Monl hl y
VOLUME 98, NUMBER 3
MARCH 1970
UDC 651.613.1:566.1:551.509.313
1.
2.
3.
4.
5.
6.
7.
SIMULATED CLIMATOLOGY OF A GENERAL CIRCULATION MODEL
WITH A HYDROLOGIC CYCLE
111. Effects of Increased Horizontal Computational Resolution
SYUKURO MANABE, JOSEPH SMAGORINSKY, J. LEITH HOLLOWAY, JR., and HUGH M. STONE
Geophysical Fluid Dynamics Laboratory, ESSA, Princeton, N.J.
ABSTRACT
The results of a numerical time integration of a hemispheric general circulation model of the atmosphere with
moist processes and a uniform earths surface has already been published by Manabe, Smagorinsky, and Strickler.
I n this study, the integration is repeated after halving the midlatitude grid size from approximately 500 t o 250 km.
This increase in the resolution of the horizontal finite differences markedly improves the features of the model
atmosphere. For example, the system of fronts and the associated cyclone families in the high resolution atmosphere
is much more realistic than that in the low resolution atmosphere. Furthermore, the general magnitude and the
spectral distribution of eddy kinetic energy are in better agreement with the actual atmosphere as a result of the
improvement in resolution.
I n order to explain these improvements, an extensive analysis of the energetics of both the low and high resolution
atmospheric models is carried out. It is shown that these improvements are due not only to the increase of the accuracy
of the finite differences but also to the shift in the scale of dissipation by the nonlinear lateral viscosity toward a smaller
scale resulting from the decrease in grid size. In the low resolution atmospheric model, the transfer of energy from
eddy to zonal kinetic energy is missing because of excessive subgrid scale dissipation a t medium wave numbers,
whereas i t has significant magnitude in the high resolution atmospheric model. It is speculated that further increase
of resolution should improve the results because it tends t o separate the characteristic scale of dissipation from that
of the source of eddy kinetic energy.
The analysis of the energetics in wave number space clearly demonstrates the differences between the energetics
of the different parts of the atmosphere. In middle latitudes there are essential differences between the energetics
of the model troposphere and that of the model stratosphere. I n the model troposphere, the eddy kinetic energy is
produced by the conversion of eddy potential energy in the range of wave numbers from 2 to 8. Part of the energy
thus produced is dissipated by the subgrid scale dissipation, and most of the remainder is decascaded to zonal kinetic
energy. I n the model stratosphere, where very long waves predominate, the eddy kinetic energy is generated in the
range of wave numbers from 2 to 3 by the energy supplied from the troposphere. Most of this energy is then
decascaded barotropically to zonal kinetic energy.
I n the Tropics, eddy kinetic energy is mainly produced by the release of eddy available potential energy generated
by the heat of condensation. Although the rate of conversion is maximum a t very low wave numbers, the conversion
spectrum extends to very high wave numbers.
A box diagram of the energetics of the high resolution moist model shows that the eddy available potential energy
is generated by the heat of condensation as well as by energy transfer from the zonal available potential energy.
Furthermore, i t is noteworthy that the zonal kinetic energy is maintained not only by the barotropic exchange from
the eddy kinetic energy but also from the conversions of zonal potential energy. The intensification of the direct
tropical cell and the weakening of the indirect Ferrel cell in middle latitudes caused by the moist processes are respon-
sible for this positive zonal conversion.
One of the highlights of the results from the integration of the high resolution moist model is the successful
simulation of the evolution of fronts and the associated cyclone families. The influence of moist processes upon frontal
structure as well as other synoptic features is investigated by comparing the moist model atmosphere with the dry
model atmosphere without the effect of the selective heating of condensation. It is found that the heat of condensation
significant,ly reduces the width of fronts and the characteristic scale of cyclones in the lower troposphere.
CONTENTS
Introduction ....................................... 176
Brief description of the model.. ...................... 177
Time integration.. .................................. 178
Time mean field .................................... 178
Front and cyclone families. .......................... 182
Budget of eddy kinetic energy.. ...................... 185
Budget of angular momentum, heat, and moisture. ..... 198
8. Budget of energy. .................................. 204
9. Summary and conclusions.. .......................... 208
Appendix I-Scheme of numerical time integration. ....... 210
Appendix 11-Brief description of the dry general circulation
model .............................................. 211
References ............................................ 211
175
vel. 98, No. 3 176 MONTHLY WEATHER REVIEW
1. INTRODUCTION
This study is a continuation of Sim$ated Climatology
of a General Circulation Model With a Hydrologic Cycle.
Hereafter, we shall refer to part 1 (Manabe et al. 1965)
as study M and part 2 (Manabe and Smagorinsky 1967)
as study M2 for ease of identification.
I n study M, a numerical experiment of the general
circulation of the atmosphere was carried out taking into
consideration interactions with the hydrologic cycle. The
state of quasi-equilibrium was obtained by numerical
integration of the primitive equations of motion, the
thermodynamical equation, and the prognostic equation of
specific humidity. The radiative flux was computed from
the equation of radiative transfer assuming the climato-
logical distribution of water vapor, carbon dioxide, ozone,
and cloud. The solar radiation at the top of the model
atmosphere was assumed to have the annual mean
distribution. The hydrological part of the model consisted
of the advection of water vapor by the large-scale motion,
evaporation from the earths surface, precipitation, and a
convective adjustment of temperature and moisture. The
temperature of the earths surface was determined from a
condition of heat balance at the earths surface, which was
assumed t o be a completely wet and flat surface with zero
heat capacity. For further details of the model structure,
see also the study of the general circulation model without
the hydrologic cycle (Smagorinsky et al. 1965). Hereafter,
we shall refer to this paper as study S.
Quasi-equilibrium was obtained as an asymptotic state
emerging from the numerical time integration of the model
atmosphere. The initial condition for this time integration
was an isothermal and dry atmosphere at rest. In order
to economize computer time, a very coarse grid size was
chosen at the beginning of the time integration, and the
resolution was refined stepwise as the integration pro-
ceeded. It was our original intention to keep refining the
resolution until the large-scale motion field was sufficiently
resolved and the result insignificantly altered by further
refinement. However, in these studies we stopped at a grid
size of 500 km because of the limitation of computer time
despite the, as yet, significant dependence of the results
upon refinement. Nevertheless, the state of quasi-equilib-
rium which emerged from this integration has various
features common with the actual atmosphere, such as the
tropical rain belt, the subtropical dry belt, and the
middle-latitude rain belt. Furthermore, the thermal
structure of the joint troposphere-stratosphere system and
the latitudinal gradient of the tropopause height were
successfully simulated. There are, however, various
features of the mode1 which require further improvement.
For example, the finite-difference mesh with a grid size of
500 km is not capable of satisfactorily resolving frontal
structure and associated cyclones. According to study
M, the heat of condensation tends t o reduce the scale
of the flow patterns near the earths surface. With respect
to the characteristic scale of surface cyclone systems,
the moist model atmosphere more closely matches the
real atmosphere than does the dry model atmosphere,
Detailed analysis of synoptic systems of the moist model
atmosphere reveals^+that the 500-km grid is far from
sufficient to resolve the flow patterns near the earths
surface and indicates the need for further increases in the
resolution of the horizontal finite differencing.
Another unrealistic feature of the low resolution moist
model atmosphere was the small magnitude of the eddy
kinetic energy. The magnitude of the eddy kinetic energy
in the moist model atmosphere was much smaller than in
the actual atmosphere. An analysis of experimental
weather predictions performed with a similar model
(Smagorinsky and Staff Members 1967, Miyakoda et al.
1969) indicated that improvement in horizontal resolution
increased the level of eddy kinetic energy and markedly
improved the quality of the dynamical prediction. Since
the nonlinear viscosity proposed by Smagorinsky (1963)
is proportional not only t o the absolute value of the
vertical component of deformation but also to the square
of the grid size, it is probable that the improvement in
resolution of the horizontal finite differences affects the
spectral distribution of dissipation as well as the accuracy
of horizontal finite differencing, that is, it decreases the
scale of dissipation. As a result of the availability of a
faster computer, it was decided t o repeat study M with
double horizontal resolution in an attempt t o improve
the simulation of the dynamical structure of the atmos-
phere. In the new study, special emphasis is placed upon
an analysis of the budget of eddy kinetic energy in wave
number space in order to determine how the change of
resolution affects the energetics and the energy spectrum
of the model atmosphere. Since many earlier numerical
simulations of the general circulation have been conducted
with a coarse grid system, the present study can be useful
for reevaluating the previous results of these experiments.
Moreover, a study of the energetics in wave number
space can be useful for establishing how the spectrum of
eddy kinetic energy is maintained in a moist atmosphere.
Special emphasis will be plticed upon a comparison between
the energetics in the stratosphere vis-&vis the troposphere
and in the middle latitudes vis-%-vis the Tropics.
Another objective of this study is the construction of a
box diagram of the energy budget in the high resolution
moist model atmosphere. The energetics of the dry model
atmosphere has been extensively investigated by Phillips
(1956), Smagorinsky (1963), Smagorinsky et al. (1965,
study S). However, the energetics of a moist model
atmosphere has hardly been explored. I n a dry atmosphere,
the eddy available potential energy is solely supplied by
energy transfer from zonal to eddy available potential
energy. Wornever, the preliminary study of the moist
model with coarse resolution (M2) indicated that the heat
of condensation could also be responsible for the genera-
tion of eddy available potential energy. I n a dry model
atmosphere, there is a small conversion of zonal kinetic
energy into zonal potential energy. According t o the
March 1970
preliminary study (Ma) , the sense of the conversion could
be reversed in the moist atmosphere because of the
intensification of the direct tropical cell of the meridional
circulation resulting from the releas$ of the heat of
condensation. Since it is very d a c u l t t o determine the
Syukuro Manabe, Joseph Smagorinsky, 1. Leith Holloway, Jr., and Hugh M. Stone
PERFECT GAS IAW
the meridional circulation in the actual atmosphere, the
177
THERMODYNAMICAL EQUATION OF MOTION EQUATION FIE1o w
magnitudes of the zonal conversion and the eddy genera-
these circumstances, it is very useful t o evaluate the
contribution of these effects in the moist model atmosphere
provided that the model atmosphere is sufficiently similar
tion of available potential energy are not certain. Under 9 : ;
$8 2 z z: ;
2 -
.
c
E O
I 2: f
t o the actual atmosphere in other verifiable characteristics. MUATION OF
WATER VAPOR RADIATIVE TRANSRR
9. BRIEF DESCRIPTION OF THE MODEL
The structure of this high resolution model is identical
with that of the low resolution model described in the
previous study M. Therefore, i t will be explained only HEAT BALANCE OF
EARTHS SURFACE
a - P*=-gj3(l). at
x
2
$
Y
::
and
The hydrostatic relation is
ih RT a coseax au a COS eah
z=-=* U
(5 )
The prognostic equation of the mixing ratio of water vapor a
at - (P,v)=-B3(u)-sin . (i+29) . (P,u)
where a is the radius of the earth, R is the gas constant,
P is pressure, P, is the surface pressure, u=a cos e .i ,
v=ab, a=P/P,, !J is the angular velocity of the earth,
T is temperature, @ is the geopotential of a a-surface, FA
and Fe are frictional forces in the zonal and the meridional
directions, respectively, and
The first law of thermodynamics is
where c p is the specific heat capacity a t constant pressure,
w is vertical P-velocity (dP/dt), and a is heat added per
unit mass by radiation and condensation, and FT is the
effect of thermal diffusion. The continuity equation may
where r is the mixing ratio of water vapor, C is the rate
of condensation, and F, is the change of mixing ratio due
to the subgrid scale eddy diffusion. These equations were
rewritten in the stereographic projection system and nu-
merically integrated in a hemispheric domain. A free-slip
insulated wall is placed at the Equator, and a flat and wet
earths surface without any heat capacity is imposed as
the lower boundary condition. Figure 1 is a schematic
diagram of the structure of the moist model and indicates
that computation of the radiative transfer and of hydro-
logic changes yield the values of @ and C, respectively. The
details of these computations and the description of the
coupling among the various components of the model can
be found in M. In order to identify the interactions of
the hydrologic process, the moist model atmosphere is
often compared with the dry model in this study. Further
details of the structure of the dry model which lacks
moisture advection and condensation can be found in
study S.
478 MONTHLY WEATHER REVUEW 401. 98, No. 3
Since we shall discuss how the horizontal (subgrid scale)
eddy viscosity affects the circulation of the moist model
atmosphere depending upon the re?olblntio-n of horizontal
finite differencing, we need to briefly describe the formula-
tion of the nonlinear viscosity proposed by Smagorinsky
(1963). The zonal and meridional components of the
frictional forces may be written as the sum of the deriva-
tives of stress tensors in the following way:
where p is the density of air, rxx, rxe, rex, and P denote the
stress tensors resulting from horizontal mixing, and + and re* denote those from vertical mixing. Assuming that
the stress tensors can be expressed by a linear combination
of strain tensors and imposing the hydrostatic require-
ment, the following relationship for two-dimensional
isotropic turbulence results:
-__- -
T -a ~~~e a ~ a a8 coso
cOse a 3 D -
and
The assumption of an inertial subrange yields through
dimensional analysis that
?,If= l2 ID I (14)
where [Dl=[D$+Di] l I 2 and I is a characteristic length
corresponding to the scale defining the exchange coefficient.
This length is assumed t o be
where d is the grid size and k , is the Ktirmtin constant.
Based upon spectral measurements of Pond et al. (1963),
Lilly (1966) estimated the value of I t o be 0.23 d, which
is close to what is used here.
In the following discussions, the horizontal resolution
will be identified by N , that is, the number of grid points
between the Pole and the Equator. Accordingly, the grid
size of the high resolution grid will be identified as
N4O and that of the low resolution grid as N20
(table 1 ). The capital letters M and D will denote the
moist and the dry model, respectively. See appendix I1
for a brief description of the dry model.
3. TIME INTEGRATION
The initial condition for the numerical time integration
of the N40M model atmosphere was the state which was
TABLE 1.-Latiludinal dependence of grid size for the high and low
resolution grids
Grid size (km) N.40 N.90
Equator
40 latitude Pole
160 320 270 540
320 640
TABLE 2.-Horizontal resolulion used in the integration oflhe N4OM
atmosphere
Resolution Period of integration
N=5 N= 10 N=20 N=40
0 4 t h day 50-70th day 70-210th day 150430th day
reached after 150 days in the time integration of the
models with coarser resolutions. Table 2 shows the
specific horizontal resolution a t various stages of the
integration. The value of N was increased stepwise from
5 to 10, to 20, and then to 40. Since the computer time
required for the numerical time integration is proportional
to N3, this method of integration significantly reduces the
amount of computation for reaching the state of quasi-
equilibrium. Further details of the method of numerical
time-integration may be found in appendix I.
The initial condition for the N6Mmodel is an isothermal,
completely dry, and resting atmosphere. The time evolu-
tion of the hemispheric average value of the total potential
energy, and of the gross static stability are shown in
figure 2. Similar graphs for precipitable water, rate of
precipitation, and rate of evaporation are shown in figure
3. The time evolution of total kinetic energy and of relative
and absolute angular momentum are shown in figure 4.
These figures clearly demonstrate that the atmosphere is
sufficiently close to a state of quasi-equilibrium toward
the end of the integration. The period of analysis chosen
for this study is the last 30 days of the N4OM integration,
that is, from the 200th to the 230th day. The data, which
are shown in the following section are obtained by taking
the time mean for this 30-day period.
4. TIME MEAN FIELD
A. ZONAL WIND FIELD
In figure 5 , the latitude-height distribution of the zonal
wind of the N4OM model and N2OM model are compared
with that of the actual atmosphere. General features of
the distributions of the two models are very similar, and
the intensity of the westerlies in both model atmospheres
is much stronger than in the actual atmosphere. A similar
tendency for strong westerlies was evident in the model
atmospheres of Phillips ( 1956) and Smagorinsky ( 1963).
The reason for such a tendency is not obvious. This seems
to be one of the most fundamental discrepancies between
the features of model atmospheres and those of the actual
atmosphere.
March 1970 . Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 179
POTENTIAL ENERGY
I
I
I
I J
FIGURE 2.-Time variation of various hemispheric mean quantities.
Upper graph, total potential energy; lower graph, gross static
stability. See appendix 2c of study S for the definition of gross
static stability.
A detailed comparison between the zonal wind fields of
the two models (N2OM and N4OM) reveals the following
changes which resulted from the increase in the resolution
of the horizontal finite differencing:
1) A poleward shift of 5O latitude of maximum tropo-
spheric zonal wind. This is still equatorward of the position
of the maximum in the actual atmosphere.
2) The development of surface easterlies in the polar
region.
B. MERIDIONAL CIRCULATION
The stream functions of the meridional circulation in the
N4OM atmosphere and in the N2OM atmosphere are
shown in figure 6. In the tropospheres of both models,
there are three cells, that is, the direct cell of the Tropics,
the indirect cell in middle latitudes, and the direct cell of
the polar region. These cells are characterized by a shallow
layer of intense mass transport in the planetary boundary
layer and a return flow mainly in the upper troposphere. In
both model stratospheres there are two major cells and a
very weak indirect cell a t the Equator. With information
based upon a study of the budget of angular momentum,
Teweles (1963) computed the meridional circulation in
the actual atmosphere, obtaining two major circulation
cells similar to those shown in figure 6. Miyakoda (1963)
also obtained two major cells for the winter season by
solving the u-equation. The existence of a weak indirect
cell in the actual equatorial stratosphere is not certain,
although Wallace (1967) suggested the existence of down-
ward motion in the tropical stratosphere based upon a
study of the budget of angular momentum.
PRECIPIIAW WAnR
m
*t vuv 1
EVAPORATION
FIGURE 3.-Time variation of various hemispheric mean quantities.
Upper graph, precipitable water; miadle graph, rate of precipi-
tation; lower graph, rate of evaporation.
If one compares the intensities of the meridional circu-
lation in the two model atmospheres, one finds that the
indirect cell of middle latitudes in the N4OM atmosphere
is about one and a half times stronger than in the N20M
atmosphere. According to figure 7, which shows the lati-
tudinal distributions of the zonal mean surface pressure
in both models, the meridional gradient of the surface
pressure in middle latitudes increases with increased reso-
lution. This increase probably is responsible for the differ-
ence in the intensity of the indirect cell. Phillips (1954)
has shown that baroclinically unstable waves are respon-
sible for driving the indirect cell. As it will be shown in
the following section, the magnitude of the eddy kinetic
energy in the N4OM atmosphere is larger than that in
the N2OM atmosphere. Therefore, it is reasonable that
180 MONTHLY WEATHER REVIEW vol. 98, No. 3
KINETIC ENERGY
I, I I ,
-
---
i .-
D4Y
RELATIVE ANGULAR MOMENTUM
' 0 7 5 100 123 150 '____it,- 2W 225
DAY
ABSOLUTE ANGULAR MOMENTUM
FIQURE 4.-Time variation of various hemispheric mean quantities.
Upper graph, total kinetic energy; middle graph, relative angular
momentum; lower graph, absolute angular momentum.
the indirect meridional circulation in the former is more
intense than that in the latter.
The latitude-height distributions of the meridional com-
ponent of the wind and vertical P-velocity (w ) in the
N4OM atmosphere are shown in figure 8.
C. EDDY KINETIC ENERGY
The latitudeheight distributions of the zonal mean
value of the eddy kinetic energy in both the N4OM and
the N2OM atmospheres are shown in figure 9. In both
models, the height of the maximum eddy kinetic energy
lies in the upper troposphere in agreement with the features
of the actual atmosphere. The magnitude of the eddy
kinetic energy of the N4OM model, however, is auite
ZONAL WIND
'. Iw" 70" MIo 500 IO" 30" 20" 10' 0"
u r n
UrmDt
FIQURE 5.-Zonal mean of zonal wind in the various atmospheres.
Upper distribution, actual atmosphere (data source, Rasmusson
and Oort 1970 and Batten 1964); middle distribution, the N4OM
model; lower distribution, the N2OM model. Units, m sec-l.
tudes, the former is about twice as large in the stratosphere.
As we shall see in the following section, this difference is
different fro; that of the N2OM model. In middle i a t t caused partly by the decreasi of the effective viscosity
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 181
20
1 f B
1 I
10
n
20
I s
10
n .-- ,991' 90" 80" 70" fOO 50" IO" 30" 20" loo 0" - u r n
FIGURE 6.-Stream functions of meridional circulation; area of
counterclockwise circulation is shaded. Upper half, the N4OM
model; lower half, the N2OM model. Units, 1014 gm sec-1.
and partly by the increase of the conversion of eddy
potential energy. Both of these changes result from the
increase of the horizontal resolution.
Figure 10 compares the distributions of eddy kinetic
energy in the model atmospheres at 500 mb with those in
the actual atmosphere estimated by Saltzman ( 1962) .l
Figure 10 clearly indicates that the magnitude of the eddy
kinetic energy of the N4OM model is closer to observed
than that of the N2OM model.
D. ENERGY SPECTRUM
In figure 11, the spectra of the eddy kinetic energy of
both models at various isobaric surfaces are shown
together with the annual mean spectrum in the actual at-
mosphere which was estimated by Teweles ( 1963). Accord-
ing to this figure, the magnitudes of the spectral com-
ponents of the N4OM model are significantly higher than
Note that Saltzman (1962) used geostrophic wind for his study. Therefore, it is prob-
able that his estimate may be too high.
97s t 9701 I 90 80 70 60 50 40 30 20 10 0
LATITUDE
FIGURE 7.-Latitudinal distribution of the zonal mean of surface
pressure. Without mountains these models have mean sea level
pressures about 25 mb lower than the actual atmosphere.
those of the N2OM model at all wave numbers and are
closer to those estimated for the actual atmosphere.
Originally, it was speculated that the energy spectrum of
the lower wave numbers depended only slightly on the com-
putational resolution because the nonlinear viscosity
should only influence the higher wave numbers. This
figure, however, indicates that energy of the lower wave
numbers increased more than did that of the higher wave
numbers as a result of the increase in the resolution.
Hence, the general increase of the kinetic energy with
decreasing wave number evident in Teweles' result is
successfully simulated by the N4OM model but not by
the N2OM model.
In order to compare the characteristic scale of the
motion in the two models, we define an effective wave-
length
(16) 2~ COS e Le,=
where
a=radius of the earth, and O=latitude.
Here, E(n) denotes the vertical mean kinetic energy of
the component with wave number n. In figure 12, LEY/ is
plotted as a function of latitude. We see that the charac-
teristic scale of motion definitely increases in middle
latitudes and in the subtropics, resulting from the in-
crease of horizontal resolution. This increase is consistent
with the increase of the kinetic energy of the larger scale
motion mentioned above. In subsection 6B, the reason
for the difference between the energy spectra of the two
models will be discussed extensively.
E. TEMPERATURE AND RELATIVE HUMIDITY
The latitude-height distributions of the zonal mean
temperature and the zonal mean relative humidity in the
N4OM atmosphere are shown in figures 13 and 14, respec-
tively. They differ slightly from the corresponding distri-
d 82
N40M
MONTHLY WEATHER REVBEW
,009 - - 30
,074-
d 1
.lI- 1
,336-
SOO-
,664 - .Il-
.W .YYI LA?lTUOE
.. .'LY,
- - -
,991 90" 80'. 70" 60" 50" 80" 30" 20" 10'
bm5I
vel. 98, No. 3
L A r n E
.20
1 i E i?
-10
0
- I
0 _-_
,991' 90" 80" 70" 60" 50" 40' 30" 20' 10' 0'
ULTlNOE
FIGURE %--Latitudeheight distribution of the zonal mean currents
in the N4OM atmosphere. Upper half, vertical P-velocity (cm
sec-1) ; lower half, meridional component of wind (m sec-1).
FIGURE 9.-Latitudeheight distribution of the zonal mean vtalue
of eddy kinetic energy. Upper half, the N4OM model; lower half,
the N2OM model. Units, 10-3 joules ern+ mb-1.
butions in the NZOM atmosphere. The NZOM distributions
were already discussed in study M.
5. FRONT AND CYCLONE FAMILIES
In order to discuss the synoptic features of both moist
atmospheric models, for each resolution the maps of
daily precipitation, surface pressure, surface temperature
at level 9 (P-991 mb), and geopotential height at the
500-mb level on model day 173 are shown in figures 15
through 18. I n the N4OM atmosphere, narrow spiral bands
of frontal rain extend from high to very low latitudes.
I n figure 19, which shows a TIROS picture of the Southern
Hemisphere, spiral bands of cloud extend counterclockwise
toward the Equator, suggesting the existence of rain
bands with similar features in the actual atmosphere. As
figures 16 and 17 indicate, fronts in the N4OM atmosphere
are also characterized by a very dense concentration of
isotherms and are accompanied by cyclone families near
the earth's surface.
In the low resolution NZOM atmosphere, the rain bands
are much wider and less distinct than in the N4OM
atmosphere. Also, the concentration of the isotherms is
somewhat less in the NZOM than in the N4OM model, as
is evident in figure 17. I n short, the general features of
the surface front become much more realistic as a result of
doubling the resolution of horizontal finite differencing.
Obviously, the coarse horizontal resolution of the NZOM
model is not sufficient to resolve the frontal structure.
We have shown the synoptic maps of both the NZO
and the N4O atmosphere and pointed out that the general
features of the frontal systems in the N4OM atmosphere are
much more realistic than those in the N2OM atmosphere.
It would now be instructive to follow the time evolution
of these simulated fronts in the N4OM atmosphere and
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 183
I
8 0 -
I U
3 6 0 -
3
4 0
'p AO-
20
1201 I I I I I I I 1
-
A SALTZMAN
c
A
IO0 A A
A
!b 80 70 60 SO A0 30 20 I O 0
LATITUDE
I I I I I I I 1 I
FIGURE 10.-Mean latitudinal distribution of eddy kinetic energy
a t the 500-mb level. (Actual atmosphere, Saltzmsn 1962; geo-
strophic wind approximation is used for this estimate.)
COMPUTED N20M
I
I 3 5 7 9 I 1 1315
WAVE NUMBER
COMPUTED N40M
P/& a ,009
-&--
OBSERVED
1 3 5 7 9 1 1 1 3 I t
WAVE NUMBER
I 3 5 7 9 I I 1315
WAVE NUMBER
FIGURE 11.-Mean spectra of eddy kinetic ebergy of the region
from 15' to 81" latitude a t various isobaric surfaces in the NdOM
and N4OM models and in the actual atmosphere (Temeles 1963).
The spectra of eddy kinetic energy of the zonal and meridional
components of the wind are shown by solid and dashed lines,
respectively .
carry out a detailed analysis of their structure. A standard
synoptic frontal analysis was made daily for the N4OM
model atmosphere during the period from day 167 to
day 177. Pressure, temperature, and wind fields were
used to determine frontal locations. Figures 20 and 21
show part of this analysis for alternate days. They give
maps of surface pressure, precipitation, wind, and tem-
perature at level 9 (P/P*=O.991). All scalar fields in this
figure were analyzed by machine, and the wind vectors
were drawn by machine. The frontal analysis is the only
operation done subjectively by hand.
The development of two frontal systems can be followed
during this period. Frontogenesis is indicated for one sys-
70001
LATITUDE
FIGURE 12.-Latitudinal distribution of the average mean effective
wavelength.
I B
f
UllNtE
FIGURE 13.-Latitude-height distribution of the zonal mean
temperature of the NdOM atmosphere. Units, O K .
tem in the upper left-hand corner of the map at the 167th
day; the other system on this day is a weak open wave.
Along the front in the upper left-hand corner, one can
identify three cyclones denoted by LA, LB, and LC,
which resemble a typical cyclone family in the actual
atmosphere. These cyclones move northeastward and go
through a typical life cycle. For example, cyclone LC is
barely detectable on the 167th day. It keeps developing
until the 173d day. On the 175th day, the occlusion
process is completed, and the cyclone starts decaying
very rapidly. Another frontal system on the right-hand
side undergoes a similar evolution. The evolution of the
front and the associated cyclone described here is quite
similar to that observed in the actual atmosphere (see
also, Miyakoda et al. 1969). Furthermore, the temperature
gradient, wind shifts, and rainfall in the vicinity of
these model fronts are highly realistic. I n general, the
change of direction of the surface wind vector is not
very abrupt across a warm front. However, large wind
375-632 0 - 70 - 2
184 MONTHLY WEATHER REVIEW vol. 98, No. 3
u r n
FIGURE 14.-Latitudeheight distribution of the zonal mean relative
humidity of the N4OM atmosphere.
FIGURE 15.-Stereographic map of daily precipitation (cm day-1)
on model day 173. Shading indicates areas of precipitation;
slashed-line shading, area of daily precipitation in excess of 1.0
cm day-1; dotted shading, area of daily precipitation in excess of
0.1 cm day-1; circular boundary, Equator.
shear is evident across the cold front, in agreement with
the features of the actual atmosphere.
The cross sections presented in figure 22 were made
through the model fronts on day 173. Their locations
are indicated in figure 21 by heavy lines marked A-B,
C-D, and E-F. The top, middle, and bottom rows of figure
22 show the wind component normal to the cross section,
the vertical motion, and the relative humidity, all
superimposed upon temperature.
I n cross-section A- 3, which cuts the cold and warm
fronts through the warm sector of the wave, upward
vertical motion is strong in the warm sector with weak
downward motion in the cold air behind the cold front and
ahead of the warm front. Upward-moving air is moist,
and downward-moving air is dry. The slope of the cold
front is steeper than the slope of the warm front, in agree-
ment with characteristics of the actual atmosphere. The
jet stream is located above the cold front, where the
horizontal temperature gradient is strongest. It is similar
to cross sections through the real atmosphere (for example,
Bjerknes 1951). Fronts on the cross sections were drawn
FIGURE 16.-Stereographic map of surface pressure on model day
173. Contour interval is every 5 mb. Shading indicates area of
precipitation in excess of 0.1 cm day-'; circular boundary,
Equator.
on the basis of temperature gradient only. They are not
as distinct as real fronts because of the still rather coarse
spatial resolution of the model. A temperature inversion
is analyzed through the warm front because the inversion
does exist a t the adjacent grid points.
Cross section @-D is placed through the cold high-
pressure region and connects the two frontal systems.
Upward vertical motion is located over all three fronts in
this cross section, with weak downward motion directly
above the cold dome of air and between the primary and
secondary cold fronts. The secondary cold front began
forming on day 171 when a cold mass of polar air began
pushing southward. By day 173, the surface temperature
gradient is quite strong behind this secondary front, even
stronger than behind the primary front. The cross section
shows that the primary front, although weak at the sur-
face, extends high into the troposphere, above the 500-mb
level, while the secondary front is rather shallow. This
results in strong upward motion and heavy rain associated
with the primary front. The two distinct rain bands can
be seen on the map for day 173. Note that the slope of the
cold front is steeper than that of the warm front, as was
previously pointed out.
The cross section E-F intersects the cold front at about
20' latitude, which is quite far south for a front to be
located; however, the surface temperature gradient is still
strong here. The cross section shows that the front is
shallow with a gradual slope; the vertical motions associ-
ated with it are weak. Weak downward motion appears in
the cold air behind the front.
All three cross-sections shown here resemble cross sec-
tions through fronts in the real atmosphere. Although
further increase in the resolution of the vertical and
horizontal finite differences is desirable, it is clear that the
NdOM model is capable of reproducing the evolution and
the structure of fronts and associated cyclone families.
Edelman (1963) has shown that it is possible to simulate
the formation of fronts by a time integration of a baro-
clinic model without considering the effects of condensa-
tion. In order to verify his conclusion, the maps of surface
March 1970 Syukuro Manabe, Joseph Srnagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 185
FIGURE 17.-Map of temperature at the lowest level (991 mb) on
model day 173. Contour interval is 3K.
FIQURE 18.-Stereographic map of geopotential height of the 500-
mb level on model day 173. Contour interval is 60 m. Shading
indicates area of precipitation in excess of 0.1 cm day-1; circular
boundary, Equator.
pressure, temperature at level 9, and geopotential height
of the 500-mb level on day 58.6 of the integration
of the dry N4OD model are made and are shown in figures
23 and 24. (See appendix I1 for a brief description of the
N4OD model.) These maps clearly show that the dry
general circulation model is capable of producing frontlike
systems.
Comparing the structure of the frontal system in the
N4OM atmosphere and in the N4OD atmosphere, (see
figs. 16,17, 18,23, and 24) we find the following differences:
1) The scale of cyclones associated with fronts at the
earth's surface in the dry N4OD atmosphere is generally
larger than in the moist N4OM atmosphere.
2) The widths of fronts in the former are significantly
wider than in the latter.
3) The fronts in the N4OM atmosphere extend toward
lower latitudes than in the N4OD atmosphere.
Such a scale reduction in the presence of condensation
was suggested by Smagorinsky (1957) particularly by a
reduction in the area of upward motion and an increase in
the area of downward motion. In study M, it was demon-
strated that the heat of condensation indeed produced the
narrow area of intense convergence and reduced the scale
of cyclones in the lower troposphere. This characteristic
was also discussed in connection with prediction experi-
ments (Smagorinsky and Staff Members 1967). In the dry
N40D model, the scale of predominant cyclones near the
earth's surface is similar t o that at the 500-mb level, and
the distribution of surface pressure is highly unrealistic;
whereas in the moist N4OM model, the cyclones with
smaller scale develop near the earth's surface. These
results reflect the dynamical consequence of the reduced
static stability due to water phase changes, that is, a
smaller scale of maximum baroclinic instability. The
reduction of the area of convergence caused by the con-
densation process is probably responsible for the nauow-
ness of the width of fronts in the moist model atmosphere.
has a controlling effect on local weather. Since these
cyclones are not always resolved su5ciently by the N4O
grid, a further reduction in grid size is desirable for the
successful forecast of local weather.
Recently, a numerical weather prediction model with
very high horizontal resolution has been developed by
Bushby and collaborators (Bushby and Timpson 1967).
They obtain partially successful results in predicting
frontal rain.
6. BUDGET OF EDDY KINETIC ENERGY
The major difference between the NdOM and N2OM
atmospheres lies in the spatial distribution, particularly
in the spectral distribution, of eddy kinetic energy. I n
this section, we shall briefly describe how the eddy
kinetic energy is maintained in the N4OM atmosphere.
We shall discuss why the kinetic energy spectrum of the
N4OM atmosphere is different from that of the N2OM
atmosphere based upon an analysis of the energy budget
in wave number space. An extensive study of the budget of
eddy kinetic energy in the various parts of the model
atmosphere will also be described in terms of wave
number space.
A. SOURCES AND SINKS OF EDDY KINETIC ENERGY
Outside the tropical region, the distribution of sources
and sinks of eddy kinetic energy in the N4OM model
atmosphere is very similar to that in the N2OD atmosphere
which was extensively described in study S. Therefore,
we shall cover this subject only very briefly here.
Subtracting the prognostic equation of zonal kinetic
energy from the equation of total kinetic energy, one gets
the following equation describing the rate of change of
eddy kinetic energy:
a ----A The results shown in this section indicate that it is
necessary to take into consideration the effect of the heat
of condensation for a successful simulation of fronts and
associated cyclone families. Each of the small cyclones
a z k
7- --[L 2 (cos e . a COS e ae
(W K ~) ] -- +(K,-&)--Y'*V~'~+V' *FtA (18)
186 MONTHLY WEATHER REVIEW vel. 98, No. 3
FIGURE 19.-Satellite, ESSA 3, photographic mosaic of the Southern Hemisphere on Sept. 22, 1967.
Kz=+ ((;ii)((VX)2).
where
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 187
and
I n these equations, a, 6, A , P, and t are the radius of the
earth, lati tude, longitude, pressure, and time, respectively.
Here, u, v, and V are the northward and eastward com-
ponents of the wind and the horizontal wind vector,
respectively; w is the vertical P-velocity and 4 is the geo-
potential height of an isobaric surface. F is the frictional
force due to vertical and horizontal subgrid scale mixing;
( ) indicate the zonal mean and the deviation
from zonal mean, respectively.
In equation (18), the first term on the right-hand side
of the equation represents the contribution of flux diver-
gence, and the second term the contribution of energy
exchange between zonal and eddy kinetic energy. The third
term computes the rate of the production of eddy kinetic
energy by the pressure gradient force, and the fourth
term gives the rate of dissipation. The production term
may be broken down into two parts, that is
-A
) and (
where a is specific volume. Thc first term on the right-
hand side of this equation represents the energy exchange
with the surroundings through the pressure interaction,
and the second term represents the conversion of the eddy
potential energy. Using equation (19), and taking a hemi-
spheric average, one can convert equation (18) , into
-H H H
a -A H __ -__ - ap (w K 6 ) +(Kz.KE) -ap (w+) a Z at
- According to study S, the eddy conversion (-w t a l X ) is at a
maximum in the midtroposphere of the model atmosphere.
The pressure interaction term (-mx) in turn transfers
energy both upward and downward. The upward flux
converges in the upper troposphere and is responsible
for the eddy kinetic energy maximum there. On the other
hand, the downward flux converges in the planetary
boundary layer and compensates for the large dissipation
there. The contribution of flux divergence of the eddy
kinetic energy by the large-scale eddies, qwK,) /aP,
is negligible.
Figures 25 through 27 show the latitude-height distri-
butions of the eddy conversion, eddy pressure interaction,
and eddy dissipation in the N40M atmosphere. (Figure 28
was found to be in error and was deleted in press.) Figures
25-27 indicate that the general features of the distributions
of energy budget terms in the N4OM atmosphere are
similar to those of the N2OD atmosphere, which we
discussed in study S. There are, however, some important
differences. In the dry NZOD model, an area of large eddy
conversion lies in middle latitudes, whereas in the N4OM
atmosphere, centers of large conversions are located
in the Tropics as well. As pointed out in study M2,
the tropical maximum of eddy conversion is produced by
the generation of eddy potential energy from released heat
of condensation. The cyclone in the model Tropics has a
cold core near the earths surface and a warm core in the
upper troposphere. The level of maximum eddy conversion
in the Tropics coincides with that of the warm core located
around the 300-mb level. This is somewhat higher than
the level of the maximum in middle latitudes. The upward
flux of energy by the pressure interaction in the Tropics
converges mainly around the 200-mb level and is respon-
sible for the eddy kinetic energy maximum there. Except
for the differences in the height of the maximum, the
qualitative features of the distribution of the conversion
term and of the pressure interaction term in the Tropics
are very similar to those in middle latitudes. The major
mechanism for the generation of eddy available potential
energy in the Tropics, however, is different from that in
middle latitudes. In a study of the Tropics of the NZOM
atmosphere, Manabe and Smagorinsky (M2) found that
the flow of energy that predominates in the model Tropics
is as follows: heat of condensation-eddy available
potential energy+eddy kinetic energy-dissipation. In
middle latitudes, it is well known, however, that
the following process plays a major role: meridional
heating contrast-+zonal available potential energy-eddy
available potential energy-eddy kinetic energy-dis-
sipation. For further details of the energetics of a cyclone
in the model Tropics, refer to the study M2.
B. SPECTRAL DISTRIBUTION OF THE BUDGET
OF EDDY KINETIC ENERGY
Hemispheric spectrum--In this subsection, we shall dis-
cuss how the spectrum of eddy kinetic energy, which is
described in subsection 4D, is maintained in the model
atmosphere, and why the spectra of the N4OM atmosphere
are more realistic than those of the NZOM atmosphere.
Assuming the hydrostatic relationship, and adopting
pressure as a vertical coordinate, we may write the
equation of motion in the spherical coordinate system as
follows :
aV/& = - a 3 (V ) - sin 0. (i+2n) (k X V) - vd+ F (21)
where
188 MONTHLY WEATHER REVIEW vol. 98, No. 3
h
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 189
190 MONTHLY WEATHER REVIEW Vol. 98, No. 3
U
u1
BW
BW
BW
BW
BW
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holioway, Jr., and Hugh M. Stone 191
FIGURE 23.-Distribution of surface pressure (contour interval, 5
mb) and surface air temperature at the lowest level of 991 mb
(contour interval, 3OK) on day 8.5 of the time integration of
the N4OD model are shown on the left- and right-hand side,
r espectively. The circular boundary indicates the Equator. FIGURE 25.-Latitudeheight distribution of the rate of the conver-
sion of eddy potential energy (-mA) in the N4OM atmosphere.
Units, joules cm-2 mb-I day-1.
N40D
L
20
1
I
10
0
FIGURE 24.-Stereographic map of the geopotential height of the
50CLmb level on day 8.5 of the time integration of the N4UD
model. The contour interval is 60 m. The circular boundary
indicates the Equator.
Here, k is the unit vector pointing upward, and 0 is the an-
gular velocity of the earth. For further details of the nota-
tion, see the explanation of equation (18). Analyzing each
term of this equation into a Fourier series along latitude
circles, we get
(%),= - (B3(V))n- (sine.(X+2Q> .(kxV)) ,- (~d ,+
(22)
where ( ), denotes the Fourier components with wave
number n. For example, if one expresses a function f( 2)
in terms of its Fourier components,
cos Nx , +aN - J;;
UrmmE
FIGURE 26.-Latitudeheight distribution of the pressure interaction
term in the N4OM atmosphere. The region of the
downward flux of energy is shaded. Units, joules
-
day-'.
and
Multiplying each term on both sides of equation (22)
by (V), and applying the zonal mean operator, we obtain
where
(23)
375-632 0 - 70 - 3
Vol. 98, No. 3 I 192 MONTHLY WEATHER REVIEW
30
20
I s
I
10
0
FIQURE aP.-Latitude-height distribution of the rate of eddy dissi-
pation (-w) in the N40M atmosphere. Units, 10-3 joules
cm-8 mb-1 day-l.
and
D(n)=-(V), 0 (F),'.
E(n) denotes the kinetic energy of eddies with longitudinal
wave number n; T(n), P(n), and D(n) are the nonlinear
inertia term, the production term, and dissipation term,
respectively. These terms are computed along each latitude
circle. The values of V, a 3 (V ), vd, and F at eachlatitude
circle are obtained by interpolating between the grid
points on the stereographic map. The Fourier coefficients
are then obtained. Because of this interpolation, the sum
of the contributions from all wave numbers is not neces-
sarily correct. For example, CT(n) should be equal to
the magnitude of the energy transfer from the zonal to
the eddy kinetic energy. However, the former turned out
to be significantly larger than the latter in the moist
model atmospheres due t o the interpolation. Since we are
analyzing the period of quasi-equilibrium, the following
approximate relation should hold:
N H
n =l
_I- - T(n)t+P(n) -D(n>t =O .
In our analysis, this is not necessarily satisfied.
In figure 29, the hemispheric mean values of these
terms are shown as a function of longitudinal wave
number. The distributions for the N4OM model and for
the N2OM model are given in the upper and lower parts
of the figure. In the N40M atmosphere, there are si&-
cant differences between the production P(n>" and the
dissipation D m H , particularly in the wave number range
of 4 to 8, where baroclinic instability prevails, and the
nonlinear term T(n)" plays a signiiicant role in com-
pensating for this imbalance. On the other hand, the
difference between the production and the dissipation is
very small in the N2OM low resolution model, and the
nonlinear term does little in redistributing energy from
one wave number to others. (The net transfer of energy
toward higher wave number components at wave number
n can be estimated by the following integration with
respect to wgve pumber k:
since S( a) = 0.)
Further examination of the nonlinear effect in the
N40M atmosphere reveals that there is very little net
transfer of energy from medium wave numbers to higher
wave numbers. Instead, most of the energy, which is con-
verted by baroclinic instability in the medium wave num-
ber range, is transferred t o the low wave number range.
Although the rate of change of the zonal kinetic energy
is not shown in this figure, it is evident that the rate of
energy transfer to zonal kinetic energy is significantly
large as demonstrated later by the energy box diagram
of figure 59. According to Starr (1953), this net transfer of
energy from synoptic scale eddies to the zonal current is
the main agent for maintaining the existing level of zonal
kinetic energy in the actual atmosphere. Such a net
transfer is missing in the N2OM atmosphere.
I n recent discussions of two-dimensional turbulence,
Eraichnan (1967) and Leith (1968) noted that two kinds
of inertial ranges could formally be defined, that is, that
of Eolmogoroff in which energy is transferred through
wave number space, and a new one in which mean square
vorticity, but not energy, is transferred. The first leads
to the familiar minus 513 power law. Eraichnan predicts
the development of a minus 513 power range at wave
numbers lower than those of the energy source and the
development of a minus 3 power range at wave numbers
higher than those of energy sources. In the former,
kinetic energy is transferred to large scale, whereas in
the latter, no kinetic energy is transferred toward small
scales. The recent results from a numerical time integra-
tion of the barotropic vorticity equation by Lilly (1968)
seem to substantiate the theory of Eraichnan and Eeith.
Figure 30 shows the spectrum of eddy kinetic energy
in the latitude belt ranging from 15" to 81" of the N40M
model. In the high-frequency parts of the spectrum, the
slope of the plotting indicates approximately minus 3
power which is not substantially different from
Kraichnan's. This result is not inconsistent with the
smallness of the energy cascade in the high-frequency
range. It is interesting that such a slope emerges despite
the depletion of kinetic energy by the nonlinear viscosity
in the model atmosphere. The spectral distribution of the
eddy kinetic energy of the geostrophic wind in the actual
atmosphere, which was computed by Wiin-Nielsen (1967),
is plotted for the sake of comparison. His spectrum also
indicates a slope of approximately minus 3 power for the
range of wave numbers larger than 8.
Dissipution spectru-In order to examine the difference
between the N4OM model and the N2OM model further,
we shall compare in detail the spectrum of kinetic energy
dissipation in the two moist model atmospheres. I n
figure 31 are shown the rates of the dissipation at various
wave numbers by horizontal and vertical mixing. This
figure clearly indicates that the horizontal dissipation of
kinetic energy takes place not only at large but also a t
March 1970 Syukuro Manabe, Joseph Smagor'insky, J. Leith Holloway, Jr., and Hugh M. Stone
- i 7.s-
E
U
7
P 7.0- Y Y
193
HEMISPHERIC AVERAGE DAY 200-230
k - PRODUCTION
TOTAL DISSIPATION
CI
0
ONLINEAR GAIN
-200
4 8 12 16 20 24 28 32 36 40 44 48 52 $6 60 64 68 72 76 80
WAVE NUMBER
HEMISPHERIC AVERAGE DAY 174-210
4 -
PRODUCTION
- TOTAL DISSIPATION
1 I I I I I I I I I I I I I I I I I I I I-100%
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
WAVE NUMBER
FIGURE 29.-Spectral distribution of the hemispheric mean values
of the nonlinear term T(n), production term P(n), and dissipa-
tion term -D(n). Note that the zero line of T(n) is different from
the zero lines for the other terms.
small longitudinal wave numbers. In other words, the
rate of horizontal dissipation is significantly large at the
wave numbers of baroclinic instability or at the wave
numbers where energy is mainly transferred from high
to low longitudinal wave numbers. This figure also
reveals that the relative magnitude of dissipation by
horizontal mixing and by vertical mixing depends upon
the resolution of the model. Although the partitioning
differs, the total dissipation does not, being mainly
controlled by the energy made available through genera-
tion. In the NZOM atmosphere, the rate of vertical
dissipation is much less than the rate of horizontal
dissipation; whereas in the N4OM model, both rates have
comparable magnitudes. In NZOM, the eddy kinetic
energy of medium wave numbers is dissipated mainly
in situ by horizontal mixing; whereas in NdOM it is
transferred to lower longitudinal wave numbers where
it is dissipated by both vertical and horizontal mixing.
The significant decrease of horizontal dissipation
accompanying the increase in the resolution of the horizon-
tal grid indicates that the effective viscosity acting on the
A WIN-NIELSEN
(2O0-9Oo Latitude)
k*
WAVE NUMBER
FIGURE 30.-Spectral distribution of the vertical integral of eddy
kinetic energy of the N4OM model (latitude 15'-81'). The slanted
lines have a-3.0 slope. The observed spectrum by Wiin-Nielsen
(1967) is also shown.
synoptic scale disturbances decreases with increasing
resolution. Since the scale at which the nonlinear dissipation
is very effective decreases with decreasing grid size, it
is reasonable that dissipation of large-scale motion by
horizontal mixing is larger for the low resolution model
than for the high resolution model. This is the major
reason why the N4OM atmosphere has larger eddy kinetic
energy, larger decascade of energy from medium wave
numbers to low wave numbers, and more dissipation by
vertical mixing than the N20M atmosphere. It is expected
that further increase of horizontal resolution should re-
sult in a more realistic atmosphere, because of the greater
separation between the synoptic scale and the scale of
Spectra of kinetic energy production (-V v+ ).
Another important factor controlling the spectrum of
eddy kinetic energy is the production of eddy kinetic
energy resulting from the conversion of potential energy.2
In figure 32, the spectra of kinetic energy production of
both models are shown. This figure indicates that magni-
tude of the production in the N4OM atmosphere is sig-
nificantly larger than in the NZOM atmosphere in the range
of wave numbers less than 8: The increase of kinetic
energy production at lower wave numbers resulting from
the improvement in the horizontal resolution seems to be
one of the major reasons why the kinetic energy of the
long waves in the N4OM atmosphere is much larger than
in the NZOM atmosphere.
It is rather surprising, however, that the maximum
difference between the production spectra of the two
dissipation by horizontal subgrid scale mixing. -H
2 Note that the hemispheric mean spectrum of the production term is identical with
that of conversion. In practice, there is some difference mainly because of the error caused
by interpolation and finite differencing.
1 94 MONTHLY WEATHER REVIEW vol. 98, No. 3
I
WAVE NUMBER
FIGURE 31.-Hemispheric mean rates of dissipation by subgrid
scale mixing as a function of wave number for the two models.
Solid and dashed lines indicate the dissipation by horizontal and
vertical mixing, respectively.
models appears at wave numbers 2 and 3. Since such a
long wave can be represented by a relatively coarse grid,
the doubling of the resolution should hardly improve the
accuracy of representation. Therefore, the improvement
of the finite-difference representation may not be the
direct cause for this difference in the production spectrum.
As we have pointed out, horizontal dissipation accounts
for most of the dissipation in the N2OM atmosphere;
whereas in the N4OM atmosphere, the contribution of
vertical dissipation is increased and the contribution
of horizontal dissipation is decreased. It is probable that
the differences in the magnitude and in the vertical dis-
tribution of dissipation between the two model atmos-
pheres cause the differences in amplitudes and in the struc-
ture of the long waves and accordingly cause the differences
in the spectra of production and kinetic energy at small
wave numbers.
Spectra in middle and low latitudes--So far, we have
discussed the hemispheric mean spectrum of various
components of the kinetic energy budget. As previously
mentioned, the energetics of middle latitudes are some-
what different from the energetics of the Tropics. There-
fore, it is worthwhile to examine the kinetic energy budget
for middle latitudes and the Tropics separately.
The spectra of the eddy kinetic energyin both the middle
latitudes and the Tropics of the N4OM model are shown
in figure 33. In middle latitudes, the eddy kinetic energy
of the meridional component of the wind is maximum in
the range of wave numbers 4 to 8, whereas that of the
zonal component of the wind generally increases with
decreasing wave numbers. In the Tropics, the eddy kinetic
energy of the zonal component of the wind is a maximum
at low wave numbers ranging from 1 to 5 and is much
larger than that of the meridional component due probably
to the effect of the equatorial boundary. The relative
magnitudes of the two, however, approach each other
- N40M
N20M ---_
WAVE NUMBER
FIGURE 32.4pectral distribution (hemispheric average) of the
rate of change of eddy kinetic energy by the production term
P (4 .
with increasing wave numbers. The spectral distribution
of the energy balance components in middle latitudes and
in the Tropics of the N4OM model are shown in figure 34.
In middle latitudes, the production term P(n) is maximum
in the range of wave numbers 5 to 7 where the eddy kinetic
energy of the meridional wind is maximum. The eddy
dissipation term D(n) is maximum in the range of wave
numbers 1 to 8. The magnitude of the production term
exceeds the magnitude of the dissipation term in the wave
number range of 4 to 10 and constitutes a net supply of
eddy kinetic energy in this range. This excess of kinetic
energy may be mainly transferred to lower wave numbers
(less than 4) barotropically. Although it is not explicitly
shown in this figure, a large amount of energy is also
transferred to zonal kinetic energy by this process. This
subject will be discussed again in section 8. There seems
to be a small cascade of energy toward higher wave
numbers.
In the Tropics, both the dissipation and production
terms have s i m c a n t magnitude up to very large wave
numbers, suggesting the predominance of small-scale dis-
turbances. As pointed out in study M2, the generation of
eddy available potential energy from released heat of
condensation is responsible for such disturbances with
wave lengths of 2000 to 3000 km. It is also interesting that
the largest production of kinetic energy takes place in
the range of wave numbers 1 to 5 where the eddy kinetic
energy is maximum. Since the magnitude of eddy kinetic
energy is very small in the Tropics of the dry model, as
the results of study M2 indicate, it is clear that the heat
of condensation is responsible for the development of such
long waves. I t is not clear, however, why such long waves
can develop in the Tiopics of the moist models. Further
study is necessary in order to establish the mechanism of
the growth of the long waves. I n the range of wave num-
bers 1 t o 13, the production of kinetic energy is larger
than dissipation. As was the cme in middle latitudes, the
excess energy may be mostly transferred to the zonal
current, and the energy transfer toward higher wave
numbers seems to be very small.
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 195
KINETIC ENERGY 1.0 I I I I I I I I
IN 40M f?IDLATITUDES (3 6 O -5 4 O ) DAY 200-230 ,9
WAVE NUMBER
1.0 I I I I I I I I I N40M TROPICS (O O -6 " LATITUDE) DAY 200-230 .9 -
.a-
$.?-
i .5 -
n 6-
p .2,
WAVE NUMBER
FIGURE 33.-Kinetic energy spectrum of the N4OM model for
middle latitudes and the Tropics.
C. VERTICAL DISTRIBUTION OF THE ENERGY SPECTRUM
According to the comparison between the surface
pressure map and the geopotential height map of the 500-
mb level of the N4OM atmosphere (figs. 16 and 18), the
characteristic scale of waves at the 500-mb level is signifi-
cantly larger than that of the disturbances near the
earth's surface. Also, as figure 38 indicates, the character-
istic wavelength in the model stratosphere is longer than
in the model troposphere. These results indicate that the
characteristic scale of disturbances of the moist atmos-
phere generally increases with altitude in the stratosphere
as well as in the troposphere. In order to determine how
the vertical distribution of the energy speckrum is main-
tained, we shall examine the vertical distribution of the
various componenk of the energy balance in wave number
space.
Applying the hemispheric mean operator and using the
relation
we can transform equation (22) to
w H '
H H H
The first term on the right-hand side of this equation
represents the contribution of the nonlinear term to the
- TOTAL DISSIPATION
WAVE NUMBER
- PRODUCTION
- TOTAL DISSIPATION
LINEAR GAIN
- PRODUCTION
- TOTAL DISSIPATION
.xu 0
-100 r
.I f
NONLINEAR GAIN
IF
-100"
i i 1 -200-
1 1 1 1 1 1 1 1 1 1 l I l I I I 1 I 1
4
WAVE NUMBER
8 12 16 2 0 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
FIGURE 34.-Spectral distribution of the rate of change of eddy
kinetic energy due to the nonlinear term, the production term,
and the dissipation term. Upper half, middle latitudes (36O to 54'
latitude) ; lower half, Tropics (0" to 6" latitude).
change of the kinetic energy of the wind component with
wave number n. The second and third terms represent
the contributions of the pressure interaction and conver-
sion terms, respectively, and the fourth term represents
the contribution of dissipation. Figure 35 shows the spec-
tral distributions of the conversion terms at various levels
in the N4OM atmosphere. These distributions represent
the area averages of each term in the latitude belts rang-
ing from 15' to 81' N. The tropical area is not included in
this averaging because the vertical distribution of the
energy budget in the Tropics is somewhat different in
middle latitudes.
According to this figure, the characteristic scale of eddy
conversion in the N4OM atmosphere increases with in-
creasing altitude throughout the troposphere. In the
middle troposphere, the eddy conversion is a maximum
around wave numbers 3 to 7; whereas in the upper tropo-
sphere it is maximum at wave number 2. Also, the eddy
conversion at higher wave numbers (n>lO) has signscant
magnitude in the lower troposphere partly due to conden-
sation, whereas it does not in the upper troposphere. This
increase of the characteristic scale of conversion with
increasing altitude is consistent with the increase of the
scale of the eddies with height.
Another factor that determines the change of kinetic
energy is the pressure interaction; figure 36 shows the
spectral distribution of this term. I n the upper tropo-
1 94 MONTHLY WEATHER REVIEW VoI. YJ3, No. 3
N40M
P I P , = ,009
,074
,189
.99t
-
I I I I I I I I I I
2 4 6 8 10 12 14 16 18 20
WAVE NUMBER
FIGURE 35.-Spectral distribution of the conversion term
(- (w),(a),) at various levels in the N4OM atmosphere. Here,
( )A indicates the average for the area between 15 to 81
latitude.
sphere, that is, above the 500-mb level, the pressure
interaction transfers energy upward. On the other hand,
it transfers energy downward at very small wave numbers
below the 500-mb level. Accordingly, the convergence of
energy takes place above the 300-mb level and below the
the 800-mb level. I n the intermediate layer, between the
300-mb and 800-mb levels, divergence of energy flux takes
place. Above the 300-mb level, upward energy flux by the
pressure interaction increases its scale with increasing
altitude and helps maintain very long waves in the strato-
sphere. I n the layer of energy divergence (the 300-mb to
800-mb layer) , the characteristic scale of the pressure
interaction is very large, and the energy of very long waves
is depleted. In short, the pressure interaction contributes
to the increase of the scale of eddies with increasing
altitude above the 850-mb level.
Summing up the contributions of the pressure inter-
action and of the conversion, we obtain the net rate of the
production of kinetic energy by the pressure gradient
force. The vertical distribution of the spectrum of the
rate of the net production is shown in figure 37. This
figure shows that the baroclinic effect as a whole is respon-
sible for the increase in the scale of eddies with increasing
altitude above the planetary boundary layer. Figure 37
also indicates that the nonlinear transfer counterbalances
N40M
PIP, = ,009
,074
,189
,664
2 4 6 8 10 12 14 16 18 20
WAVE NUMBER
FIGURE 36.-Spectral distribution of the pressure interaction term
(- (w),(+)) at various levels in the N40M atmosphere. Here,
( ,) indicates the average for the area between 15 to 81
latitude.
this baroclinic effect. In general, the nonlinear transfer
depletes the energy of medium wave numbers and mainly
increases the energy of the zero wave number, that is,
the zonal kinetic energy. I n the free atmosphere the char-
acteristic scale of depletion by the nonlinear transfer
increases with increasing altitude.
In the planetary boundary layer or below the 850-mb
level, the large convergence of downwaid energy flux at
very low wave numbers contributes to the production of
the energy of very long waves. According to figure 37,
the energy of such long waves, however, is depleted very
rapidly by the dissipation due to the vertical mixing and
surface drag which predominate in the planetary boundary
layers. The contribution of the dissipation due to the
horizontal mixing is relatively small. Since the dissipation
by surface drag is proportional to the cube of the wind
speed and since the kinetic energy of the surface wind
increases with decreasing wave number, the dissipation
by vertical mixing and surface drag increases with decreas-
ing wave number. Therefore, vertical mixing dissipates
the kinetic energy of very long waves and counterbalances
the production in the planetary boundary layer.
The basic cause for the increase of the scale of the dis-
turbances with height in the troposphere of the moist
March 1910 Syukuro Manabe, Joseph Smogorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 197
NONLINEAR TRANSFER
P/P, = .009
-
~
-400-1
- ,811 -200-
--- .926 -200 -
,991 -200 -
0 -- fC.
-lo00 - -
I I I 1 I 1 L-- 2 4 6 8 10 12 14 16 18 20
400
200
0
200
0
800
600
400
200
0
200
.O
600
400
200
0 _-
DISSIPATION
.074 -200 -
0-
-200 .189 -
0
-200
-400 1 .336
0 ....... ~-;~.-::::.:...-.....~ .... ............................. ........................... .........
-=-=--
-200
r y
..... ....................
-400
0
-200
-400
0
-200
0
-200
0
-200
,, ; -2000 ,/A4 r
I , e I I 1 I I ,, I
2 4 6 8 10 12 14 16 18 20
WAVE NUMBER WAVE NUMBER WAVE NUMBER
FIQURE 37.-Spectral distribution of the contribution of each term to the rate of change of eddy kinetic energy at various levels of the
N4OM atmosphere. The contribution of the nonlinear term (- (V )..(V .~V ~O (~V /~P )).), the production term (- (V)..(V+)/),
and the dissipation term ((V),,.(F)/) are shown in the left, middle, and right diagrams, respectively. Here, ( )A indicates the
average for the area between 15 and 81 latitude.
___-
model becomes evident if we compare the characteristic
scale of disturbances of the moist atmosphere with that of
the dry atmosphere. Figure 38 shows such a comparison.
We note that the dependence of the effective wave number
on height is hardly detectable in the N4OD troposphere,
whereas it is evident in the N@M troposphere. One can
also identify the effect of condensation on the spectral
distribution of the eddy conversion by comparing the
conversion spectra of the N4OD atmosphere shown in
figure 39 with those of the N40M atmosphere shown in
figure 35. In the N4OD troposphere, the conversion spectra
have a distinct maximum at wave number 5, whereas in
the moist NJOM troposphere, it spreads over a wider
range of wave numbers. These results explain why intense
large-scale cyclones dominate the map of surface pressure
of the N4OD atmosphere and less intense, smaller scale
cyclones are evident in the corresponding map of the
N4OM model. Also, the comparison between figures 35
and 39 seems to show that the effects of condensation
tend to develop ultralong waves particularly in the upper
troposphere.
In order to compare the energetics of stratospheric and
tropospheric motion, spectral distributions of various
components of the kinetic energy budget for the model
stratosphere and those for the whole model atmosphere
are given in figure 40. I n the model stratosphere (levels
1 and 2), eddy kinetic energy is mainly produced at wave
numbers 2 and 3 by the pressure interactions. This eddy
kinetic energy is transferred in turn to the kinetic energy
of the zonal flow inertially by nonlinear transfer. On the
other hand, the eddy kinetic energy of the troposphere is
mainly fed by the conversion from eddy potential energy
in the range of wave numbers between 2 and 8. A sig-
nificant part of this eddy kinetic energy is transferred to
zonal kinetic energy by the decascade. I n summary, the
zonal kinetic energy of the model stratosphere is main-
tained by the transfer of energy from very long waves;
whereas in the troposphere the energy of the zonal currgnt
is acquired from the waves of medium wave numbers
between 2 and 8. Similar results have been obtained by
Teweles (1963) and Saltzman and Teweles (1964) for the
actual atmosphere. They computed the energy exchange
198 MONTHLY WEATHER REVIEW VOB. 98, NO. 3
EmcTNE WAVE NUMBER
FIQURE 38.--Vertical distribution of the effective wave number in
the N4OM and N4OD atmospheres are shown by solid and dashed
lines, respectively. (See subsection 4D for the exact definition of
the effective wave number n.) In order to exclude the tropical
eddies from the analysis, the effective wave number is computed
for the area between 15' and 81' latitude.
among the waves with different wave numbers by the
barotropic processes and obtained results similar to those
described here.
7. BUDGET OF ANGULAR MOMENTUM,
HEAT, AND MOISTURE
Before discussing the budget of various quantities, it
is desirable to define the transport of any quantity q
by the large-scale eddies and the meridional circulation.
The poleward transport of q by large-scale eddies is
given by the expression
and the poleward transport of q by meridional circulation
is given by the expression
where PMAx is maximum surface pressure and v is the
meridional component of the wind.
A. ANGULAR MOMENTUM
Transport-The latitudinal distributions of the vertical
integral of poleward transport of angular momentum by
lasge-scale eddies, meridional circulation, and subgrid
scale Reynolds stress in the NdOM atmosphere are con-
trasted with those in the N2OM atmosphere in figure 41.
This figure shows that one of the important differences
between the two models lies in the magnitude of the sub-
grid scale transport. I n the low resolution N2OM atmos-
1 I I I I I I "I
2 4 6 8 IO 12 14 16 18 20
WAVE NUMBER
FIQURE 39.-Spectrd distribution of the conversion term
(-.(&(a)/) at various levels in the N4OD atmosphere. Hem,
( indicates the average for the area between 15' and $1'
latitude.
phere, this "transportJJ is almost as large as the large-
scale eddy transport in low latitudes. In the high
resolution model (NdOM), it has negligible magnitude
indicating the drastic decrease in the effective Reynolds
stress as a result of the doubling of the horizontal reso-
lution. On the other hand, the magnitude of the poleward
transport by large-scale eddies markedly increased 8s a
result of the change of the resolution and coincides very
well with the estimates obtained by Stam and White
(1954).
The latitude-height distribution of the poleward trans-
port of angular momentum by large-scale eddies in the
N4OM atmosphere is shown in figure 42. In general,
this distribution is realistic and reasonable. (For the
observed distribution see fig. 6B1 of study S.) There is,
however, one curious feature, that is, the equatorward
transport of angular momentum by the large-scale eddies
in the Tropics. Although such equatorward transport is
missing in the annual mean statistics of the angular
momentum transport in the actual atmosphere which
were obtained by Starr and White (1954), it is evident
in the distribution recently obtained by Rasmusson and
by Oort (1970). This equatorward transport in the model
atmosphere is probably accomplished by tropical dis-
turbances which were discussed in previous sections and
in study M2.
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 199
LEVEL I
lo(
NONLINEAR WVJ
- -7
---(
I
-la
-?XI
-M(
0ISSIPATK)W
-
-101
I I I I I I I I I I
4 a 12 16 20
WAVE NUMBER
NONLINEAR GAIN
DISSIPATIOIOI OISSIPATION
-2 o o w -
I E l l l l l l l l I l l l l l l l l l l
4 a IT 16 m ' 4 8 12 16 20
WAVE NUMBER WAVE NUMBER
FIGURE 40.-Spectral distributions of &he contribution of each term to the rate of change of eddy kinetic energy. Level I (P/P*=0.009),
level 2 (P/P*=0.074), and the vertical integral of the whole atmosphere. Units, 10-3 ergs cm-2 mb-1 sec-1. Results shown are hemispheric
averages.
Budget-The latitudinal distributions of the rates of
change of angular momentum in the N4OM model atmos-
phere by various factors are sho.wn in figure 43. Poleward
of 35' latitude, the contribution of the large-scale eddies
and of the surface torque have opposite signs and almost
cancel each other. Equatorward of 35' latitude, the
meridional circulation plays a significant role and affects
the budget of angular momentum together with the large-
scale eddies and surface torque.
In order to find out how the distribution of momentum
exchange between the earth's surface and the atmosphere
is affected by the horizontal resolution of the finite differ-
ences, the latitudinal distributions of surface torque in
the NAOM and N2OM models are compared in figure 44.
375-632 0 - 70 - 4
The corresponding distribution of the N2OD dry model is
also shown. This- figure demonstrates that the intensity
of momentum exchanges between the earth's surface and
the atmosphere is much larger for the N4OM model than
for the N2OM model at extratropical latitudes. This
difference in intensity is consistent with the difference in
distributions of zonal mean surface pressure of the two
models. As figure 7 indicates, the surface pressure has a
much larger latitudinal variation in the N4OM model
than in the N2OM model. This difference accounts for the
difference in the intensity of the surface zonal wind and,
accordingly, accounts for the difference in the magnitude
of surface torque. As aforementioned, the magnitude of
poleward transport of momentum by large-scale eddies
MONTHLY WEATHER REVIEW voi. 98, No. 3
7 3s-
5 30-
0
2s-
20-
5 15-
L 0 10-
0
0
0
-5-
POLEWARD TRANSPORT OF ANGULAR MOMENTUM
I I I I I I I I
45 N U
40
nSTAlllnWHm -
-
EDDY
5-
I I I I I I I
W I N D E
POLEWARD TRANSPORT OF ANGULAR MOMENTUM I I I I I I I I
H2@M
30
-'80 80 70 60 50 40 30 20 10 0
LATITUDE
FIGURE 41.-Latitudinal distributions of the poleward transport
of angular momentum by large-scale eddies, meridional circula-
tion, and subgrid scale Reynolds stress (H.D.). The rates of the
poleward transport of angular atmosphere, estimated by Starr
and White (1954), are also plotted.
increases markedly with increased horizontal resolution.
In middle and high latitudes, this increase compensates
for the corresponding increase in the magnitude of surface
torque mentioned above.
The latitude-height distribution of the rate of change
of angular momentum by the meridional circulation,
large-scale eddies, and subgrid scale mixing in the N4OM
atmosphere is shown in figure 45. In the boundary layer,
the contributions of vertical mixing and meridional circu-
lation almost compensate each other, and the effect of
the large-scale eddies is relatively small. In the free atmos-
phere, large-scale eddies play a major role and have almost
the same magnitude but the opposite sign as the meri-
dional circulation. Furthermore, the effects of vertical
and horizontal mixing are very small there. A similar
feature of the momentum balance was evident in the
N20D model atmosphere described in study S. Recently,
Wiin-Nielsen and Vernekar ( 1967) investigated the angu-
lar momentum balance and found, similar characteristics
of compensation in the actual atmosphere. Contrasting
figure 45 with the upper half of figure 5, which shows the
distribution of the zonal mean of the zonal current, we
find the following: The momentum transfer by the large-
scale eddies contributes toward intensifying the northern
half of the tropospheric jet, and the momentum transfer
by the meridional circulation helps intensify the southern
half of the jet. In the stratosphere, the convergence of
angular momentum transport by the large-scale eddies
.=."I
.991 90 80' 10' EO0 50" 40" 30' 20" 10' 0'
lnm
FIQURE 42.-Latitude-height distribution of the poleward transport
of angular momentum by large-scale eddies in the N4OM at-
mosphere. The region of equatorward transport is shaded.
Units, 1028 gm cm2 sec-1 day-1 mb-1.
7 , I I I 1 I I I I I
I I I I I I I I I 90 80 70 60 50 40 30 20 10 0
LATITUDE
FIGURE 43.-Latitudinal distributions of the rate of change of the
angular momentum in the N4OM atmosphere by meridional
circulations, large-scale eddies, subgrid scale horizontal Reynolds
stress (H.D.), and surface torque.
is mainly responsible for the intensification of the jet,
while the effect of the meridional circulation weakens the
intensity of the jet stream.
B. HEAT
Transport-The latitudinal distributions of the pole-
ward transport of heat by the large-scale eddies and by
subgrid scale mixing in the N4OM and the N20M atmos-
pheres are shown in figure 46. Again, the relative magni-
tude of the transport by the subgrid scale mixing is
significantly reduced by the increase of the horizontal
resolution, while the heat transport by the large-scale
eddies is increased. The percentage increase, however, is
not as large as in the case of the angular momentum
transport.
In figure 47, the latitude-height distribution of the
poleward transport of heat by the large-scale eddies in the
N.OM atmosphere is compared with that in the actual
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 201
6 I 5 -
4 -
3-
- 2 -
ir 1-
; O -
5 -2-
v)
n -1-
-3
-4
-5 -
-6 -
-
-
N40M
UTITUDE
FIGURE 44.-Latitudinal distributions of surface torque for various
models.
atmosphere, which was estimated by Starr and White
(1954). Although the latitude of maximum poleward
transport in the N4OM model is located to the south of
the maximum observed in the atmosphere, there are
various features of the model atmosphere distribution
that agree with those of the actual atmosphere. For ex-
ample, the poleward transport in middle latitudes has
maxima a t two different levels. The primary maximum is
located a t about the 900-mb level, and a secondary maxi-
mum a t about the 200-mb level, in agreement with obser-
vation. Another interesting feature of the model distribu-
tion is the weak equatorward transport of heat in the
subtropics. Preliminary analysis of 5 yr of data by
Rasmusson and Oort (1970) shows that the actual atmos-
phere has similar features. A poleward transport of heat,
however, appears in the model Tropics.
The latitude-height distribution of the upward transport
of heat by the large-scale eddies in the N4OM atmosphere
is shown in figure 48. It has two tropospheric maxima, one
in middle latitudes and another in the Tropics. The lati-
tudes of these maxima correspond well with those of the
maximum poletvard transport of heat which may be
identified in the upper part of figure 47. In the subtropics,
the vertical transport is very small or even negative. This
region of small vertical heat transport corresponds to the
region of weak equatorward heat transport by the large-
scale eddies mentioned earlier. In the stratosphere of the
NdOM model, downward transport appears, but its mag-
nitude is very small.
Heat baEance-The latitudinal distributions of the verti-
cal integral of various heat balance components of the
N4OM atmosphere are shown in figure 49. In the Tropics,
the heating by condensation and convection is compen-
sated through the cooling by meridional circulation and
radiation. In the subtropics, the heating due to the down-
ward-moving branch of the meridional circulation is counter-
(balanced mainly by radiative cooling. I n middle latitudes,
the heat of condensation is mostly compensated by radia-
tive cooling. In high latitudes, the heating resulting from
the convergence of the poleward flux of heat by large-scale
eddies is compensated mainly by radiative cooling.
n
n . I
umuor
FIGURE 45.-Latitude-height distributions of the rate of change
of angular momentum by various components in the N4OM
atmosphere. Units, 1012 gm cm2 sec-1 day-1 mb-1. Upper third,
meridional circulation; middle third, large-scale eddies; lower
third, Reynolds stress by vertical and horizontal subgrid scale
mixing.
202 MONTHLY WE vol. 98, No. 3 ATHER REVIEW
1 1 1
A STARR + WHITE
5 t
:E
0 90
1 -
-
-
I I I I I
80 70 60 50 40 30 20 10 0
LATITUDE
FIGURE 46.-Latitudinal distributions of the poleward transport
of sensible heat by the large-scale eddies (Eddy) and by the
subgrid scale horizontal mixing (H.D.). Upper half, the N4OM
atmosphere; lower half, the NgOM atmosphere. The eddy trans-
port values obtained by Starr and White (1954) are also plotted.
The latitude-height distributions of the rate of temper-
ature change due to radiation, condensation and convec-
tion, meridional circulation, and large-scale eddies are
shown in figure 50. This figure clearly indicates how the
heat balance is maintained in the various parts of the
atmosphere. For example, the contribution of radiative
transfer and condensation is relatively small in the lower
stratosphere, while the effect of large-scale eddies has
almost the same magnitude but opposite sign as that of the
meridional circulation. As previously discussed, the strato-
spheric flow field is not driven by the meridional heating
contrast but by the energy flow from the troposphere.
This is why a significant level of eddy kinetic energy is
maintained there, despite the weak heating contrast. The
relationships between the thermal structure of the strato-
sphere and the various thermodynamical factors were
discussed extensively by Manabe and Hunt (1968) and
will not be repeated here.
In the troposphere of the N4OM model, the rate of tem-
perature change due to condensation and convection has
FIQURE 47.-Latitude-height distribution of the polewmd trans-
port of heat by the large-scale eddies in units of 10'' joules mb-*
day-l. Lower half, actual atmosphere (Starr and White 1954) ;
upper half, the N4OM atmosphere.
a distinct double maximum, one in the Tropics and another
in middle latitudes. The former is caused by the tropical
rain belt, and the latter is maintained by the middle-
latitude rain belt. As one would expect, the heating due
to moist convection and condensation penetrates much
higher in the Tropics than in middle latitudes.
The mass integrals of various heat balance components
have been embodied into box diagrams for both the N40M
and the N2OM atmospheres. These are shown in figure 51.
We see that the heat balance of the hemisphere as a whole
depends slightly upon the resolution of the horizontal
finite differences. However, the rate of heat released by
condensation is larger in the N4OM atmosphere than in
the N2OM atmosphere.
C. WATER
Transport-The latitudinal distribution of the poleward
transport of moisture by the meridional circulation, large-
scale eddies, and by subgrid scale mixing in both the N40M
and the NZOM atmosphere are compared in figure 52. The
poleward transport of moisture by large-scale eddies is
affected slightly by the change of horizontal resolution,
although in middle latitudes the eddy transport is slightly
March 1970 Syukuro Manabe, Joseph Srnagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone 203
.OM 30
N40M
20
1
I 10
0
FIGURE 48.-Latitude-height distribution of upward transport of
heat by large-scale eddies in the N4OM atmosphere. Units,
joules ern+ day-.
larger in the N4OM than in the N2OM atmosphere. I t is
rather interesting that the degree of the dependence upon
the horizontal resolution is not the same for momentum,
heat, or moisture transport by the large-scale eddies. The
moisture transport is least affected.
The poleward transport of moisture by the indirect
meridional circulation cell in middle latitudes increases
significantly as a result of the intensification of this cell
caused by the change of resolution; whereas, the equator-
ward transport by the tropical Hadley cellis affected only
slightly. The poleward transport of moisture by the subgrid
scale mixing decreases significantly as a result of the change
of the horizontal resolution, as was also the case with heat
and momentum.
The latitude-height distributions of the poleward
transports of moisture by the large-scale eddies and by
the meridional circulation are shown in figure 53. It is
interesting that the direction of the eddy transport of
moisture in the model stratosphere is opposite in the
model troposphere. In the stratosphere, moisture is
transported equatorward. The moisture balance of the
stratosphere has already been discussed in the study M.
Refer to study M for further details of water balance in
the model stratosphere.
Budget-The latitudinal distributions of precipitation
and evaporation of the N4OM and the N2OM atmospheres
are shown in figure 54. This figure shows that the dif-
ference between these distributions in the two models is
rather small. Therefore, we shall describe the water
balance of the N4OM model only. The latitudinal distri-
butions of various water balance components are shown
in figure 55. In the Tropics, the moisture supplied by the
meridional circulation and by evaporation is consumed
by the precipitation of the tropical rain belt. In the sub-
tropics, the moisture gained by evaporation is removed
mainly by meridional circulation. In middle latitudes,
moisture supplied by evaporation and large-scale eddies
d
3l
2
- ,L CONVECTION a CONJDENSAT
-Zt
-3 90 80 70 60 50 40 30 20 10 0
LATITUDE
FIGURE 49.-Latitudinal distributions of the vertical mean tem-
perature changes, which are due to radiation, heat of condensa-
tion and convection, meridional circulation, largescale eddies,
and subgrid scale diffusion (H.D.) in the NdOM atmosphere.
is precipitated in the middle-latitude rain belt. In high
latitudes, the influx of moisture by the large-scale eddies
is counterbalanced by precipitation.
Finally, the hemispheric water balance of both models
is tabulated in figure 56. Again, this table indicates that
the intensity of the hydrologic cycle in the N4OM model
is slightly greater than in the N2OM model. In general,
the magnitude of vertical motion is larger in the N4OM
than in the N2OM atmosphere. The difference in the in-
tensity of vertical mixing of moisture by large-scale eddies
may be responsible for the difference in the intensity of
the hydrologic cycle mentioned above.
D. EDDY TRANSPORT IN DRY MODELS AND MOIST MODELS
In the preceding subsections, we have compared the
distributions of the poleward transport of angular mo-
mentum, heat, and moisture by large-scale eddies in the
N4OM atmosphere with those in the N2OM atmosphere.
In this subsection we shall compare some of these distri-
butions with those in the dry model (N2OD and N4OD)
atmospheres, because dry models are widely used for
various purposes. Figure 57 shows the latitudinal dis-
tributions of the poleward transport of angular momen-
tum by large-scale eddies in the four different model
atmospheres. This figure demonstrates that the magni-
tude of the poleward transport of angular momentum is
influenced very markedly by the incorporation of the
hydrologic cycle as well as by the change in the horizontal
resolution. The magnitude of the poleward eddy trans-
port of angular momentum in the N2OD model atmosphere
happens to be very realistic because the poor horizontal
resolution compensates for the absence of the hydrologic
cycle. Accordingly, further improvement of the hori-
zontal resolution results in a very large and highly un-
realistic magnitude of transport in the N4OD atmosphere.
In short, we obtained superficially realistic results from
204 MQ NTHLY W EAT" ER R EVB EW 'dol. 98, No. 3
T I t I
,009
,074
n.
. n.
,189 I
336
,500
,664
,811
,9269 .99l SOe BO" 70' 60" 50" 40" 30' 20" 10' 0"
LAll'NOE
I s
t
RADIATION
30
20
t f
iij I
10 ___-_------
.DO9
.074
e n.
.189 1
336
SO0
,664
.mi
CONDENSATION AND CONVECTION
30 Nmm
PIQURE iiO.-Latitude-height distributions of the rates of temperature change which are due to various components of the heat balance
("C day+) in the N40M atmosphere.
an unrealistic model. Under these circumstances, it is
highly desirable to perform integrations with still higher
resolutions (say N8OM) to make sure that the N4OM
results are asymptotically stable.
The latitudinal distributions of the poleward transport
of heat by large-scale eddies in the various model at-
mospheres are also shown in figure 58. Note that the
incorporation of the hydrologic cycle significantly affects
the magnitude of the poleward transport of heat. The
poleward transport of latent energy contributes to a
reduction in the meridional temperature gradient in the
moist atmospheres compared with dry models and,
accordingly, the rate of poleward eddy transport of
sensible heat is reduced. I t is noteworthy, however, that
the doubling of the resolution from NZO to N4O has as
much effect upon the poleward transport of heat as the
moist process does.
8. BUDGET OF ENERGY
It is known that the zonal kinetic energy in the atmos-
phere is maintained against dissipation mainly by the
nonlinear barotropic transfer of energy from the large-
scale quasi-two-dimensional eddies to the zonal current
(Starr 1953). Such a transfer of energy is evident in the
energy diagram for the actual atmosphere, which has been
compiled by Oort (1964b) and shown in figure 60. The cor-
responding energy diagram for the N2OM atmosphere
(fig. 59) shows that the flow of kinetic energy is in the
opposite direction, that is, the zonal kinetic energy is
transferred into eddy kinetic energy. The zonal kinetic
energy is maintained against dissipation by the conversion
of the zonal potential energy into zonal kinetic energy. As
discussed in section 6, this unrealistic feature of the energy
transfer is attributed mainly to the excessive subgrid scale
dissipation operating in this coarse resolution moist model.
205 March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Holloway, Jr., and Hugh M. Stone
1s I I I I I I I I
,o 0 PEIXOTO N40M -
A PEIXOTO + CRlSl
NSR N40M NLR NSR N20M NLR
IS I I I I I I I 1
APEIXOTO t CRlSl
0 PEIXOTO N20M - 10 -
NSR SH NLR
I& .2>7 .oil .oia ,257 .Q2l .0+4
FIGURE 51.-Box diagrams of heat balance of the N4OM and the
NdOM atmosphere. The lower and upper boundaries of the boxes
indicate the earth's surface and the top of the atmosphere,
respectively; NSR, net solar radiation; NLR, net long wave
radiation; SH, sensible heat flux from the earth's surface; C,
heat of condensation. Units, ly min-1.
5 -
'> 0
d
*o -5 -
z
z
-10 -
.I5 -
I -
EDDY I
z
*o -5 -
z
-10 -
.I5 -
-
5-
>
2 0 =
$-5- \ \ I
E 0
c \ I
-' O b -15
-20
d, do 70 60 50 40 30 20 10 0 I I I I I I
LATITUDE
I I
FIGURE 52.-Latitudinal distributions of the poleward transport
of moisture by meridional circulation, large-scale eddies, and
subgrid scale diffusion (H.D.). The estimates of the annual mean
poleward transport in the actual atmosphere obtained by Peixoto
(1958) and Peixoto and Crisi (1965) are also plotted. Upper
half, N4OM atmosphere; lower half, NdOM atmosphere.
In the N4OM atmosphere, where the effect of the hori-
zontal subgrid scale mixing is less dominating and where
.a&", . ,991 90" 80" 70' 60" 50" 40' 30' 20' 10" 0"
UlllUOE
FIGURE 53.-Latitude-height distribution of the poleward trans-
port of water vapor in the N4OM atmosphere. Upper half,
transport by largescale eddies; lower half, transport by merid-
ional circulation. Units, 1012 gm mb-1 day-l.
the barotropic net energy exchange is of significant magni-
tude, energy is transferred from eddy to zonal kinetic
energy and contributes to the maintenance of zonal kinetic
energy in agreement with the behavior of the actual at-
mosphere. This is one of the most fundamental irnprove-
ments resulting from the increased horizontal resolution.
One of the interesting features of the energy balance of
the moist model atmosphere is the conversion from zonal
potential to zonal kinetic energy. Although the zonal con-
version in the N4OM atmosphere is smaller than that of
the N2OM model, there is no doubt that it has a positive
value with significant magnitude. In the dry model
atmospheres (Phillips 1956, Smagorinsky 1963, N2OD and
N4OD atmospheres), energy is transferred from zonal
kinetic energy to zonal available potential energy. This
negative conversion is particularly large in the N4OD
206 MONTHLY WEATHER REVIEW vel. 98, No. 3
60
55
50-
45
- 40-
L
LATITUDE
FIGURE 54.-Latitudinal distributions of the rates of precipitation
and evaporation.
I I I I 1 I I I
- -
........ N40D ." r - .. .. .. i :
- -
-
i
L A
M.C. ,I 'j
EVAPORATION
\ I
/.-I
/f
-0.5 -
-
-
-1.0 -
.- -
-
9 0 8 0 7 o ~)5 0 4 o x )2 o t o o
LATITUDE
FIGURE 55.-Latitudinal distributions of the various water balance
components of the N4OM atmosphere, that is, condensation,
evaporation, meridional circulation, large-scale eddies, and sub-
grid scale horizontal mixing (H.D.).
atmosphere. As it is evident in table 3, which shows the
intensity of meridional circulations in both the dry and
the moist models, the condensation process helps to in-
tensify the direct tropical cell and weaken the indirect
Ferrel cell in middle latitudes. As pointed out in study M,
the heat of condensation released in the upward branch of
.2980
FIGURE 56.-Table of hemispheric water balance at the earth's
surface. Units, cm day-'.
-101
-201 I I I I I I I I
90 80 70 60 50 40 30 20 10 b 1 AT IT U D E
FIQULE 57.-Latitudinal distributions of poleward transport of
angular momentum by the large-scale eddies in various models.
The rates of transport in the actual atmosphere, which were
obtained by Buch (1954), are also plotted.
the Hadley cell is expected to increase the intensity of the
cell. Also, the poleward transport of latent energy acts to
reduce the meridional temperature gradient and, accord-
ingly, the degree of baroclinic instability which drives the
indirect Ferrel cell. Both of these changes caused by moist
processes help to increase the release of zonal available
potential energy into zonal kinetic energy and prevent the
zonal conversion in the moist atmospheres from becoming
negative.
It is very difficult to estimate the value of the zonal
conversion in the actual atmosphere because the wind
components involved in the meridional circulation are
very small. According to the energy diagram, compiled
by Oort (19643) and shown in figure 60, this conversion
has a small positive value, although the accuracy of his
estimate of this determination is so unreliable that small
negative values are within the limits of the probable dis-
persion in the estimate. As described previously, the zonal
conversion of potential energy in the N4OM atmosphere
has a positive value of significant magnitude and con-
tributes to the maintenance of zonal kinetic energy. Al-
though it is not certain whether the positive conversion
of such magnitude takes place in the actual atmosphere, it is
March 1910 Syukuro Manabe, Joseph Smagorinsky, J.
r
I I I I I I
I
I.,
.. .. .. .. .. .. .. .. ..
.i
LATITUDE
FIGURE 58.-Latitudinal distributions of poleward transport of
sensible heat by the large-scale eddies in various models. The
rates of transport in the actual atmosphere, which were obtained
by Starr and White (1954), are also plotted as solid triangles.
highly unlikely that the zonal conversion in the actual at-
mosphere has such a large negative value as does the N4OD
atmosphere because of condensation. In view of the signifi-
cant difference between the magnitudes of the zonal
conversion in the N40M and the NZOM atmosphere and
in view of the highly idealized nature of the lower boundary
conditions chosen for the present model, it is desirable to
reevaluate this term after further improvement of the
model.
Another notable feature of the energy diagram of the
moist model is the large generation of the eddy available
potential energy due to condensation and convection.
In the moist model atmospheres, the eddy available
potential energy is maintained despite its large conversion
into eddy kinetic energy by the following processes:
1) The transfer of the zonal available potential energy
into the eddy available potential energy.
2) Generation of the eddy available potential energy
by heat of condensation and convection.
Comparing the energy diagram of the dry N4OD model
with the energy diagram of the moist N4OM model, we
find that the conversion of zonal to eddy available poten-
tial energy in the N4OD atmosphere is larger than in N4OM
and N20M, compensating for the lack of eddy generation
by the heat of condensation and convection.
Figure 61 shows the latitude-height distribution of the
rate of generation of available potential energy by the
Leith Holloway, Jr., and Hugh M. Stone 207
heat of condensation and convection in the N40M atmos-
phere. According to this figure, there are two regions of
large generation, one in middle latitudes and another in
the Tropics. Figure 62 is a schematic illustrating a typical
developing cyclone in middle latitudes. In such a cyclone,
a thermal trough is located to the west of .a pressure
trough. Since condensation usually takes place eastward
of the trough, a positive correlation exists between the
heat of condensation and temperature. Accordingly, the
heat of condensation generates eddy available potential
energy in middle latitudes. The mechanism for gen-
erating eddy available potential energy in the model
Tropics is somewhat different in middle latitudes and is
extensively discussed in study M2. In the model Tropics,
the eddy available potential energy is geneiated by the
heat of condensation, which is released in tropical cyclones
with warm cores in the upper troposphere. According to
our crude estimate, the generation in the middle latitudes
accounts for two-thirds of the total eddy generation and
the generation in the Tropics covers the remainder. I t is
probable that the eddy generation due to condensation
has a significant magnitude in the actual atmosphere,
although the energy diagram of Oort suggests otherwise.
Finally, the energy budget of the stratosphere of the
N4OM astmosphere is shown in figure 63. The major
source of eddy kinetic energy in the stratosphere comes
from the troposphere through the pressure interactions
(- ut@tA). The major sink of eddy kinetic energy is the trans-
fer to zonal kinetic energy which in turn is converted into
zonal potential energy. The zonal potential energy is
destroyed by radiative damping (QR) or transferred into
eddy potential energy. Qualitatively, similar features were
found for the dry general circulation model discussed in
study S. In summary, the major source of energy in the
stratosphere is the troposphere rather than an in situ
conversion of potential energy, and the major consumption
of energy is accomplished by the radiative damping of the
zonal available potential energy as well as by the dissipa-
tion of kinetic energy. There is one significant difference
between the energy budget of the N4OM stratosphere and
the energy budget of the actual stratosphere which was
determined by Oort (1964a). In the model atmosphere,
energy is transferred from zonal to eddy available potential
energy, whereas the reverse is true in the actual atmos-
phere. This discrepancy is related to insufficient poleward
heat transport by large-scale eddies and, ,accordingly, to
the insufficient increase of temperature with increasing
latitude in the lower stratosphere. Recently, Manabe and
Hunt (1968) performed an integration with a dry general
circulation model with a high vertical resolution and
successfully obtained a sufficient latitudinal increase of
temperature. In their model atmosphere, a counter-
gradient poleward flux of heat by the large-scale eddies
predominates, and energy transfer of significant magni-
tude from eddy to zonal available potential energy is
maintained in agreement with observation. Therefore,
an increase of vertical resolution seems to be necessary
for a satisfactory simulation of the energetics of the
stratosphere.
208
31.9
h40M
A A
8.9 15.8 3.3
MONTHLY
+
WEATHER REVIEW
c
\/oP. 98, bdo. 3
b 18.2
~
P,=499.2 " 1.7
N40D
6.2 P,=43.1
3.4 25.2
N20M
2.0
8.6
nF4
VF *
9.4
13.6
' HF
K ,=65.0 K,=154.0 + 16.6
* VF
0.8
3.1
HF 4
V F 4
QE
13.3
14.5
10.3
H'
- K,=54.5 1.6 Kp132.6 *
e VF
I
31.8 7.5 19.3
I 7.6 I 35.5
3.2 31.8 7.5 19.3
M20D
3.2
Q:
11.9
4.4
FIGURE 59.-Box diagrams of energetics of the various model atmospheres. QC, generation of available potential energy by condensation
and convection (here, convection includes the sensible heat flux from the earth's surface) ; QR, generation of available potential energy
by radiation; Q;, sum of QR and Qc; H , destruction of available potential energy by the horizontal subgrid scale mixing; HF, destruc-
tion of kinetic energy by the horizontal, subgrid scale mixing; vF, destruction of kinetic energy by the vertical subgrid scale mixing.
Units of exchange, See study S for the mathematical expression of
the transfer term.
joules cm-2 mb-1 day-'; units of energy (in boxes), joules
1 0 .9 b P~z41.3 - 15.5 Pz=498.9
9. SUMMARY AND CONCLUSIONS
The decrease of the horizontal grid size from 500 t o 250
km increases the accuracy of the dynamical computation.
For example, in the low resolution N2OM model, the
fronts are not very sharp and the bands of frontal rain
are wide and diffuse. On the other hand, the surface fronts
in the high resolution N40M model have highly realistic
structures. They are characterized by dense concentra-
tions of isotherms, narrow bands of frontal rain, and
associated cyclone families. Further reduction in the
horizontal grid size seems to be desirable in order to resolve
sufficiently the structure of cyclone families.
The comparison between the at'mospheres with and
without the effects of the selective heating of condensation
indicates that the moist processes are responsible for
sharpening the fronts and reducing the scale of cyclones
4.4
in the lower troposphere. In the dry model atmosphere,
the characteristic scale of the cyclones near the earth's
surface is comparable with that a t the 500-mb level;
whereas in the moist model atmosphere, the scale of
lorn-level cyclones is significantly smaller than that of
the disturbances in the upper troposphere.
According to an harmonic analysis, the dissipation of
eddy kinetic energy by horizontal subgrid scale viscosity
prevails not only at the grid scale but also a t larger scales.
Since the scale a t which the horizontal dissipation is most
effective decreases with decreasing grid size, the eddy
kinetic energy of the large-scale disturbances is dissipated
more effectively in the low resolution NROM atmosphere
than in the high resolution NJOM atmosphere. In the
N2OM atmosphere, the total dissipation, which is the
sum of the horizontal and the vertical dissipation, is
almost equal to the rate of conversion from potential to
kinetic energy at each wave number. Accordingly, there is
1 0 .9 b P~z41.3 - 15.5 Pz=498.9
March 1970 Syukuro Manabe, Joseph Smagorinsky, J .
Hadley cell, Tropics- __. . . . - -. . . . . . -. . . . . . . . .
Ferrell cell, midlatitudes-. . -. . -. -. . . . . - -. - -. .
Polar cell-.. -. . -. - -.. __ - -... ___._ .__ _._. . - -..
Leith Holloway, Jr., and Hugh M. Stone 209
N40M N@M 1 N@D 1 NBD
_____________
153 125 77 67
36 22 71 42
5 2 8 G
Q: H Q ~H
t 6.9 ? 8.6 I 26.8f 8.6
25.9 ? 8.6 400 ? 100 lSO? 50
I
0.9 ? 1.7 19.0 ? 8.6
r k
80 ? 30 3.5 ? 1.7 70 30
4.3 t i.7 15.6? 8.6
F J F I
FIGURE 60.-Box diagram of energetics of the actual atmosphere
compiled by Oort (1964b). QgH, sum of QR, Qc, and H ; for a
further explanation, see figure 59.
TABLE 3.-Intensiiy of meridional circulations (units, 10'8 gm sec-1)
I Model
little net transfer of energy between eddies of different
scales. I n the N4OM atmosphere, the rate of total conver-
sion is somewhat greater than the dissipation rate at
medium wave numbers. The excess energy converted at
medium wave numbers is for the most part transferred
barotropically to very long waves and to the zonal current.
This comparison between the energetics of the two model
atmospheres indicates that the horizontal viscosity tends
to dissipate the kinetic energy of large-scale eddies which
should otherwise be transferred to eddies with lower
wave numbers through the barotropic process. Further-
more, it should be noted that the rate of conversion of
potential energy a t lower wave numbers is much larger
in the N40M atmosphere than in the N2OM atmosphere
which probably results from the difference in the character
of the horizontal dissipation mentioned above. Owing to
the differences in the spectral distributions of dissipation,
conversion, and baro tropic energy exchange, the kinetic
energy is larger and more realistic in the N4OM atmos-
phere than in the N2OM atmosphere a t low as well as
high wave numbers.
Since the scales of horizontal dissipation overlap with
those of conversion from potential to kinetic energy as
indicated above, the horizontal nonlinear viscosity is not
operating in the inertial subrange for which it was
designed. Furthermore, the typical inertial subrange of
umrt
FIGURE 61.-Latitude-height distribution of the rate of generation
of available potential energy by condensation and convection
in the N4OM atmosphere. Units, ergs ern+ sec-1 rnb-I.
GEOPOTENTIAL HEIGHT
CONTOURS
REA OF CONDENSATION
FIGURE 62.-Schematic illustrating a typical developing cyclone
in middle latitudes.
14.40 1.30
FIGURE 63.-Box diagram illustrating the energetics of the strato-
sphere of the N4OM model. Units of energy (in boxes), joules
cm-2 mb-1 (see fig. 59 for details).
three-dimensional eddies should not be realized a t grid
scale in any event because the horizontal mesh interval
is much larger than the equivalent depth of the atmos-
phere. Recently, Leith (1969) proposed a formulation of
a horizontal eddy viscosity based upon the dimensional
analysis of two-dimensional turbulence. Since his version
of nonlinear viscosity is more scale-selective than the one
adopted for this study, the choice of his formulation
should decrease the dissipation of synoptic scale eddies.
Whichever formulation we may use, i t seems to be highly
desirable to increase the resolution of horizontal finite
differencing further. Such an increase not only improves
Pi 0 MONTHLY WEATHER REVIEW voi. 98, No. 3
the accuracy of the horizontal finite differencing but also
separates the scale of subgrid scale dissipation from the
scales of baroclinic instability.
Another alternative is to represent the equation of
motion by spherical harmonics as Blatzman (1960) pro-
posed and systematically truncate the higher wave number
components, thereby avoiding the explicit use of hori-
zontal viscosity. One can show that such systematic
truncation of the series does not deplete kinetic energy
from the system. According to Eraichnan (1967), a two-
dimensional cascade of energy at wave numbers higher
than those of the energy source does not exist when the
energy spectrum has attained the equilibrium distribu-
tion. It was shown that such a cascade is very small in the
N4OM atmosphere. Therefore, the use of orthogonal
functions may be one of the attractive ways t o avoid this
di5culty.
A wajor emphasis of this study was a detailed analysis
of the various components of energy transfer in the moist
model atmospheres. In the N2OM model, the decascade
of eddy to zonal kinetic energy is missing because of the
dominatioh of the horizontal eddy viscosity. On the other
hand, in the N40M model energy is transferred from the
eddy to the zonal kinetic energy in agreement with the
actual atmosphere. It is noteworthy that the energy
conversion from zonal available potential energy also
plays a significant role in maintaining the zonal kinetic
energy of the moist model atmosphere? I n the dry model
atmospheres, the zonal conversion is negative and de-
pletes the zonal kinetic energy. The difference between the
dry and the moist models is due t o the difference in the
intensity of the Wadley cell in the Tropics and the Berrel
cell in middle latitudes.
Another interesting outcome of the energy analysis of
Lhe moist model atmosphere is the large generation of the
eddy available potential energy by the heat of condensa-
tion. I n the dry model atmospheres, the eddy available
potential energy is produced solely by the transfer of
energy from the zonal available potential energy. On
the other hand, in the moist model atmosphere the eddy
generation by the heat of condensation has u magnitude
comparable with Ghe supply of energy from the zonal
available potential energy.
A detailed analysis of the energetics of the N4OM
atmosphere in wave number space reveals essential
differences between the energy budgets of various parts
of the atmosphere. I n the model troposphere, eddy kinetic
energy is produced by the conversion of eddy potential
energy in the range of wave numbers 2 to 8. Some of the
energy thus produced is dissipated, but most of the
remainder is transferred to zonal kinetic energy. I n the
stratosphere, where very long waves predominate, eddy
kinetic energy is generated in the 2 to 3 wave number
range, mainly as a result of the pressure interaction with
* Since the free-slip equatorial wall could affect the intensity of the Hadley cell, it is
desirable t o reevaluate this conclusion by use of a model without such a wall.
the underlying troposphere. Some of the eddy kinetic
energy is then transferred to the zonal kinetic energy by
means of a barotropic nonlinear decascade. It is very
interesting that the magnitude of the transfer toward
higher wave numbers is extremely small as compared with
the magnitude of the transfer toward lorn wave numbers.
In the Tropics, the eddy kinetic energy is produced
mainly by the release of eddy available potential energy
generated by the heat of condensation. Although the rate
of conversion is a maximum at very low wave numbers,
the conversion spectrum extends to very large wave
numbers.
In conclusion, the dynamical structure of the moist
model atmosphere becomes highly realistic as a result of
the reduction of the grid size, which in midlatitudes is
approximately from 500 t o 250 km, although further
increase of the resolution of horizontal finite differences
seems to be desirable.
The time step for integration of the N4OM model was
5 min, half that used in the N2OM and N2OD models. I[io
order to prevent the growth of the computational mode,
the wind, temperature, humidity and pressure fields at
three consecutive time steps were averaged by the weights
0.25, 0.5, and 0.25 every 53 time steps. After such a
smoothing time step, one noncentral time extr2polation
was performed before the normal leapfrog time integra-
tion was resumed.
In spite of this periodic time smoothing, the N4OM
model developed high-frequency external gravity waves
in the Tropics which were not present in the low resolution
model experiments. Apparently, the irregular boundary a t
the Equator excited these surface-pressure disturbances
because they appeared first as meridional bands radiating
poleward from the roughest sections of the boundary. The
reason why these occur only at high resolution has never
been determined.
The most effective technique for suppressing these
gravity waves was to use the so-called Euler backward
time-differencing method (see, for example, Matsuno
1966) periodically in place of the normal leapfrog pro-
cedure until these unwanted disturbances were reduced to
insignificant amplitude. This method consists of a two-pass
procedure in which tentative next-time step values of the
predictands are obtained from which definitive rates of
change are derived for computing the final values by non-
central time extrapolations. Let V represent a typical
predictand, then
av V* =V+ At (first pass)
and
av * ~+l = V+ ~t +
(=- (second pass)
where the superscripts denote the time level, 7 is the time
March 1970 Syukuro Manabe, Joseph Smagorinsky, J. Leith Hclloway, Jr., and Hugh M. Stone 21 1
step number, the asterisk indicates the tentative estimate
of the predictand, and At is the length of the time step.
During the time integration of the high resolution
models, this Euler backward procedure was carried out
every 1,008 time steps for a series of 143 consecutive time
steps. Furthermore, during the application of this method,
the time step was reduced from 5 min to 135 sec because
this At was found by trial and error to be the longest time
step that was effective in smoothing out the gravity waves,
which in the N4OM model had a mean period of about
15 min. While this method mas being used, forecast time
thus advanced at less than one-quarter of the normal speed.
APPENDIX II-BRIEF DESCRIPTION OF THE DRY
GENERAL CIRCULATION MODEL
In view of the frequent reference to the dry model in
the manuscript, a brief description of the dry model is
appropriate. The basic difference between the dry and
moist model is that the former does not contain moisture
processes, that is, transport of moisture, condensation and
evaporation, whereas the latter model does. For example,
FIGURE 64:-Upper half, latitudoheight distribution of the zonal
mean of zonal wind in the N4OD atmosphere (units, m sec-l);
lower half, latitude-height distribution of cddy kinetic energy
in the N4OD atmosphere (units, joules cm-2 mb-I).
the effects of selective heating by the condensation process
and the polemard transport of latent energy are absent in
the dry model but are present in the moist model.
The effect of the moist convection, however, is incorpo-
rated into the dry model in a very crude manner in order
to prevent the lapse rate from becoming too unrealistic. It
consists of adjusting the lapse rate to the moist adiabatic
value wherever the lapse rate exceeds this critical value.
For further details of the structure of the dry model,
see study S.
The initial conditions for the numerical time integration
of the N4OD model is the state of the moist atmosphere
at the 200th (moist) model day. The period of integration
is 70 model days. The last 30 days of this integration
period is chosen for the analysis. Various quantities of the
N4OD model shown in this study represented the average
value for this 30-day period unless we specified otherwise.
The latitude-height distributions of the zonal mean of
zonal mind and eddy kinetic energy are shown in the upper
and lower parts of figure 64. According to this figure, the
intensity of the zonal current in the N4OD model is un-
realistically strong. The eddy kinetic energy of this model
is much larger than that of the N4OM model. This result
partially explains why the eddy transports of the various
quantities in the N4OD atmosphere are very large. (See
study S for the description of the N,OD atmosphere.)
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Batten, E. S., A Model of the Seasonal and Latitudinal Variation
of Zonal Winds and Temperatures in the Stratosphere Above 30
Km, Memorandum RM-4144-PR, The Rand Corporation,
Santa Monica, Calif., June 1964, 28 pp.
Bjerknes, Jakob A., Extratropical Cyclones, Compendium of
Meteorology, 1951, pp. 577-598.
Buch, H. S., Hemispheric Wind Conditions During the Year 1950,
General Circulation Project, Final Report, Part 2, Contract No.
AH19-122-153, Department of Meteorology, Massachusetts In-
stitute of Technology, Cambridge, 1954, 126 pp.
Bushby, F. H., and Timpson, Margaret S., A 10-Level Atmos-
pheric Model and Frontal Rain, Quarterly Journal of the Roval
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Edelman, W., O n the Behaviour of Disturbances in a Baroclinic Chan-
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Weather Service, Offenbach, West Germany, Apr. 1963, 35 pp.
Kraichnan, Robert H., Inertial Ranges in Two-Dimensional Tur-
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Leith, Cecil E., Numerical Simulation of Turbulent Flow, Prop-
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New York, 1969, pp. 267-271.
Lilly, Douglas K., On the Application of the Eddy Viscosity Con-
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No. 123, National Center for Atmospheric Research, Boulder,
Colo., 1966, 19 pp., (unpublished manuscript).
Lilly, Douglas K., Numerical Simulation of Two-Dimensional
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published manuscript).
1417-1423.
pp. 671-672.
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[Received July 2.2, 1969; revised October 17, 19691