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A Proposal for the
Intercomparison of the
Isaac M. Held”
and Max J. Suarez**
Dynamical Cores of Atmospheric
General Circulation Models
Abstract
‘Abenchmarkcalcuation|s proposedtor evaluatingthe dynamical
cores of mosphere general cveuiation modalsindependenty ofthe
physical parameterizations. The test focuses onthe long-term statis
‘ical properties of a fuly developed general circulation; thus, i
pariculaly appropriate fr intercomparing the dynamics used in cl
mato models. To ilusratetheuse ofthisbenchmark, wo very diferent
“atmospheric dynamical cores—one spectral, one fnte iferance—
‘ae compared, tisfound thatthe long term statistos produced bythe
!womodels are very similar, Selected results from these calculations
_are presented inate the intercomparison,
1. Introduction
The careful evaluation and comparison of atmo-
spheric general circulation models (GCMs) is essen-
tial if we are to move methodically toward improved
climate models. A comparison of the climatic simula-
tions produced by various GCMsusing realisticbound-
ary conditions is currently being conducted by the
‘Atmospheric Model intercomparison Project (Gates
1992). While evaluation of full GCMs is essential, the
difficulty of interpreting such intercomparisons is well
appreciated by the modeling community. The number
ofchoices that must be made about poorly understood
parts of the system—particularty in the parameteriza-
tion of subgrid-scale processes—is large enough that
it is difficult to exhaustively compare even closely
related models. Furthermore, important differences
often arise from the details of the implementation or
from the way in which different parameterizations
interact. By the same token, errors in the simulations,
as identified by comparison with observations, are
difficult to attribute to specific modeling assumptions.
“GeophysicalFluldDynamicsLaboratoryINOAA, Princeton University,
Princeton, New Jersey.
“Goddard Space Fight Conter/NASA, Greenbelt, Maryland.
Corresponding authoradaress: Dr. IsaacM. Held, Geophysical id
Dynamics Laboratory/NOAA, Princeton Universiy, P.O. Box 208,
Princeton, NJ 08542.
In final form 17 May 1984.
(©1904 American Meteorological Society
Bulletin of the American Meteorological Society
Indeed, itis possible for models to produce realistic
simulations of some climatic features for the “wrong”
reasons, of for the simulation of some features to
‘worsen when other parts of the model are improved.
A logical reaction to these difficulties to try to think
of GCMs as being decomposed into modules that can
be tested and compared independently. A good ex-
ample of this approach is provided by the Intercom-
parison of Radiation Codes for Climate Models
(ICRCCM) project (Ellingson et al. 1991; Fouquart et
al. 1991), in which the focus is on testing the broad-
band parameterizations used in GCMs against the
more precise, but expensive, line-by-line calculations.
A similar effort is under way comparing land surface
parameterizations (Henderson-Sellers et al. 1992).
We foe! that narrow, well-controlled comparisons like
these are the most productive way to proceed and that
they should be expanded to all aspects of GCMs,
including the dynamics. In this article, we are propos-
ing a way of carrying out such a comparison for what
we refer to as the “dynamical core” of the model.
‘All GCMs solve discrete forms of the equations of
motion for the time evolution of the three-dimensional
flow field and of the thermodynamic state of the air.
There is a need for a careful reexamination of the
standard numerical methods used to discretize these
equations and, given the advent of parallel computer
architectures, for the evaluation of novel approaches.
Williamson et al. (1992) propose a series of tests
using the nonlinear shallow water equations on the
sphere as surrogates for the three-dimensional equa-
tions of motion used in GCMs. They provide analytical
or high-resolution solutions to several initial value
problems, aiming, for example, at testing the ability of
agiven numerical scheme to simulate the propagation
of a planetary wave without loss of amplitude or to
transporta tracer across the pole. While shallow water
solutions provide valuable indications of the perfor-
mange of horizontal discretization schemes, there are
‘two important limitations to this approach: 1) three-
dimensional atmospheric flows, in addition to intro-
ducingproblemsassociated with vertical discretization,
may place distinctive demands on the numerics; and
18252) for climate models, itis important to directly evalu-
ate the long-term statistics rather than focus on the
accuracy of short-term, deterministic solutions. Tests
like those of Williamson et al. are more applicable to
numerical weather prediction models, for which the
relevant trade-offs between accuracy, efficiency, and
conservation properties maybe quite different than for
climate models.
We are attempting to define a set of benchmark
calculations for the evaluation of statistically steady
states produced by the atmospheric dynamical cores
used in climate models. In these tests, we replace the
detailed radiative, turbulence, and moist convective
parameterizations with very simple forcing and dissi-
pation. There are problems with this approach as well.
‘The most obvious is that the true solution is unknown.
We presume that different modeling approaches will
converge to the true solution as resolution is in-
creased, keeping in mind that if convergence is not
Clearly obtained in these idealized problems, it may
not be easy to obtain in realistic GCMs either. Another
problem is that sampling errors, due to very low
frequency variability, may inter-
tere with our ability to accurately
define the statistically steady
state, given the finite time inter-
vals over which the models are
integrated, but once again, this
problemisaiso presentin realis-
tic climate simulations.
‘An additional motivation we
have in proposingtests focusing
on the dynamics is to nudge the
modeling community toward the
creation of modular dynamical
cores that are easy to inter-
change. The importance of us-
ing modular, or “plug-compat-
ible,” codes was discussed by
Kalnay etal. (1989) forthe physi-
‘calparameterizations. We would
like tosee these ideas extended,
as much as possible, to the de-
sign of dynamical cores. In the
spirit of having a free exchange
ofallcodes usedin climate mod-
els, we aremaking publicly avail-
able the Fortran codes for the
dynamical cores used to pro-
duce the results described below.
The first in our proposed se-
ries of benchmark calculations
is described in this report. As an
example, the benchmarkisused
to compare two GCM dynamical
1826
1p, = 1000 mb
=7.292x10" s
cores, one spectral and one finite difference, Both are
closely related to codes that are being used for climate
studies at our laboratories. We present here only a
‘sampling of the two models’ climate statistics, but we
can make available much more complete results to
anyone interested in making a more detailed compari-
son. In a separate study, we will be looking at this
proposed calculation in more depth. In particular, we
will study the behavior of these two models as a
function of horizontal and vertical resolution and dis-
cuss their sensitivity to the choice of dissipation.
2. A first benchmark calculation
In designing the forcing and dissipation, we use
simple Newtonian relaxation of the temperature field
toa zonally symmetric state and Rayleigh damping of
low-level winds to represent boundary-layer friction
Forcing GCMs in this way is relatively common,
especiallyin two-layer models [Hendon and Hartmann,
(1985) and Suarez and Duffy (1992) are two ex-
-k, (ov.
—kr($.0)[T ~Tog(P)]
a 315K -(AT), sin? ¢ (00) Joost 2
2
Po
ky = ky + (ke ~k,)mar{o
ky may{0.2=22)
op
1 day,
(aT),
1004 Jk" K*
6.371 x 10m
Vol. 75, No. 10, October 1994tive equilibrium” T,,, which is a
function oflatitude and pressure.
Thesetemperaturesandthe cor-
responding potential tempera-
tures are shown in Figs. 1a,b.
This radiative equilbriumisgiven
some positive static stability,
relatively large in the Tropics
and decreasing to zero at the
Fic. 1. The upper panels contain the prescribed radiative equlibium temperature (a) and
‘potential temperature (b) distributions. Thelower panels contain 1000-day averages ofthe zonal
‘mean temperature (c) andppotential temperature (d} cistrbuions producedby the G72 grigpoint
mode
amples on the sphere]. Recent examples of its use in
models with more vertical resolution are James and
James (1989) and Yu and Hartmann (1993). Our
specifications are detailed in the box on the opposing
page.
We start with an ideal gas atmosphere over a
rotating spherical surface. There is no topography, in
the sense that the surface is at constant geopotential.
Nothing is said as to whether the flow is or is not
hydrostatic, While most global models assume hydro-
static balance, this is not considered part of the
specification of the problem but rather a modeling
choice. The choice of upper boundary condition is also
left open. The inclusion of a rigid lid at some height or
pressure, for example, isagain considered a modeling
choice. Aside from the forcing parameters, we need to
specify only the gas constant A, the specific heat of air
at constant pressure c,, the acceleration of gravity g,
the radius of the sphere a,, and the total mass of the
atmospherep,/g. The acceleration of gravityis needed
only in a nonniydrostatic model.
The only specified dissipation is a simple linear
‘damping of the velocities. The strength of the damping
k, isa function of «= pip,, where pis the pressure and
p, is the instantaneous surface pressure. We use o
rather than pressure in this expression so that this
“boundary layer" will follow the topography in future
calculations in this series. This damping is nonzero
only in layers near the surface (a > 0.7).
‘Temperatures are relaxed to a prescribed “radia-
Bulletin of the American Meteorological Society
poles. One canthinkof this tropi-
cal static stability as taking into
accountthe effects of moist con-
vection, but this is potentially
misleading, and it is better to
think of it as simply an artifact
that helps minimize the ocour-
rence of gravitational instability
The radiative relaxation time is
also a function of latitude and o.
If one uses a long relaxation
time everywhere, an unrealistic
thincold layer develops near the
surface, particularly in the Trop-
ics. The relaxation time is in-
creased in this region to reduce
this effect, but some vestige of this shallow stable
layer tilremainsin the solutions, Thisis clearly visible
inthe time-mean temperature and potential tempera-
ture distribution (Figs. 1¢,d) produced by the"G72”
finite-difference model (described below). The poten-
tial temperatures also show that the dynamics is
maintaining the static stability well above its radiative
equilibrium value in the extratropics.
No explicit diffusion is included in the specification
of the model. We prefer to think of subgrid-scale
diffusivity as part of the numerical scheme. (For the
sake of well-posedness, one can think of molecular
viscosity and diffusion as being present, but negligibie
on all scales that we can hope to resolve in a global
climate model.) This allows us to treat, evenhandedly,
conservative schemes that require explicit subgrid-
scale mixing, and schemes that are dissipative by
design. By specifying a large diffusivity that produces
‘smooth large-scale solutions, one would be creating a
very different problem from the effectively infinite
Reynolds number problem that we have posed. Our
goal is not simply the accurate simulation of the
evolution of smooth flows but the optimum treatment
of flows that generate motions on all resolved scales
through turbulent cascades, as does the atmosphere.
In the two models described below, we have in-
‘oluded only a very scale-selective horizontal mixing
and have omitted vertical mixing (diffusion or convec-
tive adjustment) altogether. Although gravitationally
Unstable regions do form on occasion, particularly in
1827(ums)
untios
Fi. 2. The zonal-mean zonal wind produced the T63 spectral
‘madeland G72 gridpoint adel. Bothare 1000-day moans. Sioa the
forcing is symmetric about the equator, ferences between the
hemispheres are indicative o sampling erors.
low latitudes, the models can be integrated stably
without enhanced vertical mixing in such regions.
3. A sample intercomparison
To begin the process of developing standards for
dynamical cores of atmospheric climate models, we
performed long-term integrations with our own mod
els. The models use very different discretization meth-
ods and were coded independently by the two authors.
a. The models
The spectral model is a standard hydrostatic, o
coordinate, serni-implicit, spectral transform model, in
the vorticity-divergence form described by Bourke
(1974), The transform grid is chosen to ensure alias-
free computation of quadratic products, in the usual
way. The vertical differencing uses the simplest cen-
tered differences, and the hydrostatic equation is
integrated analytically assuming that temperature is
constant within each layer. This differencing is not
energy conserving. There are 20 vertcallevels, equally
spaced in sigma, with the top of the model formally at
zero pressure. A leapfrog scheme is used for time
1928
stepping, using the time filter described by Robert
(1966) to control the computational mode. The hori-
zontal mixing of vorticity, divergence, and tempera-
ture takes the form of a Laplacian raised to the fourth
power, with the strength set so that the e-folding time
forthe smallestwave in the system is always.0.1 days.
‘The truncation is triangular. We made integrations at
four resolutions: 721, T30, T42, and T63, where the
numeral refers to the maximum number of zonal
waves present. In this note, we present results only
from T63. The code for this model is available from
Isaac Held (e-mail: in@ gfdl.gov)..
The finite-difference model is also hydrostatic and
uses o coordinates, with 20 layers equally spaced in
sigma. The vertical differencing is that proposed by
‘Arakawa and Suarez (1983). Alatitude-longitude grid
with Arakawa’s C-grid staggering is used for the
horizontal discretization. The horizontal finite-
differencing scheme is second order in all respects
‘except the horizontal advection of vorticity, which is
fourth-order accurate for nondivergent flow, reducing
to the fourth-order Arakawa (1966) Jacobian in this
case. A Fourier filter is applied to all tendencies to
damp short zonal scales poleward of 45° latitude. An
explicit leapfrog time step is used, with the pressure
gradient averaging suggestedby Brown andCampana
(1978). The computational mode of the leapfrog is
controlled in exactly the same way as in the spectral
model. An eighth-order Shapiro (1970) filter controls
gridpoint noise. The filter damps the 2A wave with an
effective e-folding time of 1.5 h. As with the spectral
model, we have made integrations at four resolutions:
6° latitude x 7.5° longitude, 4°x 5°, 3°x 3.75, and 2°
x 25°. We will show results only for the highest
resolution, which we label G72, the numeral referring
to half of the number of grid points around a latitude
Circle. The finite-difference dynamical core is de-
scribed in Suarez and Takacs (1995). That report and
the code are available from Max Suarez (e-mail
suarez@nino.gsfc.nasa.gov)
b. Results
All cases presented were integrated 1200 days.
The model's climate in each case is obtained by
averaging the last 1000 days of integration. The
integration of the spectral model was started from an
isothermal state atrest, with some small perturbations
added to break the symmetry. The gridpoint integra-
tion was started from an earlier run of a lower-resolu-
tion model that had already equilibrated to the forcing,
By discarding 200 days, we are reasonably certain
that we eliminate significant differences due to the
different initialization schemes.
Aspects of the climates produced by the two mod-
els are displayed in Figs. 2-4: the zonal-mean zonal
Vol. 75, No. 10, October 1994\\
}
\
1 iM
oo»
Fr 8. Asin Fig. 2 but for the eddy variance ofthe temperature.
wind (Figs. 2) as a function of latitude and sigma; the
eddy temperature variance (Fig. 3), also as a function
of latitude and sigma; and the zonal spectrum of the
eddy zonal wind (Fig. 4), as a function of zonal
wavenumber and latitude. Since the forcing is sym-
metric about the equator, we can use the difference
between the climates of the two hemispheres for a
quickestimate of sampling errorin the 1000-day means.
We see from these figures that our simple forcing
and dissipation produce a reasonably realistic zonal
mean circulation. A single jet is generated with maxi-
mum strength of roughly 30 ms“ near 45° latitude.
The jet closes off, with a well-defined reversed shear
above o = 0.2. The surface westerlies reach almost
&8ms"'near45° latitude. There are equatorial easterlies
in the model stratosphere. The eddy temperature
variance shows two midiatitude maxima, one in the
lower troposphere and another above the tropopause.
‘An unrealistic feature of the results is the penetration
of the temperature variance near the surface well into
the Tropics. The spectrum of the eddy zonal winds has
two peaks at zonal wavenumber 5 on the flanks of the
storm track (with the peak on the polar side somewhat
stronger), a third peak at wavenumbers 1 and 2 at the
center of the track, and yet another peak, associated
with cross-polar flow, in wavenumber 1 at the pole.
Bulletin of the American Meteorological Society
Fra. 4.Asin Fig.2butforthe vertically averaged zonal spectra of
the eddy variance of zonal wind
These figures indicate an impressive degree of
agreement between the two models, despite their very
different numerical algorithms. There are large differ-
ences in the eddy temperature variance in the strato-
sphere, which we attribute to the different vertical
discretizations and the coarse stratospheric resolu-
tion. Another difference, not evident in the figures
shown, is that the grid model is noisier than the
spectral model near the pole. After completing these
integrations, we found this noise couldbe significantly
reduced but not eliminated by a simple modification of
the treatment of the momentum equations at the pole.
This change has no noticeable effect on other statis-
tics. Details ofthe finite differencing at the poles are
given in Suarez and Takacs (1993)
‘The agreement between these particular spectral
land gridpoint models should not be interpreted as
implying that the true solution to our problem has been
‘oblained. Additional calculations, desoribedelsewhere,
show that the climates of both models are sensitive to
resolution and have not yet converged at the resolu-
tion presented here. The solutions may also be more
sensitive to the choice of parameters and to the model
formulation than the agreement between these two
cores suggests. Asan example, the same calculations
repeated with a second-order version of the finite-
1929difference dynamical core produce a general circula~
tion quite different from the one presented here: the
jets are narrower and shifted equatorward, and the
surface winds and eddy momentum transports are
considerably weaker. Thus, it may be that the current
parameters are near a transition in the flow regime.
(Hints of the existence of such a transition are also
seen when the surface friction is varied.) If the flow
regime is sensitive to the model parameters, this test
can be a very sensitive indicator of the quality of the
mode's numerics, although it may be necessary to
study the solution as a function of a parameter to
understand the differences between models
4. Future directions
We are interested in the development of tests or
standards for the intercomparison of the dynamical
cores of atmospheric climate models and have de-
cided that a test should satisfy two conditions to be
Useful: it should isolate the dynamics as much as
Possible from the complexity of the physical
Parameterizations, particularly those for moist con-
vection, clouds, and the planetary boundary layer, and
it should evaluate the long-term statistical properties
of a realistic three-dimensional global circulation. We
feel the benchmark calculation we have proposed
satisfies these requirements and will be a good start-
ing point for such an intercomparison. We do recog-
nize that there are deficiencies in the current formula-
tion. For example, as vertical resolution is increased,
some vertical mixing may be required, and it is not
clear what lower boundary condition is most appropri-
ate given the current drag formulation. Despite these
misgivings, the circulation produced is sufficiently
meteorological, and the solutions we have obtained
with two very different models are sufficiently similar
that we feel it provides a useful test. We hope other
groups wil join in this effort by repeating the calcula
tion with their own models, experimenting with other
simple forcing functions, and sharing their results.
The statistics presented here only touch the sur-
face of interesting quantities for intercomparison. Sev-
eralthatinterestusare measures of coherentbarociinic
wave packet structure, the amplitude and spectral
width of the westward propagating external mode
resonances, and the gravity wave activity in the model
stratosphere. The Lagrangian statistics of the flow are
most easily probed by adding passive tracers, which
can then be used to evaluate transport algorithms as
well. It should also be of interest to test the model's
treatment of the pole by rotating the coordinate sys~
tem or the radiative equilibrium temperature so that
the numerical pole resides in the storm track and by
1830
checking for asymmetries in the resulting climate. We
have begun tests with a second benchmark calcula-
tion, in which an idealized midiatitude mountain is
introduced. This calculation should be useful for com-
paring alternativesto sigma coordinates andor evalu-
ating the sensitivity of low-frequency planetary wave
activity to the choice of model.
‘Acknowtedgments. NUS was supported in part by the Global
“Atmospheric Modelingand Analysis Program ofthe Ofce of Mission
to Pianet EarthiNASA Headquarters and by the Earth and Space
‘Science Applications Project of the NASA HPCC Program,
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Vol. 75, No. 10, October 1994