2206 MONTHLY WEATHER REVIEW VOLUME 119Semi-Lagrangian Integration Schemes for Atmospheric Modeis--A Review ANDREW STANIFORTH AND JEAN C(~T~Recherche en pr~vision nurndrique, Service de l'environnernent atmosph~rique, Dorval, Quebec, Canada(Manuscript received 24 September 1990, in final form I March 1991) The semi-Lagrangian methodology is described for a hierarchy of applications (passive advection, forcedadvcction, and coupled sets of equations) of increasing complexity, in one, two, and three dimensions. Attentionis focused on its accuracy, stability, and efficiency properties. Recent developments in applying semi-Lagrangianmethods to 2D and 3D atmospheric flows in both Cartesian and spherical geometries are then reviewed. Finally,the current status of development is summarized, followed by a short discussion of future perspectives.1. Introduction Accurate and timely forecasts of weather elementsare of great importance to both the economy and topublic safety. Weather forecasters rely on guidanceprovided by numerical weather prediction (NWP), acomputer-intensive chain of operations beginning withthe collection of data from around the world and culminating in the production of weather charts and computer-worded messages. At the heart of the system arethe numerical models used to assimilate the data andto forecast future states of the atmosphere. The accuracy of the forecasts depends among other things onmodel resolution. Increased resolution, given the realtime constraints, can only be achieved by judiciouslycombining the most efficient numerical methods onthe most powerful computers with the most appropriateprograming techniques. A long-standing problem in the integration of NWPmodels is that the maximum permissible time step hasbeen governed by considerations of stability rather thanaccuracy. For the integration to be stable, the time stephas to be so small that the time truncation error ismuch smaller than the spatial truncation error, and itis therefore necessary to perform many more time stepsthan would otherwise be the case. The choice of timeintegration scheme is, therefore, of crucial importancewhen designing an efficient weather forecast model,and this is also true when designing environmentalemergency response models. Early NWP models usedan explicit leapfrog scheme, whose time step is limitedby the propagation speed of gravitational oscillations.By treating the linear terms responsible for these os Corresponding author address: Dr. Andrew Staniforth, Rechemheen pr~vision num~rique, 2121 Route Trans-Canadienne porte 508,Dorval, Qu(~bec H9P I J3, Canada.dilations in an implicit manner, it is possible tolengthen the time step by about a factor of 6, at littleadditional cost and without degrading the accuracy ofthe solution [e.g., Robert ( 1969); Robert et al. ( 1972)].Such a scheme is termed semi-implicit. Nevertheless,the maximum stable time step still remains muchsmaller than seems necessary from considerations ofaccuracy alone (Robert 1981 ). Discretization schemes based on a semi-Lagrangiantreatment of advection have elicited considerable interest in the past decade for the efficient integration ofweather forecast models, since they offer the promiseof allowing larger time steps (with no loss of accuracy)than Eulerian-based advection schemes (whose timestep length is overly limited by considerations of stability). To achieve this end it is essential to associatea semi-Lagrangian treatment of advection with a sufficiently stable treatment of the terms responsible forthe propagation of gravitational oscillations. By associating a semi-Lagrangian treatment of adveaion witha semi-implicit treatment of gravitational oscillations,Robert ( 1981, 1982) demonstrated a further increaseof a factor of 6 in the maximum stable time step, atsome additional cost. This idea was demonstrated inthe context of a three-time-level shallow-water finitedifference model in Cartesian geometry, and resultedin the time truncation errors that were finally of thesame order as the spatial ones. Since Robert's seminal papers, the semi-Lagrangianmethodology for advectionqlominated fluid flowproblems has been extended in several important ways.The purpose of this paper is to summarize the fundamentals of semi-Lagrangian advection (section 2), todescribe its application to coupled sets of equations(section 3), to review recent extensions of the method(section 4) not covered in the discussions of the previous sections, and to draw some conclusions (section 5).c 1991 American Meteorological SocietySEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COT~ 22072. Semi-Lagrangian advection In an Eulerian advection scheme an observerwatches the world evolve around him at a fixed geographical point. Such schemes work well on regularCartesian meshes (facilitating vectorization and parallelization of the resulting code), but often lead tooverly restrictive time steps due to considerations ofcomputational stability. In a Lagrangian advectionscheme an observer watches the world evolve aroundhim as he travels with a fluid particle. Such schemescan often use much larger time steps than Eulerianones, but have the disadvantage that an initially regularly spaced set of particles will generally evolve to ahighly irregularly spaced set at later times (Welander1955), and important features of the flow may consequently not be well represented. The idea behindsemi-Lagrangian advection schemes is to try to get thebest of both worlds: the regular resolution of Eulerianschemes and the enhanced stability of Lagrangian ones.This is achieved by using a different set of particles ateach time step, the set of particles being chosen suchthat they arrive exactly at the points of a regular Cartesian mesh at the end of the time step. This idea gradually evolved from the pioneering work of Fjortoft(1952, 1955), Wiin-Nielsen (1959), Krishnamurti(1962), Sawyer (1963), Leith (1965) and Purnell(1976). Of the formulations introduced prior to thatof Purnell (1976 ), those of Krishnamurti (1962) andLeith (1965) are perhaps the most similar to those usedin present-day semi-Lagrangian advection schemes;however, as formulated they are only valid for Courantnumbers (C = I UI At/Ax) less than unity.a. Passive advection in 1D To present the basic idea behind the semi-Lagrangianmethod in its simplest context, we apply it to the 1Dadvection equationdF OF dx OFdt Ot +dt Ox - O, (1)where--= U(x,t), (2)dtand U(x, t) is a given function. Equation (1) statesthat the scalar F is constant along a fluid path (or trajectory or characteristic). In Fig. 1, the exact trajectoryin the (x-t) plane of the fluid particle that arrives atmesh point Xm at time t, + At is denoted by the solidcurve AC, and an approximate straight-line trajectoryby the dashed line A'C. Let us assume that we knowF(x, t) at all mesh points Xm at times tn - At and tn,and that we wish to obtain values at the same meshpoints at time t, + At. The essence of semi-Lagrangianadvection is to approximately integrate ( 1 ) along theapproximated fluid trajectory A'C. Thus,F(xm, ~ + at) - F(xm - 2am, tn -2At= o, (3)where am is the distance BD the particle travels in x intime At, when following the approximated space-timetrajectory A'C. Thus if we know am, then the value ofFat the arrival point Xm at time t, +/xt is just its valueat the upstream point Xm - 2Cirri at time t, - At. However, we have not as yet determined O/rn even if wehad, we only know F at mesh points, and generally itstill remains to evaluate F somewhere between meshpoints. To determine Otm, note that U evaluated at the pointB of Fig. 1 is just the inverse of the slope of the straightline A'C, and this gives the following O( At2) approximation to (2) (Robert 1981 ):am = AtU(Xm - am, in).(4)tt n+At X X X X .X..~ B '' Dtn X X X. (l m ~t n-At XA'~: X X ^ X X ~.,,,X x [2 x mm FIG. 1. Schematic for three-time-level advection. Actual (solid curve) and approximated (dashedline) trajectories that arrive at mesh point Xm at time tn + At. Here a,, is the distance the particleis displaced in x in time At.2208 MONTHLY WEATHER REVIEW VOLUME 119Equation (4) may be iteratively solved for the displacement am, for example by am(k+l) = AtU[Xm -- am(k), tn], (5)with some initial guess for amr, provided U can beevaluated between mesh points. To evaluate F and Ubetween mesh points, spatial interpolation is used. Thesemi-Lagrangian algorithm for passive advection in 1Din summary is thus: (i) Solve (5) iteratively for the displacementsfor all mesh points Xm, using some initial guess (usuallyits value at the previous time step), and an interpolationformula. (ii) Evaluate Fat upstream points Xm -- 2am at timetn -- At using an interpolation formula. (iii) Evaluate F at arrival points xm at time tn +using (3). We defer the discussion of interpolation details tosection 2d, and first generalize the above three-timelevel algorithm to forced advection in several spacedimensions (section 2b), and to two time levels (section 2c).b. Forced advection in multidimensions Consider the forced-advection problem dF -~- + G(x, t) = R(x, t), (6)where dF OF - + V(x, t). VF, (7) dt at dx dt V(x, t), (8)Here, x is the position vector (in 1-, 2- or 3D), V isthe gradient operator, and G and R are forcing terms.A semi-Lagrangian approximation to (6) and (8) isthen: F+ - F- 1 2/xt + 5 [G+ + G-] = R-, (9) ot = AtV(x -- a, t), (10)where the superscripts "+", "0" and ..... , respectively,denote evaluation at the arrival point (x, t + At), themidpoint of the trajectory (x - or, t) and the departurepoint (x - 2a, t - /xt). Here, x is now an arbitrarypoint of a regular (1-, 2- or 3D) mesh. The above is a centered O( At2) approximation to(6) and (8), where G is evaluated as the time averageof its values at the end points of the trajectory, and Ris evaluated at the midpoint of the trajectory. The trajectories are calculated by iteratively solving (10) forthe vector displacements ot in an analogous manner tothe 1D case for passive advection [Eq. (5)]. If G isknown (we assume that R is known since it involvesevaluation at time t), then the algorithm proceeds inan analogous manner to the 1D passive advection oneand is thus, (i) Solve (10) iteratively for the vector displacements a for all mesh points x, using some initial guess(usually its value at the previous time step), and aninterpolation formula. (ii) Evaluate F - AtG at upstream points x - 2aat time t - At using an interpolation formula. Evaluate2AiR at the midpoints x - a of the trajectories at timet using an interpolation formula. (iii) Evaluate F at arrival points x at time t +usingF(x, t + fit) = (F - AtG)l(x-2,,t-a,) + 2AtRI (x-~,t) - AtG[(x,t+at) =(F- AtG)- + 2AtR-- AtG+ (9')If G is not known at time t + fit (for instance if itinvolves another dependent variable in a set of coupledequations), then this leads to a coupling to other equations (more on this in section 3).c. Two-time-level advection schemes (and a pollutant transport application) Present semi-Lagrangian schemes are based on discretization over either two or three time levels, andthus far we have restricted our attention to three-timelevel schemes. The principal advantage of two-timelevel schemes over three-time-level ones is that theyare potentially twice as fast. This is because three-timelevel schemes require time steps half the size of twotime-level ones for the same level of time truncationerror (Temperton and Staniforth 1987). It is, however,important to maintain second-order accuracy in timein order to reap the full benefits of a two-time-levelscheme (since enhanced stability with large time stepsis of no benefit if it is achieved at the expense of diminished accuracy). Early two-time-level schemes forNWP models unfortunately suffered from this deficiency (e.g., Bates and McDonald 1982; Bates 1984;McDonald 1986). The crucial issue is how to efficientlydetermine the trajectories to at least second-order accuracy in time (Staniforth and Pudykiewicz 1985;McDonald 1987). This problem arises in the context of self-advectionof momentum. To see this we reexamine the algorithmof section 2a for 1D advection. Provided U is knownat time t,, independently off at the same time, thenit is possible to evaluate the trajectory, and then leapfrogthe value of F from time t~ - At to t~ + At, withoutknowing any value of F at time tn. Proceeding in thisway, F(t~ + 3/xt) is then obtained using values of F(t~+ At) and U(t~ + 2At). Thus we have two decoupledindependent integrations, one using values off at evenSEPTEMBER1991 ANDREW STANIFORTH AND JEAN C(bT~ 2209time steps and U at odd time steps, the other usingvalues off at odd time steps and U at even time steps.Either of these two independent solutions is sufficient,thus halving the computational cost, and we obtain atwo-time-level scheme (for the advected quantity F)by merely relabeling time levels t~ - At, tn and tn + At,respectively, as tn, t, + At/2, and tn + At (see Fig. 2).Note that values of U (assumed known) only appearat time level tn + At/2, and they are solely used toestimate the trajectories. This is the essence of the 2D advection-diffusionalgorithm described and analyzed in Pudykiewicz andStaniforth (1984). It led to the development of a threedimensional pollutant transport model (Pudykiewiczet al. 1985), where a family of chemical species areadvected and diffused in the atmosphere using windsand diffusivities: these are either provided by a NWPmodel (for real-time prediction) or from analyzed data(for postevent simulations). This model is designed toprovide real-time guidance in the event of an environmental accident and has been used to successfully simulate the dispersion of nuclear debris from the Chernobyl reactor accident (Pudykiewicz 1989). It hasevolved into Canada's Environmental Emergency Response Model (Pudykiewicz 1990). Returning to the problem of self-advection of momentum, the above argument breaks down in the special case where F = Uin ( 1 ) orF = V in (6); i.e., whenthe transported quantity U or - is advected by itself,as is the case for the momentum equations of fluiddynamic problems in general, and NWP models inparticular. This problem was addressed simultaneouslyand independently by Temperton and Staniforth(1987) and McDonald and Bates (1987), opening theway toward stable and accurate two-time-level schemes.The key idea here is to time extrapolate the winds [withan O( At2)-accurate extrapolator] to time level t + At/2 using the known winds at time levels t and t - At:these winds are then used to obtain sufficiently accurate[O( At2)] estimates of the trajectories, which in turnare used to advance the dependent variables from timelevel t to t + At. Thus, the two-time-level algorithm tosolve (6)-(8), analogous to the three-time-level onegiven by (9)-(10), is (see Fig. 2) F+ - F- 1 -- + [G++ GO] =Rl/2, at ~ (l~)where a = AtV*(x -- or/2, t + At/2), (12)V*(x, t + At/2) = (3/2)V(x, t) -- (l/2)V(x, t -- At) + O(At2); (13)the superscripts "+", "1/2", and "0" now, respectively,denote evaluation at the arrival point (x, t + At), themidpoint of the trajectory (x - a/2, t + At/2), andthe departure point (x - a, t), and a is still the distancethe fluid particle is displaced in time At. In the above formulation the evaluation of R(1/2)involves extrapolated quantities and, therefore, couldpotentially lead to instability. Temperton and Staniforth (1987) did not find this to be a problem whensome weak nonlinear metric effects were evaluated inthis way in a shallow-water model integrated on a polarstereographic projection, but it seems preferable toevaluate all nonadvective terms (i.e., G in the above)as time averages along the trajectory whenever possible.[Subsequently C6t6 ( 1988 ) showed how to avoid evaluating the above-mentioned metric terms in terms ofextrapolated quantities.] However, Higgins and Bates(1990) report that evaluating the product term (of thegeopotential perturbation and divergence) in the continuity equation of a global shallow-water model usingtime-extrapolated quantities [ as in Bates et al. (1990) ]leads to the growth of computational noise. An alternative solution is to discretize the continuity equationin logarithmic form as in C6t6 and Staniforth (1990), tt n+At C X X X X X ~li ~ ~t n+At/2- X X x ~,t n x~'- - X X X X X AA' ', ~ ~ m ~'~ 12 I > -- X Xm m Xm ~G. 2. Schematic for two-time-level advection. Actual (solid cu~e) and approximated (dashedline) trajectories that a~ve at mesh point x~ at time t~ + ~t. Here a~ is the distance the panicleis displaced in x in time ~t.2210 MONTHLY WEATHER REVIEW VOLUME 119at the price of making the elliptic boundary-valueproblem of the semi-implicitly treated terms mildlynonlinear: this has the advantage that it retains O(/xt2)accuracy because it is still a centered approximation.A further possible alternative is discussed in section4d. Note that when all nonadvective terms are evaluated implicitly as time averages along trajectories, thenextrapolated quantities are used solely for the purposeof obtaining a sufficiently accurate estimate of the trajectories. Temperton and Staniforth (1987) examined severalalternative ways of extrapolating quantities for thepurpose of estimating trajectories. They found thatthose methods that keep a particle on its exact trajectoryfor solid-body rotation seem to give better results forthe more general problem than those that do not. Inparticular, they found it advantageous to use a threeterm extrapolator (using winds at times t, t - At, andt - 2zXt to obtain an extrapolated wind at t + At/2)instead of the two-term extrapolator (13). They alsofound that time-extrapolating winds along the trajectory (their method 4) is less accurate than time-extrapolating winds at mesh points as in (13).d. Interpolation A priori, any interpolation could be used to evaluateF and U (or V) between mesh points in the above algorithm. In practice the choice of interpolation formulahas an important impact on the accuracy and efficiencyof the method. Various polynomial interpolations havebeen tried including: linear; quadratic Lagrange; cubicLagrange; cubic spline; and quintic Lagrange. For step (ii) of the algorithm, it is found (see e.g.,Purnell 1976; Bates and McDonald 1982; McDonald1984; and Pudykiewicz and Staniforth 1984, for analysis) that cubic interpolation is a good compromisebetween accuracy and computational cost. While quadratic Lagrange interpolation is viable and was usedin most of the early studies (e.g., Krishnamurti 1962,1969; Leith 1965; Mathur 1970, 1974; Bates andMcDonald 1982), cubic interpolation has been widelyadopted in recent studies (e.g., Robert et al. 1985;McDonald 1986; Bates and McDonald 1987; Ritchie1988; Ctt6 and Staniforth 1988; Bates et al. 1990).Cubic interpolation gives fourth-order spatial truncation errors with very little damping (it is very scaleselective, affecting primarily the smallest scales),whereas linear interpolation (see McDonald 1984 fordiscussion) has unacceptably large damping (it is alsoscale selective, but has a much less sharp response).Cubic spline interpolation has the useful property thatit conserves mass for divergence-free flows (Bermejo1990). Purser and Leslie ( 1988 ) recommend using atleast fourth-order (i.e., cubic) interpolation, and haveused quintic interpolation in their recent work (Leslieand Purser 1991 ). Improving the order of the interpolation formally increases the accuracy, but at additional cost, and the law of diminishing returns ultimately applies. For step (i), the order of the interpolation is muchless important. Theoretically, McDonald (1987) hasshown that one should use an interpolation of orderone less than for step (ii); e.g., quadratic interpolationof U when using cubic interpolation of F. In practicehowever, in the context of both passive advection andcoupled systems of equations in several spatial dimensions, it is found (Staniforth and Pudykiewicz 1985;Temperton and Staniforth 1987; Bates et al. 1990) thatit is sufficient to use linear interpolation for the computation of the displacements, when using cubic interpolation for F, which is very economical. It is alsofound that there is no advantage in using more thantwo iterations for solving the displacement equation[step (i)]. McDonald (1987 ) has shown theoreticallythat it is not necessary to use the same order of interpolation for each iteration. For example, it is moreeconomical and no less accurate to perform the firstiteration using linear interpolation and the second usingquadratic, than to use quadratic interpolation for both. Pudykiewicz et al. (1985) have shown that a sufficient condition for convergence of the iterative solutionof step (i) is that At be smaller than the reciprocal ofthe maximum absolute value of the wind shear in anycoordinate direction. Thus At < [max(I uxl, l uyl, I vxl,Ivyl)]-x for 2D flow, where u and v are the two windcomponents, and the time step of semi-Lagrangianschemes is not only limited by accuracy considerations(i.e., temporal discretization errors) but also by properties of the flow (i.e., wind shear). They estimated forNWP and long-range transport of pollutants problemsthat convergence is assured provided/xt is less than 3h, which is an order of magnitude larger than the maximum time step permitted by Eulerian advectionschemes in analogous circumstances.e. Stability and accuracy (and connection with other advection methods) Analyses of the stability properties of the semi-Lagrangian advection scheme (e.g., Bates and McDonald1982; McDonald 1984; Pudykiewicz and Staniforth1984; Ritchie 1986, 1987) show that the maximumtime step is 'not limited by the maximum wind speed,as is the case for Eulerian advection schemes, and consequently it is possible to stably integrate with Courantnumbers (C = I UI At/ax) that far exceed unity. Toillustrate this point we reproduce (with permission)the results of Bermejo (1990) for the slotted cylindertest of Zalesak (1979). In Fig. 3a we show the slottedcylinder at initial time, and in Fig. 3b the correspondingresult after six revolutions of solid-body rotation atuniform angular velocity about the domain center. Theexperiment was conducted using a cubic-spline interpolator at a Courant number of 4.2, which is considSEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COTI~ 2211FIG. 3. The "slotted" cylinder, (a) at initial time and (b) after six revolutions.erably larger than that of an Eulerian advection scheme.This is recognized as being a challenging test, and theresult is remarkably good. In particular the results illustrate the scheme's ability to handle sharp discontinuities without disastrous consequences (even thoughit was not designed specifically to do so) and the absence of noticeable dispersion problems (which aretypically present for Eulerian advection schemes). This behavior in the presence of discontinuities ornear discontinuities was also observed in the study ofKuo and Williams (1990) for a scale collapse problem.They concluded that semi-Lagrangian schemes are tobe preferred to Eulerian schemes for this kind of problem since they have much smaller dispersion errors(which are localized around the shock) and can beintegrated with significantly longer time steps. This isan important finding since, as pointed out by one ofthe reviewers, there is a mistaken belief that semi-Lagrangian schemes are only good for smooth flows. Theirfindings clearly show that this is not the case. In a similar vein, Ritchie (1985) argued that the localizationof errors to the regions where the gradients are strongestwhen semi-Lagrangian advection is used is a desirableproperty that may be advantageously exploited for thetreatment of moisture transport in NWP models, sincelarge local gradients frequently occur in moisture fields(e.g., at fronts). He reported that semi-Lagrangian adovection led to better results than Eulerian advection inthe context of a 48-h forecast. In general it is found that semi-Lagrangian advectionis competitive with Eulerian advection with respect toaccuracy, but it has the added advantage that this ac2212 MONTHLY WEATHER REVIEW VOLUME II9curacy can be achieved at less computational cost, sincemodels can be integrated stably with time steps thatfar exceed the maximum possible time steps of Eulerianschemes. The aforementioned stability analyses showthat semi-Lagrangian advection schemes have verygood phase speeds with little numerical dispersion, butcontrary to some Eulerian schemes (e.g., leapfrog-basedschemes) there is some damping due to interpolationas discussed in section 2d. This damping is fortunatelyvery scale selective (at least when using high-order interpolators). McCalpin ( 1988 ) has theoretically compared this damping with more traditional forms suchas Laplacian and biharmonic dissipation, and derivedsome criteria to ensure that the damping due to semiLagrangian advection is less than that due to the moretraditional forms. In practice Ritchie ( 1988 ) and C6t6and Staniforth (1988) have found that semi-Lagrangianintegration schemes have three times less damping thana typical Eulerian global medium-range forecast modelrun at typical resolution with a typical biharmonic dissipation. Semi-Lagrangian advection is intimately connectedwith several other advection methods that have appeared in the literature over the years, including particle-in-cell (e.g., Raviart 1985 ) and characteristic Galerkin (e.g., Morton 1985; Karpic and Peltier 1990)methods. Indeed for uniform advection in 1D, thesimplest semi-Lagrangian advection scheme (usinglinear interpolation, and not recommended) is equivalent to both classical upwinding and to the simplestcharacteristic Galerkin method; and semi-Lagrangianadvection using cubic-spline interpolation is equivalentto the higher-order characteristic Galerkin methods ofMorton (1985) and Karpic and Peltier (1990), andalso to a particle-in-cell method described in Eastwood(1987). Further, under more general conditions (including nonuniform advection in 2- and 3D), Bermejo(1990) has shown that semi-Lagrangian advection using cubic-spline interpolation can be viewed as a particle-in-cell finite-element method. Several well-known Eulerian methods can also beinterpreted as being special cases of semi-Lagrangianones. Thus the Lax-Wendroff, Takacs (1985) thirdorder, and Tremback et al. (1987) schemes are, respectively, equivalent for 1D uniform advection tosemi-Lagrangian schemes with quadratic-Lagrange,cubic-Lagrange and n th-order Lagrange interpolation.Note, however, that these Eulerian methods are restricted to Courant numbers less than unity and areconsequently less general than their semi-Lagranglancounterparts. Although the semi-Lagrangian method is equivalentfor uniform 1D advection to several other methods,what distinguishes it from other methods is that it generalizes differently to nonuniform advection in multidimensions. The principal difference is the use of(10), introduced in Robert (1981), for the trajectorycalculations. Of particular importance is that the approximation of the trajectory equation (8) is O(At2)accurate. It is possible to use a simpler, and cheaper,O(zXt) accurate method to approximate the displacement equation (8) (as in e.g., Mathur 1970; and Batesand McDonald 1982) but this can dramatically deteriorate the accuracy of the scheme, as shown by Staniforth and Pudykiewicz (1985) and Temperton andStaniforth (1987), and analyzed by McDonald (1987).Consequently most, if not all, recent semi-Lagrangianschemes use an O(At2) method for discretizing thetrajectory equation.3. Application to coupled sets of equations To illustrate how semi-Lagrangian advection can beadvantageously used to solve coupled systems of equations, we describe its application to the discretizationof the shallow-water equations: dU -- + ckx -fV = 0, (14) dt dV dt + qSY + fU= O' (15) d ln- --~-- + Ux + Vy = 0, (16)where U and Vare the wind components, qb (=gz) isthe geopotential height (i.e., height multiplied by g) ofthe free surface of the fluid above a flat bottom, and fis the Coriolis parameter. These equations are often used in NWP to test newnumerical methods, since they are a 2D prototype ofthe 3D equations that govern atmospheric motions[they can be derived from them under certain simplifying assumptions (Pedlosky 1987) ]. They share several important properties with their progenitor. A linearization of the equations reveals that there are twobasic kinds of associated motion, slow-moving Rossbymodes (most of which affect the large-scale weathermotions, and which move to leading order at the localwind speed) and small-amplitude fast-moving gravitational oscillations (which are inadequately represented at initial time due to the paucity of the observational network). From a numerical standpoint thishas the important implication that the time step of anexplicit Eulerian scheme (e.g., leapfrog) is limited bythe speed of the fastest-moving gravity mode. Since foratmospheric motions this speed is six times faster thanthose associated with the Rossby modes that governthe weather, this leads to time steps that are six timesshorter than those associated with an explicit treatmentof advection. A time-implicit treatment of the pressuregradient term of the vector momentum equation [ second terms of (14) and ( 15 ) ] and horizontal divergenceof the continuity equation [ second and third terms of(16)], introduced in Robert (1969) and termed thesemi-implicit scheme, allows stable integrations withSEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COTI~ 2213no loss of accuracy using time steps that are six timeslonger than that of the leapfrog scheme. The price tobe paid for this increase in time-step length is the needto solve an elliptic boundary-value problem once pertime step; nevertheless this improves efficiency by approximately a factor of 5. Analysis shows that the maximum possible time-step length is then limited by theEulerian treatment of advection. Early applications of semi-Lagrangian advection tocoupled sets of equations (e.g., Krishnamurti 1962,1969; Leith 1965; Mathur 1970, 1974; Mahrer andPielke 1978) didn't take advantage of the enhancedstability properties of the method, since the modelswere formulated in such a way that they were not, inthe terminology of Bates and McDonald (1982),"multiply upstream" and so the Courant number (associated with the treatment of advection) was alwaysless than unity. Nevertheless these studies did demonstrate that semi-Lagrangian advection is an acceptably accurate method for advection. Robert (1981)reasoned that since semi-Lagrangian advection is stablefor Courant numbers significantly larger than unity, itshould be possible to associate a semi-Lagrangiantreatment of vorticity advection with a semi-implicittreatment of the terms responsible for gravitational oscillations, and thereby obtain stable integrations withtime steps four to six times longer than that of a corresponding semi-implicit model employing an Euleriantreatment of advection. Using such a strategy he wasable to obtain a computationally stable solution witha 2-h time step (approximately four times longer thanthat of a corresponding semi-implicit Eulerian model),although there was some evidence of a small noiseproblem at the western inflow boundary. It was alsonoted that there was an inconsistency in the formulation inasmuch as the advecfion terms in the divergence and continuity equations were not evaluated using the semi-Lagrangian technique, and the questionof accuracy (as opposed to stability) was deferred to alater study. It turned out [Robert ( 1982)] that the Robert ( 1981 )integrations included a divergence diffusion term anda time filter, and that when these were removed aninstability was observed. This was attributed to twofactors: the explicit treatment of the Coriolis terms,and the application of the semi-Lagrangian techniqueto only the vorticity equation. To remedy these twodeficiencies, Robert (1982) introduced a revised formulation using the primitive (instead of the differentiated vorticity/divergence) form of the equations together with a semi-Lagrangian treatment of all advectedquantities and an implicit treatment of the Coriolisterms. This was done in the context of a three-timelevel scheme where the metric terms of the momentumequation were treated explicitly at the midpoint of thetrajectories [cf., R in (9)] and all other nonadvectiveterms as time averages of values at the end points ofthe trajectories [cf., G in (9)]. A stability analysis wasgiven to demonstrate that this scheme sh6uld be stablewith time steps that exceed those of the gravitational,advective, and inertial limits, and this was verified insample integrations. To illustrate the application of the semi-Lagrangianmethod we discretize the shallow-water equationsusing a two-time-level semi-implicit semi-Lagrangianscheme, which permits a further doubling of efficiencywith respect to the Robert (1982) algorithm at no extracost. For simplicity we describe the scheme in planegeometry, and it is then formally equivalent to that ofTemperton and Staniforth (1987) with the map-scalefactor set to unity. In spherical geometry the discretization is a little more complicated due to the appearance of metric terms in the momentum equations.These can be trivially absorbed into the formulationgiven below, using either the approach of Ritchie(1988) or that of C6t~ (1988) and Bates et al. (1990)[see section 4c for a further discussion of this point].Thus,U+ - U- -x+ + qbx- 1+ [(fv)+ + (fv)-] = 0, At 2 2 (17)V+ V0 + 4~Y+ + ~bY- + z1 [(fU)+ + (fU)-] = 0, At 2 2 (18)ln~b+ -- ln~b- + 1 At ~ [( Ux + Vy)+ + ( Ux + Vy)-] = 0, (19)where ( 14)-(16) have been discretized using ( 11 ) withR set to zero. Here advection terms are treated as timedifferences along the trajectories and all other termsare treated as time averages along the trajectories, leading to an O(At2)-accurate scheme. Where traditional(three-time-level) semi-implicit time discrefizafionshave an explicit time treatment of the Coriolis terms,the above discretization employs a time-implicit treatment [as in Robert (1982)] in order to achieve anO( At2)-accurate scheme: note that explicitly evaluatingthese terms at time t would not only reduce the accuracy to O (At) but would also lead to instability. Thetrajectories are computed using the discretized Eqs.(12)-(13) introduced by Temperton and Staniforth(1987) and McDonald and Bates (1987). For the 1D shallow-water equations it can be shown that there are three characteristic velocities in the cou pled set, one being the local wind speed that is asso ciated with the slow Rossby modes that govern weather motions, the other two being associated with the prop agation of gravitational oscillations. Thus the coupling of a semi-Lagrangian treatment of advection with a semi-implicit treatment of gravitational oscillations corresponds to integrating along the most important characteristic direction of the problem (i.e., that as2214 MONTHLY WEATHER REVIEW VOLUME 119sociated with the local wind speed); this is somewhatsimilar in spirit to a suggestion given on page 860 ofMorton (1985). Equations (17) and (18) can be manipulated to give U+ = At -- -~- [a~x+ + bey+] + known, (20) V+ _ At 2 lacy+ -- bqbx+] + known, (21)where a = [1 + (fAt/2)2] -~ and b = (fAt/2)a. Takingthe divergence of (20)-(21 ) and eliminating this in(19) then leads to the elliptic boundary-value problem: [(aqbx)~,+ (aqbv)y + (b-y)x - (bCx)y - 4 lnqb][ At2Jl(~,t+at) -- known. (22)We now summarize the above as the following algorithm: (i) Extrapolate - using (13) and solve (12) iteratively for the displacements am for all mesh points Xm,using values at the previous time step as initial guess,and an interpolation formula. Note that it is only neeessary to perform this computation once per time step,since the same trajectory is used for all three advectedquantities. (ii) Compute upstream (superscript 0) quantitiesin ( 17 )-(19) by first computing derivative terms (e.g.,Ux) and then evaluating quantities upstream (these twooperations are not commutative). Here it is more efficient to collect together all terms to be evaluated upstream in a given equation before interpolating (thedistributive law applies). (iii) Solve the elliptic boundary-value problem (22)for 4~(x, t + zXt). (iv) Back substitute 4~(x, t + At) into (20)-(21) toobtain U(x, t +/xt) and V(x, t + fit). The above elliptic boundary-value problem is weaklynonlinear and is solved iteratively using ~O at the previous time step as a first guess. It is only marginallymore expensive to solve than the Helmholtz problemassociated with traditional three-time-level semi-implicit Eulerian discretizations. The multigrid methodis particularly attractive for solving such ellipticboundary-value problems because of its relatively lowarithmetic operation count. Such a solver is describedin Barros et al. (1990) and was successfully employedin the global model of Bates et al.'(1990) using a discretizafion scheme very similar to that described above. Semi-Lagrangian advection has also been successfully coupled with the split-explicit method (Bates andMcDonald 1982) and the alternating-direction-implicitmethod (Bates 1984; Bates and McDonald 1987 ). Bothof these approaches have the virtue of being simplerthan the semi-implicit semi-Lagrangian one (there isno elliptic boundary-value problem), but unfortunatelythey do not perform as well. The split-explicit-basedmodel is less efficient (Bates 1984) than the alternatingdirection-implicit-based one, 'which in turn performsless well (Bates and McDonald 1987) than the semiimplicit semi-Lagrangian model of McDonald (1986).This latter scheme was adopted in the study of McDonald and Bates (1989), and it was subsequentlyfound (Bates et al. 1990) that its performance withlarge time steps was not as good as had been hoped.This was attributed (McDonald 19893 Bates et al. 1990)to a time-splitting error introduced in the momentumequation associated with the Coriolis terms. To date itappears that the best schemes arise from associatingsemi-Lagrangian advection with a semi-implicitscheme, and that time splitting is best avoided since itintroduces unacceptably large truncation errors forlarge time steps.4. Further advances When Robert (1981 ) proposed associating a semiLagrangian treatment ofadvection with a semi-implicittreatment of gravitational oscillations, it was thoughtthat this approach was restricted to three-time-levelschemes in Cartesian geometry using a finite-differencediscretization. This has happily proved not to be thecase, and in this section we discuss some importantextensions of the approach. Although important, theextension to two-time-level schemes has already beendiscussed in some detail, and will, therefore, only bebriefly discussed in this section in the context of otherextensions.a. Finite-element discretizations and variable resolu tion Pudykiewicz and Staniforth (1984) coupled semiLagrangian advection with a uniform-resolution finiteelement discretization of the diffusion terms in the solution of the 2D advection-diffusion equation, and thiswas extended to the 3D case in Pudykiewicz et al.( 1985 ). Staniforth and Temperton (1986) extendedthe methodology in the context of a coupled system ofequations (the shallow-water equations) in two ways.First they showed that in this context the semi-Lagrangian method can be coupled to a spatial discretization scheme other than a finite-difference one--viz.,a finite-element discretization--and second that it canalso be applied on a variable-resolution Cartesian mesh.A set of comparative tests demonstrated that with atime step six times longer it is as accurate as its analogous semi-implicit Eulerian version (Staniforth andMitchell 1978) when run with its maximum possibletime step (which in turn uses a time step six timeslonger than an Eulerian leapfrog scheme). A further doubling of efficiency was then demonstrated in Temperton and Staniforth (1987) by reSEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COT~ 2215placing the three-time-level scheme of the Staniforthand Temperton (1986) model with a two-time-levelone. Both these models use a differentiated (vorticitydivergence) form of the governing equations. This hasthe advantage of easily allowing variable resolution,but has the disadvantage of incurring additional interpolations and the need to solve two Poisson problems,resulting in an approximately 20% overhead whencompared to the ideal. This overhead can be eliminatedby the use of the pseudostaggered scheme proposed inCft6 et al. (1990) with no loss of accuracy.b. Noninterpolating schemes The interpolation in a semi-Lagrangian scheme, asmentioned previously, leads to some damping of thesmallest scales. While this damping is very scale selective, it may be argued that it would unacceptably degrade accuracy for very long simulations (e.g., manydecades in the context of a climate model). To addressthis problem Ritchie (1986) proposed a noninterpolating version of semi-Lagrangian advection. The basicidea here is to decompose the trajectory vector intothe sum of two vectors, one of which goes to the nearestmesh point, the other being the residual. Advectionalong the first trajectory is done via a semi-Lagrangiantechnique that displaces a field from one mesh pointto another (and, therefore, requires no interpolation),while the advection along the second vector is donevia an undamped three-time-level Eulerian approachsuch that the residual Courant number is always lessthan one. Thus the attractive stability properties of interpolating semi-Lagrangian advection are maintainedbut without the consequent damping. The noninterpolating scheme is also more efficient than a threetime level interpolating one, since there are only halfthe number of interpolations per time step (i.e., thereare no longer any interpolations associated with themiddle time level). Ritchie (1986) demonstrated thenoninterpolating scheme for a gridpoint shallow-watermodel on a polar-stereographic projection, and foundit to be more efficient and slightly more accurate thanan interpolating scheme run at the same resolution. The noninterpolating methodology is not restrictedto gridpoint discretizations and has also been successfully applied to spectral discretizations (Ritchie 1988,1990). This offers the possibility of retrofitting a noninterpolating semi-Lagrangian scheme into existingspectral models; there is, however, a minor technicalcomplication inasmuch as present spectral modelsgenerally use the differentiated vorticity-divergenceform of the equations whereas noninterpolating (andinterpolating for that matter) semi-Lagrangian spectralmodels employ the primitive form, and this necessitatessome changes to the spectral part of the formulation.There are, however, a couple of disadvantages of thenoninterpolating approach for problems where thesmall damping of the interpolating scheme is acceptable; this is generally the case for NWP applications,but probably not so for climate models (since they aregenerally run at much lower resolution and the damping is consequently more severe). First, the noninterpolating method has the dispersive properties of its Eulerian component, which are not generally as good asthose of interpolating semi-Lagrangian advectionschemes. Second, being based on a three-time-levelscheme it is potentially twice as expensive as a twotime-level interpolating scheme. The scheme proposed in Ran~i6 and Sindji6 (1989)is advertised as being a noninterpolating one, but thisis not in fact the case (Dietachmayer 1990; Bates 1990).Two schemes are derived for uniform advection in 1Dbased on the Lax-Wendroff and Takacs (1985)schemes. A close examination of these schemes revealsthat the Lax-Wendroff-based scheme is identical to asemi-Lagrangian one with quadratic Lagrange interpolation, whereas the Takacs-based scheme is identicalto a semi-Lagrangian one using cubic Lagrange interpolation. A simple and interesting idea, somewhatburied in the detail of the Ran~i6 and Sindji6 (1989)paper, is to show how to make a two-time-level Eulerianadvection scheme stable for Courant numbers greaterthan one. The idea, however, unfortunately seems tobe limited to the 1D case, since it is predicated on theassumption that a particle passes over a mesh point atsome time during the time interval of the time step,which assumption does not hold for multiply-upstreamparticles in 2D. It is of course possible to split the 2Dadvection problem into two passes of the 1D algorithm,but this then has the disadvantage that it usually introduces significant splitting errors for large time steps(see e.g., Williamson and Rasch 1989). An alternative way of viewing the noninterpolatingformalism of Ritchie (1986) is presented in Smolarkiewicz and Rasch (1990). They showed that it is possible to convert any advection algorithm into a semiLagrangian framework, thus permitting the use ofmuch larger time steps with the scheme for little additional cost. This interesting realization is of potentialbenefit for models whose maximum time step is limitedby an Eulerian treatment of advection. To demonstratethis idea they successfully extended the stability limitof the Tremback et al. (1987) family of algorithms. Inso doing they obtained a family of schemes that isequivalent to using a time-split semi-Lagrangianscheme with Lagrange interpolation. They also successfully extended the stability limit of a family of positive-definite monotone advection algorithms. However, after comparing results with those of semi-Lagrangian algorithms, they concluded that for problemswhere small undershoots and slight lack of conservationare acceptable, this family of positive-definite monotone algorithms cannot compete.2216 MONTHLY WEATHER REVIEW VOLUME 119c. Spherical geometry The convergence of the meridians at the poles of anEulerian finite-difference model in spherical geometryleads to unacceptably small time steps being requiredin order to maintain computational stability. The usualapproach to this problem is to somehow filter the dependent variables in the vicinity of the poles. Whilethis procedure does relax the stability constraint, it unfortunately deteriorates accuracy (e.g., Purser 1988).Ritchie (1987) demonstrated that it is possible to passively advect a scalar over the pole using semi-Lagrangian advection with time steps far exceeding the limiting time step of Eulerian advection schemes. Thispaved the way to applications in global spherical geometry. The first such application was to couple semiLagrangian advection with a spectral representation(i.e., expansion in terms of spherical harmonics) of thedependent variables to solve the shallow-water equations over the sphere (Ritchie 1988). A new problemarose here associated with the stable advection of avector quantity (momentum). The solution proposedin Ritchie (1988) is to introduce a tangent plane toavoid a weak instability due to a metric term. The diagnosis of this problem, which led to the tangent planealgorithm, is described in Desharnais and Robert(1990). An alternative solution, proposed by C6t6 (1988),is to use a Lagrange multiplier method. In this approachthe horizontal momentum equations of the shallowwater equations on the sphere are written in 3D vectorform using the undetermined Lagrange multipliermethod. These equations are time discretized directly,and the Lagrange multiplier is then determined fromthe discretized equations to ensure that motion is constrained to follow the surface of the sphere. This is incontrast with the usual approach where the Lagrangemultiplier is first determined from the continuousequations, followed by a discretization of the resultingequations. The procedure is applicable to any coordinate system and can also be extended to multilevelmodels. Both methods give good results that are almost indistinguishable in practice. More recently Bates et al.(1990) have described an approach based on the discretization of the vector form of the momentum equation. Although they state that their vector discretizationis somewhat different from the Lagrange multipliermethod of C6t6 (1988), it can be shown that the resulting algorithms are identical. It also turns out thatthe tangent-plane algorithm of Ritchie (1988) is identical to the Lagrange-multiplier one in the context ofa two-time-level scheme. Ritchie (1988) successfully integrated his shallowwater model with a time step six times longer than thelimiting time step of the corresponding Eulerian semiimplicit spectral model (which in turn uses a time stepsix times longer than that of an Eulerian leapfrogmodel). C6t~ and Staniforth (1988) then further doubled the efficiency of the Ritchie (1988) model by replacing its three-time-level scheme by a two-time-levelone analogous to that of Temperton and Staniforth(1987) for Cartesian geometry. The spectral method (i.e., expansion of the dependent variables in terms of spherical harmonics) hasbeen the method of choice during the past decade forthe horizontal discretization of global NWP models.However, the spectral method ultimately becomes veryexpensive at high enough resolution, due to the O(N3)cost of computing the Legendre transforms, where Nis the number of degrees of freedom around a latitudecircle. Finite-difference and finite-element methods onthe other hand have a potential O(N2) cost. This, andthe success of the semi-Lagrangian method in addressing the pole problem, suggests that it would be highlyadvantageous to use a semi-Lagrangian treatment ofadvection in a finite-difference or finite-element globalmodel for medium-range forecasting. A first tentative step in this direction was taken byMcDonald and Bates (1989), who introduced semiLagrangian advection into a two4ime-level global semiimplicit shallow-water model using the time discrefization of McDonald (1986). Although their schemewas stable with time steps that exceeded the limitingtime step of an Eulerian treatment of advection, theenhanced stability was unfortunately achieved at theexpense of accuracy. The degradation of accuracy isattributable to a time-splitting error introduced in themomentum equation associated with the Coriolisterms. The solution to this problem is to avoid timesplitting altogether and then the algorithm (Bates et al.1990) is very similar to that employed in Ritchie (1988)and C6t6 and Staniforth (1988), and results in significant improvements in accuracy for large time steps.Nevertheless, Bates et al. (1990) found it necessary touse divergence damping (with what appears to be arather large coefficient) in order to integrate to fivedays, suggesting that some accuracy and/or stabilityproblems still remain. C6t6 and Staniforth (1990) replaced the spectraldiscretization in the C6t6 and Staniforth (1988) modelby a pseudostaggered finite-element one [analogous tothat described in C6t6 et al. ( 1990)], to obtain a twotime-level semi-implicit semi-Lagrangian global modelof the shallow-water primitive equations. Its performance at comparable resolution matched that of theircorresponding 1988 model based on a spectral discretization, and this performance was achieved withoutrecourse to any divergence damping. By evaluating the product term (of the geopotentialperturbation and divergence) in the continuity equation using quantities at time t rather than at time t+/xt/2, but still evaluating it at the trajectory midpoint,Higgins and Bates (1990) show that it is possible toSEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COT~ 2217integrate the Bates et al. (1990) model with no divergence damping, although this formally reduces the accuracy of the treatment of this term to O(At) (but itis not a very important term in a shallow-water model).This result strongly suggests that the source of weakinstability observed in the Bates et al. (1990) results(without divergence damping) is somehow due to thisterm, but it still remains to explain why. We believethe explanation may be found in a stability analysisgiven in C6t6 and Staniforth (1988) for a somewhatsimilar time discretization, which analysis is valid forthe Bates et al. (1990) formulation. C6t6 and Staniforth (1988) showed that such a timediscretization is only stable provided -* > t~max, where~b* is the reference geopotential of the semi-implicitscheme and ~bmax is the maximum-possible value of thegeopotential. Thus where this condition is violated,such a time discretization is likely to be unstable, andthis is most likely to occur in the tropics where thegeopotential is generally largest. We believe that theqb* of the Bates et al. (1990) integrations (without divergence damping) is probably an average value of thegeopotential (rather than its maximum value) and thusviolates this stability criterion. An examination of thedivergence-damping-free result (given in Higgins andBates 1990) of the Bates et al. (1990) formulation reveals that the forecast is unstable in the tropics, butstable in the extratropics, consistent with the aboveargument. We, therefore, speculate that the Bates et al.(1990) formulation could be stabilized by merely increasing the value of the reference geopotential, andthat this solution would be preferable to the one proposed by Higgins and Bates (1990) since it is O(At2)[rather than O(zXt)] accurate.d. Conservation, shape preservation, and monotonicity An important issue for the discretization of globalmodels, and to a lesser extent for regional ones, is theextent to which various properties are conserved. Following the pioneering work of Arakawa (1966), manyfinite-difference schemes have been derived to maintaindifferent integral and local properties of the underlyingcontinuous equations (e.g., conservation of energy andenstrophy of the rotational part of the flow; conservation of energy and transformation between kineticand available potential energy; and prevention of spurious generation of vorticity by pressure-gradientterms). For spectral and finite-element models, someof these properties are immediate by virtue of a Galerkin framework [e.g., Bourke (1988); Yakimiw andGirard ( 1987)]. However, many of the properties thatare supposedly conserved by conservative schemes onlydo so under the assumption that there is a continuoustime discretization (which is not in general the case),and such schemes can and do go unstable, particularlyin the absence of any time filter (which damps thesolution). Nevertheless, such schemes have proven tobe quite successful for integrating models, and it is natural to ask the question (as one of the referees did) ofhow semi-Lagrangian schemes behave in this regard.While there doesn't appear to be a definitive answerto this question at this time, we summarize what ispresently known. From the theoretical standpoint, Bermejo (1990) hasshown that semi-Lagranglan schemes using cubicspline interpolation conserve mass for divergence-freeflows, and this is the only exact conservation propertyof semi-Lagrangian schemes of which we are aware.From the practical standpoint, several studies havemeasured how well certain quantities are conserved.Ritchie (1988), and to a lesser extent C6t6 and Staniforth (1988), have examined the conservation of energy in the context of 20-day integrations of three- andtwo-time-level shallow-water semi-Lagrangian globalmodels. As an indication of the energy conservationthat is considered acceptable in typical medium-rangeforecast models, an Eulerian spectral control modelwas run at T106 resolution with a X74 diffusion havingthe same coefficient as that used operationally at thisresolution by the European Centre for Medium RangeWeather Forecasts. It was found that this control modellost 3% of its total energy after 20 days, whereas thetwo semi-Lagrangian models at T 126 resolution onlylost 1%. At lower (T63) resolution, the three-time-levelinterpolating semi-Lagrangian model lost 5%, while thenoninterpolating version again only lost 1%. These results suggest that (i) interpolating semi-Lagrangian schemes conserve energy acceptably well atresolutions typical of state-of-the-art medium-rangeforecast models, but don't necessarily do so at resolutions typical of general circulation models, and (ii)noninterpolating semi-Lagrangian schemes conserveenergy acceptably well at both resolutions. Ritchie(1988) also compared the enstrophy spectra of variousT 126 integrations with that of a control T213 integration, and the results suggest that interpolating andnoninterpolating semi-Lagrangian models perform aswell or better than Eulerian models in this regard.However, these results and conclusions remain to beconfirmed for baroclinic models. Although most authors have adopted polynomialschemes for the interpolatory steps of semi-Lagrangianschemes, other interpolators are also possible. Williamson and Rasch (1989) and Rasch and Williamson(1990a) have examined several different possible interpolators, designed to better preserve the shape ofadvected fields and to maintain monotonicity. Theyperformed experiments in both Cartesian and sphericalgeometry, and concluded that the approach is viable.The principal difficulty with the shape-preserving andmonotonic approaches appears to be to decide how toprecisely determine the required attributes of the interpolator, and how to tailor it to respect them, sincethere is no universal best choice.2218 MONTHLY WEATHER REVIEW VOLUME 119 Williamson and Rasch (1989), Williamson (1990),and Rasch and Williamson ( 1990a, 1990b, 1991 ) havepursued this approach and thoroughly compared spectral and semi-Lagrangian schemes for the transport ofwater vapor in otherwise Eulerian, hydrostatic primitive-equation models, in the context of both mediumrange forecasting and general circulation modeling. Thetransport of water vapor is a stringent test of an advection scheme, since the moisture spectrum has muchmore variance at small scales than do wind and massspectra. Several important points have emerged fromtheir studies. Although spectral advection conserves mass well, itdoes so by producing serious undershoots (negativewater vapor) and overshoots (supersaturation), bothof which cause serious problems for the parameterization of moist processes, necessitating remedial measuresthat adversely affect conservation. Semi-Lagrangianadvection, particularly monotonic positive-definiteversions, conserves mass a little worse for passive advection, but has fewer undershoot/overshoot problemsand consequently interacts better with moist parameterizations in forced problems (having moisture sourcesand sinks). In particular (Williamson 1990) semi-Lagrangian advection leads to much smaller regions ofspurious light precipitation for medium-range forecasting. A particularly troubling result (Rasch and Williamson 1991 ) is that spectral and semi-Lagrangian advection schemes lead to different climatologies in generalcirculation models, with semi-Lagrangian advectionleading to a generally colder troposphere and warmerstratosphere. These differences are attributed primarilyto differences in the numerical treatment of verticaladvection, which subsequently lead to different cloudclimatologies. Rasch and Williamson ( 1991 ) argue thatsemi-Lagrangian vertical advection is formally fourthorder accurate, whereas Eulerian vertical advection isonly first-order accurate, and it should a priori be expected to be more accurate at fixed vertical resolution.They caution the reader, however, that it is difficult toseparate cause and effect in such comparisons, and underline the importance of continuing to improve numerical methods for climate models since they stillrepresent an important source of error. These authorshave indicated a preference for semi-Lagrangian advection of moisture to spectral advection for climatemodeling, and plan to use it in the next version of theNCAR Community Climate Model. This sensitivity ofthe climatology to the choice of advection scheme isalso of importance in NWP applications, but the problems are probably less severe due to the generally higherresolution of NWP models.e. 3D NWP applications Thus far we have mostly discussed the use of semiLagrangian advection for extending the limiting timestep of 2D applications for iNWP. To be useful themethod must also be applicable in 3D. A first step inthis direction was taken in Bates and McDonald(1982), where a semi-Lagrangian treatment of horizontal advection in a 3D (baroclinic primitive equations) model was coupled with a split-explicit timescheme in the Irish Meteorological Service's operationalmodel of the time. This was the first scheme to demonstrate the enhanced stability of semi-Lagrangian advection in a 3D model, and the first to be used operationally. However, it is only O(At) accurate and although stable with long time steps, the increase in timestep is consequently very much limited by accuracyconsiderations. The 3D model formulated in McDonald (1986) andimproved in McDonald and Bates (1987) [by modifying the trajectory calculations to make them O( At2)accurate, which improves accuracy and allows longertime steps ] is four times more efficient than the Batesand McDonald (1982) model for the same accuracyand replaced it operationally. Nevertheless, the resulting scheme still has some O(zXt) truncation errors andthe time step is, therefore, smaller than it would be foran O(At2) scheme [see discussion in the precedingsubsection of the McDonald (1986) scheme in thecontext of a global model]. Bates and McDonald(1987) have also coupled a semi-Lagrangian treatmentof horizontal advection in a 3D model with the alternating-direction method, but found in comparativeexperiments that it does not perform as well as theMcDonald and Bates (1987) scheme. Robert et al. (1985) introduced a three-time-levelO(At2)-accurate 3D limited-area gridpoint model witha semi-Lagrangian treatment of horizontal advection,and were able to successfully integrate with longer timesteps than had hitherto been possible; however, themodel had no mountains and a very simple parameterization of physical processes. This semi-Lagrangiansemi-implicit model does, however, demonstrate thepractical importance of achieving a truly O(At2)-accurate scheme. Although it employs a three-time-levelscheme, it is only marginally more costly per time stepthan the nominally two-time-level scheme of McDonald and Bates ( 1987 ) (which has several substeps)but can be integrated with longer time steps. In principle it should be possible to further double the efficiency of the Robert et al. (1985) algorithm by usinga two-time-level scheme. While such an improvementhas been achieved in 2D (e.g., C6t6 and Staniforth1988; Bates et al. 1990) the extension to 3D applications remains to be demonstrated. A somewhat similar model to the Robert et al.(1985) one, but with mountains included, is describedin Kaas (1987). It was reported that when strong windsblow over steep mountains, instabilities may appear ifthe linear part [V(~b + RTo lnps)] of the horizontalpressure-gradient term in sigma coordinates [see MesSEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COTI~ 2219inger and Janji6 (1985) for a related discussion of thisproblem in the context of Eulerian models] is evaluatedas the average of values at the endpoints [(x, t +/xt),(x - 2or, t - zXt)] of the trajectory, but the nonlinearpart [R( T - T0)X7 lnps] is evaluated at the midpoint(x - or, t). This behavior was attributed to a lack ofbalance (in the discrete approximation) between twolarge terms of opposite sign, due to their being evaluatedat different geographical points. The reported solutionto this problem is to evaluate the nonlinear part as theaverage of its values at the geographical points associated with arrival (x) and departure (x - 2a), bothvalues being taken at the intermediate time level t.This stratagem has also recently been incorporated inthe models of Robert et al. (1985), Tanguay et al.(1989), and Ritchie ( 1991 ). While this approach appreciably mitigates the problem, it is not at all clear that it resolves it completely.Coiffier et al. (1987) have studied it in the context ofa 2D linearized baroclinic model, and show that theuse ofsemi-Lagrangian, advection with large time stepsleads to an incorrect steady-state solution when themodel is orographically forced. Their analysis to explain this behavior also applies to the formulation proposed by Kaas (1987). It suggests that the seriousnessof the problem is a function of time step, wind speed,and detail (the larger the time step and wind speed,and the more detailed the orography, the worse is theproblem), and of whether the time scheme is a twoor three-time-level one (two-time-level schemes arebetter since the problem first occurs with time stepstwice as long as those of three-time-level schemes).Although this problem has not prevented semi-Lagrangian models from being integrated with larger timesteps than Eulerian ones while obtaining results ofequivalent accuracy, it does warrant further investigation. The time steps of the 3D above-mentioned modelsare limited by the stability of an explicit Eulerian treatment of vertical advection: or put another way, verticalresolution is limited when using a large time step (seeRitchie 1991 for an example). This is an importantlimitation. An ever-increasing emphasis in model development is being put on the parameterization ofphysical processes in general, and that of the moistturbulent planetary boundary layer in particular, andresults in ever-increasing demands on vertical resolution. To remove this limitation, Tanguay et al. (1989)proposed a three-time-level model that uses semi-Lagrangian advection in all three space dimensions: thisfinite-element regional model uses a time step that isthree-times longer than that of the corresponding Eulerian version (Staniforth and Daley 1979). It is currently used by the Canadian Meteorological Center tooperationally produce weather forecasts to 48 h twicedaily. Ritchie (1991) has recently introduced semi-Lagrangian advection into a three-time-level 3D globalspectral model in two different ways. The first uses aninterpolating semi-Lagrangian scheme in all three dimensions, as in Tanguay et al. (1989), whereas thesecond uses an interpolating semi-Lagrangian schemefor horizontal (2D) advection and a noninterpolatingscheme for vertical advection. These schemes are currently being introduced into the European Centre forMedium Range Weather Forecasts' spectral model. Hereports that the latter scheme is more accurate thanthe former for the experiments he conducted, due tothe former unduly smoothing fields in the verticalaround the tropopause. The seriousness of thissmoothing is a function of the resolution employedand of the order of the interpolator. The trend to highervertical resolution should diminish the importance ofthis source of error in the future. In the meantime itsuggests that higher-order vertical (i.e., quintic insteadof cubic) interpolation is possibly warranted as proposed in Leslie and Purser ( 1991 ).f Higher resolution and nonhydrostatic systems As computers become ever more powerful, it becomes possible to run models at higher and higher resolution. A time is approaching (Daley 1988) when itwill be possible to run current hydrostatic baroclinicprimitive-equation weather forecast models at resolutions for which the hydrostatic assumption can nolonger be assumed to hold. This motivates the need toefficiently integrate nonhydrostatic systems of equations for real-time forecasting applications over largedomains. Such systems admit acoustic modes, whichtravel much faster than either Rossby or gravity modes.Consequently if care is not exercised, the limiting timestep will be even more restrictive than that associatedwith an explicit primitive-equations model. This is because an explicit time treatment of the terms associatedwith the propagation of acoustic energy leads to a limiting time step that is much smaller than that associatedwith an explicit treatment of gravity-wave terms, completely eliminating the efficiency advantage of a semiimplicit semi-Lagrangian treatment of the gravityRossby mode terms. Since the acoustic modes carry very little energy, itis permissible to slow them down by the use of a timeimplicit treatment of the terms responsible for theirexistence, by analogy with the retarding of the gravitymodes by the semi-implicit scheme. This is the approach taken by Tanguay et al. (1990), who generalizethe semi-implicit semi-Lagrangian methodology for thehydrostatic primitive equations to the nonhydrostaticcase. They show that it is possible to integrate the fullycompressible nonhydrostatic equations (that are presumably more correct) for little additional cost, opening the way to highly efficient nonhydrostatic models.Note that this is a proof-of-concept study, since themodel employed has several important deficiencieswith respect to operational hydrostatic forecast models;2220 MONTHLY WEATHER REVIEW VOLUME II9it has no mountains, an extremely simple parameterization of physical processes, and very low vertical resolution (particularly in the planetary boundary layer)such that the time step is not unduly limited by theEulerian treatment of vertical advection (it is only thehorizontal advection that is treated in a semi-Lagrangian manner). Nevertheless, it represents a very important first step towards highly efficient nonhydrostatic forecast models. Increasing the resolution not only has importantimplications for the appropriate choice of governingequations, but also for the relative order of the temporaland spatial truncation errors. To date it has been foundadvantageous to couple semi-Lagrangian advectionwith a semi-implicit time scheme. This allows integration with larger time steps than would otherwise bepossible, chosen such that the temporal and spatialtruncation errors are of the same magnitude. We arethus presently in the position where the O( At2) temporal truncation errors are approximately equal inmagnitude to the O( Ax4) spatial ones (assuming cubicinterpolation in the semi-Lagrangian discretization ofadvection). For sake of argument, assume that thisis four times larger than the limiting time step of acorresponding semi-implicit model with an O(Ax4)accurate Eulerian advection scheme. We now ask theimportant question, what will be the size of the timestep (chosen such that the temporal and spatial truncation errors are of the same magnitude) of the semiimplicit semi-Lagrangian model for successive doublings of the spatial resolution, and how will this timestep compare to that of the corresponding semi-implicitEulerian model? For the first doubling of resolution, the spatial truncation errors will be decreased by a factor of 16 (=24).To ensure that the temporal truncation errors will beof the same magnitude as the spatial ones it is, therefore,necessary to reduce them also by a factor of 16 (=42),which implies reducing At by a factor of 4. Comparingthis time step now with that of the corresponding Eulerian model for the same doubling of resolution (wherethe time step is halved to respect the CFL stability criteflon), we see that it is only twice as large (whereasbefore the doubling of resolution it was four timeslarger) and the relative advantage of the semi-Lagrangian time step is thus halved. Repeating the argument for a second doubling wesee that the time step of the semi-Lagrangian modelnow equals that of the Eulerian one, and there is nolonger any advantage of time-step length for the semiLagrangian model. This is because the time step of thesemi-Lagrangian model is limited by accuracy considerations (it cannot be any larger otherwise the temporaltruncation errors would dominate), and it so happensthat the time-step of the Eulerian model is now limitedby both stability and accuracy considerations. For anyfurther increase in resolution beyond this critical resolution, the time steps of the two models will be identical since they will be determined solely by accuracyconsiderations. So we conclude that for this examplethere is no time step advantage for the semi-Lagrangianmodel at a quadrupling or more of resolution. Since the semi-Lagrangian model is somewhat moreexpensive per time step it is, therefore, debatable as towhether it would be advantageous in the above exampleto use a semi-Lagrangian treatment of advection atsuch resolutions rather than an Eulerian one [ althoughone might argue that semi-Lagrangian advection mightstill be advantageous (particularly for moisture transport) since there are fewer dispersion problems; seee.g., Ritchie (1985) and Rasch and Williamson( 1990a)]. The important implication of the above argument is that when the resolution of the Euleflanmodel is sufficiently high that the time step is governedby its O(At2) temporal truncation error (rather thanby its CFL stability criterion), it will become importantto increase the order of the time diseretization of thecorresponding semi-Lagrangian model. How this mightbe done is discussed in McDonald (1987). With that said, there is perhaps a weakness in theabove argument. We have assumed that the dominantsource of spatial truncation error is O(Ax4), and fora realistic NWP model this means that we are implicitlyassuming that the spatial truncation errors associatedwith the physical parameterization also behave asO(Ax4). In reality it is highly unlikely that a doublingof resolution reduces these errors by a factor of 16, andit is far more likely that they behave as O(/Xx2), inwhich case this would be the leading source of horizontal truncation error and the efficiency advantage ofthe semi-Lagrangian model would be maintained forall resolutions. A further important consideration is that the temporal discretization associated with the incorporationof physical processes in the models be of higher orderthan the present O(At) ones in order to benefit fromthe enhanced stability of semi-Lagrangian schemes.This is intimately connected to the intrinsic time scalesof the physical processes being parameterized (e.g.,cloud dynamics), and to what extent the chosen parameterizations are valid when using large time steps.It is also an issue for Eulerian models, but is moreserious for semi-Lagrangian models because of theirgenerally longer time steps. Present physical parameterizations appear to be rather sensitive to time-steplength, and this problem requires further work.5. Conclusions During the past decade much progress has beenachieved in using semi-Lagrangian advection to improve the efficiency of numerical models of the atmosphere. In this paper we have reviewed the semiLagrangian literature for atmospheric models, and havedrawn the following conclusions:SEPTEMBER 1991 ANDREW STANIFORTH AND JEAN COT~ 2221 1 ) The semi-Lagrangian methodology has been extended from finite-difference applications in Cartesiangeometry to finite-difference, finite-element, and spectral applications in both Cartesian and spherical geometry. 2) The extension to finite-difference and finite-element discretizations for global applications is particularly noteworthy, since such discretizations areasymptotically much cheaper at high resolution thanthe present method of choice, the spectral method. 3) At the present state of development the efficiencygains are more spectacular in 2D than in 3D. 4) Two-time-level schemes are inherently twice asefficient as three-time-level schemes. This has beenclearly shown in 2D, but the full benefits of two-timelevel schemes in 3D remain to be demonstrated. 5) Best results are obtained when coupling semiLagrangian advection to a semi-implicit treatment ofgravitational oscillations, rather than to splitting methods such as split-explicit and alternating-direction-implicit. It is crucial to avoid introducing O(/Xt) truncation errors in either the trajectory computations orthe discretization of the governing equations, in orderto fully reap the benefit (i.e., long time steps) of enhanced stability. 6) It is important to use the semi-Lagranglanmethod for vertical as well as for horizontal advection,in order to avoid unduly limiting vertical resolution. 7) Noninterpolating semi-Lagrangian schemes areattractive for climate applications, due to their lack ofdamping, a particularly important property for lowresolution simulations. 8) The semi-Lagrangian framework facilitates theincorporation of shape-preserving and monotonicschemes for moisture advection, because of the relatively small dispersion errors in the presence of discontinuities or near discontinuities. It also more easilypermits higher-order discretizations of vertical advection. 9) Semi-Lagrangian schemes do not in general formally conserve quantities, but the somewhat limitedevidence to date suggests that they perform acceptablywell in this regard. Although much progress has been achieved, thereremain several areas that warrant further investigation.The most pressing of these, in our opinion, is the incorporation of orographic (and other) forcing intosemi-Lagrangian models in such a way as to obtain thecorrect response to the stationary component. Thereis evidence to suggest that present schemes are deficientin this regard for Courant numbers greater than unity.Further research on physical parameterization is alsoneeded, to ensure that they adequately parameterizethe essential features of the underlying physical processes when using large time steps. Current parameterizations appear to be overly sensitive in this regard.As the spatial resolution is increased in a typicalsemi-Lagranglan model having, respectively, fourthorder spatial and second-order temporal truncation errors, it will become important to increase the order ofthe time discretization, otherwise there will ultimatelybe no time-step advantage with respect to an analogousEulerian model. Research on higher-order time discretizations is, therefore, desirable. A further consequence of increasing spatial resolution is that the hydrostatic approximation becomes increasingly lessvalid, thus motivating the efficient integration of nonhydrostatic systems of equations. An important firststep in this direction has already been achieved, butfurther research is needed to address several importantdeficiencies of such a nonhydrostatic model with respect to operational hydrostatic forecast models. Inparticular, it remains to demonstrate the viability ofthe method in the presence of realistic parameterizations of physical processes and a semi-Lagrangiantreatment of advection. In our review of the semi-Lagranglan literature wehave endeavored to present the strengths and weaknesses of the method, particularly with respect to accuracy, stability, and efficiency. This is not to say thatwe believe that the semi-Lagrangian approach outperforms traditional schemes in each and every respect;e.g., conservation properties. We endorse the suggestionof one of the referees that it would be valuable for thenumerical modeling community to compare the performance of various semi-Lagranglan and Eulerianschemes for the solution of a given set of problems.Such an activity is presently being organized (Williamson et al. 1991 ) within the framework of the CHAMMP(computer hardware, applied mathematics, modelphysics) initiative. Acknowledgments. 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