds. The structure and basic
mechanisms of the cloud groups are qualitatively
similar to observed cloud clusters as will be
demonstrated shortly.
Fig. 8 shows the time change of rainfall
intensity at the surface for case F. As in the
previous cases, a large number of short-lived
clouds are generated by the initial seeding in the
first few hours. After the clouds disappear, the
number of clouds decreases. The distribution of
this cloud activity in X-T space has a less regular
structure than those of the two previous cases.
Fig. 8. Time evolution
of horizontal
distribution
of
surface
precipitation
intensity
in case F. The
maximum value is 73mm/hr.
321
However, it is seen that most of the clouds are
organized into several chain-like features. This
means that most of the convective clouds are
successively generated in several regions which
migrate in space. These features are more clearly
recognizable in Fig. 9, which shows the time
evolution of the spatial distribution of low level
updraft velocities. This figure clearly shows that
most of the clouds generated in this experiment
are interconnected by lines of low level updrafts
in X-T space.
The properties of the successive cloud
formation systems can be summarized as follows:
(a) They are composed of a number of deep
convective clouds which are formed in
succession. The lifetime and the spatial scale of
the individual deep cloud are O(1 hour) and O(1
km), respectively. The typical time interval
between one deep cloud and the next in the cloud
systems is 2-5 hours. The position of new cloud
formation is 5-15km
from the older one.
Because of this horizontal shift of cloud
development, the separation of individual clouds
from one another is much clearer in this case
than in case RD.
(b) Although the beginning and end of a life
cycle is not well-defined, the lifetime of a
Fig. 9.
Time
evolution
of vertical
m in case F. Only updrafts
maximum value is 2.59m/s.
velocity
are
at Z=600
indicated.
The
322
Journal
of the Meteorological
successive-cloud-formation system can be estimated to be longer than 10 hours. This is at least
an order of magnitude longer than the lifetime of
individual clouds in the system.
(c) The successive-cloud-formation systems
migrate in space with a typical propagation speed
of 1-2m/s. This migration results from the
horizontal shift of cloud development mentioned
in (a). Because of this migration, the width of the
area which experiences rainfall from one of the
cloud systems during its lifetime becomes as
large as 30-100km. In this sense the horizontal
scale of a successive-cloud-formationsystem is an
order of magnitude larger than the horizontal
scale of the individual clouds. In this respect the
long-lived cloud systems in case RD differ from
those in the present case. The systems in case RD
practically did not move, so that the width of the
area of precipitation was no larger than the width
of a single cloud element.
Because of the points discussed above, we can
consider the successive-cloud-formation system
in this case to be a higher level structure of cloud
convection than the individual clouds. This
means that we obtained a "double-scale"
structure without any special forcings. Moreover,
the spatial and temporal scales of both the
larger-scale structure (successive-cloud-formation
systems) and the smaller-scale structure
(individual clouds) in the experiment agree with
the corresponding scales of a cloud cluster and an
individual cumulonimbus element in the real
atmosphere. The overall structure of the model
atmosphere as a whole is also qualitatively similar
to that of the real atmosphere. Overall, the
model atmosphere is unsaturated, and the
stratification is conditionally unstable. This
supports the hypothesis presented in the
introduction that the "double-scale" character of
cloud convection in the real atmosphere is a
natural property of cloud convection caused by
vertical differential heating. This is the most
important result of this study.
In order to understand the origin of the
above-mentioned double-scale property of cloud
convection, we will next examine the structure
of one of the cloud groups in greater detail. First,
we will examine snapshots of the motion field
and thermodynamic variables in X-Z sections,
Society of Japan
Fig. 10a.
Air motion
Vol. 66, No. 2
around
a deep cloud and a cold air
pool at the foot of the cloud. Plotted contours are
mass stream function at T=27.5h.
Contour interval
is 500kg
m-1 s-1. The
region
of negative
perturbation
potential temperature
in the lowest 600
m (cold air near the surface) is hatched.
Fig, l0b. Outer boundaries of clouds at the same time
as Fig. 10a defined by the contours of 0.1g/kg total
liquid water mixing ratio. Relative humidity
distribution is indicated by hatching. Saturated
region is blank. Most heavily dotted are region of
89%-99% relative humidity followed by 79%-89%,
6910-79%, 59%-690, 49%-5910. The most lightly
dotted are regions of 390-491 relative humidity.
and next we will examine
successive cloud formations
the time sequence
in X-T space.
of
Fig. l0a shows a strong convective cloud which
is
generated
at X=410km
at T=27.5h.
The motion of the air is indicated by contours of
the mass stream function.
The hatched
area
represents
negative
perturbation
potential
April 1988
K. Nakajima and T. Matsuno
323
temperature near the surface, which will be
referred to below as a "cold air pool". In the
upper half of the cloud, a strong concentrated
updraft exists. On the other hand, at levels below
1200m, a downdraft exists. At middle levels, the
updraft and downdraft coexist side by side. Fig.
l0b shows the outer boundary of the cloud at
the same time step, which is defined as the 0.1
g/kg contour of liquid water content. The spatial
distribution of relative humidity is indicated by
the hatching. In the downdraft region, liquid
water coexists with the unsaturated air, so that
rainwater is evaporating there. The evaporation
cools the air, and generates a cold air pool near
the surface.
The thickness and horizontal
extent of the pool are 300m and about 20km,
respectively, and its temperature is more than
2* cooler than the surrounding air. This cold air
pool is generated by both this cloud and by the
cloud whose life cycle ended 90 minutes earlier.
The remnant of the older cloud is seen as a
narrow moist column at about 10km to the left
of the updraft. The cold air spreads outward with
a horizontal velocity of about 2.5m/s. It is noted
that, on the left hand side, a new cloud appears
at about 10km from the main cloud. This is
triggered by the upward push of surface air at the
outer edge of the cold air pool.
Fig. 11a clearly shows the time sequence of
such triggering of new clouds by surface cold air
Fig. 11a. Time evolution of horizontal distributions of
low-level updrafts and surface cold air pools. Vertical
velocity (positive only) at Z=600m is indicated by
contours.
The maximum value is 2.53m/s.
Perturbation potential temperature (negative only)
at the surface is indicated by hatching. The area
cooler than -1.0K is heavily hatched.
pools. This figure shows the time evolution of
the upward motion at Z=600m
(contours) and
the perturbation potential temperature at Z=0
m (hatching). It is evident that a large number of
clouds are formed along the edges of cold air
pools, which may be identified as 'gust fronts'.
However, not all of these clouds grow into deep,
precipitating clouds.
Fig. 11b shows the time evolution of the
updraft at Z=3000m
in the same area. The
potential temperature distribution at the surface
is superimposed on that figure. The number of
tall clouds which reach this level is much smaller
than the number of the clouds triggered in the
lower levels seen in Fig. 11a; most of the deep
clouds are generated at the latest stages of the
life of the cold air pools. Also, note that only
such deep clouds are able to generate a
significant amount of cold air. The formation of
Fig. 11b. Time evolution of horizontal distributions of
mid-level updrafts and surface cold air pools
(hatching). Vertical velocity (positive only) at Z=
2000m
is indicated by contours. The maximum
value is 4.72m/s.
324
Journal
of the Meteorological
Society of Japan
Vol. 66, No. 2
cold air pools alternates
with the generation
of
deep clouds. From Fig. 11b, it is evident that the
life cycle of a deep convective cloud finishes as
soon as a cold air pool is formed beneath
the
cloud.
Thus
the lifetime
of a deep cloud
coincides with the time scale of surface cold air
formation.
From the above investigations,
it is clear that
cold air pools on the surface play two key roles
in generating
the "double-scale"
structure
of
cloud convection:
First, a surface cold air pool
limits the lifetime
of the individual
cloud.
Second, a cold air pool triggers new clouds in the
vicinity of the cloud which formed the cold air.
These two roles of cold air pools are responsible
for generating
the successive-cloud-formation
system, whose spatial and temporal scales are an
order of magnitude
larger than those of the
individual clouds.
The formation
of cold air pools, which we
identified
as the main mechanism
in generating
the "double-scale"
structure of cloud convection,
is a result of a series of cloud physical processes.
These are: the generation
of raindrops
from
cloud water, precipitation
of the raindrops and
the evaporation
of the rainwater into unsaturated
air. Therefore,
"double -scale"
the primary
cause
structure is the existence
of the
of cloud
microphysical
processes as was proposed earlier.
We stress
that,
in this experiment,
the
"double -scale" structure
appeared
as a natural
behavior of precipitating
cloud convection driven
by vertical
differential
heating;
no external
forcings
which
would
compel
larger-scale
organization
of clouds are included.
It is also
noted that the behavior of cloud convection in
this case is qualitatively consistent with the cloud
convection observed in the real atmosphere.
Thus
it is highly
likely
that
the "double-scale"
structure
of tropical cloud convection is also one
of the intrinsic properties of precipitating
cloud
convection
caused
by
vertical
differential
heating.
4.5
Case RE.
This experiment
was performed
in order to
examine the relative importance
of the drag force
in the spontaneous
generation
of the "doublescale" structure in case F.
Fig.12. Time evolutionof horizontaldistributionof
surface precipitation in case RE. The maximum
valueis 164mm/h.
Fig. 12 shows the time evolution of the
horizontal distribution of precipitation intensity
at the surface. A number of small, short-lived
precipitation areas are identified. Their horizontal dimensions and lifetime are similar to
those of the case F. We also note that they are
generated successivelyinto groups whose lifetime
is much longer than that of individual clouds.
They propagate horizontally by generating new
clouds adjacent to the older clouds. The above
characteristics of the clouds in this case are
similar to those in case F.
The mechanism of successivecloud formation
is the same as that was observed in case F. The
X-T section of the potential temperature at the
surface (not shown) reveals that a cold air pool
is generated at the foot of each cloud in a way
similar to case F. Furthermore the triggering of
new clouds at the edges of the cold air pools is
also a mechanism common to both cases.
The above similarities between the structures
of convection in case RE and case F confirm that
the evaporation of precipitation is of primary
importance in generating the double-scale
structure. The effect of the drag force by liquid
water loading is secondary. This is because (1)
April 1988
K. Nakajima and T. Matsuno
the evaporation of raindrops is more effective
than the drag force in terminating individual
clouds and (2) the drag force cannot trigger new
clouds; the former point was argued by Ogura
and Takahashi (1971) and Yau (1980).
325
domain. This proves that the generation of a
larger-scale circulation is also one of the intrinsic
properties of conditional instability of the first
kind (CIFK), i.e., cloud convection in a
conditionally unstable atmosphere. We believe
that the appearance of successive-cloudS. Conclusions
formation systems (i.e., cloud clusters) should be
Based on the results of the above experiments, understood as a property of CIFK at a highly
we conclude the following:
non-linear stage, where the effects of cloud
microphysics omitted in the simplified linear
(1) The double-scale structure of tropical
cloud convection is an intrinsic property of
theories (e.g., Kuo, 1961) become significant.
precipitating cloud convection caused by vertical Larger-scalecirculations, if they exist, should be
differential heating. The generation of cloud viewed as a result of the formation of such cloud
clusters is also a "free-mode" characteristic of systems rather than the cause. 5)
precipitating cloud convection. This point is
(2) The origin of the unique morphological
based upon the results of case F. Although the
character of tropical cloud convection is the
model's two-dimensionality severely limits direct existence of cloud microphysical processes in the
applicability of the results to the three- atmosphere; the difference between double-scale
dimensional atmosphere, the successive forma- cloud convection and ordinary Berard-Rayleigh
tions of connective clouds exemplified in this convection comes from the difference in the
study are expected to occur also in the
physical properties of the working fluids in each
three-dimensional case. Under such conditions
case. The thermodynamical aspect of the phase
the successiveformation of clouds may occur in changes of water substances alone cannot explain
a two-dimensional way on the horizontal plane. the behavior of tropical cloud convection. This
Thus cloud clusters with a realistic size and point is confirmed from the result of case NR,
structure may be generated.
where Benard-like cellular convection appeared
This kind of successive-cloud-formation even though the phase change of water was
system has been reported in the numerical study
included.
by Yamasaki (1975, 1979, 1983, 1984 etc). He
The cloud physical processes which are
named disturbances composed of such successive- responsible for the differences between Benardcloud-formation systems and the accompanying Rayleigh convection and double-scale cloud
large-scale circulation NF mode CISK dis- convection are (a) the generation of raindrops,
turbances (Yamasaki, 1979), which stands for (b) the precipitation of the raindrops, and (c) the
"non -frictional" CISK (Conditional Instability of evaporation of raindrops. The drag force by
the Second Kind) disturbances. However, we liquid water is of secondary importance. Their
consider his terminology of CISK in the above contribttions can be summarized as follows: The
case to be inappropriate, especially when the role of (a) and (b) is to remove liquid
condensate. This keeps the atmosphere dry, and
effect of the earth's rotation is not taken into
account. The reasons are as follows: First, in case forms conditionally unstable stratification. In
F of the present study, several such cloud such an atmosphere, the spatial structure of
systems appeared simultaneously in the com- convection is characterized by asymmetry
putational domain, and behaved more or less between a strong narrow updraft and a weak
independently. No coherent "large-scale" circula- wide downdraft; the narrow moist updraft is
tion appeared in our case. This suggests that the identified as an individual "cloud". The role of
existence of a particular large scale circulation is (c) is two-fold: First, to limit the lifetime of
not necessary for the successive formation of
5) If the effect of the earth's rotation
is considered,
clouds. Second, in our case R, where only one
this does not hold for circulations whose spatial scale is
steady cloud was realized, there appeared a
larger than the radius of deformation,
beyond which the
circulation associated with CIFK cannot expand.
circulation which covered the entire model
326
Journal
of the Meteorological
individual clouds, and second, to trigger new
clouds adjacent to the old cloud. Through the
combination of these two, the spatial and
temporal structure of convection takes the form
of longer-lived larger-scale 'clusters' which are
composed of successively formed short-lived
small-scale 'clouds'. Both of the above two roles
of (c) are accomplished through the formation of
cold air pools at the surface as seen in case F.
The importance of surface cold air pools
associated with cloud clusters in the real
atmosphere is also argued in a number of
observational studies (e.g., Learly and Houze,
1979).
(3) The appearance of the double-scale
structure of cloud convection is mutually
consistent with maintenance of mean stratification in the tropical atmosphere. The atmosphere
in the tropics is latently unstable, so that
finite-amplitude triggering is required for new
cloud development. Those finite-amplitude
triggerings are provided by the surface cold air
pools at the foot of existing convective clouds. In
the vicinity of older clouds, new clouds can be
generated even under the condition of latent
instability, whereas new cloud development is
quite difficult to achieve far away from existing
clouds. As a result clouds are almost always
formed as groups. On the other hand, the
consumption of water vapor by the successive
cloud developments keeps the low-level
atmosphere from saturating, so that latent
instability is maintained in spite of the flux of
water vapor from the warm ocean below.
(4) Finally, we remark on the possibility (or,
rather, impossibility) of explaining the generation of the double-scale structure on the basis of
linear theory. As we stated above, the generation
of the larger-scale structure (cloud cluster) is
simply a highly non-linear development of the
smaller-scale structure (individual cloud) in a
basic state which is latently unstable only to
finite-amplitude perturbations. Therefore we feel
that, if we start from the basic governing
equations, it is impossible to construct a
straightforward linear theory in closed form
which correctly predicts all of the properties of
cloud clusters (e.g., preferred scale, propagation
velocity, growth rate, etc.) under various
Society
of Japan
Vol.
66,
No.
2
conditions. The only possible way would be to
introduce some empirical ad hoc tuning parameters which must be borrowed from the very
solution that the theory intends to predict. This
approach has little appeal, and we did not pursue
it in this study.
In conclusion, we state that the generation of
both individual cumulonimbi and cloud clusters
is the intrinsic property of precipitating cloud
convection in the tropical atmosphere. Not only
(1km) cumulonimbi but also O(100km) cloud
O
clusters are intrinsic "quanta" of the convection.
Acknowledgements
The authors wish to express their sincere
thanks to Dr. Yoshi-Yuki Hayashi for his many
helpful comments on several versions of the
original manuscript. They also thank Prof.
Tsutomu Takahashi for discussions on various
aspects of cloud convection, Dr. Hajime
Nakamura for advice in the course of developing
the numerical model, Drs. MasanoriYamasaki and
Tsuyoshi Nitta for discussions on the preliminary
results of this study. Comments from Drs.
Wei-Kuo Tao, Yoshihisa Matsuda and two
anonymous referees are appreciated. Finally, the
authors extend their deep gratitude to Dr.
Priscilla Cehelsky for her kind help in improving
much of the English expressions.
Calculations were carried out mainly on a
HITAC S-810/20 at the Computer Center,
University of Tokyo. The Graphic Utility Library
of NCAR was used to draw many of the figures.
This study was partially supported by Grant-inAid for Scientific Research from the Ministry of
Education.
Appendix A
List of symbols
C
instantaneous condensation or
evaporation of cloud water
CD
surface drag coefficient
cp
specific heat of air at constant pressure
D
diffusion term
DR
coefficient for Newtonian cooling
t
time step of calculation *
x
grid spacing in
* the x direction
z
grid spacing in the z direction
*
r
evaporation of rain water
*
April 1988
F*
Fu
F*
K. Nakajima and T. Matsuno
sensible heat flux from the
boundary
momentum
flux
from the
boundary
water vapor flux from the
boundary
lower
Y
z
dummy variable for diffusion
Cartesian coordinate pointing up
lower
Ztop
height of the top boundary
lower
onv
switching parameter for the conversion
*c
from cloud water to rainwater
drag
switching parameter *for the drag force
by liquid water
switching *parameter for the evaporation
of rain water
mass stream
*
function
evap
g
KH
Kv
Prc
P0
II0
QR
qc
qr
qs
q*
q*0
sat
R
T(z)
T0
t
(z)
u
Vs
VT
w
x
321
acceleration of gravity
diffusivity in the x direction
diffusivity in the z direction
production of rainwater
from cloud
water
basic state pressure
basic state non-dimensional
pressure
(II*(P0/PR)*,
where PR=1000hPa
and *=R/cp)
* perturbation
non-dimensional pressure
specified cooling rate to simulate
the radiative effect
mixing ratio of cloud water
mixing ratio of rainwater
mixing ratio of water vapor at the
surface
mixing ratio of water vapor
basic state mixing ratio of water vapor
saturation mixing ratio of water vapor
q*
gas constant of air
basic state density
*0
horizontal
average of perturbation
temperature
basic state temperature
time
*0state
basicpotential temperature
perturbation potential* temperature
horizontal
average of
*
perturbation
potential temperature
velocity component in the x direction
magnitude of velocity at the surface
precipitation velocity of rainwater
specified transverse velocity
*0
at the
surface
velocity component in the z direction
Cartesian coordinate pointing right
Appendix
B
Treatment of cloud microphy sits
In this appendix, details of the treatment of
cloud microphysics are described. All of the
values of the parameters are expressed in MKS
units.
a.
Condensation
or eraporallon
of cloud
Condensation (or cloud evaporation) term C
in eqs. (8), (10) and (11) is determined by the
saturation adjustment of *, q*, and qc. The
adjustment scheme is as follows:
First we evaluate eqs. (1)-(12) excluding C
and Er, so that we obtain unadjusted values *1,
q*1, and qc1. Using these unadjusted values, we
calculate * defined as below:
Note that *q
is approximately conserved
when both * and q* change as a result of
reversible condensation or evaporation .
Next we adjust * and q* to the saturation
values **
and q**, keeping *q
constant .
Neglecting the effect of the pressure perturbation, the relationship between *q and ** is
one-to-one. In actual calculations, a table
containing ** as a function of *q is prepared
for each height level before solving the time
marching problem. From this table, we can get
the amount of condensation (or cloud evaporation if negative) required to attain a justsaturated condition. It is written as
If *C*<0,
evaporation
of cloud
possible. The maximum
amount
tion cannot exceed qc1, so that
water
is
of the evapora-
Using *C we adjust the values of *,
as follows:
q*, and qc
328
Journal
of the Meteorological
Society of Japan
velocity
is formulated
Vol. 66, No. 2
as
References
Note
respectively
that *
and
q*,
become **
and
q**
if
or
where the adjusted state is saturated.
where
*t is the time step for computation. A numerical
factor of 2 is required because we use the
leap-frog time integration scheme. Then, the
adjustments are carried out as follows:
c.
Conversion
from cloud to rain
As mentioned in the text,Prc is the sum of the
autoconversion A and the collection Co.
The expressions for the two components are
where *=10-3,
3=1.0*10-3, and
The values of parameters are taken from
Yamasaki (1975).
d.
Precipitation velocity of rainwater
Following Yamasaki (1975), the terminal
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熱 帯 大 気 中 の ク ラ ウ ドク ラ ス タ ー の 起 源 に 関 す る 数 値 実 験
中
島
健
介
・
松
野
太
郎
東 京 大学 理 学 部 地 球 物 理 学 教 室
理 想 的 条 件下 に お け る 雲 対 流 の 自然 な 性 質 を 調 べ る た め に,大 き な 計 算 領 域 を 持 つ〓次〓雲 対 流 モ デ
ル を 川 い て 数 値 実 験 を 行 った 、 大 気 は,そ の下 に 置 か れ た 一
様 な 暖 か い 水 面 か ら熱 と水 蒸 気 を受 け 取
り,同 時 に,一 定 の 割 合 で 冷 却 す る、,実験 は五 つ 行 った が,各 々 に お い て 異 な る 雲 物 理 過程を 導 入 した。
各 々 の 実 験 の 中 で 自然 に 実 現 した 準 定 常 状 態 に お い て,対 流 は そ れ ぞ れ 異 な る 空 間 的 ・時 間 的 構 造 を
持 っ て い た。 雨 が 生 成 しな い 実 験 に お い て は べ ナ ー ル 対 流 に 似 た 細 胞 状 構 造 が 現 れ た 。 雨 は 生 成 す る が
蒸 発 しな い 実 験 に お い て は,背 が 高 く幅 が 狭 い宍 が た だ ひ と つ 現 れ た が,そ の 寿 命 は 非 現 実 的 に 長 か っ
た 、標 準 的 な 雲 物 理 の全 て を取 り入 れ た 実 験 に お い て は〓 重 ス ケ ー ル 構 造 」 が 現 れ た。 即 ち,1㎞
の
オ ー ダ ー の 水平 ス ケ ー ル と1時 間 の オ ー ダ ー の 寿 命 を 持 つ 背 の 高 い 雲 が 多 数 生 じ,そ れ らが10時 間 以上
の 寿 命 を 持 つ い くつ か の 雲 シ ス テ ム と して 自発 的 に 組 織 化 して い た 。 各 々 の 雲 シ ス テ ム は 、 そ の 寿 命 の
問 に 幅30-100㎞
の 領 域 に雨 を 降 らせ て い た。
の 最 後 の 実 験 に お け る〓重 ス ケ ー ル 構上造 の 生 成 の主 た る メ カ ニ ズ ム は,背 の 高 い 雲 の直下 に お け
る,雨 水 の 蒸 発 に よ る寒 気 塊 の 形 成 で あ る 、 こ の 寒 気 塊 の 形 成 は,個 々 の 雲 の 寿 命 を 終 わ ら せ る こ とに
よ っ て 小 規 模 ・短 寿 命 の 構 造-即
ち 個 々 の 雲-の
特 徴 的 時 間 ス ケ ー ル を 決 め る。 同 時 に こ の 寒 気 塊
は 密 度 流 と して 周 囲 に 広 が り,そ の 周 縁 で 新 しい 雲 を 誘 発 し,大 規 模 ・長 寿 命 の 構 造
即 ち雲 シス テ
ム-を
作 り維 持 す る。
雲 物 理 の全 て を 含 む 実 験 に お け る〓重 ス ケ ー ル 構 造 は,熱 帯 海 洋上 の 対 流の〓重 ス ケ ー ル 構 造-即
ち,短 寿 命 ・小 規 模 な 積 乱 雲 が 長 寿 命 ・大 規 模 な ク ラ ス タ ー に 紐 織 化 さ れ た 構 造-と
似 て い る。 更
に,こ の 実 験の 中 の 雲 シ ス テ ム の 維 持 に お け る 寒 気 塊 の 役 割 は,実 際 の ク ラ ウ ド ク ラ ス タ ー に お い て 観
測 され る もの と 整 合 的 で あ る。 これ ら の こ と は,地 球 の 熱 帯 大気 中 の 重 ス ケ ー ル 構 造 が,鉛直 差 分 加
熱 に よ り駆 動 さ れ る 降 水 雲 対 流 の 自然 な 形 態 で あ る こ とを 強 く示 唆 す る。 実 験 結 果 は ま た,熱 帯 の 対 流
の〓重 ス ケ ー ル 構 造 の 起 源 が 雲 微 物 理 過 程 の 存 在 で あ る こ と も示 して いる。