Numerically Converged Solutions of the Global Primitive ......Solutions of the dry, adiabatic, primitive equations are computed, for the ﬁrst time, to numerical convergence. These

NOVEMBER 2004 2539P O L V A N I E T A L .

q 2004 American Meteorological Society

Numerically Converged Solutions of the Global Primitive Equations for Testing theDynamical Core of Atmospheric GCMs

L. M. POLVANI

Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University,New York, New York

R. K. SCOTT

Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York

S. J. THOMAS

Scientific Computing Division, National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 5 October 2002, in final form 22 March 2004)

ABSTRACT

Solutions of the dry, adiabatic, primitive equations are computed, for the first time, to numerical convergence.These solutions consist of the short-time evolution of a slightly perturbed, baroclinically unstable, midlatitudejet, initially similar to the archetypal LC1 case of Thorncroft et al. The solutions are computed with two distinctnumerical schemes to demonstrate that they are not dependent on the method used to obtain them.

These solutions are used to propose a new test case for dynamical cores of atmospheric general circulationmodels. Instantaneous horizontal and vertical cross sections of vorticity and vertical velocity after 12 days,together with tables of key diagnostic quantities derived from the new solutions, are offered as reproduciblebenchmarks. Unlike the Held and Suarez benchmark, the partial differential equations and the initial conditionsare here completely specified, and the new test case requires only 12 days of integration, involves no spatial ortemporal averaging, and does not call for physical parameterizations to be added to the dynamical core itself.

1. Introduction

The development of atmospheric general circulationmodels (AGCMs) is an important task among the currentefforts to understand and predict the climate. At theheart of every AGCM is the so-called dynamical core,the component of the model that deals with the nu-merical solution of the dry, adiabatic, primitive equa-tions. While in many ways a much simpler task thanthe design of other model components, the search forever more efficient numerical solvers for the primitiveequations is an important area of research, often neces-sitated by the continual evolution of computer archi-tectures.

In order to build new reliable dynamical cores andquantitatively evaluate their accuracy, it is important totest their solutions against known ones. The first stepin this direction was taken by Williamson et al. (1992,

Corresponding author address: L. M. Polvani, Department of Ap-plied Physics and Applied Mathematics, Columbia University, 500West 120th St., Arm. 216, New York, NY 10027.E-mail: [email protected]

hereafter W92), who proposed a set of ‘‘standard tests’’for the shallow-water equations in spherical geometry.While useful in their own right, the tests in the W92set present three major limitations. First, they cannot beused to validate AGCM dynamical cores directly, sincethe shallow-water equations lack the dependence on avertical coordinate. Second, most of the flows in theW92 tests are unrealistically simple compared to thekind of flows that need to be computed with typicalAGCMs. Third, and most importantly, in recent yearsit has been shown that the most dynamically interestingtests in W92 (tests 5 and 6) are in fact very difficult tocompute accurately; this severely limits their practicalutility. For a more complete discussion, see Galewskyet al. (2004).

A second important step in the development of testcases for dynamical cores was the proposal of Held andSuarez (1994, hereafter HS94). They suggested addinganalytically specified parameterizations of two simplephysical processes (Newtonian relaxation and surfacedrag) to a dynamical core, and then comparing the time-mean, zonally averaged wind and temperature fields thatresult from a 1000-day integration. The averaging over

2540 VOLUME 132M O N T H L Y W E A T H E R R E V I E W

such a long integration period is dictated by the desireto evaluate the long-term statistics of the computed cir-culation, an important objective for any AGCM. Al-though extremely idealized, the HS94 system is capableof reproducing key features of the tropospheric circu-lation and hence provides a realistic benchmark for thecomparison of dynamical cores.

One drawback of the HS94 test case is that the quan-tities to be compared involve both temporal and spatialaveraging. One important application of dynamical coretest cases is the debugging of new cores, and it is easyto imagine how subtle programming errors might bemasked by the temporal and spatial averaging. Anotherdrawback of the HS94 test case is that the authors didnot show actual numerical convergence of their solu-tions, although this issue has been partially addressedby Boer and Dennis (1997). A final minor drawback ofthe HS94 test case is the fact that new physical param-eterizations need to be added to the dynamical coreitself; ideally, one would want to test an AGCM dy-namical core by simply switching off all physics witha flag, and then integrating the dry, adiabatic, primitiveequations from a given initial condition, without theneed to add more code.

In order to address these shortcomings, we here pro-pose a new test case that involves a short integration ofthe unforced primitive equations, and we accuratelycompute the solutions to be used as benchmarks. Unlikethe HS94 test case, the new test case is a simple initialvalue problem and consists of solving a precisely spec-ified set of equations (the primitive equations), with ananalytically specified initial condition (a baroclinicallyunstable midlatitude jet), for a period of 12 days fol-lowing a small initial perturbation. In order to constructbenchmark solutions, we compute the evolution of theflow to numerical convergence. To the best of ourknowledge, this is the first set of published solutions ofthe time-dependent, atmospheric, primitive equations inspherical coordinates for which numerical convergencehas been attempted or achieved. Furthermore, we com-pute solutions with two completely distinct numericalschemes, thereby demonstrating that the solution to thenew test case is model independent and should thus bereproducible with any future scheme. We provide bothsnapshots of the actual fields, plots of key norms, andtables of a few diagnostic quantities as referencesagainst which solutions from future dynamical coresmay be contrasted.

From this it should be clear that the philosophy ofthe new test case presented here is fundamentally dif-ferent from the one originally proposed by HS94 andcommon to most ‘‘model intercomparison’’ tests. Inthose, the goal is to set up different models so as tocompute similar (if not identical) physical evolutions,and then compare the output of the different modelswith one another. For such tests, no unique or exactsolution is known or believed to exist, and the exerciseconsists in contrasting the features produced by the dif-

ferent models, presumably in an effort to determinewhich ones ‘‘do better’’ than the others.

However, for the development, debugging, and testingof dynamical cores, we suggest that the emphasis beplaced on the mathematics and not the physics. To thedegree that an identical set of partial differential equa-tions is agreed upon (e.g., the primitive equations), andthat identical initial and boundary conditions are used,any consistent numerical scheme ought to produce thesame numerically converged solution. This is the pur-pose of the present paper: to provide one such solutionto assist in the testing of future dynamical cores. Weview this test as complementary to the HS94 bench-mark. In practice, we envisage our new test case beingperformed first, to ensure that a newly developed coreis able to reproduce known solutions to within agreedtolerances over short-time integration periods; once thistest is successfully passed, an HS94 integration can beperformed to ascertain the long-time statistics of themodel.

The paper is organized as follows. In the next sectionwe carefully describe the differential equations and theinitial conditions to be used for the new test case. Insection 3 we present the evolution of the flow over a12-day period of integration, computed with a familiarpseudospectral model, and demonstrate how numericalconvergence is achieved. In section 4, we compute so-lutions to the same problem with a different numericalmodel, based on the spectral-element method, and showthat the numerically converged solutions thus obtainedare identical to those obtained with the pseudospectralmodel (up to a certain number of digits). In section 5,we discuss the effect of hyperdiffusion on the solutions,stressing the fact that identical differential equationsmust be integrated if one is attempting to reproduce thesolutions presented here. We conclude with a summaryof how our solutions can be used as a new test case,followed by a brief discussion.

2. Test case specifications

In this section we describe in detail the differentialequations and initial conditions to be used for the newtest case. It is important to note that, unlike the HS94test case, we are aiming for complete reproducibilityhere. Hence, we stress that the specifications in thissection need to be followed exactly if one is attemptingto reproduce the solutions presented in the next section.

a. The model equations

In order for this test to be of practical applicabilityto the greatest number of atmospheric models, the dif-ferential equations to be integrated were chosen to bethe dry, adiabatic, primitive equations in spherical co-ordinates, as these are at the heart of the vast majorityof present-day AGCMs. For simplicity, we confine ourattention here to the case in which s is used as the


FIG. 1. The initial zonal wind from (5) with contour intervals of5 m s21 and the balanced potential temperature from (10) with contourintervals of 5 K (the dashed line is the 300-K surface). The dash–dotted lines are the zonal winds from Thorncroft et al. (1993).

vertical coordinate. We recognize that the majority ofcomprehensive AGCMs use a hybrid vertical coordinate(i.e., one involving both s and pressure); in that case,setting the pressure coefficients to zero would be aneasy way of reproducing the solutions presented below.

Although these differential equations are well known,we reproduce them here for the sake of clarity, repro-ducibility, and completeness. The horizontal velocityfield u 5 (u, y), the temperature T, and the surfacepressure ps, satisfy the following four prognostic equa-tions:

du RT21 f k 3 u 1 =F 1 =p 5 n¹ usdt ps

dT kT22 v 5 n¹ T

dt sps

1]p1 = · (p u) ds 5 0, (1)E s]t 0

where = contains only the horizontal derivatives, s [p/ps, the intermediate variables F and v (the geopoten-tial and pressure vertical velocity) are computed diag-nostically from the definitions

s

F 5 2R T d(lns9) and (2)E1

s

v 5 su · =p 2 = · (p u) ds9, (3)s E s

0

and the material derivative is defined by

d ] ]5 1 u · = 1 s . (4)

dt ]t ]s

Our notation is completely standard and, for further de-tails, the reader is referred to Durran (1999), where theseexpressions are derived.

For this set of equations, the appropriate boundaryconditions are 5 0 at s 5 1 (the ground) and at ss5 0 (the ‘‘top’’ of the atmosphere). Note that we haveexplicitly included diffusive terms on the right-handside of the momentum and temperature equations in (1).An explanation for the need to include these terms isgiven in section 2c below.

b. The initial conditions

In order to meaningfully test an atmospheric dynam-ical core, it seems reasonable to require that it be ableto compute the short-time evolution of a nontrivial ini-tial condition, that is, one that represents a physicalevent of importance for the atmospheric circulation. Wecould think of no more obvious choice than the for-mation of synoptic-scale eddies via baroclinic instabil-ity. As such eddies are well known to play an essentialrole in the circulation of the atmosphere, it would seem

that any decent dynamical core ought to be able to ac-curately compute their development.

The initial conditions for the new test case thus con-sist of a simple zonal flow, representing a typical mid-latitude tropospheric jet, to which a small perturbationis added to induce the development of baroclinic insta-bility. Both the basic flow and the initial perturbationare analytically specified, allowing the complete initialcondition to be reproducible in testing future dynamicalcores.

The basic flow, illustrated in Fig. 1, is constructed tobe similar to the one used in the study of Thorncroft etal. (1993). This is easily accomplished by letting thezonal velocity u be a simple product of a function oflatitude (f) and a function of pressure (p)

3 2u sin (pm )F(z) for f . 00u(f, p) 5 (5)50 for f , 0,

where m [ sinf, z [ 2H log(p/p0), and the verticalstructure F(z) is taken to be of the form

1 z 2 z pz03F(z) 5 1 2 tanh sin . (6)1 2 1 2[ ]2 Dz z0 1

With the numerical values of the parameters u0, z0, Dz0,z1, H, and p0 given in Table 1, the above expressionsreproduce a zonal wind field that, below 200 hPa, isnearly identical to the one labeled LC1 in Thorncroftet al. (1993), as can be seen in Fig. 1. Of course, weset the meridional velocity y 5 0.

The temperature T is initialized by combining themeridional momentum equation,


TABLE 1. Numerical values of the parameters to be used for thenew test case.

Parameter Value Units

gaVRkcp

9.8066.371 3 106

7.292 3 1025

2872/7R/k

m s22

m1 s21

J kg21 K21

J kg21 K21

p0

Hu0

z0

Dz0

z1

105

7.345022

530

Pakmm s21

kmkmkm

Tl0

f0

abn

10

p /41/31/6

7.0 3 105

Kradiansradians

m2 s21

]F(af 1 u tanf)u 5 2 , (7)

]f

and the hydrostatic balance relation (equivalent to 2),

]F R5 T, (8)

]z H

to obtain

]T ]u215 2HR (af 1 2u tanf) , (9)

]f ]z

where f 5 2V sinf. Again, the precise numerical valuesof R, a, and V are given in the Table 1. Meridionalintegration of this equation yields the expression for theinitial temperature

f ]T(f9, z)T(f, z) 5 df9 1 T (z). (10)E 0]f9

The constant of integration T0(z) is chosen so that, ateach level z, the global average of T(f, z) is identicalto TUS(z), the U.S. Standard Atmosphere, 1976 (COESA1976) temperature. For easy reference, and to ensurereproducibility, we explicitly list in the appendix thenumerical parameters that are needed to compute TUS(z).Finally, since the initial winds in (5) are identically zeroat the ground, no surface geopotential needs to be spec-ified to balance the initial condition.

The initial, unperturbed temperature profile resultingfrom (10) is shown in Fig. 1. To construct it, the de-rivative on the right-hand side of (9) is worked outanalytically, and thus the integrand in (10) is evaluatedexactly. Only the latitudinal integral needs to be per-formed numerically, but this can be done very accu-rately. We have used Gaussian quadrature, and havefound that a mere 100 Gauss–Legendre points for theintegral in (10) are sufficient to produce an initial bal-anced T that is accurate to machine precision.

Because our initial zonal winds are very flat near theequator [u(f) } f6 for 0 , f K 1], as demonstratedby the nearly horizontal isentropes, the initial conditionin Fig. 1 is stable with respect to symmetric instability.Furthermore, we have computed local Richardson num-bers for our initial flow, and they are everywhere muchlarger than 1/4; this indicates that it is also stable toKelvin–Helmholtz instabilities. Finally, we havechecked that N 2 is everywhere positive, and thus theflow is stable to dry convective instability as well. Ofcourse, our initial condition is baroclinically unstable:this is precisely what we are trying to set up.

In order to initiate the baroclinic instability, the zonalflow just described is perturbed by adding to the basictemperature profile a function T9 in the form of a lo-calized bump, centered in the midlatitudes and on theGreenwich meridian, specifically

l 2 l f 2 f0 02 2ˆT9(l, f) 5 T sech sech1 2 1 2a b

for 2p , l , p, (11)

where l is longitude. Using an amplitude T 5 1 K, thisperturbation is applied identically at all pressure levels(the numerical values of the other parameters are givenin Table 1). Note that we have not followed Thorncroftet al. (1993) who perturbed the flow with the most un-stable linear mode; this procedure requires the cum-bersome computation of the linear modes, which is sure-ly something one would like to avoid unless necessary.Also, we have refrained from perturbing the flow witha single (or a couple) of zonal wavenumbers, such asm 5 5 or 6; while this is often done in the context ofspectral models, it is not as natural for finite-differenceor other spatial discretizations, and we have tried toavoid an initial condition that might bias or skew thesolution in favor of a particular scheme.

The specification of the initial condition is completedby the choice of a uniform initial surface pressure ps 5p0.

c. The need for explicit diffusion

Having discussed the physical characteristics of theinitial condition, we now return to discussing the needfor the explicit diffusion terms in (1). It is well knownthat the initial conditions we have chosen lead to thedevelopment of sharp fronts over short periods of time(i.e., on the order of days). While we are aware of noformal proof, for all practical purposes the solution ofthe inviscid equations develop features that look likefinite-time singularities. Therefore one cannot computethe solution of inviscid equations for very long givenour initial condition.

It may be argued that the formation of fronts, inas-much as it precludes the integration of inviscid equa-tions, disqualifies the initial condition we have chosen.However, we contend that our choice of initial condition


is, in fact, extremely relevant to climate modeling. First,baroclinic eddies are primary agents of temperature andmomentum transport in the atmosphere; hence onewould like to make sure that a given AGCM is able tocompute them accurately. Second, the formation of steephorizontal gradients is one of the distinguishing featuresof high Reynolds number, nearly two-dimensionalflows, such as large-scale atmospheric flows. Hence ourchoice of initial condition is highly realistic and of im-mediate practical interest for atmospheric modeling.

The choice of a simple diffusive operator to dissipatesubgrid-scale processes is dictated, mainly, by the factthat it is easy to implement with most numericalschemes. In principle, other forms of diffusion couldhave been chosen, but most are specialized to specificschemes, or difficult to implement in general. For in-stance, ¹2n hyperdiffusion (with n 5 2 or n 5 3) actingon the vorticity/divergence equations is popular withspectral models, but it is very difficult to implementwith finite-difference or spectral-element schemes forwhich the horizontal velocity is the prognostic variable.In section 5 we demonstrate how the choice of diffusionhas a profound effect on the solutions. For the momentwe simply caution the reader that, if the numerical modelbeing tested does not solve the primitive equations withthe diffusion terms exactly as specified in (1), it is un-likely that the computed solutions will reproduce theones presented in this paper.

Before presenting the actual solutions, a word is need-ed as to the choice of the diffusion coefficient n. Inatmospheric modeling studies, it is customary to choosethe value of n to be as small as possible for a givenhorizontal resolution, and to progressively decrease thatvalue as the resolution is increased. This is, for instance,the procedure adopted by Boer and Denis (1997). In thepresent context, however, we are trying to demonstratethe convergence of our numerical solutions for a specificset of partial differential equations. If the value of nwere changed as the resolution was increased, we wouldeffectively be changing the equations themselves, andhence would not expect convergence. In other words,if we were to decrease n with increasing resolution, wewould be trying to converge to the solution of the in-viscid equations. As already argued, such solutions arelikely to be singular, and thus not a good choice for atest case. Hence we need to keep the value of n fixedas the resolution is increased. For the new test case, wehave used the value 7.0 3 105 m2 s21.

A final caveat: before computing the evolution of theflow, it is essential that the dynamical core be rid of allfudges (e.g., filters, fixers, smoothers, etc.) that are com-monly found in AGCMs and not always explicitly doc-umented. While these may be necessary for climatemodeling, the inclusion of such procedures clearlyamounts to solving different sets of equations than theones specified here, and will thus likely lead to differentresults.

3. Numerically converged solutions

Having described how the new test case is to be setup, we now discuss how we have obtained numericallyconverged solutions. All the results in this section werecomputed using the Geophysical Fluid Dynamics Lab-oratory Flexible Modelling System (FMS) Spectral Dy-namical Core (using the ‘‘Galway’’ release version),hereafter referred to as GFDL-SDC. In a nutshell,GFDL-SDC solves the primitive equations in s coor-dinates, using a pseudospectral transform method in thehorizontal, a Simmons–Burridge finite-difference meth-od in the vertical, and a Robert–Asselin filtered semi-implicit Crank–Nicholson/leapfrog scheme for time in-tegration. These techniques are completely standard.

The evolution of the temperature field at the surface,1

for the first 12 days,2 is illustrated in Fig. 2. While thisfigure was constructed from a calculation using 20 ver-tical levels (equally spaced in s), a horizontal trape-zoidal truncation at wavenumber 341 (commonly de-noted as a ‘‘T341L20’’) and a time step of 150 s, wesubmit that it represents the numerically converged so-lution of (1), in the sense that this figure will not change(a) if the vertical, horizontal, or temporal discretizationsare further refined, and (b) if it is computed with adifferent numerical scheme at sufficient resolution. Wenext demonstrate the first fact. The second one is dis-cussed in the following section.

First, let us consider the horizontal resolution. In or-der to show that the solution in Fig. 2 is numericallyconverged, we plot in Fig. 3 a canonical quantity, the‘‘eddy kinetic energy’’ (EKE) as a function of time, forfive different horizontal resolutions: T21, T42, T85,T170, and T341, all with 20 vertical levels. The timesteps used for these computations are 2400, 1200, 600,300, and 150 s, respectively. Following Thorncroft etal. (1993), EKE is defined as follows:

1 12 2EKE(t) 5 [(u 2 u) 1 (y 2 y ) ]r dV, (12)E24pa 2

where r is the density, overbars denote zonal averages,and the volume integral is taken over the entire atmo-sphere. The fact that all the curves are basically super-imposed might suggest that even a very low T21 res-olution might be enough to compute accurate solutions.This, however, is not the case, as illustrated next.

1 We use the term ‘‘surface’’ to indicate s 5 0.975, which is verynearly the value of s at our lowest model level for a 20-level cal-culation. Since we are using a Simmons–Burridge (1981) scheme,the model levels sk are located at values logsk 5 (sk11/2 logsk11/2 2sk21/2 logsk21/2)/(sk11/2 2 sk21/2) 2 1, where sk11/2 are the level in-terfaces. Thus, for a 20-level model, with levels of equal s thickness,the lowest model level is located at s 5 exp[20.95 log(0.95)/.05 21] ø 0.974893147. . . To avoid inflicting cruel and unusual punish-ment onto those wishing to reproduce our results, we extrapolate thefields onto s 5 0.975 using a linear extrapolation.

2 At the behest of a concerned reviewer of the original manuscript,we add that we are using here the solar definition of day (i.e., 1 day5 86 400 s) as opposed to the sidereal definition.


FIG. 2. The evolution from (a) day 4 to (e) day 12 of surface (s5 0.975) temperature. The contour interval is 2.5 K. This solutionwas computed with GFDL-SDC at T341L20 resolution.

FIG. 3. The time evolution of EKE, as defined in (12), for fivecalculations of the GFDL-SDC, with 20 vertical levels and increasinghorizontal resolution.

In Fig. 4, the surface vorticity z at day 12 is plottedfor the five resolutions from T21 to T341. At T42 z isstill rather noisy, while at T85 it is nearly indistinguish-able from the one at T170. One may thus consider theT85 solution as numerically converged for most—if notall—practical purposes. It should also be clear that thereis nothing general about the fact that convergence is hereachieved around T85. This value is only applicable tothis test case and is a direct consequence of our choiceof initial condition and of the value of the diffusivity n.

Finally, contrasting Figs. 4 and 3, one may correctlyconclude that EKE is a highly inappropriate indicator

of numerical convergence. Ideally, one would like toconstruct error norms that are able to bring out differ-ences that the eye is not able to detect from a contourplot. In the case of EKE, precisely the reverse occurs:the EKE suggests convergence while the contour plotstell the opposite story. This implies that more carefulmeasures are needed to establish numerical conver-gence.

To this end we have computed several norms relatingto the vorticity field. We wish to stress that z is not adifferentiated quantity, but one of the prognostic vari-ables in the GFDL-SDC [as well as in the CommunityClimate Model Version 3 (CCM3) and many other mod-els]. It is the prognostic variables themselves that needto be looked at in order to establish numerical conver-gence, and not smoother integrated quantities (e.g., thegeopotential) as is often done. Since both T and ps arerelatively smooth fields, we concentrate on z as a sen-sitive test of convergence (the divergence d and theclosely related vertical velocity v are discussed below).

In Fig. 5a, we show the L2 norm of z at the surface(s 5 0.975). This norm is defined as follows:

p/21|z | (s, t) [ cosf df2 E5Ï4p 2p/2

1/22p

23 dl[z(l, f, s, t)] (13)E 60

and has units of vorticity. Perhaps surprisingly, in viewof Fig. 4, plots of | z | 2 at the surface are qualitativelysimilar to those of EKE, in that curves for all five res-


FIG. 4. The surface (s 5 0.975) vorticity, at day 12, for five dif-ferent horizontal resolutions ranging from (a) T21 to (e) T341. Con-tours from 27.5 to 7.5 3 1025 s21, in steps of 1 3 1025 s21; negativevalues are blue, and positive values are red. Results from GFDL-SDC, with 20 vertical levels.

olutions are superimposed. Again, this suggests that theaveraging involved in the computation of this normmasks some of the noisy features that are apparent inthe actual plots of the vorticity field.

To apply a more stringent convergence test, we com-pute the L` norm of z, defined by

| z | (s, t) [ max | z(l, f, s, t) | ,` (14)

(the maximum is over all latitudes and longitudes) and

plot its surface evolution in Fig. 5b. It is clear from thisfigure that the T21 and T42 calculations are noticeablydifferent from their higher-resolution counterparts, asthe contour plots in Fig. 4 attest.

To bring out even more differences among the low-and high-resolution calculations, it is necessary to com-pute higher derivatives. This is illustrated in Fig. 5c,where the L` norm of the gradient of z (more preciselyof its magnitude) is plotted. Note that the T42 valuesare quite far from the converged values (even the T85results are not completely identical to those at higherresolution). This is perhaps worrisome, given the dy-namical importance of vorticity gradients for the prop-agation of Rossby waves (see, e.g., Scott et al. 2004).Computation of higher and higher derivatives, of course,will show differences at any resolution; hence we seelittle point in carrying this exercise further.

For readers who might be interested in more tradi-tional measures of convergence, we offer in Fig. 6 thel2 relative error norm of the differences between thelower resolutions and the T341 solution. Following Ja-kob-Chien et al. (1995), we define this as

|z 2 z |T 2l (z) 5 , (15)2 |z |T 2

where the subscript T indicates the T341 solution. Inorder to compute the numerator of l2(z) with a maximumof accuracy, we first compute the Fourier/Legendre co-efficients of z, then pad the coefficients up to T341 withzeros, then transform back onto a T341 Gaussian grid,and finally compute the integral of the difference usingGaussian quadratures on the T341 grid. It should beclear from Fig. 6 that our numerical solutions are con-verging without problems, although the rate of conver-gence does not appear to be exponential.

In Fig. 7, a log–log plot of l2(z) versus the numberof degrees of freedom (DOFs) at day 12, it can be seenthat the convergence rate with respect to the T341 so-lution is in fact algebraic. The dotted line has a slopeof 20.955. We suspect that the algebraic convergenceis due to the nature of our initial conditions, that is, thefact that not all derivatives of u as defined in (5) arecontinuous. A more careful choice of initial conditionswould lead to exponential convergence. This, however,would change the problem definition significantly (andis left for future work). For the time being, we submitthat Figs. 6 and 7 adequately demonstrate that our nu-merical solutions are converged with horizontal reso-lution.

Second, we address the question of vertical resolu-tion. Surprisingly enough, convergence is more easilyachieved in that case. In dealing with the vertical di-rection, it will prove useful to monitor a quantity thatone might expect to be more sensitive to vertical dis-cretization (rather than the vorticity z just discussed).We have chosen to use the vertical velocity v, definedin (3), as it is closely related to the divergence. Often


FIG. 5. Vorticity norms as a function of time, computed for the GFDL-SDC solution in Fig. 4. Each frame shows five calculations ofincreasing horizontal resolution, all with 20 vertical levels. The three norms are precisely defined in the text.

FIG. 6. Norm of the difference to the T341 solution, as defined in(15), computed with GFDL-SDC. All solutions, including the one atT341, are computed with 20 vertical levels.

FIG. 7. Norm of the difference to the T341 solution, as defined in(15), at day 12, as a function of the DOFs. The four points correspond,from left to right, to the T21, T42, T85, and T170 solutions, respec-tively. The dashed line is a linear fit and has a slope of 20.955.

found to be rather noisy in AGCMs, the values of vare crucially important to coupling the dynamical coreto many physical processes, for example, convectiveparameterizations.

Using the horizontally converged resolution T85, wehave performed five calculations with 10, 20 40, 80,and 160 levels, respectively. Noting that all the inter-esting action is concentrated in the midlatitudes, wherethe baroclinicity is originally strongest (cf. Fig. 2), weplot vertical cross sections of v versus longitude at f

5 458N and t 5 12 days. Results3 obtained using theGFDL-SDC are shown in Fig. 8 for the different numberof levels.

An alternating sequence of updrafts and downdrafts

3 As the Gaussian grid does not contain a point located at precisely458N, we perform a linear interpolation in latitude to construct thecross sections of v at 458N, shown in Fig. 8.


FIG. 8. Vertical cross sections of the vertical velocity v at f 5458N and t 5 12 days, obtained with the GFDL-SDC at T85 resolutionwith (a) 10, (b) 20, (c) 40, (d) 80, and (e) 160 levels. Contours from20.22 to 0.22 in steps of 0.04 Pa s21. Red indicates upward motion,and blue indicates downward motion.

FIG. 9. The vertical velocity v, at s 5 0.5, f 5 458N, and t 512 days, obtained with the GFDL-SDC at T85. The solid line showsthe values with 160 levels, the dashed line with 10 levels. The curvesfor 20, 40, and 80 levels are contained between the two plotted curvesand have been omitted for the sake of clarity.

accompanies the positive and negative centers that de-velop in the vorticity field. Note that the features areextremely similar for all calculations, even near the sur-face. To further bring out the minuteness of the differ-ences as the number of levels is increased from 10 to160, we plot the values of v at the specific height s 50.5 in Fig. 9. For clarity, only the curves for 10 and160 levels are shown, as the curves for 20, 40, and 80levels are contained between these two.

Third, having demonstrated the convergence of ournumerical solution with vertical resolution, we finally

address the question of temporal resolution. To evaluatethis, we have computed T85L20 solutions with timesteps of 600, 300, and 150 s. Plots of the vorticity normssimilar to those in Fig. 5 show that curves for thesethree different time steps are totally indistinguishablefrom each other. For this reason, we have avoided re-producing them here. With this last item, we believe tohave convincingly demonstrated that our solutions, ob-tained with the GFDL-SDC, are numerically converged.

In summary then, solving the primitive equations asdefined in (1), with the initial conditions spelled out insection 2, our semi-implicit pseudospectral model, withsecond-order finite-differences in the vertical, and a sim-ple leapfrog time step, is able to obtain converged so-lutions with T85 horizontal resolution (roughly equiv-alent to a 1.48 grid), 20 vertical levels, and a time stepof 600 s, for this particular value of n. These resultscan now be used as a point of comparison with othernumerical schemes.

Finally, to assist modelers trying to reproduce thistest case in the future, we offer in Table 2 a number ofconverged numerical values relating to the two keyfields we have discussed, namely, the vorticity at thesurface and pressure vertical velocity at 458N after 12days of integration. We suggest that future dynamicalcores able to reproduce both the values in this table (tothe two significant digits reported) and to visually matchthe converged surface z and v at 458N fields presentedabove should be considered to have ‘‘passed’’ the newtest case.


TABLE 2. Converged numerical values for the solutions to the testcase described in section 3 and figures therein. All values are fort 5 12 days.

Quantity Value Units

EKE|z|2 at s 5 0.975|z|` at s 5 0.975|¹z|` at s 5 0.975Max (v) at 458NMin (v) at 458N

2.4 3 103

7.8 3 1026

7.4 3 1025

3.0 3 10210

1.9 3 1021

21.7 3 1021

J m22

s21

s21

m21 s21

Pa s21

Pa s21

FIG. 10. Black contours: as in Fig. 4, but computed with NCAR-SEDC at increasing horizontal resolution. Contour levels as in Fig.4. Colored patches: the GFDL-SDC solution at T341. All solutionscomputed with 20 vertical levels.

4. Computational validation of the solutions

Having demonstrated that one numerical scheme (inthis case, the GFDL-SDC) is able to numerically con-verge as the spatial and temporal discretization is re-fined, we have at this stage no guarantee this convergedsolution is independent of the scheme we have used toobtain it. To the best of our knowledge, there is norigorous general proof that solutions to the primitiveequations exist or that they are unique. However, weare working under the assumption that all consistentnumerical schemes will converge to an identical nu-merical solution.

To demonstrate that the converged numerical solutiondiscussed above is not an artifact of the pseudospectralmodel that was used to compute it, we have adoptedthe practical expedient of computing a new set of nu-merical solutions to the identical set of equations andinitial conditions but with a different numerical scheme.In what follows we show that the numerical solutionsobtained with this other scheme converge to those ob-tained with the GFDL-SDC. Hence, for most practicalpurposes, we are able to claim that the numerical so-lutions we are presenting here ought to be reproducibleby any numerical scheme, and that they establish abenchmark against which future dynamical cores canbe tested.

To compute a new set of independent solutions to theequations and initial conditions of section 2, we use adynamical core whose horizontal spatial discretizationemploys spectral elements in curvilinear coordinates ona cubed sphere, with second-order finite differences forthe vertical discretization and advection. This dynamicalcore is known as the National Center for AtmosphericResearch (NCAR) Spectral Element Dynamical Core(hereafter NCAR-SEDC) and is fully documented inThomas and Loft (2004). We emphasize that the NCAR-SEDC not only is based on a completely different nu-merical method, but was compiled and executed on adifferent machine than the one used for the GFDL-SDCcomputations of the previous section.

In Fig. 10 we demonstrate the convergence of theNCAR-SEDC solution to the GFDL-SDC solution, byconsidering the surface (s 5 0.975) vorticity after 12days of integration. The solid black contours show theNCAR-SEDC solution for five calculations with in-creasing horizontal resolution (and all with 20 equally

spaced vertical sigma levels). NCAR-SEDC computesthe solutions using a cubed-sphere spectral-elementmethod, for which the horizontal resolution is denotedby the parameter ne, the number of spectral elementsalong one edge of one face of the cube. The total numberof spectral elements in each NCAR-SEDC calculationis thus 6 . To compute the number of grid points used2ne

at a particular resolution, one multiplies this number by64 (8 3 8), the number of Gauss–Lobatto–Legendrequadrature points in each element. Thus, for instance,an NCAR-SEDC integration at ne 5 37 utilizes 64 3


FIG. 11. As in Fig. 5, but for the NCAR-SEDC solutions illustrated in Fig. 10.

6 3 372 5 525 696 grid points; this is roughly equiv-alent to a GFDL-SDC integration at T341, which in-volves 1024 3 512 5 524 288 Gauss–Legendre gridpoints. The horizontal resolutions for the five panels inFig. 10 are chosen to be roughly equivalent to theirpseudospectral counterparts in Fig. 4.

The key point in Fig. 10 is that, as the horizontalresolution is increased, the black contours of the NCAR-SEDC solutions become coincident with the color patch-es that represent the T341 GFDL-SDC solution. Theagreement between colors and contours is already ex-cellent with ne 5 5, although some tiny differences canbe detected. At ne 5 9 the colors and contours are in-distinguishably superimposed. Hence, as for the pseu-dospectral model, the spectral-element model appearsto visually converge at a resolution comparable to T85.

In order to go beyond a merely visual demonstration,we plot in Fig. 11 the same norms for the surface vor-ticity field that were computed for the GFDL-SDC so-lutions. Although the manner in which convergence isachieved with each model is different (not exactly asurprising fact), the two models converge to the samenumerical solution at sufficiently high resolution. Infact, the NCAR-SEDC curves at ne 5 37 are completelysuperimposed on the GFDL-SDC ones at T341 (whichwe have omitted for clarity). Similarly, the numericalresults in Table 2 are identical (to within the reportednumber of digits) for GFDL-SDC and NCAR-SEDC atthe highest resolutions. We claim, therefore, that thesenumerical results represent the converged solution ofsystem (1) with the initial conditions as specified.

Note that the numerical results reported in Table 2are given to two significant figures. This is the numberof digits to which the GFDL-SDC and the NCAR-SEDCsolutions agree. As can be seen from Fig. 7, convergencefor the pseudospectral GFDL-SDC is algebraic as op-

posed to exponential, likely due to the fact that the initialvelocity components are not infinitely differentiable.Thus, obtaining more digits would require much higherresolutions, well beyond the computational power avail-able to us. In any event, the main objective of the presentwork is not to examine the convergence properties of aparticular numerical method with respect to the initialcondition, but to establish that one can compute a con-verged numerical solution to the primitive equations,independent of the numerical method.

For completeness, we offer in Fig. 12 the cross sec-tions of v at 458N at day 12 computed with NCAR-SEDC, at different vertical resolutions (black contours)from 10 to 160 levels, and superimposed on the GFDL-SDC solution (colored patches). Only at the lowestNCAR-SEDC vertical resolution is one able to detectthe difference between the colors and the contours.Again, there should be little doubt that both models areconverging to the same numerical solution.

5. Effects of hyperdiffusion

We offer in this section an example to demonstratethe importance of specifying exactly the same diffusionterms as in (1) if one is attempting to reproduce thenumerically converged solutions presented above.While there is widespread practice of using a variety ofmethods to smooth out solutions in AGCMs, we wishto demonstrate here that indiscriminate use of suchmethods is likely to generate solutions that will differsubstantially from the ones we have just presented.

With this goal in mind, we here compute a numeri-cally converged solution to the dry, adiabatic, primitiveequations with a different diffusion operator. Specifi-cally, the terms on the right-hand side of (1) are replacedby the terms 2n¹4z, 2n¹4d, and 2n¹4T applied to the


FIG. 12. Black contours: as in Fig. 8, but computed with NCAR-SEDC at ne 5 19. Colored patches: the GFDL-SDC solution at aresolution of T85L160.

vorticity, divergence, and temperature equations, re-spectively (and not to the momentum equations them-selves). Such a choice is very common with spectralmethods and, in fact, is the one adopted by both NCAR’sCCM3 and GFDL’s Atmospheric Spectral Model (FMS-AM2). Hence, consideration of ¹4 hyperdiffusion mightbe of interest in illuminating the kind of baroclinic in-stabilities that are computed by these well-establishedAGCMs.

For n 5 2.5 3 1016 m4 s21, the numerical convergenceof the vorticity field at 12 days is illustrated in Fig. 13.Note how different this solution is from the one with

the regular diffusion operator in Fig. 4, especially in itsability to capture much steeper vorticity gradients. Thecontour interval in Fig. 13 is double the one in Fig. 4.

It is also interesting to note that, with this value ofn, the T42 calculation is relatively noisy. Recall that thevorticity z is not a differentiated quantity; it is one ofthe prognostic variables. Thus the noise is due to thevalue of n not being sufficiently large to produce smoothfields at T42. Perhaps surprisingly, both CCM3 andFMS-AM2, for which T42 is the standard horizontalresolution, use the even smaller value n 5 1 3 1016 m4

s21 (Kiehl et al. 1996). The T42 solution, for that value,is even noisier and further from converged than the oneshown here. These standard AGCMs, typically runningat T42, are therefore computing rather noisy fields.

From Fig. 13 it appears that T170 is the visuallyconverged resolution in this case. For completeness, wethen show in Fig. 14 the cross-sectional vertical velocityat day 12, computed at T170 for 10 to 80 vertical levels(we were unable to fit a T170 calculation with 160 ver-tical levels on our machine). Again, note that the contourinterval is double the one used in Fig. 8, indicating muchstronger updrafts and downdrafts in this much less dif-fusive solution. It is perhaps not immediately obviousthat one direct consequence of modifying the horizontaldiffusion is to alter the vertical velocity magnitude bya factor of 2. This implies that physical parameteriza-tions (e.g., convection schemes) as well as the verticaltransport of constituents (e.g., water vapor) are likelyto be profoundly impacted by the choice of horizontaldiffusion in AGCMs.

Finally, while we do not expect the solutions withhyperdiffusion to be easily reproducible by models otherthan those based on the spectral method, we report acouple of values associated with the numerically con-verged solutions in Figs. 13 and 14. At 12 days, themaximum and minimum values of z at s 5 0.975 are15 3 1025 and 28.4 3 1025 s21, respectively, and themaximum and minimum values of v at 458N are 0.37and 20.32 Pa s21.

6. Summary and discussion

We have computed and presented numerically con-verged solutions to the dry, adiabatic, primitive equa-tions in spherical coordinates. The solutions were ob-tained with two completely independent numerical mod-els. To the best of our knowledge, this is the first set ofsuch solutions to have been computed. These solutionsare meant to serve as benchmarks to help in the de-bugging and evaluation of future dynamical cores ofAGCMs.

Specifically, we wish to propose a new test using theresults presented here. It consists in three steps. First,starting from the initial conditions spelled out in section2, one would produce numerically converged instanta-neous fields of z at the surface and v at 458N after 12days of integration. At the risk of stating the obvious,


FIG. 13. As in Fig. 4, but for solutions with ¹4 hyperdiffusion.Contours from 217 to 17 3 1025 s21, in steps of 2 3 1025 s21.

FIG. 14. As in Fig. 8, but for solutions with ¹4 hyperdiffusion (andexcluding the 160-level solution). Contours from 20.44 to 0.44 Pas21, in steps of 0.08 Pa s21. Results from GFDL-SDC, with 20 verticallevels.

we stress that there is little point in considering non-converged numerical results since one is then unable totell if the differences are due to truncation errors orcoding errors.

The second step would consist of performing a visualcomparison of z and v with the fields presented above.The key qualitative features, number of positive andnegative maxima, and their relative positions should vi-sually match those presented in this paper. Finally, thethird step would consist of comparing the values givenin Table 2 with those computed with future dynamicalcores. Agreement with the values in those tables should

suffice to ensure that a model is free of serious codingerrors or other deficiencies.

As pointed out in the introduction, we believe thatthe new test proposed here should serve a complemen-tary role to the by-now classic HS94 benchmark. Onceour test is passed, one may feel reasonably confidentthat a dynamical core is free from major bugs, and thenext obvious step would then be to compute the statisticsof the model with a 1000-day HS94 integration.

Finally, one might rightly ask whether our choice ofincluding explicit diffusivity as part of the specificationsof our new test is likely to limit its applicability. Spe-cifically, the objection might be raised in the context oftesting primitive equation models whose horizontal dis-cretization does not require the explicit use of diffusivity(e.g., semi-Lagrangian schemes). Such schemes, typi-cally constructed with conservation properties in mind,often involve implicit diffusion, and can thus be stablyintegrated without the need of explicit diffusivity. Anexample of current interest is the Lin–Rood (1996)


TABLE A1. Numerical values of the bases zi and lapse rates (dT/dz)i for the eight layers used to construct our piecewise linear ex-pression for the U.S. Standard Atmosphere, 1976 temperature.

zi (km) (dT/dz)i (K km21)

0112032

26.50

11.012.8

47517180

022.822.0

0

scheme. One might argue that it is foolish to add explicitdiffusion if a numerical scheme does not necessitate it.

Our answer to this objection is twofold. First, itshould be clear that no obstacles exist to adding explicitdiffusion to any scheme: to the degree that an identicalset of partial differential equations is being solved, withidentical initial and boundary conditions, and that nu-merically converged solutions are computed, anyscheme should be able to reproduce the results presentedin the paper. Second, if our initial conditions are inte-grated with no explicit diffusivity, it is highly unclear,in our opinion, what an implicitly diffusing scheme willconverge to as the resolution is increased. As our initialcondition is likely to generate infinite gradients in finitetime in the absence of diffusivity, an implicitly diffusingscheme would presumably try to converge to a singularsolution as the resolution is increased (and the implicitdiffusion becomes negligible). Our suspicion, therefore,is that, for these initial conditions, numerical conver-gence may not be achievable with any scheme that doesnot contain explicit diffusion.

Acknowledgments. We gratefully acknowledge in-sightful conversations with Isaac Held, Paul Kushner,and Rich Loft. We also wish to thank Mr. ConstantineNenkov for a careful reading of the original manuscript.During the writing of this manuscript we have becomeaware of a somewhat similar test case being developedindependently by Christiane Jablonowski and DavidWilliamson; we wish to thank them for sharing theirresults with us prior to publication. This work is sup-ported, in part, by an NSF grant to Columbia University,and by the generosity of the David and Lucile PackardFoundation. Last, and not least, we express our gratitudeto Chief Editorial Assistant Mary Golden for her un-derstanding and support.

APPENDIX

The U.S. 1976 Standard AtmosphericTemperature Profile

To compute the U.S. Standard Atmosphere, 1976(COESA 1976) temperature, we use the canonical def-inition of TUS(z) as being piecewise linear in each ofeight layers. The base zi of each layer and the lapse rate(dTUS/dz) i in each layer are given in Table A1. Fromthese, for zi , z , zi11, the U.S. Standard Atmosphere,1976 temperature is evaluated using the expressionTUS(z) 5 TUS(zi) 1 (dTUS/dz)i(z 2 zi). For z 5 0, we

use TUS(0) 5 288.15 K. Note that the formulas are beinghere applied to the log pressure height z, and not the‘‘real’’ height.

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Numerically Converged Solutions of the Global Primitive ......Solutions of the dry, adiabatic, primitive equations are computed, for the ﬁrst time, to numerical convergence. These