Table Of ContentTHE REPRESENTATION OF CUMULUS
CONVECflON IN NUMERICAL MODElS
Emanuel
Kerry A. Emanuel
David J. Raymond
Pubtished by the Americam Metecrolgical Society
METEOROLOGICAL MONOGRAPHS
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METEOROLOGICAL MONOGRAPHS
VOLUME 24 DECEMBER 1993 NUMBER 46
THE REPRESENTATION OF CUMULUS
CONVECTION IN NUMERICAL MODELS
Edited by
Kerry A. Emanuel
David J. Raymond
American Meteorological Society
45 Beacon Street, Boston, Massachusetts 02108
© Copyright 1993 by the American Meteorological Society. Per
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ISSN 0065-940 I
ISBN 978-1-878220-13-4 ISBN 978-1-935704-13-3 (eBook)
DOI 10.1007/978-1-935704-13-3
Softcover reprint of the hardcover 1st edition 1993
Published by the American Meteorological Society
45 Beacon St., Boston, MA 02108
TABLE OF CONTENTS
Preface v
Part I. General Considerations
Chapter I. Closure Assumptions in the Cumulus Parameterization Problem
-AKIO ARAKAWA ............................. .
Chapter 2. Observational Constraints on Cumulus Parameterizations
-DAVID J. RAYMOND .......................... 17
Chapter 3. Trade Cumulus Observations
-MARCIA BAKER 29
Chapter 4. Impacts of Cumulus Convection on Thermodynamic Fields
-MICHIO Y ANAl AND RICHARD H. JOHNSON 39
Chapter 5. The Nature of Adjustment in Cumulus Cloud Fields
-CHRISTOPHER S. BRETHERTON 63
Chapter 6. Momentum Transport by Convective Bands: Comparisons of Highly Idealized Dy
namical Models to Observations
-MARGARET A. LEMoNE AND
MITCHELL W. MONCRIEFF 75
Chapter 7. Cumulus Effects on Vorticity
-STEVENK.EsBENSEN 93
Part II. Schemes for Large-Scale Models
Chapter 8. Convective Adjustment
-WILLIAM M. FRANK AND JOHN MOLINARI ..... '" 101
Chapter 9. The Betts-Miller Scheme
-ALAN K. BETTS AND MARTIN J. MILLER. . . . . . . . .. 107
Chapter 10. The Arakawa-Schubert Cumulus Parameterization
-AKIO ARAKAWA AND MING-DEAN CHENG ........ 123
Chapter 11. Implementation of the Arakawa-Schubert Cumulus Parameterization with a Prog-
nostic Closure
-DAVID A. RANDALL AND DZONG-MING PAN ...... 137
Chapter 12. The Kuo Cumulus Parameterization
-DAVIDJ. RAYMOND AND KERRY A. EMANUEL 145
Part III. Representation of Convection in Mesoscale Models
Chapter 13. A Hybrid Parameterization with Multiple Closures
-WILLIAM M. FRANK 151
Chapter 14. An Overview of Cumulus Parameterization in Mesoscale Models
-JOHN MOLINARI ............................. 155
Chapter 15. Convective Parameterization for Mesoscale Models: The Fritsch-Chappell Scheme
-J. MICHAEL FRITSCH AND JOHN S. KAIN . . . . . . . . .. 159
Chapter 16. Convective Parameterization for Mesoscale Models: The Kain-Fritsch Scheme
-JOHN S. KAIN AND J. MICHAEL FRITSCH. . . . . . . . .. 165
Chapter 17. A Method of Parameterizing Cumulus Transports in a Mesoscale Primitive Equa-
tion Model: The Sequential Plume Scheme
-DONALDJ. PERKEY AND CARL W. KREITZBERG 171
Part IV. Representation of Convection in Oimate Models
Chapter 18. Efficient Cumulus Parameterization for Long-Term Climate Studies: The GISS
Scheme
-ANTHONY D. DELGENIO AND MAO-SUNG Y AO .... 181
Chapter 19. A Cumulus Representation Based on the Episodic Mixing Model: The Importance
of Mixing and Microphysics in Predicting Humidity
-KERRY A. EMANUEL .......................... 185
Part V. Representation of Slantwise Convection
Chapter 20. Parameterization of Slantwise Convection in Numerical Weather Prediction Models
-THOR ERIK NORDENG ........................ 195
Chapter 21. A Parameterization Scheme for Symmetric Instability: Tests for an Idealized Flow
-SIN CHAN CHOU AND ALAN J. THORPE . . . . . . . . . .. 203
Part VI. Use of Explicit Simulation in Formulating and Testing Cumulus Representations
Chapter 22. Cumulus Ensemble Simulation
-KUAN-MAN Xu 221
References 237
VOL. 24, NO. 46 METEOROLOGICAL MONOGRAPHS
PREFACE
Of the many subgrid-scale processes that must be tions of cumulus clouds and the effects of ensembles
represented in numerical models of the atmosphere, of cumulus clouds on mass, momentum, and vorticity
cumulus convection is perhaps the most complex and distributions. A review of closure assumptions is also
perplexing. It is by now well known that the simulation provided. A review of "classical" convection schemes
of many individual phenomena, ranging from tropical in widespread use is provided in Part II. These schemes
and extratropical cyclones to the Madden-Julian os include the convective adjustment scheme developed
cillation, is sensitive to the way convection is repre by Manabe, the Kuo parameterization, the Betts-Miller
sented. It has also been recognized that the water vapor scheme, and the Arakawa-Schubert parameterization.
content of large parts of the atmosphere is strongly
The special problems associated with the representation
controlled by cloud microphysical processes, including
of convection in mesoscale models are discussed in
those operating within cumulus clouds, yet scant at
Part III, along with descriptions of some of the com
tention has been paid to this problem in formulating
monly used mesoscale schemes. Part IV covers some
most existing convection schemes. Given that water
of the problems associated with the representation of
vapor variability is the strongest feedback in climate
convection in climate models, while the parameteriza
simulations, it would seem timely to reconsider this
tion of slantwise convection is the subject of Part V.
and other aspects of the cumulus parameterization
The monograph concludes with a single paper describ
problem.
ing some recent and very promising efforts to use ex
As a step in this direction, a workshop on cumulus
plicit numerical simulations of ensembles of convective
parameterization was held on 3-5 May 1991, at Key
clouds to test cumulus representations.
Biscayne, Florida, bringing together many of the lead
ing specialists in convection and convective parame No single chapter is devoted to the issue of validation
terization. The main objectives of the workshop were of cumulus parameterizations. In the opinion of the
to promote a vigorous discussion of the major issues editors, this is an issue that needs far more attention
in representing cumulus convection in numerical and will, we trust, be the subject of future publications.
models and to produce a monograph suitable both for Many organizations and individuals contributed to
describing the state of the art of the field and for use this volume. In particular, we would like to thank Mr.
in graduate education. This volume is the fruit of the Joel Sloman of MIT, who was responsible for a major
labors of most of the individuals involved in the work part of the editing, and Dr. Ronald Taylor of the Na
shop. Each chapter has been reviewed by one or more tional Science Foundation, which in part subsidized
of the authors of other chapters, and an attempt has both the workshop and this monograph.
been made to make the material accessible to nonspe
cialists. Kerry A. Emanuel
The monograph is divided into six parts. Part I pro David J. Raymond
vides an overview of the problem, including descrip- Editors
v
I
PART
General Considerations
CHAPTER I ARAKAWA
Chapter 1
Closure Assumptions in the Cumulus Parameterization Problem
AKIO ARAKAWA
Department ojA tmospheric Sciences. University ojCalifornia. Los Angeles. Los Angeles. California
1.1. Introduction ferent large-scale conditions. The problem of cumulus
parameterization, therefore, is analogous to that of cli
Physical processes associated with condensation of mate dynamics. In the latter problem, we are concerned
water vapor are inherently nonlinear and, therefore, with time and space means and their statistical signif
their collective effects can directly interact with larger icance, identification of external and internal param
scale circulations. But most individual clouds, in which eters for different temporal and spatial scales, free fluc
condensation takes place, are subgrid-scale for the tuations under given external conditions, interactions
conventional grid size of general circulation and nu between low-and high-frequency variations, and pos
merical weather prediction models. Then, for a set of sible multiple equilibria of the overall regime. All of
model equations to be closed, we must formulate the these may have their counterparts in the cumulus pa
collective effects of subgrid-scale clouds in terms oft he rameterization problem. Difficulties in a cumulus pa
prognostic variables ofg rid scale. This is the problem rameterization can then be compared with those in
of cumulus parameterization in numerical modeling parameterizing transient processes in the climate
of the atmosphere. system.
It should be emphasized that the need for cumulus Since cumulus parameterization is an attempt to
parameterization is not limited to numerical models. formulate the collective effect of cumulus clouds with
Understanding the interaction between moist-convec out predicting individual clouds, it is a closure problem
tive and large-scale processes is one of the most fun in which we seek a limited number of equations that
damental issues in dynamics of the atmosphere, and govern the statistics of a system with huge dimensions.
cumulus parameterization is needed for a closed for The core of the parameterization problem is, therefore,
mulation of that interaction regardless of whether we in the choice of appropriate closure assumptions. When
are using numerical, theoretical, or conceptual models. we have global models with comprehensive physics in
Even if we had a numerical model that resolved all mind, rather than idealized models with a more limited
scales, understanding its results inevitably requires
scope, closure assumptions must meet the following
simplifications that involve various levels of "param requirements:
eterization." In Fig. 1.1, the upper half of the loop
represents the effect of large-scale processes on moist (i) Closure assumptions must not lose the predict
convective processes, while the lower half represents ability ofl arge-scale fields. This is an obvious require
that of moist-convective processes on large-scale pro ment since we need to parameterize clouds for pre
cesses. For the loop to be closed, it must include the dicting the time evolution of large-scale fields. If we
segments shown by heavy curves, a formulation of wish to assume that a certain variable is in an equilib
which is precisely the objective of a cumulus param rium, the variable must be one whose prediction is not
eterization. (We refer to the upper half as "control" intended by the model.
and the lower half as "feedback," although there is no (ii) Closure assumptions must be valid quasi-uni
need to identify which is the first in the loop. These versally. This is also an obvious requirement because
terminologies are convenient only when we are con comprehensive global models must be valid for a va
centrating on the parameterization problem repre riety of synoptic and surface conditions.
sented by the right half of the loop.)
There are a number of cumulus clouds in virtually One may then ask, Can we really find closure as
all tropical and most extratropical disturbances. Some sumptions satisfying these requirements? In other
of these clouds may be developing, some may be fully words, To what extent is it possible to parameterize
developed, and some may be decaying. In the cumulus cumulus clouds? These are difficult questions to an
parameterization problem we are concerned with the swer. The difficulty is amplified by the existence of in
statistical behavior of such cloud ensembles under dif- termediate scales in cloud organization, which are gen-
2 METEOROLOGICAL MONOGRAPHS VOL. 24, No. 46
Obviously, Ql and Q2 are different from the true
heat source and moisture sink, because ( 1.1 ) and ( 1.2 )
are applied to averaged fields. When the large-scale do
main is a grid box of a model, Ql and Q2 include the
collective effects of subgrid-scale processes. Using the
usual assumptions in Reynolds averaging and those in
the anelastic approximation, and neglecting the hori
zontal transports due to subgrid-scale processes, we may
write
FIG. 1.1. A schematic figure showing the interaction between
large-scale and moist-convective processes.
(1.4 )
erally termed "mesoscale." We will discuss problems
- dw'Lq'
associated with this situation in a later section. Here -Q2 = - LC - -- . ( 1.5 )
dp
we emphasize that, even when there was no mesoscale
organization, or even when the model grid resolves Here QR is the radiation heating, QIC is the part of Ql
such organization, the answer to the question of par due to condensation and associated transport processes,
ameterizability is by no means obvious. C is the rate of net condensation (per unit mass of dry
The purpose of this chapter is to review the concep air), s is the dry static energy, CpT + gz, and a prime
tual basis for cumulus parameterization in view of clo denotes the deviation from the horizontal area average.
sure assumptions. We will concentrate on the ther As (1.4) and ( 1.5) show, the difference of QIC and Q2
modynamical aspect of parameterizing deep cumulus from the net heat of condensation, LC, is due to the
clouds, for which most of the conceptual difficulties transport of sensible and latent heat, respectively, b~
exist. In section 1.2, we define the thermodynamical convective (and turbulent) motions. Eliminating C
aspect of the cumulus parameterization problem. Sec from ( 1.4) and ( 1.5), we obtain
tion 1.3 presents a classification of closure assumptions.
Sections 1.4-1.6 review some basic examples of closure -d-w--;';hp' ,
QIC - Q2 = - ( 1.6)
assumptions and their logical consequences. Section
1.7 reviews some of the current studies on parameter
izability of cumulus convection. Finally, section 1.8 where h is the moist static energy, s + Lq. For further
presents summary and further discussions. discussion of ( 1.4 ), ( 1.5 ), and ( 1.6), see the chapters
by Yanai and Johnson (chapter 4) and by Arakawa
1.2. The thermodynamical aspect of the cumulus and Cheng (chapter 10) in this monograph.
parameterization problem Terms QIC and Q2 represent the "feedback" shown
In this section we define the objective of cumulus in Fig. 1.1. If we interpret observed and subs~quently
parameterization more specifically, concentrating on spatially smoothed values of v, (), and q as v, ()-2 and q,
and if we have a time sequence of observed () and q,
its thermodynamical aspect. With the pressure coor
then all terms on the left-hand sides of ( 1.1 ) and ( 1.2),
dinate, the large-scale potential temperature and water
including the time derivative terms, are known. Then
vapor budget equations may be written as
we can calculate Ql and Q2 from ( 1.1) and ( 1.2) as
cp [ddOt + v_ • V (_) + w_ ddPO ] = (PPO) RICP Ql ( 1.1 ) nreossitdiuca slstu. dTiehsi so pf rcoucmeduulures, ewnsheimchb lies ss, tfaonlldoawrds tihne dtihagin
segment of the loop in Fig. 1.1 backward from "large
and
scale processes." Since the path does not follow the
heavy segments shown in the figure, the values of Ql
(1.2 )
and Q2 obtained in this way are independent of cu
mulus parameterization. Figure 1.2 shows an example
where an overbar denotes the Reynolds average with
of such calculations for the GATE [GARP (Global
respect to a large-scale horizontal area. The quantities
Atmospheric Research Program) Atlantic Tropical
Ql and Q2 are the apparent heat source and apparent
Experiment] phase III period. Here and in all subse
moisture sink (Yanai et al. 1973), respectively. All
quent figures for Ql, QIC, and Q2, the unit we have
other symbols are standard except that subscript p for chosen is equivalent to degrees Celsius per day when
time and horizontal derivatives is omitted. The area
divided by
cpo
averaged continuity equation,
From Fig. 1.2 we see that
dw
V . v + dP = 0, ( 1.3) (i) both QIC and Q2 highly fluctuate in time;
(ii) the fluctuations of QIC and Q2 are strongly cou
determines w from v. pled;
