efﬁcient conservative global transport schemes for climate

AUGUST 2002 2059N A I R E T A L .

q 2002 American Meteorological Society

Efficient Conservative Global Transport Schemes for Climate andAtmospheric Chemistry Models

RAMACHANDRAN D. NAIR

Department of Marine, Earth and Atmospheric Sciences and Department of Mathematics, North Carolina State University,Raleigh, North Carolina

JEFFREY S. SCROGGS

Department of Mathematics, North Carolina State University, Raleigh, North Carolina

FREDRICK H. M. SEMAZZI

Department of Marine, Earth and Atmospheric Sciences and Department of Mathematics, North Carolina State University,Raleigh, North Carolina

(Manuscript received 23 April 2001, in final form 20 December 2001)

ABSTRACT

A computationally efficient mass-conservative transport scheme over the sphere is proposed and tested. Thescheme combines a conservative finite-volume method with an efficient semi-Lagrangian scheme based on thedimension splitting ‘‘cascade’’ method. In the regions near the poles where the conservative cascade procedurebreaks down, a globally conservative, but locally approximate scheme is used. This procedure is currentlyrestricted to polar meridional Courant numbers less than one. The resulting conservative cascade scheme isevaluated using a solid-body rotation test and deformational flow test, and found to be both accurate and efficient.Compared to the traditional semi-Lagrangian scheme employing a bicubic-Lagrange interpolator, the proposedscheme is considerably more accurate and almost twice as fast while conserving mass exactly.

1. Introduction

Current climate models require efficient schemes fortransport of humidity, liquid and solid water variables,and a number of chemical constituents. Monotonic ver-sions of semi-Lagrangian (SL) transport schemes haveproved to be an efficient numerical method for treatingthe advection process in climate and atmospheric chem-istry models (Williamson and Olson 1998). However, aserious disadvantage of most SL schemes is that theydo not formally conserve integral invariants as totalmass or total energy (Laprise and Plante 1995; Mach-enhauer and Olk 1997). The total mass, in particular,has been found to drift significantly if no correctionsare applied during longer integrations of the SL climatemodel (Moorthi et al. 1995). A posteriori correction isoften employed to enforce conservation, but may requiread hoc mass restoration or correction (Priestley 1993).Such a posteriori mass-fixing after upstream interpola-

Corresponding author address: Dr. Jeffrey S. Scroggs, Departmentof Mathematics, North Carolina State University, Campus Box 8205,Raleigh, NC 27695-8205.E-mail: [email protected]

tion is not only computationally expensive but also lacksrigor.

In SL methods, it is customary to use backward tra-jectories (Staniforth and Cote 1991) with arrival pointsat regular (Eulerian) grid points. The departure (La-grangian) points are, in general, not distributed uni-formly in the domain. Backward-trajectory SL schemescalculate the values at upstream Lagrangian points fromthe surrounding values at regular grid points by meansof interpolation. These values are the new estimates ata future time level. This idea can be extended for finite-volume based conservative schemes. Such schemes arebased on grid cells rather than grid points. Finite-volumebased SL schemes have gained prominence during therecent years (e.g., Rancic 1992; Laprise and Plante1995; Scroggs and Semazzi 1995; Leslie and Purser1995). However, only a few conservative transportschemes are available for spherical geometry (global)application (e.g., Prather 1986; Smolarkiewicz andRasch 1991; Rasch 1994), and (most of ) such schemesare computationally expensive. Hourdin and Armen-guad (1999) have reviewed various finite-volumeschemes for global atmospheric advection.

Recent developments in global conservative transport

2060 VOLUME 130M O N T H L Y W E A T H E R R E V I E W

schemes include the methods developed by Li andChang (1996), Lin and Rood (1996), Machenhauer andOlk (1996), and Rasch (1998). Recently, Nair andMachenhauer (2002), hereafter referred to as NM02,generalized the conservative global advection schemeintroduced by Machenhauer and Olk (1998). As imple-mented, this scheme was restricted to meridional Cour-ant numbers less than 2 near the poles. Among thechanges to the original Machenhauer and Olk (1998)scheme, NM02 introduced a new formulation in the sin-gular Lagrangian latitude rows, which removed anyCourant number restrictions. The main drawbacks ofthis approach, at least in the formulation of NM02, arethat it requires storage of global information for the 2Dremapping algorithm used and its computational effi-ciency is comparable to traditional SL schemes. Theconservative scheme presented in this paper does notrequire storage of global information; furthermore, theconservative remapping is based on the computationallyefficient dimension splitting ‘‘cascade methods,’’ intro-duced for interpolation in an SL scheme by Purser andLeslie (1991).

In this paper, a global conservative advection schemeis developed by combining a computationally efficientSL scheme with a mass-conservative 1D finite-volumemethod, and utilizing the cascade interpolation frame-work. The cascade interpolation replaces computation-ally expensive multidimensional interpolation associ-ated with the SL scheme by an efficient sequence of 1Dinterpolations. For example, a typical tensor productinterpolation in 3D, with the basic 1D Lagrange inter-polator, requires O(p3) operations per grid point perfield, where p is the formal order of accuracy of theinterpolator. However, a 3D cascade scheme can use thesame 1D interpolator three times, and needs only O(p)operations, a significant saving (Purser and Leslie 1991;Nair et al. 1999a,b).

Cascade interpolation is applicable to both backward-and forward-trajectory SL schemes (Purser and Leslie1991). Nair et al. (1999b), significantly reduced thecomputational overhead of cascade interpolation bysimplifying the intermediate grid point calculations.Nair et al. (1999a), hereafter referred to as NCS99, fur-ther extended the cascade interpolation scheme forspherical geometry application. The remarkable com-putational efficiency of the cascade scheme is exploitedfor designing efficient conservative SL schemes in thenew method presented here.

Leslie and Purser (1995) and Rancic (1995), hereafterreferred to as R95, have developed mass-conserving SLschemes (forward trajectory) in Cartesian geometry,based on cascade interpolation. However, their ap-proaches have serious limitations for conventionalspherical geometry. As in the present paper, R95 com-bines the accurate 1D finite-volume remapping methodbased on the piecewise parabolic method of Colella andWoodward (1984), with the computationally efficientcascade interpolation procedure of Purser and Leslie

(1991). However, the Lagrangian cells were not explic-itly defined, and as in the scheme of Leslie and Purser(1995) the values of the mass is assigned at the nodes(or ‘‘mass points’’) of the Lagrangian grid. In bothschemes, these values were transferred to the regularEulerian grid using their respective cascade scheme. Onthe surface of the sphere with a uniform latitude–lon-gitude grid, the area of the cells changes with latitudeand an application of both schemes is not obvious. An-other difficulty applying to both schemes at the polearises because some Lagrangian cells (i.e., cells boundedby Lagrangian grid lines) may not be well defined; thusin these regions, the conservative option of cascade in-terpolation procedure may break down.

The conservative cascade scheme presented in thispaper uses explicitly defined intermediate grid cells.This provides a complete geometric basis for the trans-port. The cell-averaged density is transferred cell to cellby employing 1D conservative remapping through thisintermediate grid cell system. For example, in 2D, two1D conservative remapping operations are requiredthrough an intermediate grid cell system to completethe transport.

The change in grid cell area with latitude is takeninto account by performing the remappings in a Car-tesian (l, m 5 sinu) coordinate system, where l is thelongitude and u is the latitude. This idea was introducedby Machenhauer and Olk (1996) and was used in Mach-enhauer and Olk (1998) and in NM02. In the regionnear the poles where the conservative cascade procedurebreaks down, we use the globally conservative but lo-cally approximate scheme like the one used in NM02.This procedure is currently restricted to polar meridionalCourant numbers less than one.

The layout for the remainder of the paper is as fol-lows. In section 2, the formulation of conservative semi-Lagrangian advection and the basic 1D finite-volumerepresentation of grid cells for cascade remapping isdescribed. The computational procedure for interme-diate grid generation and the conservative cascadescheme are introduced in section 3. Details of extensionto spherical geometry and special treatment for polarcaps are also discussed in section 3. Section 4 deals withthe numerical experiments using the global conservativecascade scheme, followed by a concluding discussionin section 5.

2. Conservative semi-Lagrangian advection

To introduce the basic idea of a conservative La-grangian scheme, consider the Eulerian flux form ofmass continuity and tracer conservation equations in theabsence of source or sink,

]r1 = · (rV) 5 0, (1)

]t

]rq1 = · (rqV) 5 0, (2)

]t


where r is the density of the fluid, q is the passive scalar(e.g., mixing ratio) per unit mass and V is the three-dimensional flow velocity. From (1) and (2), in the ab-sence of discontinuities, the Lagrangian form of the ad-vection equation

Dq5 0 (3)

Dt

can be derived, where D/Dt [ ]/]t 1 V · = is the total(Lagrangian) derivative. However, a semi-Lagrangiandiscretization of (3) cannot, in general, guarantee themass-conservation or other properties such as tracermonotonicity.

To formulate the conservative scheme, consider theintegral form (Chorin and Marsden 1993) of Eq. (1),

dr dV 5 0. (4)Edt A(t)

Here, the integral is over A(t), an arbitrary referencevolume of fluid (moving with the fluid), and dV is theelement of volume. Similarly for Eq. (2), the integralform is

drq dV 5 0. (5)Edt A(t)

The properties of integral forms (4) and (5) are discussedin Dukowicz and Baumgardner (2000). Since the so-lution procedure for these equations are similar, we fo-cus our attention on (4). The integral in Eq. (4) may beinterpreted as the mass of a quantity in the referencevolume to be advected along a trajectory. Discretizationof (4) can lead to a mass-conservative advection scheme(Laprise and Plante 1995; Dukowicz and Baumgardner2000). Moreover, Eq. (4) is suitable for a computation-ally efficient semi-Lagrangian time integration scheme.

a. Semi-Lagrangian formulation

The method is described in one spatial dimension,then extended to multiple dimensions. Let xj11/2, j 5 0,1, 2, . . . be the regularly spaced Eulerian grid pointswith grid spacing Dxj 5 xj11/2 2 xj21/2, and let tn, n 50, 1, . . . be the discrete times at which we will computethe solution. For a single time step, we define the de-parture point at time tn to be the location of ma-x*j11/2

terial originally located at xj11/2 at time tn11. That is, thedeparture point is obtained by setting 5 B(tn),x*j11/2

where B is obtained by solving the ordinary differentialequation

dB(t)n115 u(B(t), t), B(t ) 5 x (6)j11/2dt

backward in time. Here u(x, t) is the 1D velocity. Thesemi-Lagrangian time discretization of (4) in 1D leads to

r dx 5 r dx. (7)E En11 nA(t ) A(t )

The Eulerian and Lagrangian grid cells may be de-fined in terms of the Eulerian and Lagrangian gridpoints, respectively. Here, the jth Eulerian cell is definedby the grid points xj21/2 and xj11/2 as the corner pointssuch that the cell width is Dxj. The corresponding La-grangian cell width is D . Note that, the scalar fieldx*j(density) to be advected is cell centered and its initialvalue is a cell-averaged quantity. The grid points for-mally define each cell, and the values at these pointscan be estimated from the cell averages.

Stability and conservation of finite-volume basedconservative SL methods (hereafter referred to as cell-based methods) require choosing the time step Dt insuch a way that the trajectories do not cross in one timestep (Machenhauer and Olk 1997). This is analogous tothe Lipschitz criterion \]u/]x\Dt # 1 of traditional SLadvection schemes (Smolarkiewicz and Pudykiewicz1992). Moreover, this condition implies that the linesegments forming a cell do not cross as we traversetrajectories backward in time to the departure points(Scroggs and Semazzi 1995). This is a criterion for thecell to be ‘‘well-defined,’’ that is, cell walls should notintersect nor should the cell collapse at a single timestep (Lin and Rood 1996; NM02).

The computational domain is assumed to be spannedby nonoverlapping and well-defined Lagrangian cells(except at the polar regions, a topic covered later in thispaper). Thus, the cell-based method reduces to find themass in the Lagrangian cells using (7), that is, estimatingmass from the known Eulerian cells intercepted by theLagrangian cell. Now, representation of scalar fields inthe Eulerian cells is discussed.

b. Representation of density distribution

Cell-based methods can be constructed using a seriesof cell-averaged quantities rather than discrete point val-ues. The conservative cascade scheme (CCS) uses cellaverages to construct a unique piecewise parabolic poly-nomial for each dependent variable within a cell. Thearea beneath each curve represents the amount of mass,energy, or other fluid property. A judicious represen-tation was provided by Colella and Woodward (1984),the piecewise parabolic method (PPM). Apart from con-servation, other desirable properties such as monoto-nicity and positive-definiteness can be further satisfiedby the polynomials. The PPM has been utilized in me-teorological modeling [see, Carpenter et al. (1990); Ran-cic (1992, 1995); Laprise and Plante (1995); Lin andRood (1996); Machenhauer and Olk (1996, 1998);NM02], but our new scheme is the first application inthe global cascade framework.

Piecewise parabolic functions represent the densitydistribution in the Eulerian cells. Specifically, when x∈ [xj21/2, xj11/2], the density for the jth Eulerian cell

(x) is defined to benhj

n 2h (x) 5 a 1 a x 1 a x ,j 0 1 2 (8)


where the dependency of the coefficients a0, a1, and a2

on j and n has been suppressed. For efficient imple-mentation, these coefficients are modified by introduc-ing a normalized local variable (Carpenter et al. 1990),

x 2 x 1j21/2j 5 2 ,Dx 2j

for each cell such that j ∈ [21/2, 1/2]. Thus, the par-abolic function (8) representing the density distributionin the jth cell can be written as,

n 1n 2h (j) 5 h 1 dhj 1 h 2 j , (9)j j S [ ]12

where is the cell-averaged density,n

h jn

dh 5 h 2 h , h 5 6h 2 3(h 1 h ),R L S j L R

and hL 5 (21/2) and hR 5 (1/2) are the left andn nh hj j

right cell boundary (edge) values, respectively. The cell-averaged density (or simply average density) , is de-

njh

fined as,x 11/2j11/2

n 1n nh 5 h (x) dx 5 h (j) dj. (10)j E j E jDxj x 21/2j21/2

Using the definition of average density, the discreteanalogue of the conservation Eq. (7) is

n11 *h Dx 5 h Dx*,j j j j (11)

wherex*j+1/21* nh 5 h (x) dx, (12)j EDx*j x*j21/2

is the average density in the departure cell and D 5x*j2 is the width of the departure cell respec-x* x*j11/2 j21/2

tively. Thus the scheme reduces to determine the newcell average at time tn11, from Eq. (11) using the

n11h j

values at time tn.The PPM uses cubic interpolation of (including

nh j

values in neighboring cells) to obtain the cell boundaryvalues hL and hR (Colella and Woodward 1984). More-over, modifications of the coefficients similar to Car-penter et al. (1990) can be made to create a monotonicor positive-definite version.

c. Conservative remapping scheme

The calculation of the average density, , is begunn11

h j

by calculating the mass enclosed in the upstream La-grangian cell [ , ] at the departure time tn. Thex* x*j21/2 j11/2

mass is [cf. Eq. (12)]x*j11/2

* nh Dx* 5 h (x) dxj j Ex*j21/2

x* x*j11/2 j21/2

n n5 h (x) dx 2 h (x) dx, (13)E Ex xr r

where xr is a common reference grid point on the grid

line such that xr # . The Eq. (13) is convenient forx*j21/2

numerical evaluation. The mass integrated from xr to aLagrangian point is the accumulated mass (or cu-x*j21/2

mulative mass) towards the wall of the Lagrangian cellat the point , and can be computed as follows,x*j21/2

x* x*k21j21/2 j21/2n

n nh (x) dx 5 h Dx 1 h (x) dx, (14)OE l l E kl51x xr k21/2

where the reference point, xr, may be chosen as the firstgrid point from the left, and ∈ [xk21/2, xk11/2]. Thex*j21/2

mass in the Lagrangian cell is the accumulated masstowards the left wall subtracted from the accumulatedmass towards the right wall of the Lagrangian cell. Thus,the remapping process consists of the determination ofaccumulated mass for each Lagrangian cell wall using(14). The average density at new time level is given by(11).

A schematic illustration of the 1D remapping con-sidered here is given in Fig. 1 of NM02, also, the same1D remapping scheme will be used as a basic componentof the 2D CCS remapping in the following section. Thisremapping scheme has been used by R95, Machenhauerand Olk (1996, 1998), NM02 and others for 1D advec-tion. The cumulative mass approach used is similar tothe one introduced by Dukowicz (1984) for remapping.

3. The global conservative cascade scheme

In order to introduce the basic geometry of the CCSin 2D, consider an Eulerian grid system (orthogonalCartesian coordinates), defined by the m 3 n grid points(li, mj) with grid spacing Dli and Dmj in l and mdirections, respectively. For every grid point (li, m j),there exists a unique upstream Lagrangian point (lij,mij) in the domain. Four neighboring Eulerian gridpoints on the vertices of a rectangular region boundedby Eulerian grid lines defines an Eulerian cell, and thecorresponding exact Lagrangian cell is a region whosecorner points are the Lagrangian points. Figure 1 showssuch a (l, m)-grid system, where the Eulerian points arethe grid intersections and the Lagrangian points aremarked by filled circles. Consider the Lagrangian cellABCD shown in Fig. 1 with the corner points, (lij, mij),(li11,j, mi11,j), (li11,j11, m i11,j11), and (li,j11, mi,j11), re-spectively (our notation implies a small Courant number,however, the methods does not have this limitation). InFig. 1, horizontal curves are Lagrangian grid lines cor-responding to horizontal Eulerian grid lines. Hereafter,the deformed Lagrangian grid lines are referred to asthe Lagrangian latitudes. The vertical grid lines (l 5li) shown in Fig. 1 are Eulerian longitudes. Intersectionsof the Lagrangian latitudes and the Eulerian longitudescreate a hybrid grid system and the resulting grid pointsare defined as the intermediate grid points. In Fig. 1,the intermediate points are marked with 3.

The computational procedure is first described for a


FIG. 1. Schematic illustration of the cascade remapping scheme in 2D, where the filled circlesrepresent the Lagrangian grid points corresponding to the Eulerian grid points (li, mj). The curvesrepresent Lagrangian latitudes, and their intersections with the vertical grid lines (l 5 li) are theintermediate grid points, denoted by 3. The crosshatched rectangular regions show the intermediatecells. An approximated Lagrangian cell (computational cell) corresponding to the Lagrangian cellABCD is shown as the shaded polygon, whose sides are parallel to the Eulerian grid lines.

(l, m) Cartesian plane and then extended for sphericalgeometry.

a. Computational procedure

The CCS consists of approximating mass enclosed inthe Lagrangian cells by two conservative 1D remap-pings along each of the coordinate directions, respec-tively. For the computational procedure, the Lagrangiancells are approximated as polygons. The generation ofan intermediate grid is a key component of the CCS,because the local conservation and efficiency of thescheme depend upon the grid generation. First, we de-scribe the intermediate grid generation followed by theapproximation of the intermediate and Lagrangian cellsused for the conservative remapping.

1) INTERMEDIATE GRID GENERATION

Let (l,, ,j) be the point formed by the intersectionmof a Lagrangian latitude and an Eulerian longitude. Forexample, in Fig. 1, consider the Lagrangian latitudepassing through the Lagrangian points A(lij, mij) andB(l i11,j, m i11,j), and the Eulerian longitude l 5 li. Thenthe intermediate grid point lies on segment AB of theLagrangian latitude such that l, 5 li ∈ [l ij, l i11,j].However, the value of ij interpolated from the neigh-mboring m coordinates of A and B. A linear interpolationmethod for generating intermediate points (Nair et al.1999b) is an obvious choice, however, this approachmay not be sufficiently accurate for the nonuniform grid

considered here. To compute the location of the inter-mediate point, we employ a cubic-Lagrange interpola-tion, requiring four Lagrangian points along the La-grangian latitude for source points. Thus, to find (li,

ij), Lagrangian points (li21,j, m i21,j) and (li12,j, mi12,j)mto the immediate left of A and right of B, respectively,(see Fig. 1), can be used. By performing a 1D cubicinterpolation with {l i21,j, lij, li11,j, li12,j} as the nodesof interpolation, provides the value of ij correspondingmto the value l, 5 li. The collection of all such (l,,

,j) uniquely defines an intermediate grid system.m

2) INTERMEDIATE AND LAGRANGIAN CELLS

For computational efficiency, Lagrangian latitudesare approximated by lines drawn parallel to the Eulerianl axis. In Fig. 1, these are shown as dashed horizontallines. In the ith column, the south wall of the jth inter-mediate cell is defined to be j 5 ( ij 1 i11, j)/2. Sim-m m milarly, the north cell wall of the intermediate cell j11mis defined. The cell width in the m direction of the in-termediate cell is D j 5 j11 2 j. Thus, the inter-m m mmediate cells are rectangles and they fall in the columnsof the Eulerian cells. In Fig. 1, an array of intermediatecells corresponding to the region bounded by two ad-jacent Lagrangian latitudes is shown as the crosshatchedcontiguous rectangles. Note that the intermediate cellsspan the entire computational domain without overlap-ping or holes.

Now, the Lagrangian cells may be approximated interms of rectangular regions carved from adjoining in-


termediate cells. These approximated Lagrangian cellsare hereafter referred to as the computational cells. Eachcomputational cell is defined as a union of nonoverlap-ping adjacent rectangles from the contiguous interme-diate cells. In Fig. 1, a computational cell correspondingto the Lagrangian cell ABCD is schematically depictedby the shaded polygon located among an array of in-termediate cells (crosshatched regions). The east andwest boundary walls of the computational cell are de-fined as the straight line segments parallel to the m axis.For example, the west wall of the computational cell asshown in Fig. 1 (shaded polygon) lies on the verticalline ı 5 (lij 1 li,j11)/2. This approximates the curveljoining AD. Similarly, the east wall of the computationalcell is defined, and the horizontal width of the com-putational cell is D ı 5 ı11 2 ı. Note that, the northl l land south walls of the computational cells, respectively,are defined by the segments of the north and south wallsof the intermediate cells. With this construction, thecomputational cells are polygons with sides parallel toEulerian grid lines. The idea to approximate Lagrangiancells by polygons with sides parallel to either l or mEulerian grid lines was originally introduced in Mach-enhauer and Olk (1998) and was then later used inNM02 to increase the efficiency of their 2D remapping.

3) CONSERVATIVE CASCADE REMAPPING

For our 2D scheme, a cascade of two 1D remappingsare performed in a consistent way so that the mass trans-fer is conservative. In the present context, a 1D remap-ping in m direction followed by another 1D remappingin l direction is briefly described as follows with theaid of Fig. 1.

The 1D remapping in m direction transfers mass fromthe Eulerian cell to the intermediate cell. Consider theith column in Fig. 1, let the initial mass in the jth Eu-lerian cell be Mij 5 DmjDli, where is the givenn nr rij ij

cell-averaged density at time tn, and Dm jDl i is the cellarea. The first step for the conservative remapping iscalculation of the average density j from Mij such thath

j 5 Mij/Dmj, for each cell in the ith column. Here, wehhave suppressed the dependency of j on i. Next, 1Dhpiecewise parabolic profiles are constructed along thevertical column, utilizing j as the cell-averaged densityhfor each 1D cell having cell width Dmj. The mass inthe intermediate cell (here it may be logically consideredas a 1D cell with cell width D j) is determined usingmthe relation (11) combined with (13). This remappingscheme is employed for each column in the domain, andthe mass enclosed in the intermediate cells are deter-mined.

Remapping in l direction is performed in an analo-gous manner, where the mass from the intermediate cellsare transferred to the computational cells. Consider ahorizontal array of intermediate cells between two ad-jacent Lagrangian latitudes (as shown as the cross-hatched cells in Fig. 1). The width in the l direction of

the intermediate cell is Dl i and that of the computationalcell is D i. In order to perform the 1D remapping in lldirection, the average density per Dl i is computed foreach of the intermediate cells from the known valuesof mass, as described in the case of 1D remapping inm direction. Piecewise parabolic profiles are then fittedalong the rows in such a way that the area below theparabolas represent the mass. The mass in the compu-tational cell having cell width D i is computed usinglthe relation (11) combined with (13), as the differenceof the accumulated mass from a reference wall. Thus,the new average density at the arrival cell can ben11r ij

determined from the known mass, since the area of thearrival cell is already known, using Eq. (7).

The cascade procedure described above can be sum-marized as follows:

• Generate an intermediate grid system by finding theintersection of Lagrangian latitudes and Eulerian lon-gitudes.

• Determine intermediate and computational cells bydetermining the midpoint of the east–west interme-diate points and north–south Lagrangian points.

• Perform a 1D conservative remapping in m directionto transfer mass in the Eulerian cell to the intermediatecell.

• Perform a 1D conservative remapping along the ldirection to transfer mass in the intermediate cells tothe computational cells. This step completes the cas-cade cycle.

b. Extension to spherical geometry

A major difficulty in extending cell-based methodsto spherical geometry is the convergence of meridiansat the polar singularities of the spherical coordinate sys-tem. A special treatment is often needed to obviate thepole problems (e.g., Rasch 1994; Li and Chang 1996;Lin and Rood 1996). Here we restrict our attention tothe cases where the polar meridional Courant numberCu # 1 (i.e., the Lagrangian pole point is located withinor on the first latitude circle from the North or SouthPole).

It is convenient to introduce an area-preserving trans-formation from the latitude–longitude (l, u) grid withl ∈ [0, 2p], and u ∈ [2p/2, 1 p/2], to the (l, m) gridsystem with m 5 sinu. The variable m can be treatedas an independent variable such that m ∈ [21, 1]. Underthis transformation, the global integral for any arbitraryfunction f over the sphere (with radius a 5 1) takesthe following form

2p 1p/2

f (l, u) cosu dl duE E0 2p/2

2p 11

5 f (l, m) dl dm. (15)E E0 21


The resulting (l, m) coordinate system may be consid-ered similar to the Cartesian form with north and southpoles as grid lines, and periodic in the east–west direc-tion. This application of a Cartesian (l, m) coordinatesystem for conservative remapping on the sphere wasoriginally introduced by Machenhauer and Olk (1996)and was then later used in Machenhauer and Olk (1998)and in NM02. The (l, m) coordinates introduced herehave nonuniform resolution in the m direction. Note thatin the (l, m) grid system, S j Dm j 5 2 5 S j D j and SimDli 5 2p 5 S ı D ı, where D j and D ı, respectively,l m lare the grid spacing for the intermediate cell in the mdirection and that for the computational cell in l direc-tion.

c. Intermediate grid generation over the sphere

The intermediate grid points can be computed by us-ing the cubic interpolation method described in the pre-vious section. However, a second method based on agreat-circle approach (NCS99) is more computationallyefficient, but is less accurate. In NCS99, an exact meth-od for the determination of intermediate grid intersec-tions (for nonconservative SL schemes) over the sphereis given. However, this approach may not be suitablefor the present conservative cascade scheme since thismethod does not isolate polar regions containing thesingularity. We use a cascade cycle that isolates the polarregions.

In most SL global models, 3D Cartesian coordinates(X, Y, Z) equivalent to the spherical polar coordinates(l, u) are computed for solving trajectory equationsanalogous to Eq. (6) (Cote et al. 1998). In NCS99, La-grangian longitudes and Eulerian latitude circles areused for determining intermediate intersection, utilizingthe (X, Y, Z) coordinates. However, for the present ap-plication, intermediate points are defined as the inter-section of Lagrangian latitudes with Eulerian longitudes.The advantage of this definition is that the Lagrangianpole cap (that includes the singular point) can be isolatedby a region with known exterior boundary.

The intermediate grid point (l,, ,j) in spherical co-uordinate can be determined by using the great-circleapproach (NCS99), and then transformed into compu-tational domain such that (l,, ,j 5 sin ,j). The great-m ucircle plane that passes through the two points (lij, uij)and (li11,j, ui11,j) is given by

aX 1 bY 1 gZ 5 0, (16)

where the formulas for coefficients ( , , ) can be founda b gin NCS99. The trigonometric equivalent to (16) is

a cosu cosl 1 b cosu sinl 1 g sinu 5 0. (17)

From (17), the value ,j corresponding to l, of the in-utermediate point (l,, ,,j) that lies on the Lagrangianulatitude can be determined by using the following equa-tion,

2(a cosl 1 b sinl ), ,21u 5 tan . (18),, j [ ]g

Thus, a set of points (l,, ,, j) [or equivalently (l,, ,, j)]u mdefines an intermediate grid system over the surface ofthe sphere or on the (l, m) plane. The cell-to-cell masstransfer is performed through the intermediate grid dur-ing the cascade cycle.

d. Special treatment for polar caps

Since the Lagrangian cell which includes the singularpoint (Eulerian pole) is not well defined in the (l, m)plane, a special treatment is needed for determinationof mass in such singular cells. Here, a polar cap is aregion on the surface of the sphere that includes the polepoint. The exterior boundary of the Eulerian polar capis a latitude circle; in the (l, m) plane, this correspondsto a rectangular region. Both the exterior boundary ofthe Lagrangian polar cap and the intermediate polar cap,by design, are the same deformed Lagrangian latitude.In the (l, m) plane, for the northern hemisphere, this isa region bounded by the lines m 5 1, l 5 0, l 5 2pand a curve marking the south boundary (see NM02 forrelevant illustrations). Thus, the mass enclosed in theintermediate polar cap is the same as that in the La-grangian polar cap. This means that the total mass Tm,enclosed in the Lagrangian polar cap is computed duringthe first sweep of conservative remapping in m directionand does not change in the second phase. Therefore, Tm

can be redistributed to each of the constituent compu-tational cells in the Lagrangian polar cap. Only the totalmass enclosed in the Lagrangian pole cap is neededduring the cascade cycle; thus, the east and west wallsof the computational cells in this zone are approximatedas straight line segments parallel to the m axis, andjoining the south boundary and the grid line m 5 1.Now, the redistribution method remains to be discussed.

The total mass Tm in the Lagrangian polar cap isredistributed to each of the constituent cells by usingthe method considered by NM02. To find the weightsfor redistribution, a regular SL scheme combined witha cubic-Lagrange interpolation is used (NM02). La-grangian cell centers are treated as upstream points andthe values of the mass in the cells are considered as theadvecting field. Then the values mk at the upstream po-sitions are estimated with cubic-Lagrange interpolation,and then used for constructing weights of mass redis-tribution. The weight is defined to be

|m |kw 5 . (19)k |m |O kk

Computed mass in the constituent Lagrangian cell isthen 5 Tmwk, such that Sk 5 Tm (NM02), andm mC Ck k

leading to a conservative scheme.The accuracy near the pole may be improved by in-

troducing additional points in the m direction on the east


and west walls. Thus, a series of Lagrangian cells alongthe latitude circle on the polar zones, hereafter referredto as Lagrangian belt (NM02), may be subdivided inm direction at a known ratio, by linearly interpolatingthe Cartesian coordinates (X, Y, Z) of the cell wall’snorth and south corner points. The values of (l, m) atthe newly introduced positions are calculated from theCartesian coordinates. This avoids computing additionaltrajectories for the new points (NM02).

4. Numerical experiments

The computational efficiency, accuracy, and robust-ness of the algorithm presented earlier are demonstratedthrough a series of advection experiments that utilizesolid-body rotation and deformational flow. A scalarfield c, that may be considered as density or tracer dis-tribution, is advected.

For the present study, the Lagrangian belts enclosingthe Lagrangian polar cap have been subdivided in them direction as described previously. In the north polarzone, the belt surrounding the Lagrangian polar cap hasbeen subdivided into three sub-belts. This is done byintroducing two sets of points on the east–west wallsaround the latitude. Similarly, the belt next to it (south)has been subdivided into two sub-belts. Also in the southpolar zone, new sets of sub-belts have been symmet-rically introduced.

a. Solid-body rotation test

The details of the solid-body rotation experiment arein Williamson et al. (1992). The parameters of the ex-periment used here are the same as NCS99, with velocitycomponents

u 5 u (cosa cosu 1 sina cosl sinu), (20)0

y 5 2u sina sinl, (21)0

where a is the angle between the axis of solid-bodyrotation and the polar axis of the spherical coordinatesystem (Williamson et al. 1992). When a 5 0, the axisof rotation is the polar axis, and when a 5 p/2 it is inthe equatorial plane. The initial scalar distribution is thecosine bell,

1/2[1 1 cos(pr/R)], if r , Rc(l, u) 5 (22)50, ifr $ R

where,21r 5 cos [sinu sinu 1 cosu cosu cos(l 2 l )],c c c (23)

is the great-circle distance between (l, u) and the bellcenter, initially taken as (lc, uc) 5 (3p/2 2 Dl/2, 0).The bell radius R is 7p/64 as in Rasch (1994) andNCS99. The exact solution has the same geometry asthe initial condition. Accuracy is measured as relativeerror in the l1, l2, and l` norms, and max and min,consistent with Williamson et al. (1992).

The domain consists of a 128 3 65 grid where thefirst and last latitudinal grid lines are actually the southand the north poles, respectively. Thus, there are 1283 64 grid cells spanning the entire spherical domain.The scalar field is initially distributed at the cell centers(cell average). The time step and the value of the max-imum wind speed u0 are chosen such that the angularvelocity of the rotational flow is v 5 2p/256 radiansper time step. That means 256 time steps are requiredfor one complete revolution around the globe. When thewind is in the pole-to-pole direction (a 5 p/2), themeridional Courant number is Cu 5 0.5.

b. Results of solid-body rotation

Numerical experiments have been performed for a 50, p/2 2 0.05 and p/2 with exact trajectories. Experi-ments were conducted for intermediate grids generatedwith either cubic-Lagrange interpolation or the great-circle approach (see section 3). The great-circle ap-proach is potentially more efficient since it uses theknown Cartesian coordinate (X, Y, Z); however, the in-termediate grid generated with 1D cubic-Lagrange in-terpolation was found to be more accurate. For the sakeof brevity, only the results from the intermediate gridgenerated with cubic-Lagrange interpolation are includ-ed. Three different options of the CCS scheme are com-pared to the results with others, these options are themonotonic (CCS-M), the positive-definite (CCS-P), andthe CCS without any filtering (CCS-N).

The results from solid-body rotation of a cosine bellafter one revolution when flow is in the pole-to-poledirection are displayed in Fig. 2 (the contour valuesvaries from 0.1 to 0.9 with uniform increment 0.1). Theexact solution is displayed as dashed contours. In Fig.2, the top-left panel shows the numerical solution withCCS-N and the solution with CCS-P is shown in thetop-right panel. The bottom-left panel of Fig. 2 showsthe solution with the CCS-P, but the rotation parametera is set to p/2 2 0.05 so that the center of the cosinebell deviates from the poles to avoid symmetry (offsetpolar flow). The numerical solution with monotonic op-tion, CCS-M is shown in bottom-right panel of the Fig.2. No smoother or filter is applied to suppress the noisegenerated at the poles. Table 1 shows the global erroras a function of rotation parameter a for the CCS sim-ulations.

For the polar flow, the cosine bell has undergone asmall stretching along the flow direction, and the heightof the cosine bell is slightly reduced as compared to theexact solution. The monotonic constraint has resultedin slight degradation of the shape of the cosine bell,particularly at the center (Fig. 2, bottom-right panel).Also, error is larger for the CCS-M, as compared to theCCS-N or CCS-P (Table 1). For a comparison, the re-sults with cell-integrated semi-Lagrangian (CISL)schemes of NM02 is included in Table 1. The l1 and l2

errors are smaller for the CCS schemes as compared to


FIG. 2. Results on an orthographic projection for solid-body rotation of a cosine bell after onerevolution (256 time steps) when the flow is along the pole-to-pole direction (a 5 p/2). The exactsolution is shown as the dashed contours. The conservative cascade scheme with positive-definiteoption (CCS-P), monotonic option (CCS-M) and without any filter option (CCS-N) are used foradvection. Top-left shows the numerical solution with the CCS-N and top-right shows that withthe CCS-P. The bottom left and right show numerical solutions with the CCS-P (a 5 p/2 2 0.05)and the CCS-M, respectively.

TABLE 1. Error as the function of rotation angle a for solid-body rotation of a cosine bell after one revolution using the conservativecascade schemes (a 5 p /2 corresponds to the flow along pole-to-pole direction). For comparison, results with CISL schemes (NM02) arealso included. The leters N, P, and M denote no filter, positive-definite, and monotonic options, respectively. Note that the values of errorare set to 0.0 where they reflect round-off errors.

a Scheme l1 l2 l` Max Min

p /2p /2p /2p /2 2 0.050

CCS-NCCS-PCCS-MCCS-PCCS-P

0.0540.0510.0760.0550.036

0.0420.0410.0820.0430.034

0.0650.0650.1290.0640.042

20.05520.05520.12520.04920.042

20.006920.0066

0.020.0068

0.0

p /2p /2p /2

CISL-NCISL-PCISL-M

0.0630.0590.084

0.0460.0450.084

0.0480.0480.109

20.01620.01620.052

20.004120.003420.0001

the corresponding CISL scheme, but l` errors are smallerfor the CISL scheme. Also, the max and min errors forthe CISL schemes are slightly smaller than that of theCCS schemes. For a similar set of experiments with

conservative schemes, Lin and Rood (1996) and NM02have reported the slight degradation of numerical so-lution with monotonic filter. Lin and Rood (1996, seetheir Table 6, p. 2060) have compared the accuracy of


FIG. 3. Polar stereographic projection for the offset (a 5 p/2 20.05) solid-body rotation of a cosine bell crossing over the NorthPole, at time steps 32, 64, and 96 (starting from bottom to top of theframe). The CCS-N is used for the numerical integration, and theexact solutions are shown as dashed contours.

their conservative and positive-definite scheme flux-form semi-Lagrangian-5 (FFSL-5) with Rasch (1994),for the solid-body rotation (a 5 p/2). Results with theCCS-P (Table 1) shows that errors are slightly higherthan that of FFSL-5, but much better than that of Rasch(1994) for the same experiment. For example, the l1, l2,and l` errors for FFSL-5 are 0.047, 0.041, and 0.053;for the CSS-P, these values are 0.051, 0.041, and 0.065,respectively.

The most challenging aspect of a solid-body rotationexperiment on a sphere is the accuracy of simulation ofcross-polar advection. Figure 3 shows a polar stereo-graphic projection of the cosine-bell advection for theoffset polar flow (a 5 p/2 2 0.05) with the CCS-N asthe bell approaches, passes over and leaves the NorthPole, at time steps 32, 64, and 96 respectively. Thereis a small distortion for the numerical solution when thepattern crosses the pole; however, after one revolutionthe numerical solution appears very similar to the casewhen a 5 p/2, as shown in Fig. 2, top-left panel.

The results with the CCS scheme are comparable tothat of the CISL scheme (NM02), however, the CCSscheme has additional advantages. The CISL scheme(Machenhauer and Olk 1998; NM02) requires globalinformation of the coefficients of the quasi-biparabolicfunctions involved in the remapping algorithm. This po-tentially increases the memory requirements as the co-efficients must be stored. Since the CCS remappingneeds only a few operations (as compared to the CISLremapping), it is more computationally efficient thanthe CISL scheme.

The computational efficiency of the CCS has beencompared with the CISL scheme, and a standard (non-conservative) SL scheme combined with bicubic-La-grange interpolation (SL-BCL) is used for a reference.The solid-body rotation of a symmetric slotted-cylinderin 2D Cartesian geometry, is examined with after onerevolution with parameters identical to those used in theexperiments of Nair et al. (1999b). The CCS with lin-early interpolated intermediate grid and monotonic filterare found to be 2.4 times more efficient than SL-BCLand about 2.1 times more efficient than CISL withmonotonic filter. For a similar experiment, the conser-vative piecewise biparabolic method introduced by Ran-cic (1992) is 2.5 times more expensive than the SL-BCLscheme. In spherical geometry, computational efficiencyof CCS without any filter is compared with that of SL-BCL. An intermediate grid for CCS scheme is generatedwith cubic-Lagrange interpolation and additional pointsare used around the polar zones as described earlier. Inthis case, CCS is found to be approximately 2 times asefficient as SL-BCL for solid-body rotation around theglobe (a 5 p/2 with 256 time steps).

c. Deformational flow test

To examine the accuracy and robustness of the CCSscheme, we consider two deformational flow (cyclo-genesis) tests in spherical geometry. NCS99 gives thedetails of the idealized cyclogenesis problem of Doswell(1984) on the surface of the sphere. In the first test, aflow with smoothly varying positive-definite field isused (NM02) for advection. The second test is the nons-mooth flow problem as given in NCS99. For these ex-periments, the intermediate grid is generated by usingthe 1D cubic-Lagrange interpolation.

1) SMOOTH DEFORMATIONAL FLOW

Here the vortex problem considered by NM02 is used.There are two vortices in the domain, one near eachpole, and the CCS positive-definite is used for transport.A nonconservative SL scheme with bicubic-Lagrangeinterpolation is used for comparison (NCS99).

Let (l9, u9) be the rotated spherical coordinate systemwith respect to the regular spherical coordinate system(l, u), such that the north pole of the rotated sphere islocated at (l0, u0) in (l, u) system. Then two circularvortices may be created with centers at the poles of the(l9, u9) system. Normalized tangential velocity of thevortex is defined as

3Ï32V 5 sech (r9) tanh(r9), (24)t 2

where r9 5 r0 cosu9 is the radial distance of the vortexand r0 is a constant. The angular velocity v9(u9) is de-fined as (NM02),


FIG. 4. Results on an orthographic projection for the polar vortex simulations (smooth defor-mational flow). Top-left shows the initial flow field and top-right shows exact solution at t 5 3.The vortex center is located near the pole. Numerical solutions are produced by using the CCS-N (bottom-right) and the SL scheme with bicubic Lagrange interpolation (SL-BCL, bottom-left).

0 if r9 5 0,v9(u9) 5 (25)Vt if r9 ± 0.r9

The exact solution at time t is

r9c(l9, u9, t) 5 1 2 tanh sin(l9 2 v9t) , (26)[ ]d

where d is the parameter that defines the characteristicwidth (NCS99).

With these conditions two symmetric vortices (withrespect to the equator of the rotated sphere) can beformed. The vortex centers are approximately locatedat 818N and 818S latitudes, respectively, if the param-eters values are chosen to be (l0, u0) 5 (p 1 0.025,p/2.2), r0 5 3 and d 5 5. This configuration keeps thevortex centers away from the poles of the (l, u) system,to avoid possible symmetry.

The initial values of the field (or cell averages) aregenerated at the cell centers on a 128 3 64 grid, and

integrated for 3 time units (nondimensional) with 64time steps. The corresponding Courant number in l, udirections are Cl 5 19.1 and Cu 5 0.82, respectively.The exact upstream positions and other details of theexperiment are given in NM02. Measures of error forthe polar vortex problem are the same as that for thesolid-body rotation test.

Orthographic projection of the smooth deformationalflow simulation are shown in Fig. 4, where contour val-ues range from 0.5 to 1.5. The flow fields are visuallyindistinguishable for the CCS-N and CCS-P, and thecorresponding error measures are almost identical. Fig-ure 4 shows the initial and the exact solution at the topleft and right panels, respectively, while the bottom pan-el shows the numerical solution with SL-BCL withoutany filter, and the bottom-right panel shows the nu-merical solution with the CCS-N. The center of the vor-tex is captured well in the CCS-N simulation and isvery similar to the exact solution. Moreover, the vortexcenter is more accurately represented in the CCS-N sim-ulation as compared to that of the SL-BCL. Table 2


TABLE 2. Error at t 5 3.0 for the smooth deformational flow problem (polar vortices) using the conservative cascade schemes and anonconservative semi-Lagrangian scheme combined with bicubic Lagrange interpolation (SL-BCL).

Scheme l1 l2 l` Mass Max Min

CCS-PCCS-MSL-BCL

0.00110.00130.0072

0.00270.00330.0143

0.01540.02200.0585

0.00.00.000241

0.00013620.000016

0.0

20.00018820.000073

0.0

shows the error for different versions of the CCS andthe SL-BCL. The l1, l2, and l` errors are smaller for theCCS as compared to the SL-BCL. However, max andmin errors for the CCS are slightly different from thezero errors of the SL-BCL.

2) NONSMOOTH DEFORMATIONAL FLOW

The robustness of the CCS is further demonstratedby using the nonsmooth vortex (cyclogenesis) problemconsidered by NCS99. Here the vortex is defined on apolar-stereographic plane tangent to the north pole ofthe rotated coordinate system (l9, u9) (NCS99). Theresults of the vortex simulation is projected onto thetangent plane, rather than using orthographic projectionas in the previous case.

The exact solution at time t is given as

r9c(l9, u9, t) 5 2tanh sin(l9 2 v9t) , (27)[ ]d

where r9 is the distance from the pole of projection toa point (l9, u9) in the tangent plane, and d defines thecharacteristic width of the frontal zone. Specific detailsof other parameters of projection, exact trajectory, andderivation velocity filed are given in NCS99 and willnot be discussed here.

The vortex center is located approximately 658N andthe isopleth of maximum wind speed passes over thenorth pole of the computational domain. This makes theproblem more challenging, since the polar Courant num-bers are higher as compared to solid-body rotation testsconsidered. The smoothness parameter for this problemis set to d 5 0.01 (a much smaller value than that usedin the smooth deformational flow) and t 5 2.5 timeunits are used to create the exact solution. The com-putational domain consists of 128 3 64 cells as in theprevious cases, and the values of the initial scalar fieldc(l9, u9, 0) at the cell centers are used as the initial cellaverages. The monotonic version of the conservativecascade scheme is used with 64 time steps equivalentto t 5 2.5 time units. Figures 5a and 5b show the exactsolution projected on to a tangent plane, at initial timeand at t 5 2.5 time units respectively. The numericalsolution at t 5 2.5 produced by using the CCS-M isshown in Fig. 6.

The numerical solution shown in Fig. 6, with CCS-M is very similar to that with the monotonic cascade-spline SL shown in Fig. 8b of NCS99. The l1, l2, andl` errors for CCS-M are 0.035, 0.13, and 1.06, respec-

tively, and the corresponding errors for monotonic cas-cade-spline SL in NCS99 are slightly better. The maxand min error measures for CCS-M are 0.0052 and20.0029; for the monotonic cascade-spline SL scheme(NCS99) these values are zero. Note that, in NCS99, anonconservative SL scheme with cascade cubic-splineinterpolation is used for vortex simulation with 128 365 grid points in the computational domain. This ex-perimental setup is slightly different from the presentone, and therefore not strictly comparable.

A reason for generating small spurious values for themonotonic cases could be the rounding error associatedwith the subtraction [see Eq. (13)]. In Eq. (13), the massat the new time level is estimated by subtracting theaccumulated masses obtained after applying the mono-tonic filter. Also, max and min are generated in theimmediate vicinity of the quasi discontinuity of the so-lution. Since the monotonic filter is not completely re-moving the spurious values at every time step and thewind field used is uniform, the error accumulates duringthe integration. The creation of tiny spurious values evenafter applying monotonic filter is reported in Lin andRood (1996) and NM02 for their respective conserva-tive schemes. A 1D monotonic short-wave filter appliedafter remapping operation could remove the noise.

5. Summary and conclusions

An efficient mass-conservative 2D advection schemeappropriate for global applications is presented. Thescheme combines a mass-conservative finite-volume ad-vection with semi-Lagrangian method in a computa-tionally efficient cascade interpolation framework. Thishas been tested previously (R95), but only for Cartesiancoordinates. Following Machenhauer and Olk (1996,1998), the extension to spherical geometry is facilitatedby performing the integrations over Lagrangian cells,which are approximated by polygons with sides parallelto the coordinate axis in a Cartesian (l, m 5 sinu) co-ordinate system (where l is longitude and u is latitude).As in R95 the accurate piecewise parabolic represen-tation of Colella and Woodward (1984) is used. Overthe polar caps, the areas near the poles where the con-servative cascade procedure breaks down, we have in-troduced the globally conservative, but locally approx-imate, scheme as one used in NM02. This procedure iscurrently restricted to polar meridional Courant numbersless than one. The resulting conservative cascadescheme (CCS) is demonstrated to be accurate for the


FIG. 5. (a) Exact solution of the nonsmooth deformational flow problem (see text), projected onto a plane tangentto the vortex center at initial time. (b) Exact solution of the nonsmooth deformational flow problem (see text), projectedonto a plane tangent to the vortex center at time t 5 2.5 units.


FIG. 6. Numerical solution of the nonsmooth deformational flow problem at t 5 2.5 units (64 time steps), projectedonto a plane tangent to the vortex center. The conservative cascade scheme with monotonic filter (CCS-M) and withexact trajectories are used for the numerical integration.

tests of solid-body rotation of a coherent structure anddeformational flow over the poles.

The simple redistribution of mass enclosed in the po-lar cap to the constituent cells, using the SL method(NM02), is found to be accurate. The numerical sim-ulation of cross-polar transport is accurate and numer-ical solutions are comparable to the accurate results ob-tained in similar conservative advection experiments inLin and Rood (1996). The intermediate grid generationmethods, using cubic interpolation of the Lagrangiancoordinates and the great-circle approach, are found tobe effective for conservative transport. However, thecubic interpolation method is more accurate while thegreat-circle approach is more computationally efficient.

Computational efficiency of the conservative cascadescheme is compared with a standard bicubic-Lagrangeinterpolation SL scheme and the CISL scheme (Mach-enhauer and Olk 1997; NM02). In the present imple-mentations CCS is found to be twice as efficient as ourimplementation of a full 2D SL scheme and more thantwice as efficient as the CISL scheme. Results producedwith the CCS are comparable to that of the CISLscheme; however, the memory requirement for the CCSis less than that of the CISL scheme, at least in thepresent implementations. Computational efficiency ofthe CCS is evident even without optimizing the code.

In 3D, with multiple tracer fields used as in typicalglobal chemistry models, the CCS can save a significantamount of computer time while being remaining veryaccurate.

The monotonic and positive-definite filter options arefound to be very effective for removing the overshootsand undershoots of the numerical solution. However,the filter constraints are applied before the remapping[see Eq. (13)], and this may result in creating small newspurious values due to truncation error. A 1D conser-vative monotonic (or positive-definite) filter may beused after each cascade cycle to obviate this problem.

The CCS approach presented is valid for meridionalCourant number Cu # 1, and it is independent of zonalCourant number Cl. Therefore, the CCS can be used asan efficient semi-Lagrangian option for any globalchemistry–climate Eulerian models. However, for amore general SL application (Cu . 1), the special polartreatment developed by NM02 may be used for the CCS.Generalizing CCS without Courant number restrictionand application to realistic 3D models are beyond thescope of present study. Work is in progress to implementthe CCS into a global nonhydrostatic SL model (Qianet al. 1998) developed at North Carolina State Univer-sity.


Acknowledgments. The authors are grateful for thesupport by the NSF Grant ATM-95-25786 and the sup-port of the Department of Marine, Earth and Atmo-spheric Sciences (MEAS), North Carolina State Uni-versity. The authors are also thankful to the North Car-olina Supercomputing Center (NCSC) for the computertime. We appreciate the indepth comments by the anon-ymous reviewers.

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efﬁcient conservative global transport schemes for climate

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