JULY 1999 1551D R I T S C H E L E T A L .

q 1999 American Meteorological Society

The Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations

DAVID G. DRITSCHEL

Mathematics Institute, University of St. Andrews, Fife, United Kingdom

LORENZO M. POLVANI

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

ALI R. MOHEBALHOJEH

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

(Manuscript received 17 February 1998, in final form 23 June 1998)

ABSTRACT

A new method for integrating shallow water equations, the contour-advective semi-Lagrangian (CASL) al-gorithm, is presented. This is the first implementation of a contour method to a system of equations for whichexact potential vorticity invertibility does not exist. The new CASL method fuses the recent contour-advectiontechnique with the traditional pseudospectral (PS) method. The potential vorticity field, which typically developssteep gradients and evolves into thin filaments, is discretized by level sets separated by contours that are advectedin a fully Lagrangian way. The height and divergence fields, which are intrinsically broader in scale, are treatedin an Eulerian way: they are discretized on an fixed grid and time stepped with a PS scheme.

In fact, the CASL method is similar to the widely used semi-Lagrangian (SL) method in that material con-servation of potential vorticity along particle trajectories is used to determine the potential vorticity at each timestep from the previous one. The crucial difference is that, whereas in the CASL method the potential vorticityis merely advected, in the SL method the potential vorticity needs to be interpolated at each time step. Thisinterpolation results in numerical diffusion in the SL method.

By directly comparing the CASL, SL, and PS methods, it is demonstrated that the implicit diffusion associatedwith potential vorticity interpolation in the SL method and the explicit diffusion required for numerical stabilityin the PS method seriously degrade the solution accuracy compared with the CASL method. Moreover, it isshown that the CASL method is much more efficient than the SL and PS methods since, for a given solutionaccuracy, a much coarser grid can be used and hence much faster computations can be performed.

1. Introduction

The shallow water equations (SWE) are widely usedin idealized studies of atmospheric and oceanic dynam-ics. They are the simplest equations able to describeboth slow, balanced flows and fast, gravity wave oscil-lations, the two main categories of fluid motion presentin the more complicated primitive equations, which arecommonly used for atmospheric, oceanic, and climatemodeling. For this reason SWE have been proposed asan appropriate test bed for new numerical algorithms(Williamson et al. 1992).

In this paper we present a new algorithm for the so-lution of the SWE. The guiding idea behind the new

Corresponding author address: Dr. Lorenzo M. Polvani, Depart-ment of Applied Physics and Applied Mathematics, Columbia Uni-versity, 209 S W MUDD, Mail Code 4701, New York, NY 10027.E-mail: polvani@columbia.edu

algorithm rests on the well-established observation that,even in the presence of relatively smooth, large-scaleflows, tracer fields in the atmosphere and the ocean read-ily develop extremely fine scales, often by simple ki-nematic stirring. The stratospheric circulation, longthought to be dominated by large-scale features, pro-vides a case point; recent aircraft observations combinedwith trajectory filling techniques (Waugh et al. 1994;Plumb et al. 1994) have demonstrated the existence ofextremely fine structure in the constituent distributions.In the ocean, the recent observation of Wunsch andStammer (1995) of very small-scale features on theocean surface is another striking example. Furthermore,recent studies suggest that fine scales can play an im-portant role in stratospheric chemistry (McIntyre 1995;Edouard et al. 1996; Tan et al. 1997).

For inviscid SWE, a conserved tracer of particulardynamical significance (Hoskins et al. 1985) is the po-tential vorticity. The main novelty of the new algorithmrests in that the potential vorticity is represented by level

1552 VOLUME 127M O N T H L Y W E A T H E R R E V I E W

sets separated by contours, which are advected in a fullyLagrangian way. This technique allows one to computepotential vorticity scales well below the grid scale, whilethe fields that are not materially conserved by the floware numerically represented and time stepped on an Eu-lerian grid. Hence the name of the new algorithm: thecontour-advective semi-Lagrangian (CASL) method forSWE. As we demonstrate below, advecting potentialvorticity in a Lagrangian way allows one not only toresolve features on scales that are much smaller thanthe grid scale but also to maintain potential vorticitygradients that are steeper than the grid resolution allows.The net effect is that the convergence of the computedsolutions substantially improves as the grid is refined.To the best of our knowledge, the CASL method pre-sented in this paper is the first implementation of acontour-based algorithm to a system of equations forwhich no exact potential vorticity invertibility exists.

In the next section we describe the new numericalmethod in some detail, as well as two other methodsthat have been popular in recent years: the pseudospec-tral method (PS) and the semi-Lagrangian method (SL).In section 3 we compare these three by examining indetail the evolution of a complex flow typical of at-mospheric and oceanic situations: the nonlinear insta-bility of a zonal jet. A brief discussion concludes thepaper in section 4.

2. The algorithm

In their simplest form, the SWE may be written interms of the velocity u [ (u, y) and surface height h.In planar Cartesian geometry, this gives

Du ]h2 fy 5 2g , (1)

Dt ]x

Dy ]h1 fu 5 2g , (2)

Dt ]y

]h1 = · (hu) 5 0, (3)

]t

where f is the Coriolis parameter (taken to be constanthere), g is the gravitational acceleration (or the reduced-gravity in the oceanic context), and the material deriv-ative is defined in the usual way,

D ][ 1 u · =. (4)

Dt ]t

Dissipative terms are not written, though they are re-quired in many numerical methods as discussed below.

Instead of the two velocity components u and y , it iscustomary to use the vorticity z and the divergence d,defined by

]y ]u ]u ]yz 5 2 and d 5 1 , (5)

]x ]y ]x ]y

as the prognostic variables. It is also useful to separate

the height h into a constant mean value h and a deviationh therefrom—that is, letting h [ h 1 h—and to use h9[ h/h as the third prognostic variable. In terms of z, d,and h9 the SWE take the form

]z5 2= · [(z 1 f )u], (6)

]t

]d2 2 21 c ¹ h9 2 f h9 5 f (z 2 fh9) 1 2J(u, y)

]t

2 = · (ud), (7)

]h91 d 5 2= · (uh9), (8)

]t

where c2 [ gh . We have explicitly segregated on theleft-hand side of (7) and (8) the terms that give rise tolinear rotating gravity waves, since these terms need tobe treated in a special way. It is worth recalling thatthese equations possess one material invariant of fun-damental dynamical importance, namely the potentialvorticity q, defined by

z 1 fq [ . (9)

h

We now describe the three numerical methods wehave used for solving the SWE (6)–(8) in a doublyperiodic domain: PS, SL, and CASL. To keep the com-parison as meaningful as possible, we have used anidentical scheme for solving the divergence (7) and con-tinuity (8) equations in all three methods. The simplestand most efficient scheme for this is the semi-implicitpseudospectral method. This scheme consists of em-ploying spectral representations for d and h9, fast Fouriertransforms (FFTs), evaluation of the nonlinear productsin physical space, semi-implicit leapfrog time stepping,and a Robert–Asselin time filter.

The semi-implicit leapfrog time stepping is carriedout in a standard way (Ritchie 1988). In (7) and (8),the partial time derivatives of spectral h9 and d are ap-proximated by centered differences, the other terms onthe left-hand sides are approximated as averages overthe previous and next time levels (t 2 Dt and t 1 Dt),and the terms on the right-hand sides are evaluated attime t. Such time-discretized implicit equations are eas-ily solved for each spectral component of h9 and d attime t 1 Dt. The practical merit of this scheme is thatit is numerically stable for large time steps, whereas anexplicit scheme would be restricted by the speed of thefastest gravity wave and the grid size. The disadvantageis that small-scale gravity wave motions are numericallyinaccurate. In atmosphere and ocean modeling, it iscommon practice to accept an inaccurate numerical so-lution of these waves for the sake of being able to uselarger time steps.

This semi-implicit leapfrog scheme is not stable un-less it is coupled with a Robert–Asselin time filter (Rob-ert 1966; Asselin 1972). That filter replaces a field, say

JULY 1999 1553D R I T S C H E L E T A L .

f, at time t by a combination of the fields at t 2 Dt, t,and t 1 Dt just after the latter has been computed; thatis,

f(x, t) ← f(x, t) 1 A[f(x, t 2 Dt)

2 2f(x, t) 1 f(x, t 1 Dt)]. (10)

This operation in effect damps high-frequency modes,since the term multiplying the filter coefficient A is afinite-difference approximation to (Dt)2]2f /]t2. Forsmall A, this filter principally damps small-scale, high-frequency gravity waves (a significant component ofwhich could be artificial). For such waves, ]/]t ; c=,showing that they are subject to an effective viscosityof Ac2Dt. The filter coefficient A, a dimensionless num-ber, is typically in the range (0.05, 0.3). For the com-putations presented in the next section, A 5 0.05; wehave found that values less than 0.03 lead to numericalinstability.

Given that the divergence and continuity equationsare solved identically for all three methods, it shouldbe clear that the key difference between the PS, SL, andCASL methods rests in the solution of the vorticityequation. We now describe each one in detail. We startby briefly reviewing the PS and SL methods, in orderto provide a context for the new CASL method.

a. The pseudospectral method

In the PS method, (6) is replaced by

]z2 n1 = · [(z 1 f )u] 5 2n(2¹ ) z. (11)

]t

The new term on the right-hand side of (11), usuallycalled ‘‘hyperdiffusion,’’ is necessary for numerical sta-bility. In this paper we have used n 5 3. The value ofn is chosen, typically, so that the smallest features (i.e.,those comparable to the grid scale) are efficiently dis-sipated. Here, we have used the expression

n 5 C(hQ)/ ,2nkmax (12)

where C is a dimensionless constant, kmax 5 ng/2 is thehighest resolved wavenumber (ng is the grid size), andQ is defined by

fQ 5 max q 2 , (13)) )hx,y

where the maximum is taken over all points in the com-putational domain. The idea behind (12) is that hQ isthe relevant timescale for the evolution of z. In spite ofthe definition (13), the actual choice of n remains some-what ad hoc, in the sense that the constant C can varysubstantially. When trying to compute solutions that areas free of dissipation as possible, one picks a value ofC as small as possible, without producing too muchsmall-scale noise. For the results presented in the nextsection, we have used C 5 1.

b. The semi-Lagrangian method

Whereas with the PS method only a slightly modifiedversion of (6) is solved, the SL and CASL are con-structed on the principle that potential vorticity conser-vation is a fundamental property of the SWE. Thereforein both the SL and CASL methods, the third prognosticvariable z is replaced by the potential vorticity q, and(6) is replaced by

Dq5 0. (14)

Dt

In practice, (14) is solved by trajectory integration; thatis,

dx5 u(x, t), (15)

dt

where x is the position of a fluid element; (15) is for-mally equivalent to (14) since q does not change fol-lowing x. The same method can be used to advect anyconserved tracer.

In the SL method, in order to determine q at t 1 Dt,two distinct steps are required: for each grid point xa

(the ‘‘arrival’’ point), one first needs to integrate (15)backward in time to determine the location xd (the ‘‘de-parture’’ point) of that same fluid element at time t.Since xd typically will not fall on a grid point, the secondstep consists in interpolating q at xd at time t and finallyreplacing that value at xa at time t 1 Dt. For both steps,we use the simplest and most commonly used schemes[for more details the reader is referred to Staniforth andCote (1991) and Gravel (1996)].

The back trajectory computation is done using themidpoint method (Temperton and Staniforth 1987; Bateset al. 1995),

x 1 xa dx 5 x 2 Dtu , t , (16)d a n11/21 22

together with a linear time extrapolation,

3 1u(x, t ) 5 u(x, t ) 2 u(x, t ), (17)n11/2 n n212 2

and a bilinear spatial interpolation for the evaluation ofthe velocity at departure points from (17) at tn11/2. Thesecond step is carried out using the so-called bicubicLagrange interpolation, which is commonly used in me-teorological modeling (Bates et al. 1995; Ritchie et al.1995). It is worth noting that no explicit dissipationneeds to be incorporated in the SL method. It is im-plicitly provided, as the results of the next section willdemonstrate, by the interpolations of u and q that areperformed at each time step (Gravel 1996).

c. The contour-advective semi-Lagrangian method

The new CASL method for the SWE is also basedon the idea that potential vorticity conservation is es-

1554 VOLUME 127M O N T H L Y W E A T H E R R E V I E W

sential, but it carries that idea much farther than the SLmethod: q is represented in a fully Lagrangian way.Specifically, q is discretized by level sets of uniformvalue qj separated by contours across which it jumps byDq, each contour being represented by a set of nodes.This discretization is at the heart of the contour dynam-ics method, which has been implemented to solve avariety of two-dimensional and quasigeostrophic prob-lems (Dritschel 1989).

The original contour dynamics method rested on theexistence of a linear inversion relation (Dritschel 1989)giving u directly and solely in terms of q. For q rep-resented as a piecewise-uniform function, this permitsone to calculate u from integrals over the q contours.However, for all but the simplest flows and linear in-version relations, this process is computationally ex-pensive, proportional to the square of the number ofpoints representing the q contours. A major improve-ment in the computation of u for general flows wasrecently introduced by Dritschel and Ambaum 1997(hereafter DA), who developed the CASL method formultilayer quasigeostrophic dynamics. Within theCASL method, the computation of u at each time stepis carried out by first interpolating q onto a grid andthen performing a spectral inversion. The contour-to-grid conversion, a very fast operation, not only elimi-nates the need for the existence of an inversion relationfor potential vorticity (hence allowing a full general-ization of the contour approach to the primitive equa-tions), but also provides a very substantial improvementin computational performance, typically of several or-ders of magnitude.

The novelty of this paper is the implementation, forthe first time, of a contour-based method to a system ofequations for which no inversion relation exists. Sincethe CASL method for the SWE presented in this paperis a direct carryover from the CASL method for qua-sigeostrophic dynamics, we limit the discussion here toa broad sketch, and we refer the reader to DA for alldetails.

Briefly, given q in terms of contours and the fields dand h9 on a grid of size ng 3 ng at time t, each node xi

on the q contours needs to be advected to t 1 Dt. Thefirst step consists in computing the velocity u so that(15) may be used. This is accomplished, as in the orig-inal CASL method, by projecting the contoured poten-tial vorticity onto a grid finer than ng (specifically, ofsize mgng, where mg is typically equal to 4) and per-forming an iterative averaging so as to obtain a smoothq field on a grid of size ng. Once q is thus constructed,the vorticity z is readily obtained from (9), since h isalso known on the grid. Finally, the gridded velocityfield u is computed directly from z and d using, as iscustomary, the streamfunction c and the velocity po-tential x defined by

z [ ¹2c and d [ ¹2x, (18)

and related to the velocity components via

]x ]c ]x ]cu 5 2 and y 5 1 . (19)

]x ]y ]y ]xThese operations are performed with spectral trans-forms.

Once the velocity u is known on the grid, each nodeon the potential vorticity contours can be advected for-ward to time t 1 Dt. This is done by solving (15) in amanner identical to the one described above for the SLmethod, the only difference being that in the CASLmethod the advection is done forward rather than back-ward. Hence, together with bilinear spatial interpolation,(16) and (17) are used directly, exchanging xd with xa

and letting Dt → 2Dt. Once all the nodes on the qcontours are stepped forward, the nodes may be redis-tributed; this is necessary because, while the area en-closed by each q contour is approximately conserved,its perimeter may increase drastically as the flow be-comes complex, and thus nodes need to be added topreserve accuracy (see DA for details).

The key difference between the SL method and theCASL method is now apparent: in the CASL method,once the nodes are time-stepped forward no interpo-lation of q is necessary. In contrast, the SL methodrequires that, once each grid point has been time-steppedbackward, the potential vorticity itself be interpolated(e.g., with the bicubic Lagrange scheme). This inter-polation of q results in substantial diffusion, as the re-sults of the next section will show, and degrades theaccuracy of the method.

Of course, the extremely small scales in q that areinevitably generated by the forward enstrophy cascadein complex rotating, stratified flows pose a problem forany method, whether Eulerian or Lagrangian, and theyneed to be removed. In the CASL method this is ac-complished with contour surgery. This procedure [fullydocumented in Dritschel (1989) and with some refine-ments added in DA] effectively acts only on the verysmallest scales by topologically reconnecting contoursand eliminating very finescale filamentary structures.

While contour surgery may rightly be viewed as anad hoc procedure, it is no more ad hoc than the familiarhyperdiffusion used in the PS method. The key differ-ence is that contour surgery does not diffuse q gradients,a severe drawback of hyperdiffusion (Mariotti et al.1994; Jimenez 1994; Macaskill and Bewick 1995; Yaoet al. 1995). Moreover, the surgery scale ds—belowwhich q ceases to be conserved—can be chosen to bemuch smaller than the grid scale on which the d, h9,and u fields are held, typically 10 times as small.1 Hence

1 The application of contour surgery at a tenth of the scale of thegrid on which the advecting winds are represented was originallymotivated by recent contour advection studies (Waugh and Plumb1994; Norton 1994) using both observed and model winds. Thesestudies show that potential vorticity features down to a tenth of thewind grid scale are negligibly influenced by the subgrid-scale velocityfield. Hence very thin potential vorticity filaments behave passively,and can therefore be removed with little effect on the dynamics (seealso Methven and Hoskins 1998).

JULY 1999 1555D R I T S C H E L E T A L .

FIG. 1. (a) The unperturbed potential vorticity q, (b) the corre-sponding u velocity, and (c) perturbation height h9, all used for thetest case.

the CASL method is able to capture potential vorticitystructures much smaller than the grid scale, and thisconsiderably improves its convergence over both the PSand SL methods.

One final CASL procedure needs to be discussed be-fore presenting numerical test results. Assuming that theinitial condition (whether from a model or from data)is given on a grid, one needs to generate q contours tocarry out the Lagrangian advection. The simplest wayis to first determine the minimum and maximum valueqmin and qmax in the computational domain. The range(qmin, qmax) is then divided into nc 1 1 levels. Each levelcorresponds to a region R j (possibly multiply connect-ed) where q assumes a spatially uniform value qj definedby

1q [ q 1 j 2 Dq, j 5 1, . . . , n 1 1, (20)j min c1 22

with

q 2 qmax minDq 5 (21)n 1 1c

representing the potential vorticity jump across eachcontour. This simple scheme yields a potential vorticityrepresentation consisting of exactly nc 1 1 regions sep-arated by nc contours.

3. Numerical tests

To test the new CASL method, and to compare itdirectly with the PS and SL methods, we have chosena relatively simple initial flow—a perturbed unstablezonal jet—which rapidly becomes very complex. Webelieve that complexity is generic to geophysical flows,and is precisely what makes the design of numericalalgorithms for the solution of the primitive equationssuch a challenging task. Moreover, the nonlinear evo-lution of unstable jets is commonly observed in boththe atmosphere and the ocean, and therefore this test ishighly relevant. We have considered using simpler testcases, for example, the propagation of a single linearRossby wave or the advection of a height anomaly. Suchflows have been proposed by Williamson et al. (1992)as test cases for the SWE in spherical geometry. Webelieve, however, that many of those cases lack the com-plexity necessary to adequately test the performance ofnumerical algorithms that are being proposed for prac-tical applications.

Hence, our initial flow is specified by prescribing thepotential vorticity as follows:

q(x, y, 0) 5 q 1 Q sgn(y)(a 2 | |y| 2 a|) (22)

for |y| , 2a, and q 5 q otherwise; Q is the amplitudeof the potential vorticity anomaly, q is the mean po-tential vorticity determined by the requirement of zeromean relative vorticity, 2a is the distance from the min-imum to the maximum potential vorticity, and

y 5 y 1 cm sinmx 1 cn sinnx (23)

is a displaced y coordinate (which preserves the area ofdifferential elements) used to perturb the jet. For thenumerical integrations, we have chosen the scalings h5 1, LR 5 c/f 5 0.5, a 5 0.5, hQ/f 5 1, and f 5 4p,the latter implying that a unit time interval correspondsto a day. Our doubly periodic domain spans the range(2p, p), and thus covers about 12.5 deformation radiiin each direction, the jet itself being about four defor-mation radii wide.

The unperturbed q profile (i.e., with cm 5 cn 5 0),the associated zonal flow u, and the corresponding bal-anced height field anomaly h9 are show in Fig. 1. Noticethat the h deviates by up to 40% from its mean value,while max |u/c| ø 0.45; moreover, max |z/f | ø 0.9 (notshown), demonstrating that this flow is strongly ageo-strophic. The q profile in Fig. 1a is perturbed by choos-ing m 5 2, n 5 3, c2 5 20.1, and c3 5 0.1 in (23).With q thus specified, the initial depth h9 and divergenced fields are then initialized using the balance conditions]2h9/]t2 5 ]2d/]t2 5 0. It is more common to balancea flow by setting ]nd/]tn 5 0 and ]n11d/]tn11 5 0, for(small) integer n (cf. Norton 1988), but because thereis no rigorous definition of balance, our conditions areequally acceptable; moreover, they give results that dif-fer little from those obtained with zero second and thirdtime derivatives of divergence. This will be discussedin a forthcoming article.

We start by illustrating the complexity that emergesduring the time evolution of such an apparently benigninitial condition. In Fig. 2a, the potential vorticity q isshown for the first 10 days of a high-resolution CASLcalculation, using a grid size ng 5 256 for d and h9,and with nc 5 20 contours used to discretize q. Noticehow steep potential vorticity gradients form very rapidly(cf. the t 5 2 frame). We stress that such high gradientsare not peculiar to this initial condition, but are a genericproperty of geophysical flows. The nonlinear evolution

1556 VOLUME 127M O N T H L Y W E A T H E R R E V I E W

FIG. 2. (a) The evolution of the potential vorticity q at 2-day intervals for the perturbed unstablejet of Fig. 1. This CASL solution is obtained with nc 5 20 contours to represent q, and a grid ofsize ng 5 256 for d and h9. (b) The corresponding h9 field.

leads to the breakup of the initial jet into a number ofvortices, each eventually composed of a rather flat coresurrounded by a very complicated jumble of filamentarystructure (cf. the t 5 10 frame in Fig. 2a).

It is important to contrast this very complex q fieldwith the corresponding height field h9, shown in Fig.2b. The deceptively smooth height field masks all thereal complexity of the flow evolution. For this reasonwe consider this field (and for similar reasons thestreamfunction c) to be an inadequately sensitive mea-

sure with which to test the accuracy of a numericalmethod.

For the reader perplexed as to whether the perhapssurprisingly complex evolution of the above initial con-dition may be an artifact of the contour representationand surgery in the CASL method, we present in Fig. 3a direct comparison of the potential vorticity field at day10 at even higher resolution (ng 5 512) computed withthe three methods described in the previous section, andusing the same initial condition as for Fig. 2. First notice

JULY 1999 1557D R I T S C H E L E T A L .

FIG. 2. (Continued )

how, at this very high resolution, the CASL method(Fig. 3a), the PS method (Fig. 3b), and the SL method(Fig. 3c) are all in rather good agreement. The com-plexity of the evolution, therefore, is intrinsic to theflow, not a numerical artifact.

Second, the key differences relate, as expected, to thesteepness of the potential vorticity gradients and to thesmall-scale features. The CASL method is able to re-solve scales one-tenth the grid size (where surgery isapplied) and hence can support much steeper gradientsthan the other two methods. Moreover, since the CASLmethod controls the cascade to small scales throughsurgery, it does not diffuse the potential vorticity field.

This unwanted yet inevitable diffusion in both the PSand SL method is readily apparent in Figs. 3b and 3c.Consider for instance how, in both cases, the complexfilamentary structure surrounding most vortices is sub-stantially smoothed out in the PS and SL solutions. Inthis respect, it would seem that the SL method, in whichthe numerical diffusion occurs through repeated inter-polations and is thus not directly controllable, doesworse that the PS method. Contrast the two largest vor-tices in the flow and notice how the PS solution hascaptured steeper potential vorticity gradients. This maybe related to the fact that the PS method allows a moredirect control of the numerical diffusion via the param-

1558 VOLUME 127M O N T H L Y W E A T H E R R E V I E W

FIG. 3. (a) The potential vorticity q at t 5 10 days for a CASL computation identical to the one in Fig. 2 but for a finer grid of size ng

5 512. (b) As in (a) but for a PS computation. The grid size is ng 5 512, as for the CASL case. (c) As in (a) but for a SL computation.Again the grid is of size ng 5 512.

eter n, which was chosen according to (12) for the com-putation in Fig. 3b.

Third, and most importantly, it could be objected thatthe kind of complexity that the computations in Fig. 3are able to resolve is of little practical interest. In typicalgeophysical flows the generation of small scales is suf-ficiently rapid that the resolution of successively finer

scales by grid refinement might start to resemble thequest for the Holy Grail. Moreover, in realistic circum-stances (e.g., for general circulation models) a largeportion of the computational resources need to be ded-icated to physical processes other than the fluid dynam-ics (e.g., chemical reactions or radiation schemes). Inpractice, it could be argued, only modest grid resolutions

JULY 1999 1559D R I T S C H E L E T A L .

TABLE 1. A comparison of the accuracy and efficiency of the CASL, PS, and SL methods for integrations of the initial condition (22)to t 5 5 and t 5 10.

Method ng Dt e(5) e(10) c(5) c(10)

CASLCASLCASLCASLCASL

3264

128256512

0.040.040.040.040.04

0.0031970.0019120.0012300.0007370.000454

0.0043640.0031950.0021190.0015680.001253

13.3330.9486.45

266.83922.24

35.5889.31

278.66931.44

3436.82

PSPSPSPSPS

3264

128256512

0.020.010.0050.0020.001

0.0204510.0112550.0065020.0045910.002964

0.0242650.0156600.0102770.0083470.006714

6.3431.41

234.362287.62

18 933.88

12.6862.81

468.724575.25

37 867.75

SLSLSLSLSL

3264

128256512

0.040.040.040.040.04

0.0187390.0131220.0080600.0050930.003386

0.0243770.0181550.0122120.0084760.006752

5.6616.8661.79

244.21999.26

11.3133.72

123.57488.42

1998.51

are affordable on which to compute the evolution of theflow.

If one is willing to subscribe to this point of view,the question then becomes: for a given grid size, whichnumerical method yields the ‘‘best’’ solution? Or, moreinterestingly yet: which method converges faster as theresolution is increased? To answer these questions wepresent, in Figs. 4 and 5, a direct comparison of thenumerical solutions of the initial condition (22) com-puted with the CASL, PS, and SL methods and withgrid resolutions ng 5 32, 64, 128, and 256.

The potential vorticity field at day 5 is shown in Fig.4a. At the highest resolution (cf. top row) all three meth-ods are in good agreement, as expected. However, atvery low resolution (cf. bottom row) only the CASLmethod manages to capture all the key features of theflow, that is, the number, size, and location of the vor-tices that result from the instability. The principal sourceof error in the CASL method, at low resolution, is thepoor estimation of the advecting velocity field, whichis interpolated on a coarse grid. This affects the PS andSL solutions as well, but they additionally suffer fromexcessive diffusion: hyperdiffusion in the PS case, andinterpolation errors in the SL case. Such numerical dif-fusion is the principal cause for the slower convergencewith increasing resolution of the PS and SL methodscompared with the CASL method. This is quantifiedbelow.

Moreover, only at the higher resolutions do the PSand SL solutions start to develop tight potential vorticitygradients similar to those in the CASL solution. The SLmethod appears to be diffusing more than the PS method(contrast the PS and SL solutions at ng 5 32 and 64),though the very small n we are employing for the PSmethod (to allow it to diffuse as little as possible) isbarely marginal for numerical stability (hence the tinyGibbs phenomena). Of course at the highest resolution(ng 5 256) all three methods yield nearly identical fea-

tures on the large scales, but the point here is that theCASL method can afford a much coarser grid.

The corresponding height fields at day 5 are shownin Fig. 4b. Here, except for the lowest resolution (cf.bottom row), the PS and SL solutions appear to do ratherwell. This is not surprising since h9 is a smooth fieldand is thus much easier to compute. However, tracersare not smooth fields and thus the comparison of Fig.4a is more appropriate. Moreover, it is worth noting thatat our lowest resolution (ng 5 32) the initial jet spansapproximately 10 grid points. For most current atmo-spheric general circulation models used for climate stud-ies, a resolution of 18 is considered high; at such res-olution, key features such as the subtropical jet spanonly a few grid points (in the oceans, key features suchas the gulf stream tend to be an order of magnitudesmaller in scale and are immensely difficult to resolve).Therefore, the CASL method, with its ability to capturethe basic large-scale features of smooth fields at verylow resolution, offers a substantial practical advantage.

Figure 5a shows the potential vorticity fields at day10. Here the results in all cases converge more slowly,but the flow field is in this case extremely complex.Still, the CASL algorithm converges significantly morerapidly than the other two. And here, the effect of nu-merical diffusion in the PS and SL algorithms is par-ticularly evident. With ng 5 64 (cf. third row) the CASLmethod has managed to capture the number, size, andlocation of the key features in the flow, whereas the PSand SL methods have still not converged to the correctnumber of vortices and their positions. The correspond-ing height fields (shown in Fig. 5b) confirm the key ideabehind these tests, that is, that the CASL method is ableto capture accurate solutions with coarse grid resolutionsbecause it does not suffer from numerical diffusion.

We now quantify this statement in a precise way. Wemeasure the accuracy of each simulation by its abilityto conserve mass between isolevels of potential vorticity

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FIG. 4. (a) The potential vorticity q at t 5 5 days, for four different grid resolutions ng 5 32,64, 128, and 256 and for the CASL, PS, and SL methods. (b) The corresponding height field h9at t 5 5 days.

(the mass between any two material contours in shallowwater flows is conserved in the absence of dissipation).The degree of mass conservation is important for a prop-er assessment of transport properties; it is crucial, forexample, to ozone chemistry in the stratosphere(Edouard et al. 1996), and it is no doubt equally im-portant in the oceans.

The mass error is computed as follows. The initial qfield is divided into regions R j, j 5 2N, . . . , N, eachcorresponding to a potential vorticity level qj as definedin section 2c above. The mass in each region is mj 5

h dx dy. Ideally mj does not change in time, but in##R j

practice the numerical approximations inevitably leadto changes in mj, and these changes are used to measurethe numerical error of each simulation.

The error at time t is defined as the rms differencebetween mj(t) and mj(0), normalized by the product ofh and the domain area Adom; that is,

1/21 1

2e(t) 5 [m (t) 2 m (0)] , (24)O j j5 6hA 2N jdom

where the sum is over all j except j 5 0. The regionR 0 includes all of the fluid with zero anomalous po-tential vorticity and is much larger than the others. In-cluding this region in the sum above approximately dou-bles the value of e in the PS and SL simulations, butnegligibly increases it in the CASL simulations. Thisdifference is due to the diffusion of q in the PS and SLsimulations.

JULY 1999 1561D R I T S C H E L E T A L .

FIG. 4. (Continued )

To compute the mj, R j must be known sufficientlyaccurately. Following Yao et al. (1995), we interpolateq and h to a finer grid, in this case eight times finer ineach direction.2 We then determine j at each point onthis fine-resolution grid from the nearest integer valueof (q 2 q)/Dq and add to mj the value of h at this point,multiplied by the area of a grid square. This procedureis employed, with no variations, for all three simulationmethods. In the results presented next, we choose N 510, so that each region R j effectively lies between con-tours in the CASL simulations. Note, however, that we

2 Using a grid only four times finer leads to differences on the orderof 1%.

do not take advantage of this fact in computing the mj’sfor the CASL method; as in the PS and SL simulations,we compute the set of grid points lying within eachregion R j and sum the h values.

The accuracy of any algorithm, as all would agree,has little practical value per se. It needs to be contrastedwith its efficiency, since the best algorithm is not simplythe most accurate, but also the fastest. Hence we present,in Table 1, the error e(t) at the two reference times, t5 5 and t 5 10 (cf. Figs. 4 and 5), plotted versus thecost c(t) (in seconds) needed to integrate the SWE tothose times with the three different methods. All thesimulations were performed on a vector-processingCray-J90 supercomputer, with great care taken to vec-torize and, generally, to optimize the performance ofeach algorithm.

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FIG. 5. (a) As in Fig. 4a but for t 5 10 days. (b) As in Fig. 4b but for t 5 10 days.

To provide an immediate visual understanding of howthe three algorithms compare, we plot in Fig. 6 the errorand efficiency data of Table 1, with the cost c(t) on theabscissa and the error e(t) on the ordinate. The threesymbols used in that figure represent the three methods:squares for CASL, circles for PS, and triangles for SL.The filled points correspond to the values at the refer-ence time t 5 5 in Table 1, the empty ones at t 5 10.The size of the plotted points indicate the resolution.For each method and each reference time, the data pointsfor the five grid resolutions employed (ng 5 32, 64, 128,256, 512) are connected together. This makes the effectof increasing resolution immediately clear: for all threemethods, increasing ng results in smaller errors and, ofcourse, greater cost.

Because of the coordinate choice in Fig. 6, it is readily

apparent that the CASL method outperforms both thePS and SL method. The curves for the CASL solutions(squares) are located closer to the lower-left corner ofthe plot (where fast and accurate solutions lie) than thecurves for the PS and SL solutions. The comparison ofthe latter two is quite interesting. The SL method is seento outperform the PS one despite the fact that, at thesame grid resolution, the PS method is slightly moreaccurate (cf. Table 1 and Figs. 4 and 5 where, at thesame resolution, the SL method shows more diffusion).The superiority of the SL methods, however, lies in thatthe PS method is much more costly, due to the CFLconstraint on the time step. For a given accuracy, it costsless to use an SL method than a PS one. However, itcosts even less to use the CASL method.

Consider furthermore that, as can be seen in Fig. 6,

JULY 1999 1563D R I T S C H E L E T A L .

FIG. 5. (Continued )

the CASL curves lie almost entirely below the corre-sponding PS and SL curves. At t 5 5 (filled curves),one needs to go out to the finest grid ng 5 512 with thePS and SL methods to achieve an accuracy similar tothe one that the CASL method achieves with the coarsestgrid ng 5 32. With reference to Table 1, this means thatfor a comparable accuracy (say for an error around 0.003at t 5 5), the CASL method speeds up the computationsby a factor of 75 over the SL method and by a factorof 1450 over the PS method.

At t 5 10, the comparison is even better. As can beseen directly in Fig. 6 (empty curves), the error of thecoarsest CASL computation is smaller that that of thefinest PS and SL computations. Thus, despite the factthat with the coarsest grid (ng 5 32) the advecting ve-locity field is very crudely represented, the contour dis-

cretization of q in the CASL method still manages toconserve mass between q levels better than the PS andSL simulations with the finest grid (ng 5 512). The lackof mass conservation in the PS and SL simulations islargely a result of numerical diffusion, of using a grid-based scheme for the advection of q.

4. Summary and discussion

We have presented a new algorithm for the shallowwater equations in which potential vorticity, the fun-damental dynamically active tracer, is discretized by lev-el sets separated by contours that are advected in a fullyLagrangian way. This technique allows for numericaldissipation to act only on scales that are much smallerthan the scale of the grid on which the divergence and

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FIG. 6. The mass error, as defined by (24) vs the cost (in seconds)for the solutions shown in Figs. 4 and 5. The symbols indicate thedifferent methods: squares for CASL, circles for PS, and trianglesfor SL. Empty points correspond to the error-cost values at t 5 5days, the filled points at t 5 10. The point size increases with res-olution, from ng 5 32 (smallest) to ng 5 512 (largest).

height field are represented, and thus yields accuratecomputations with much coarser grids than semi-La-grangian or spectral methods.

The use of contour advection in the new algorithmfurthermore overcomes the numerical diffusion explic-itly or implicitly associated with grid-based advection.Numerical diffusion is required in grid-based methodsto avoid the buildup of artificial structure at the gridscale; hyperdiffusion is used widely in pseudospectralmethods, and implicit diffusion occurs when interpo-lating tracers in semi-Lagrangian methods. The newCASL method, by contrast, is virtually free of numericaldiffusion. For the test case reported here, this results ina speedup of nearly two orders of magnitude over thesemi-Lagrangian method, and more than three orders ofmagnitude over the pseudospectral method, for similaraccuracy.

The contour representation can be extended directlyto any conservative tracer and may be particularly ap-propriate for those circumstances (e.g., the stratosphericcirculation) where the velocity field stretches and foldstracers into extremely fine filaments. Moreover, sincethe contour-to-grid conversion is a very fast operation,the contour representation can also be extended to re-active tracers. A scheme for incorporating general dia-batic terms in the potential vorticity equation has beendeveloped (Ambaum and Dritschel 1998).

With further work, contour advection could be im-plemented in realistic atmospheric and oceanic modelsfor those fields, notably tracers, where finescales areinvariably generated and may be important for the chem-ical, moisture, or energy balances. Already, a spherical

barotropic and a multilayer, Boussinesq, primitive-equa-tion algorithm exists (manuscript in preparation), and aspherical version of the present shallow water algorithmis under development. We shall report on these in thenear future.

Acknowledgments. DGD is supported by the U.K.Natural Environment Research Council, LMP by theU.S. National Science Foundation, and ARM by a schol-arship from MCHE of Iran and a U.K. Universities ORSaward.

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