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AUGUST 2000 2711B E C K E R S E T A L .

Numerical Discretization of Rotated Diffusion Operators in Ocean Models

J.-M. BECKERS*

University of Liege, Liege, Belgium

H. BURCHARD1

Joint Research Centre, Ispra, Italy

E. DELEERSNIJDER* AND P. P. MATHIEU#

Unite ASTR, Universite Catholique de Louvain, Louvain-La-Neuve, Belgium

(Manuscript received 5 October 1998, in final form 13 October 1999)

ABSTRACT

A method to improve the behavior of the numerical discretization of a rotated diffusion operator such as, forexample, the isopycnal diffusion parameterization used in large-scale ocean models based on the so-calledz-coordinate system is presented. The authors then focus exclusively on the dynamically passive tracers andanalyze some different approaches to the numerical discretization. Monotonic schemes are designed but arefound to be rather complex, while simpler, linear schemes are shown to produce unphysical undershooting andovershooting. It is suggested that the choice of an appropriate discretization method depends on the importanceof the rotated diffusion in a given simulation, whether the field to be diffused is dynamically active or not.

1. Introduction

Ocean models typically no longer rely on parame-terizations of subgrid-scale processes involving onlysecond derivatives taken along the coordinates under-lying the numerical grid. As recognized by several au-thors (e.g., Cox 1987; McDougall 1984; Gent andMcWilliams 1990), these parameterizations are physi-cally unsound and should be replaced by more sophis-ticated formulations.

In large-scale ocean models, a widely accepted pa-rameterization is based on the concepts of Gent andMcWilliams (1990) and McDougall and Church (1986).Their arguments are based on the fact that mixing pref-erentially occurs along isopycnal surfaces and that bar-oclinic instabilities tend to flatten these surfaces. The

* Research associate at the National Fund for Scientific Research,Brussels, Belgium.

1 Current affiliation: Institute for Oceanography, University ofHamburg, Hamburg, Germany.

# Current affiliation: Fonds pour la Formation a la Recherche dansl’Industrie et dans l’Agriculture, now the Joint Research Centre, Ispra,Italy.

Corresponding author address: Dr. J.-M. Beckers GHER, Uni-versity of Liege, Sart-Tilman B5, B-4000 Liege, Belgium.E-mail: [email protected]

parameterizations that are generally retained to considerthese two aspects are then diffusion acting on isopycnalsurfaces to describe the mixing and an additional ad-vection velocity for tracer fields that has a tendency toflatten the isopycnals. This approach, combining iso-pycnal diffusion and the so-called bolus velocity (Gentet al. 1995), will subsequently be called improved is-opycnal mixing. The parameterized additional bolus ve-locity field depends on the shape of the isopycnal sur-faces and can be calculated from the prognostic vari-ables in a primitive equation model.

Though this idea was initially developed for coarse-resolution models, Roberts and Marshall (1998) showedthat the use of horizontal diffusion rather than isopycnaldiffusion still induces diapycnal transfers when reso-lution is increased beyond the deformation radius andthey concluded that adiabatic subgrid dissipationschemes are required, even in eddy-resolving models.Adiabatic dissipation schemes rely on the concept ofmixing along isopycnals, which in their high-resolutionmodel is parameterized with more scale-selective op-erators as the biharmonic diffusion operator. Here, how-ever, we will work on the original versions of the dif-fusion part of the improved isopycnal mixing based onLaplacian diffusion, since they are the most widely used.

The improved isopycnal mixing is easily imple-mented into layer models (e.g., Bleck and Boudra 1981;Oberhuber 1993; Chassignet et al. 1996; Paiva et al.

2712 VOLUME 128M O N T H L Y W E A T H E R R E V I E W

1999). This is because in such models the natural co-ordinates of the isopycnal mixing are also the coordi-nates in which the model operates. The improved iso-pycnal mixing has also been introduced into level mod-els by several authors. For the diffusion part D, thisleads to a rotated tensor K as shown in Redi (1982):

D 5 = · (AK · =C), (1)

where K is defined as2 2r 1 r 2r r 2r r y z x y x z

2 2 2 2 2(r 1 r 1 r )K 5 2r r r 1 r 2r r . x y z x y x z y z

2 22r r 2r r r 1 r x z y z x y

(2)

Here, x and y are horizontal coordinates, while z isthe vertical one—x, y, z being a Cartesian coordinatesystem; = · and = denote the divergence and gradientoperators, respectively; and A is the isopycnal diffusioncoefficient relevant to the scalar variable C.

It is also possible to include parameterizations of is-opycnal diffusion as a rotated diffusion operator intoocean models based on terrain-following coordinates(see, e.g., Hedstrom 1994; Blumberg and Mellor 1987;Beckers 1991). The inclusion of additional velocity thatflattens isopycnals is also possible but will not be an-alyzed here. Since these models are often applied toshallow seas, traditional diffusion along the coordinatesurfaces of the numerical grid is commonly retained toaccount for the impact of the topography on the subgrid-scale motions to be parameterized (see, e.g., Mellor andBlumberg 1985). But, when stratification effects aredominant, this strategy may be questionable, at least forthose regions where strong stratification occurs oversloping bottom topography. In this case, some rotateddiffusion laws could account for the preferred directionof mixing. Another situation in which rotated diffusionin terrain-following coordinates is advantageous occurswhen isopycnal surfaces are identical to geopotentialsurfaces. In this case, no baroclinic pressure gradient isgenerated. For a departure from such a situation, whenapplying a rotated diffusion of density back into geo-potential surfaces, it can then be ensured that the systemtends toward a standstill when left to itself (see Stellingand van Kester 1994), contrary to a diffusion alongterrain-following coordinates, which tends to create athermal wind.

The attractive physical features of parameterizationsin terms of mixing along isopycnals and flattening ofthe isopycnals have led to numerical implementationsin several z- or terrain-following coordinate models. Itwas then found, however, that the straightforward nu-merical discretizations of the parameterization in termsof diffusion and advection were not without problems.The first attempts by Cox (1987) and Gerdes et al.(1991) needed the addition of a background diffusionalong grid coordinate surfaces, the limitation of theslope of the computed isopycnals, or the application of

intermittent filtering. Despite these controlled numericalinconsistencies, improvements of the simulated circu-lation were generally found (Danabasoglu et al. 1994).

Several authors (e.g., Griffies et al. 1998; Beckers etal. 1998; Mathieu and Deleersnijder 1997) then ana-lyzed why the discretized diffusion term (normally asmoothing operator) needs such additional damping, po-tentially masking the desired rotated diffusion. Othersanalyzed the best way to compute the additional ad-vection (e.g., Gerdes 1993), or how to efficiently com-bine the advection and diffusion part into a general ten-sor formalism (Griffies 1998; Gnanadesikan 1999).Some improvements to the classical discretizations wereintroduced. In particular, Griffies et al. (1998) and Grif-fies (1998) designed a scheme that eliminated someproblems for dynamically active tracers. They showedhow to ensure that along isopycnal surfaces, diffusionfluxes for temperature and salinity combine to give azero density flux. This ensures that the numerical for-mulation of the rotated diffusion applied to the densityfield itself is zero as does the mathematical formulationof Eqs. (1) and (2) when applied to the density field.The violation of this constraint was a major reason forinstabilities in the first implementations of the isopycnaldiffusion. Griffies et al. (1998) also introduced new dis-cretizations of the diffusion tensor that allow a moreaccurate diffusion where the density field varies rapidlyin space. However, none of these methods solves theproblem identified in Beckers et al. (1998), who showedthat there is no linear, first- or second-order discreti-zation of the rotated diffusion operator that is mono-tonic. This means that new local extrema may be createdby the numerical version of the diffusion operator. Thiscan be disastrous when the diffusion of a dynamicallypassive tracer1 is performed: without a positive definitescheme, specially when the tracer interacts with othertracers as in biological models, the model behavior maylead to completely unrealistic fields. Typically, localnegative concentrations are both unphysical and difficultto manage in nonlinear biological laws. This was alsorecognized by Gnanadesikan (1999), who showed thedisastrous effects of spurious minima and maxima of agiven tracer field due to discretized rotated diffusion,which in principle should smooth the fields rather thanintroduce new extrema.

In the present paper we investigate in detail some ofthe remedies proposed in Beckers et al. (1998) to dealwith the violation of the monotonicity in classical linearschemes. This article is organized as follows: First, weformulate the mathematical problem in a general frame-work (section 2), then we summarize the known nu-merical problems associated with the discretizations ofthe rotated diffusion operator (see section 3). In section

1 Hydrodynamically speaking; they could be active in the sense ofa biological growth, for example.


FIG. 1. Grid and naming convention.

4, some remedies will be described and then comparedin section 5. Finally, their application in a general 3Dmodel is discussed.

2. Problem formulation

In this work, we investigate only the case of a vertical2D section, by assuming that the 3D problem can betreated as two 2D problems. This implies that the small-slope approximation (Cox 1987) is adopted, leading tosimplifications of the tensor K [see Eq. (2)] such thatthe fluxes in the x direction are not influenced by thegradients of the field in the y direction.

Our understanding is that isopycnal diffusion is oftenused with small-slope approximation and that it is rea-sonable to follow a step-by-step procedure. This consistsof first designing numerical schemes that perform wellin 2D and then attempting to generalize them to 3D withrespect to the full tensor.

We will also restrict our analysis to the case of adynamically passive tracer, since the problem of dy-namically active tracers leads to a complex couplingand feedback with circulation, and depending on thescales at which the model is applied, the conclusionsdrawn from the study of dynamically active tracers maybe different. In any case, since studying dynamicallypassive tracers is the final purpose of an increasing num-ber of applications, a proper discretization of their dif-fusion terms is necessary (e.g., Gnanadesikan 1999).

Though the main objective of this study is the designof appropriate discrete algorithms of isopycnal diffu-sion, a slightly more general problem can also be easilyaddressed.

In order to encompass a wide range of model imple-mentations and coordinate systems (not necessarely or-thogonal), we take the general case in which diffusionof a field C along the coordinate line s(j, h), on whichthe other coordinate n(j, h) remains constant, can bewritten as

] ]CD 5 A9 . (3)1 2]s ]s

Here, (j, h) is a local coordinate system along thenumerical grid [see Fig. 1, in which the coordinate sys-tem in which the parameterization is formulated is ref-erenced by (s, n)].

In the case of isopycnal diffusion, the coordinate svaries along the isopycnal, and the generalized (posi-tive) diffusion coefficient A9 depends on the isopycnalslopes and the distance between isopycnals.

For the purpose of a conservative formulation, thediffusion term can be reformulated as follows:

]D 5 (2F), (4)

]s

where F is the diffusion flux, which can be rewritten

in terms of the derivatives along the numerical coor-dinates:

]C ]j ]C ]h ]CF 5 2A9 5 2A9 1 . (5)1 2]s ]s ]j ]s ]h

Since (e.g., Anderson 1995)

] ]j ] ]h ]5 1 , (6)

]s ]s ]j ]s ]h

]h 1 ]n5 2 , and (7)

]s J ]j

]j 1 ]n5 , (8)

]s J ]h

we can easily find an equation that will enables us toformulate a conservative equation from Eq. (3):

] ]j ] ]hJ 1 J 5 0. (9)1 2 1 2]j ]s ]h ]s

From this general tensor analysis (e.g., Aris 1962)Eq. (5) can be transformed to

] ]j ] ]h2JD 5 J F 1 J F , (10)1 2 1 2]j ]s ]h ]s

where J 5 ](s, n)/](j, h) is the Jacobian of the coor-dinate transformation. Evidently this formulations easilyleads to a conservative type of discretization.

By using a classical integration over the finite volumebox in (j, h) space, Eq. (10) can be translated into aconservative finite-difference scheme, provided that theintegrated fluxes J(]h/]s)F and J(]j/]s)F are known atthe interfaces.

As we will see below, an important factor of the for-mulation is

]h/]s ]n /]jr 5 5 2 , (11)

]j /]s ]n /]h

which measures the ratio of the slope of the lines on


FIG. 2. Representation of an isopycnal line in physical space, witha numerical grid superimposed. Depending on the discrete grid, theisopycnal line can significantly change its vertical position relativeto the numerical grid from one horizontal grid point to the next.

which diffusion occurs (e.g., isopycnals) compared tothe numerical grid aspect ratio.

From here on, it is clear that even if the small-slopeapproximation holds in the physical space, we cannotassume that the coordinate s along which the diffusiontakes place is almost flat in the numerical coordinatesj, h. Those coordinates take into account the anisotropyof the numerical grid. If the line were flat in the discretespace, then that would mean that in no event can thenumerical grid correctly resolve the isopycnal slope.This is analogous to the z-level models, which cannotcorrectly resolve topographic slopes that are below thegrid aspect ratio. In other words, if the isopycnals arealmost flat in the discrete coordinates, then there is nopoint in trying to use isopycnal diffusion since the nu-merical scheme will not be able to resolve the slopecorrectly. We thus have to assume that the numericalgrid is such that the relative height of the isopycnal mayvary significantly from one horizontal position to thenext. This translates mathematically as parameter r be-ing at least of O(1). A graphical representation of thisis given in Fig. 2, where the physical slope may beweak, but where depending on the horizontal and ver-tical discretization, the isopycnal vertical positionchanges significantly from one horizontal grid point tothe next.

To illustrate how a specific rotated diffusion may beretrieved from the generic formulation, we shall showtwo classical applications.

a. Horizontal diffusion in uniformly discretized scoordinates

In this case, the s coordinate ranges from s 5 0 atthe bottom, z 5 2d, to s 5 1 at the surface, z 5 z.Assuming uniform discretizations along the x and scoordinate with increments Dx and Ds, respectively, wemay define the coordinates j, h to vary in the discrete

space exactly as the indices of the grid points so thatDj 5 Dh 5 12 (see Fig. 1):

jDx 5 x and (12)

z 1 dhDs 5 5 s. (13)

z 1 d

The assumption made here that the grid spacing isconstant is not required but simplifies the presentation.

The coordinates that define the line on which diffu-sion takes place are thus (s, n), where the diffusion takesplace for constant n along the s line:

s 5 x 5 jDx and (14)

n 5 z 5 s(z 1 d) 2 d. (15)

From these definitions, we can easily compute themetric coefficients of the transformations,3

]j 15 and (16)

]s Dx

]h 1 d z 1 dx x x5 2 s , (17)5 6]s Ds z 1 d z 1 d

and the Jacobian is21

](s, n) ](j, h)J 5 5 5 (d 1 z)DxDs. (18)[ ]](j, h) ](s, n)

For small surface slopes (neglecting the sea surfaceslope compared to the bottom slope), parameter r reads

dxDxd

r 5 (19)1

Ds1 2 s

and can be readily interpreted as the measure for thehydrostatic consistency in terrain-following coordinates(e.g., Blumberg and Mellor 1987; Haney 1991; De-leersnijder and Beckers 1992). This gives the interestinginterpretation of the hydrostatic consistency requirement|r| # 1 in terms of variation of the z coordinates whenseen in discrete space. These lines of constant z shouldnever vary more than one discrete vertical unit whenmoving one discrete horizontal unit.

b. Isopycnal diffusion in uniformly discretizedz-coordinates

If again, we assume a uniformly discretized spacewith constant horizontal grid size Dx and vertical grid

2 This does not mean of course that the physical grid space isisotropic, but that we made the arbitrary choice of using a numericalcoordinate system varying by unit steps from one grid point to thenext.

3 Here, dx stands for ]d/]x.


size Dz, the numerical grid is defined by the coordinatesj, h, which vary as the gridpoint indices:

xj 5 and (20)

Dx

zh 5 . (21)

Dz

The coordinates that define the direction of the dif-fusion are those for which n is constant:

s 5 x and (22)

n 5 r (x, z). (23)

The vertical coordinate is density r in this case, andthe metrics of the transformations can be calculated us-ing properties of Eqs. (7) and (8), so that they read

]j 15 and (24)

]s Dx

]h r 1x5 2 , (25)]s r Dzz

with the Jacobian

J 5 rzDxDz. (26)

In this case, it may be shown that we retrieve theformulation of Redi (1982) [see Eq. (2)] by using thegeneralized diffusion coefficient A9 given by

21 rzA9 5 A . (27)2 2r r 1 rz x z

Here, the parameter r can be identified as the physicalisopycnal slope S 5 2rx/rz multiplied by the grid aspectratio.

3. Problems identified

As already mentioned in the introduction, various au-thors realized that the straightforward discretization ofthe reformulated diffusion in the grid coordinates leadsto numerical problems, which are summarized in thefollowing sections.

a. Dynamically active tracers

The major problems for large-scale applications werethe noncancellation of density flux contributions of tem-perature and salinity on isopycnal surfaces (Griffies etal. 1998). This led to inaccuracies that could explainsome of the instabilities observed in the studies ofGough and Welch (1994) and Gough (1997). This prob-lem is of a dynamic nature, but any additional problemthat exists for dynamically passive tracers is also a po-tential problem for temperature and salinity, althoughthe dynamical effect may be controlled by the methodof Griffies et al. (1998). Therefore we focus now on theproblems identified for dynamically passive tracers.

b. Dynamically passive tracers

For dynamically passive tracers, one problem iden-tified in Griffies et al. (1998) as well is the inappropriatecomputation and averaging of products of grid slopesand gradients in situations where the density field ex-hibits rapid horizontal variations with a spatially highwavenumber. Griffies et al. (1998) show how this prob-lem appears in the classical discretization (Cox 1987)and how it can be solved by using different averagingtechniques.

The focus of the present paper is on a problem thatremains unsolved even for very slowly varying or con-stant slopes: as shown in Beckers et al. (1998), no mono-tonic scheme can be obtained when classical consistentand linear schemes are used, unless the slope and thegrid is such that r 5 0 or r 5 61 or r 5 6`. Theseare cases in which diffusion occurs on a line that crossesthe grid points. In these cases a direct classical diffusiondiscretization along these grid points works well. Butif the diffusion direction does not coincide with the grid,as shown in Beckers et al. (1998), the numerical stencil,which gives the contribution of the surrounding pointto the evolution of the central point, always containsnegative coefficients. This feature is responsible for thepossibility of creating new local extrema.

Even for constant slopes, uniform grid spacing, andconstant diffusion coefficient, no well-behaved linearscheme can be found. Furthermore, it is also clear thatany method of a priori limiting the isopycnal slope toa prescribed maximum amplitude is not appropriate toeliminate this monotonicity violation. Anyway, it is notthe slope that is the important control factor of the ‘‘neg-ativeness,’’ but the slope compared to the coordinateslope, as reflected by the parameter r.

Griffies (1998) and Gnanadesikan (1999) argue in thecase of isopycnal diffusion that the problem of mono-tonicity may be overcome by the concurrent use of theGent–McWilliams advection parameterization. Theyshowed how the isopycnal diffusion and the layer thick-ness diffusion can be cast into a single asymmetric ten-sor formulation. In the numerical experiment of Gnan-adesikan (1999), this helped to stabilize a coupled bi-ological model. However, the asymmetric tensor for-mulation modifies the flux computation in such a waythat negative coefficients in the discrete stencil appearonly in the vertical fluxes. The vertical direction iswhere the biological gradients and fluxes are most im-portant and the stabilization seems to result from a bettercombination of the Gent–McWilliams advection–dif-fusion fluxes resulting from the asymmetric tensor for-mulation with these strong diapycnal exchanges and lo-cal production terms. Since the stabilization effect prob-ably depends on the biological model, it might not workin all situations. And there are further reasons why thisstabilizing effect might fail. The Gent–McWilliams ad-vection parameterization is not always relevant, for ex-ample, in the case of regional s-coordinate models with


geopotential diffusion. In other words, the tracer dif-fusion coefficient A used in the diffusion part and thethickness diffusion coefficient k used in the advectionpart may differ significantly from one situation to an-other, such that a compensation effect can generally notbe expected. Typically, A is taken as a constant param-eter, whereas the isopycnal thickness diffusivity used inthe bolus velocity calculation is calculated dependingon the local Richardson number (Visbeck et al. 1997).

Furthermore, an advection operator generally intro-duces positive and negative coefficients into the nine-point stencil unless a numerically very diffusive schemeis used. Since the Gent–McWilliams advection velocityis related to second-order derivatives of the density field,whereas the rotated diffusion is related to first-orderderivatives, there is no particular reason why the neg-ative coefficients of one operator are canceled out by apositive coefficient of the other operator. In the skewflux formulation of Griffies (1998), this is somehowhidden, since this formulation is introduced as a ‘‘dif-fusive’’ formulation with a (nonsymmetric) diffusiontensor depending on the isopycnal slopes only and noton their second derivatives. Because of the asymmetryof the tensor, the actual derivatives of the fluxes leadto an advection part and a diffusion part depending ondifferent derivatives of the density field. A striking ex-ample is the locally relevant case of a uniform slope inthe density field. This leads to isopycnal diffusion with-out any Gent–McWilliams advection both in the skewformulation and the classical formulation. In this case,there is no cancellation between the two effects. Sincethe locally uniform slope is very likely to occur, wecertainly should correctly represent the pure diffusionpart, because the advection part vanishes in this case.The development of a well-behaved discretization of therotated diffusion operator is thus discussed in the nextsection.

4. Discretization methods

A well-behaved scheme should be monotonic, and inorder to ensure such a method, one has to eliminate oneof the requirements that were shown (Beckers et al.1998) to lead to the impossibility of having a monotonicscheme: the scheme was assumed to be based on a nine-point stencil (in the vertical plane), a linear method(discretization being not a function of the solution), anda consistent scheme. At least one of these conditionscannot be satisfied by a monotonic algorithm. On theother hand, a conservative scheme could be essentialfor long-term climatic calculations and tracer dispersion.Otherwise, a simple, nonconservative method is givenby clipping overshooting and undershooting values suchthat monotonicity is obtained. However, here we do notconcentrate on nonconservative schemes (easily forcedto be monotonic), we focus rather on the behavior ofclassical linear schemes compared to certain new con-servative discretizations. Before introducing the new

schemes aiming at achieving a monotonic behavior,some classical linear discretizations are presented.

a. Linear consistent schemes

Here, we will not explain in detail the linear consistentdiscretizations used classically, because they may differstrongly in the way the isopycnal slopes are computed,averaged, and combined with the gradients of the fieldsto be diffused. Griffies et al. (1998) show that a mod-ification in this kind of averaging for variable slopesmay result in drastic changes when diffusing in a spa-tially rapidly varying density field. Since all these dif-ferent averaging techniques for the isopycnal slopes leadto the same scheme when the slope is constant, we willfocus on this case. This is also justified by the fact thatlocally uniform slopes are likely to be present in realsituations, and that such a situation is the basic situationany scheme should be able to deal with. For the linearschemes, we therefore assume a constant slope and referto Griffies et al. (1998) for a generalization to caseswith variable slopes.

For constant slopes and uniform grids s 5 Dsj, theexpression for the diffusion flux F of Eq. (5) simplifiesto

]C A9 ]C ]CF 5 2A9 5 2 1 r , (28)1 2]s Ds ]j ]h

where the r coefficient, which measures the relativeslope compared to the aspect ratio, is now a constant.The diffusion operator D given in Eq. (3) can then alsobe simplified:

2Ds ] ]C ]C ] ]C ]CD 5 1 r 1 r 1 r . (29)1 2 1 2A9 ]j ]j ]h ]h ]j ]h

This simplification for constant slopes and uniformgrid allows for understanding of the nonmonotonic be-havior more easily by analyzing the stencil obtained bythe discretization: the stencils shown in the followingfigures provide the contribution of the surroundingpoints to the evolution of the central point when per-forming flux differencing in the linear schemes; anynegative coefficient except the central point leads to anonmonotonic scheme (Beckers et al. 1998).

1) LINEAR

As an example of the classical Cox discretization ofRedi’s rotated diffusion, as well as the adaptation ofGriffies et al. (1998), the numerical stencil is shown inFig. 3 when the slope is constant. This stencil is obtainedby calculating the fluxes at the interfaces by using clas-sical algebraic averages of gradients at the interface.

To illustrate the standard way of discretization of Eq.(28), we take the calculation of the flux at the interfacebetween points i, j and i 1 1, j:


FIG. 3. Stencil for the standard LINEAR discretization in the caseof constant slopes and grid spacings. For the sake of clarity, nomultiplication constant was included. We assume that we referencefluxes and points compared to the central point of the control volume,whose coordinates are thus i, j. For fluxes, when they are not ref-erenced by coordinates, they correspond to either the interface i 1½, j or i, j 1 ½.

FIG. 4. Fluxes involved by the modified method LINEAR1. At theright interface, only two vertical derivatives are used (single lines),whereas at the top interface, only two horizontal derivatives are used(double lines). For the fluxes at the right interface, the discretizationof Eq. (33) indeed uses the vertical derivatives Ci11,j11 2 Ci11,j andCi,j 2 Ci,j21 only.

Ds2 F 5 C 2 C 1 r/4i11/2, j i11, j i,jA9

3 (C 2 C 1 C 2 Ci,j11 i,j i11, j11 i11, j

1 C 2 C 1 C 2 C ).i,j i,j21 i11, j i11, j21

(30)

Similar expressions are easily obtained for the otherinterfaces and thus involve the calculation of averagesfor gradients that are not naturally defined by a singlefinite difference.

By using a very straightforward flux differencing ofEq. (29),

2DsD 5 2F 1 F 1 r(2F 1 F ),i11/2, j i21/2, j i,j11/2 i,j21/2A9

(31)

a discretization is obtained that is simply a standarddiscretization of

2 2 2 2Ds ] C ] C ] C2D 5 1 2r 1 r , (32)

2 2A9 ]j ]h]j ]h

leading to the stencil of Fig. 3.This stencil clearly shows that the nonmonotonic ten-

dencies stem from the cross-derivative terms, since theseare the derivatives that introduce the negative coeffi-cients into the stencil and include the possibility of in-troducing new extrema into the solution. The simplestsituation for the generation of a new extrema is a con-stant field with a positive perturbation at the pointswhere the coefficients in the stencil are negative. Thiswill lead to a time tendency that will create a negative(mathematically incorrect) perturbation in the center.

2) LINEAR1

Another linear scheme, LINEAR1, can be constructedby computing the averages of the four triads defined byGriffies et al. (1998), not as a simple average, but takinginto account the direction of the slope, so as to consideronly the triads in the corresponding direction. Thismeans that if we have to calculate the gradients at aninterface, we do not compute the average of the foursurrounding gradients, but only the average of the twogradients that are crossed by the isopycnal line.

For a positive slope, this would lead to the followingevaluation of the flux:

Ds2 F 5 C 2 C 1 r/2i11/2, j i11, j i,jA9

3 (C 2 C 1 C 2 C ).i11, j11 i11, j i,j i,j21

(33)

This discretization clearly involves the fluxes shownin Fig. 4. From this and analogous formulations for theother interfaces, one obtains the stencil for the caseshown in Fig. 5.

This stencil potentially leads to a smaller monoto-nicity violation when the slope is close to one, sincethe negative coefficients are lower, but the monotonicityproblem remains. Interestingly enough, for the stencilshown in Fig. 5, the method can be rendered monotonicnot by adding a horizontal background diffusion as doneusually, but by adding vertical diffusion proportional tothe slope-dependent parameter r 2 r2. This also sug-gests that this scheme may behave better when a dia-pycnal diffusion is present, because such a diapycnaldiffusion could cancel the negative coefficients, whichwould not be the case for the classical scheme with ahorizontal background diffusion.

If we add however a vertical background diffusion


FIG. 5. Stencil for the modified linear method LINEAR1. For thesake of clarity, no multiplication constant was included and a positiveslope parameter r was assumed in this case.

FIG. 6. Stencil for a second modified linear method LINEAR2. Forthe sake of clarity, no multiplication constant was included.

that always cancels exactly the negative weighting co-efficients, then we simply obtain the inconsistent schemeCOMBI presented in section 4b(2).

3) LINEAR2

Other slope-dependent choices of the interface fluxweighting can be envisaged, as for example a linearcombination of the two vertical differences as a functionof the slope parameter:

Ds2 F 5 C 2 C 1 r/4i11/2, j i11, j i,jA9

3 [(1 2 r)(C 2 C )i,j11 i,j

1 (1 1 r)(C 2 C )i11, j11 i11, j

1 (1 1 r)(C 2 C )i,j i,j21

1 (1 2 r)(C 2 C )]. (34)i11, j i11, j21

The ensuing stencil (Fig. 6) can also be interpretedas being obtained by using the classical stencil of Fig.3 in which the vertical diffusion part of Eq. (32), ratherthan being discretized on the central vertical line, hasbeen distributed on the surrounding vertical lines (a con-sistent truncation error).

b. Inconsistent linear schemes

A first approach to render linear schemes monotonicis to relax the consistency.

1) CLASSIC

An inconsistent scheme, hereafter called CLASSIC,which is already used currently, is the scheme in whicha background diffusion in the ‘‘horizontal’’ numericalgrid is maintained. Typically, the diffusion coefficientassociated with this background diffusion is 20% of the

isopycnal diffusion coefficient. But when looking at thestencils of the linear methods, this procedure is not like-ly to reduce the nonmonotonicity problem, since it doesnot influence the cross-derivative terms.

2) COMBI

Since the problem of nonmonotonicity stems fromthe cross-derivative terms, one may try to eliminatethem by using a combination of kinds of diffusion alongthe grid lines, which mimic the directional effect of therotated diffusion: in a nonflux form this can be writtenas

2Ds| – / \D 5 aD 1 bD 1 gD 1 dD , (35)

A9

|D 5 C 1 C 2 2C , (36)i,j11 i,j21 i,j

–D 5 C 1 C 2 2C , (37)i11, j i21, j i,j

/D 5 C 1 C 2 2C , and (38)i11, j11 i21, j21 i,j

\D 5 C 1 C 2 2C . (39)i11, j21 i21, j11 i,j

This COMBI discretization is generally not consis-tent, but if the coefficients a, b, g, d are nonnegative,then the monotonicity is easily satisfied for small timesteps. Such coefficients can then at least be chosen sothat a discretization mimics the real diffusion as wellas possible. Here we used a linear combination of twoD*’s depending on the slope parameter r, such that whenr 5 1 for example, D/ is retrieved (for 0 # r # 1 oneuses for example a 5 0, d 5 0, b 5 1 2 r, g 5 r).The problem with this formulation is that the conser-vative form is more complicated due to D/ and D\ . Thoseterms written in conservative form require some aver-aging of C involving at each interface more points thanthose of the nonconservative form. When slopes anddiffusion coefficients are constant, those contributionscancel out when flux differencing is performed, but fornonuniform grids and slopes, this cannot be guaranteed


FIG. 7. Stencil for the inconsistent linear method COMBI. A slopeparameter r ∈ [0, 1] was assumed. For the sake of clarity, no mul-tiplication constant was included.

and the purely three-point stencils combination may belost.

Another problem is the clear diapycnal diffusion thatwill develop when using only positive weightings. Ifwe assume for example a moderate positive slope, theoperators D/ and D2 would be combined. This will how-ever introduce diffusion in the diagonal direction. Asignal will thus propagate between the horizontal linein the discrete space and the diagonal direction in thediscrete space, rather than propagating only in the di-rection of the slope. This will be observed later in thetest cases.

Of course some more complicated and nonlinearweighting of the two operators could be used, but thefact remains that there will always be only pure diffusionin both the diagonal and horizontal direction. One ofthese diffusions could be kept small locally, but in orderto have some kind of diffusion, at least the other onemust be present, and we will thus always tend to diffuseaway from the slope direction.

In the case of a constant slope, the implementationof the COMBI method simply combines some discretediffusion along oblique, horizontal, and vertical direc-tions weighting them in the function of the slope pa-rameter r. In this case, the scheme COMBI shown inFig. 7 clearly ensures monotonicity but is inconsistent.Compared to the consistent discretization LINEAR1 ofFig. 5, we have added a permanent vertical diffusionproportional to r 2 r2.

c. Nonlinear computations

1) AMPMIN

Stelling and van Kester (1994) presented a monotonicmethod based on a nonlinear flux minimization. In theirwork, the authors discuss the problem of diffusion alonggeopotentials in a s-coordinate model. Their approachis based on a transformation of the numerical grid,where the finite volumes (normally defined in the s

space) are first rotated into rectangular horizontal boxes.Then, since the box interfaces do not generally hori-zontally match their neighbors, a z interpolation of sca-lars is needed to compute the fluxes at interfaces. Forsmall slopes this involves only the classical nine pointsand can be efficient. However, when slopes are arbitrary,the interpolation method requires the scanning of thewhole water column for each flux computation.

On the other hand, the authors prove their scheme tobe monotonic, if the fluxes are computed by means ofa nonlinear minimization. Unfortunately, this method isby and large time consuming. The computational burdenof a hopefully small effect (‘‘horizontal’’ diffusion)should not penalize the whole ocean model. Therefore,we could adopt the approach of Stelling and van Kester(1994) if relative slopes are small (which could possiblybe enforced by slope limiting in the code) or if we findanother similar nonlinear interpolation method limitedto the local stencil rather than the whole water column.

The idea of limiting fluxes at the interfaces can beused here in the following approach. Similar to Stellingand van Kester (1994), when the slopes are weak |r| #1, two consistent flux calculations can be performed ateach interface. At the interface between i, j and i 1 1,j, two possibilities to compute F from Eq. (28) aredefined:

A91F 5 2 [(1 2 |r |)C 1 |r |C 2 C ] andi11/2, j i11, j i11, j1e i,jDs

(40)

A92F 5 2 [C 2 (1 2 |r |)C 2 |r |C ], (41)i11/2, j i11, j i,j i,j2eDs

with

e 5 sign(r). (42)

For convenience, the AMPMIN function is definedby

AMPMIN(a, b) 5 0.5[sign(a) 1 sign(b)]min(|a|, |b|),

(43)

which is an interpolating weighting function selectingthe flux whose amplitude is minimal or zero if fluxeshave different signs.

In the case |r| # 1, the flux at the interface betweeni, j and i 1 1, j is then chosen as

F 5 AMPMIN(F1, F2), (44)

with the r parameter being computed as the verticalaverage of the slopes at the grid corners above andbelow the interface.

For the interface between i, j and i, j 1 1 the methodreads

]h ]h ]hJ F 5 0.5 J F| 1 J F| .i11/2, j11/2 i21/2, j11/21 2]s ]s ]s

(45)


FIG. 8. Fluxes involved in the nonlinear method AMPMIN. At theright interface, two fluxes are involved (single and triple lines), where-as at the top interface, four (double and triple lines) are taken intoaccount for |r| # 1.

The fluxes at the corners are then again chosen bythe AMPMIN function for the two nearest available fluxcomputations,

F|i11/2,j11/2 5 AMPMIN( , ),1 2F Fi11/2,j i11/2,j11 (46)

in the case r is positive.In this case, the r parameter is again computed at the

corners and is used for the two fluxes that are surround-ing the corners.

For the case of slopes |r| # 1 the fluxes involved inthe calculation of the interface flux are those depictedin Fig. 8. Two evaluations of gradients are needed atthe right interface, whereas four are involved in the fluxat the top interface.

It is easily shown that the flux calculation itself isconsistent, because fluxes are always interpolated (non-linearly). As shown in the appendix, the flux differenc-ing for a finite volume may however lead to inconsis-tencies for the diffusion operator. This is because eachof the consistent fluxes has a different type of truncationerrors (due to the presence of the AMPMIN functionsselecting different flux discretizations and thus trunca-tion errors at the different interfaces) that, when per-forming the flux differencing, introduce the inconsis-tency.

Several tests with grids satisfying |r| # 1 everywhereshowed that the method behaved correctly concerningthe monotonicity properties, though we were not ableto demonstrate that the scheme satisfies the monotonic-ity principle for slopes |r| # 1. But even if the schemeseems monotonic for |r| # 1, the problem of largerslopes must be tackled. The generalization for largerslopes should solely be based on the local stencil inorder to avoid the expensive scanning of the water col-umn. This problem arises when the relative slope in-creases to the value where an extrapolation rather thanan interpolation is performed during the flux calculationaccording to (44). In the original version of Stelling and

van Kester (1994), a search of the two points that reallysurround the s coordinate line is carried out. But this isvery expensive and in some cases even impossible (nearthe bottom boundaries, for example).

If the slope is steeper, |r| $ 1, we suggest computing

A9 1 11F 5 2 1 2 C 1 C 2 C ,i11/2, j i,j1e i11, j1e i,j1 2[ ]Ds |r | |r |

(47)

and similarly for the other flux.While in the case of |r| # 1, two fluxes are involved

at the interface between i, j and i 1 1, j and four fluxeswere involved at the interface between i, j and i, j 11, we now have the inverse: Two fluxes must be usedat the interface above the grid point and four laterally.Not only does the number of fluxes involved in theminimization process vary depending on the slopes, butalso the decision of which two fluxes are relevant de-pends on the slope (via the sign of the slope). This thusleads to important testing sections in the algorithm.

An advantage of the method is that if formulated asa weighting of different fluxes at the interfaces, linearschemes may be recovered by the appropriate choice ofthe weighting functions. This would for example allowfor switching from the Griffies et al. (1998) scheme tothe AMPMIN scheme merely by changing the weightingfunctions.

A major practical problem of the present method ishowever the treatment of the vertical part of the fluxes,since those can lead to restrictions on the time step(Mathieu et al. 1999), because the nonlinear algorithmis not easily implemented in the framework of an ex-isting implicit treatment of vertical fluxes.

In addition, if any slope limitation is desired (for otherreasons than numerical), then this limitation can onlybe imposed at the moment when the flux is computed.However, some tests will then indicate that the methoddoes not behave correctly, because the finite volume isnot changed by the slope limiting, and the differencingof the fluxes creates problems.

As we will show later in examples, if we use theminimum amplitude flux approach proposed here forboth smaller and larger slopes, the method seems tobehave monotonically, but we are not able to prove thismathematically.

2) EQUIVALENT NONLINEAR DIFFUSION ALONG

GRID LINES

Another discretization already used is a fully implicitscheme as in Harvey (1995), but this is neither easilyimplemented into existing GCMs nor very efficient interms of CPU resources. But Harvey (1995) suggests adifferent approach. One could rewrite


JD 5 JD 1 JD , (48)j h

2] ]j r ]C

JD 5 J A9 1 2 , and (49)j 1 2 1 2[ ]]j ]s R ]j

2] ]h R ]C

JD 5 J A9 1 2 , (50)h 1 2 1 2[ ]]h ]s r ]h

where

]C/]jR 5 2 (51)

]C/]h

is the relative slope of the field to be diffused, comparedto the aspect ratio of the numerical coordinate grid.

The problem is formally equivalent to diffusion alongthe grid lines, with diffusion coefficients that depend onthe solution and can be negative locally in time andspace. A sufficient condition to ensure a monotonic so-lution for each small time step is the use of a positiveapparent diffusion coefficient along the grid lines. Usingonly positive apparent diffusion coefficients is in prin-ciple not necessary to ensure that the monotonicity prin-ciple is satisfied, since (48) is just a reformulation ofthe monotonic physical diffusion. This equivalenceshows that ‘‘negative’’ diffusion along the grid linesmay be necessary. By limiting the apparent (solutiondependent) diffusion coefficients to positive values, onecan clearly satisfy the monotonicity principle if the cho-sen time step is short enough (otherwise one can alsoimpose an upper limit on the apparent diffusion coef-ficient). This scheme is also conservative but inconsis-tent when any apparent negative equivalent diffusioncoefficient is set to zero. The advantages of the schemeare promising and, in addition, the practical implemen-tation of such a scheme is almost immediate in a modelalready including a diffusion module along grid lineswith varying diffusion coefficients. Since on the verti-cal, time stepping is generally implicit, time step re-strictions associated with the vertical flux in the case ofstrong slopes can be dealt with automatically. Further-more, it is relatively easy to ensure that for a system inwhich density depends linearly only on temperature, forexample, the isopycnal diffusion of temperature is zero.This can be easily achieved by computing r and R inan identical way so that r 5 R when temperature isconstant on s lines. There are thus numerous advantagesbut, unfortunately, when the s line and the solution havea slope of the same sign, the apparent diffusion coef-ficient must always be limited, because one of the twoequivalent diffusion coefficients is always negative inthis case. This is because 1 2 r/R and 1 2 R/r havedifferent signs when r/R . 0. For slopes of oppositesign (r/R # 0), the method is however consistent andsimply leads to a strong, physically correct diffusion inboth j and h directions.

Another problem of this equivalent diffusion is thatsometimes an upper bound limit for the equivalent dif-fusion must be set. This is due to the appearance of

gradients of the isopycnals and the tracer fields in thedenominator. In principle, this denominator should can-cel out during the final flux computation, but since someaveraging of the equivalent diffusion may be necessary,this cancellation cannot be guaranteed numerically.

We will not show results of a method in which theequivalent diffusion coefficients are always forced to benonnegative, since the next method encompasses thispossibility.

3) FLUX LIMITER APPROACH FLUXCORR4

The preceeding method of using zero equivalent dif-fusion whenever it is negative has the disadvantage thatit introduces an inconsistent scheme, whenever the slopeof the solution and the diffusion direction have the samesign. Limiting the apparent diffusion coefficient to pos-itive values is however only a sufficient condition toensure a monotonic scheme, since downgradient fluxesat interfaces are not necessary to ensure a positive def-inite scheme. What matters is the compensation of someupgradient fluxes at an interface by sufficiently strongdowngradient fluxes at another interface of the grid box.In other words, upgradient fluxes (i.e., negative equiv-alent diffusion coefficients) may be allowed as long asthe budget over the grid box remains positive definite.One possibility is thus to add positive diffusion at theinterfaces, but just the minimal quantity necessary toguarantee that the next time step does not create extremaoutside the range around the grid point.

This leads naturally toward flux limiting approaches.Though these methods were developed for advectionproblems, the basic idea is to add just as much numericaldiffusion as is necessary locally to ensure a monotonicscheme. This is, superficially at least, similar to ourproblem. Similarly to the advection problem, we definea high-order flux Fh, which is the flux based on theactual (and thus possibly negative) equivalent diffusioncoefficient, and a low-order flux Fl (assuring a positivedefinite scheme), which is either the high-order flux forpositive equivalent diffusion coefficients, or zero for theupgradient case. The low-order flux may at first glanceappear to be inconsistent, but as shown in the appendix,the limiter function itself depends upon the resolution,and when grid spacing tends toward zero, the limiterfunction gives back the high-order flux.

For the fluxes in the j direction, this method can bedescribed by the following equations in which the equiv-alent diffusion A e is used:

4 As is sometimes the case in the literature, we will also call thismethod the ‘‘flux corrected method,’’ although this terminology issomewhat incorrect since such methods are generally based on two-stage approaches.


2]j r

eA 5 J A9 1 2 . (52)1 2 1 2]s R

It should be noted that

]C ]n ]C ]n2

]h ]j ]j ]hR 2 r 5 , (53)

]C ]n

]h ]h

so that in practice the component 1 2 r/R can easilybe translated into a Jacobian in the vertical plane be-tween the solution and the isopycnal lines.

From the conservative formulations (48)–(51) and re-lation (53), it appears that the most natural way to cal-culate the equivalent diffusion coefficients A e is to com-pute first corner values (because there r/R is most nat-urally calculated) and then to take an average of thecorner values to retrieve the interface values needed forthe final flux computation.

The high- and low-order fluxes are computed as

h eF 5 2A (C 2 C ) and (54)i11/2 i11, j i,j

l e eF 5 2A (C 2 C )g(A ), (55)i11/2 i11, j i,j

where g(x) is the classical Heaviside function, which iszero for negative values of x, and takes a unit valueotherwise. In classical flux limiter methods, the limiterparameter is computed by taking the difference betweenthe two types of fluxes, and then by taking the ratio ofthis difference at the interface where the flux is to becomputed and a second interface, which is chosen to bethe left or right neighbor interface, depending on thesign of the advection velocity.

The classical flux limiter approaches were developedfor advection, and the flux differences introduced thegradients of the fields into the limiter function. The ratioof the flux differences was a measure of the ratios ofgradients and, thus, the variability of the field (whichultimately is the estimator of the need to increase thediffusion). Here we should base the ratio directly on thegradients rather than on the flux difference, because thefluxes are already based on the gradients.

The choice of the direction in which the ratio is com-puted depends on the advection direction for an advec-tion problem. Here we base it on the direction of thebasic high-order flux, which indicates in which directionthe flux should be.

The scheme is then computed as

Fi11/ 2 5 F 1 f(q)(F 2 F ),l h li11/2 i11/2 i11/2 (56)

with

h eF Ai11/22m i11/2q 5 and (57)h eF Ai11/2 i11/22m

hm 5 sign(F ). (58)i11/2

Typical limiters are (e.g., Zijlema 1996)

SMART limiter,

f (q) 5 max[0, min(4, ¾q 1 ¼, 2q)], and (59)

SWEBY limiter (Superbee for a 5 2, minmod for a5 1),

f (q) 5 max[0, min(aq, 1), min(q, a)]. (60)

In the cases shown hereafter, the SMART limiter wasused.

This method needs an additional treatment related tothe time discretization itself: an upper limit (in ampli-tude) must be set for equivalent diffusion coefficients,otherwise the time discretization itself may be unstable.In principle, the limit should be set on the time stepping,but this would penalize the overall model performancevery heavily simply because in some occasions, the de-nominator used in the computation of the equivalentdiffusion coefficient vanishes and anyway should cancelout when the flux is calculated. But since the equivalentdiffusion coefficients are most naturally computed at thegrid-box corner, they need to be averaged at the inter-faces. There however, the gradients in the denominatorof the equivalent diffusion do not cancel out with thelocal interface gradient, and for small gradients (wherediffusion is anyway small) one has to limit the time stepor diffusion coefficient.

It is also noted that from a practical programmingpoint of view, contrary to the AMPMIN method, it isnot easy to make the equivalent diffusion coefficientmethod into a linear method, because the computationof the equivalent diffusion is essentially nonlinear. Onthe other hand, the method can be easily included inany solver that allows variable horizontal and verticaldiffusion coefficients (treated implicitly or not). Prac-tically, one can apply the flux limiter approach to cal-culate the equivalent diffusion coefficient, since the gra-dients involved in the flux combination are the same forthe two fluxes (lower and higher order), so that one canin fact just combine diffusion coefficients based on thelimiter functions. These equivalent diffusion coeffi-cients can then be used very conveniently in existingdiffusion solvers (which should not enforce positive dif-fusion coefficients in the computer code).

The method presented here is based on a treatmentthat consists of analyzing two one-dimensional prob-lems when it comes to the computation of the limiterfunctions. In advection problems this is the general pro-cedure. As for advection, the rotated diffusion is also adirectional process (1D problem along the velocity orthe isopycnal direction), but there is a major difference:for the advection flux correction, when computing thelimiters, the advection flux in the other direction willnormally not introduce an important diffusion (becausethis is the goal of the method). This means that in thecase of flux limiting in one direction one can assumethat the other direction behaves correctly, but will nothelp in diffusing perturbations. In the case of isopycnaldiffusion, however, when looking at the fluxes in one


FIG. 9. Linear scheme LINEAR. Diffusion of a Dirac signal alongy 5 0.4x after 100 time steps. This standard scheme clearly introducesstrong diapycnal diffusion and dispersion with significant under-shootings (in white).

direction, limiting may in fact not be necessary, becausethe other direction could have diffusive fluxes that ren-der the 2D system monotonic. Therefore, a pure 1Dmonotone scheme for negative diffusion is too strong(because we neglect the other direction and the feed-backs on diffusion coefficients). For fixed negative dif-fusion coefficient in one direction and zero diffusion inthe other, the scheme is indeed not monotonic. We mustthus expect that the twofold monotonic 1D problem willinduce unnecessarily high diapycnal diffusion. Indeed,we may add some diffusion in one direction becausewe do not take into account the positive diffusion thatmay exist in the other direction. A promising alternativewould thus be the use of truly 2D flux limiter approaches(e.g., Thuburn 1996), which however make the numer-ical scheme increasingly complicated (specially if the3D generalization is thought of ). In addition, truly 2Dflux limiter schemes are generally designed for advec-tion schemes (e.g., Thuburn 1997; Drange and Bleck1997), and their adaptation to diffusive fluxes, as donehere in 1D, is not straightforward and some preliminarytrials of generalizations were not conclusive. Further-more, the proof of monotinicity must be two-dimen-sional and include the feedback onto the diffusion co-efficients. For general implementation and further find-ings in 2D limiters, the limiter should be parameterizedin terms of points in a stencil rather than along a co-ordinate line since the problem is really two-dimen-sional.

Though no proof is presented to mathematically en-sure that the nonlinear schemes AMPMIN and FLUX-CORR are monotonic, the heuristic explanation of theirfunctioning is similar: Both methods feature flux lim-iters that use smaller-amplitude fluxes when ‘‘problemsare expected’’: this arises when very different values ofthe different flux computations are encountered (AMP-MIN) or rapidly changing gradients (FLUXCORR). Inthis case, both methods reduce the fluxes and to someextent ‘‘freeze’’ the situation. Typically this is the casewhen a ‘‘plateau’’ is present before a jump: at the jump,for negative diffusion values, no flux can be allowedfrom the plateau into the higher value if no fluxes arepresent in the second direction. So the plateau has atendency to remain. Similarly, for a local peak and neg-ative diffusion, fluxes must be limited, otherwise thelocal peak will increase its magnitude.

5. Comparison of the new schemes

Since we already have several possible choices fornonstandard discretizations, we will now proceed to acomparison of their behavior with classical linearschemes and exact solutions.

a. Test cases and criteria

Because we have already presented the numericalstencils for the linear methods in the case of a constant

slope, we will use this case as a first test for the methodspresented here. Only those that are promising in thisframework will be further examined in the case of var-iable slopes.

1) LINEAR SLOPE CASE

Qualitative information can be gained by diffusingthe Dirac function in an uniform slope field with r 50.4 and a time stepping with DtA9 5 0.1Dx2 for 100time steps.

The analytical solution C* for a quantity Q of thetracer initially present at the origin of s is given by

2Q sC* 5 exp 2 . (61)1 24A9tÏ4pA9t

It is observed that all linear schemes produce strongdiapycnal diffusion dispersion with significant under-shootings (see Figs. 9–11). Compared to the standardscheme LINEAR, for the scheme LINEAR1 (see Fig.10), the diapycnal diffusion is visible in a narrower bandand oriented more ‘‘vertically,’’ due to the choice offluxes involved in the averaging (see Fig. 4). For LIN-EAR2 (Fig. 11) the solution is very similar to the stan-dard solution in this case with a slightly reduced prop-agation of information into the diapycnal direction. Italso seems that adding a 20% background diffusion (seeFig. 12), as it is done classically in GCM models usingisopycnal diffusion, does not change the behavior and


FIG. 10. Modified linear scheme LINEAR1. Compared to the stan-dard scheme LINEAR, the diapycnal diffusion is visible in a narrowerband and oriented more ‘‘vertically,’’ due to the choice of fluxesinvolved in the averaging.

FIG. 11. Second modified linear scheme LINEAR2. The solutionis very similar to the standard solution in this case with a slightlyreduced propagation of information into the diapycnal direction.

FIG. 12. Linear discretization CLASSIC with additional 20% hor-izontal diffusion. This scheme reduces the amplitude of the diapycnaldispersion and the associated negative values, but the pattern is stillcomparable to the standart scheme LINEAR.

pattern of the linear scheme, even if the amplitude ofthe diapycnal dispersion and the associated negative val-ues are slightly reduced. The inconsistent COMBI meth-od (see Fig. 13) clearly shows a strong diapycnal mix-ing, but remains monotonic. Clearly the diapycnal dis-persion of the linear schemes was replaced by a dia-pycnal diffusion ensuring a monotonic solution thatpropagates strongly into the diapycnal direction. Thesolution of the AMPMIN scheme (Fig. 14) is monotonicand presents less diapycnal diffusion than the COMBImethod. The staircase pattern results from the nonlinearAMPMIN function and the constant zero concentrationin the background. Finally, as in the AMPMIN solution,the solution of the flux-corrected method (see Fig. 15)remains monotonic and presents a propagation into thediapycnal direction that is reduced compared to the stan-dart scheme.

In Fig. 16, showing the evolution of the maximumvalue of C in function of time, we can clearly see thatthe AMPMIN version retains the highest values but isbelow the exact solution after 100 time steps. There isthus a diapycnal mixing present in the AMPMIN ver-sion, but it appears smaller than for the other methods.The most diffusive method is the COMBI method. Itshould also be noted that the AMPMIN version under-estimates the diffusion at the initial stage. Among thelinear versions, LINEAR1 gives the best response inlater stages, with LINEAR2 still better than the classicallinear version. It is noteworthy that the additional back-


FIG. 13. Inconsistent combination of horizontal and diagonal dif-fusion COMBI. Clearly the diapycnal dispersion was replaced by adiapycnal diffusion ensuring a monotonic solution that propagatesstrongly in the diapycnal direction.

FIG. 15. Flux-corrected method FLUXCORR. The solution ismonotonic and propagation into the diapycnal direction reduced com-pared to the standard scheme.

FIG. 16. Evolution of the maximum value of the field in functionof time (unnumbered curves in this figure and the following figuresoverlay the exact solution; the x axis corresponds to the discrete timestep).

FIG. 14. Nonlinear AMPMIN method. The solution is monotonicand presents less diapycnal diffusion than the COMBI method. Thestaircase pattern results from the nonlinear AMPMIN function andconstant zero concentration in the background.

ground diffusion does not drastically change the be-havior of the maximum. The flux-corrected method ini-tially diffuses strongly, but then slows down the dia-pycnal diffusion.

The evolution of the minimum value of C in functionof time (Fig. 17) shows that all the linear methods pro-duce a strong undershooting, especially during the initialphase when the Dirac signal is present and leads to largegradients. Here method LINEAR1 gives the worst re-sult, with the two other methods being similar. An ad-


FIG. 17. Evolution of the minimum value of the field. Due to theDirac distribution, initially, undershootings are very important buttend to decrease when the field starts to be smoother. LINEAR1produces the strongest undershootings and adding a horizontal dif-fusion (CLASSIC) only slightly reduces the undershooting.

TABLE 1. Measures of the errors.

Method I1 I2 Relative cost

AMPMINCLASSICCOMBIFLUXCORRLINEARLINEAR1LINEAR2

0.530.690.810.650.650.510.61

0.4.06007.038.447.24

1.2112111

ditional background diffusion improves the behavioronly slightly. Undershooting tends to decrease once thefield is smoothed by the overall diffusion.

In order to quantify the accuracy of the methods, wenow use two cost functions I1 and I2, which measurethe difference between the theoretical solution and thenumerical solution:

2I 5 (C 2 C*) and (62)O1 ii

extr 2I 5 (C 2 C ) g, (63)O2 ii

where C i represents the computed fields, C* the ana-lytical solution, Cextr are the extrema of the real solution,and g is a Heaviside function that is zero when the fieldCi remains between these extremal values. These costfunctions give an idea of the overall truncation error(I1) and the monotonicity (I2). We see that the normal-ized errors in Table 1 give rms errors that are lowestfor the linear methods and the AMPMIN method, fol-lowed closely by the flux-corrected method. Concerningundershootings, linear methods behave similarly, withan improvement due to the addition of the backgroundhorizontal diffusion. We also give a rough estimate5 onthe relative cost of each scheme compared to the clas-sical linear scheme. If the rotation of a diffusion operatordoes not take an important CPU fraction of a generalmodel, this indicates that all schemes are affordable.

A way to measure the diffusion of the numericalmethod is the computation of

5 The actual value in a GCM will depend on the compiler, thehardware, and the organization of the code. In addition, some of theimplementations of the linear schemes were directly based on theassumption of a constant slope, which led to some simplifications inthe code.

2X 5 j C dD and (64)ED

2Y 5 h C dD, (65)ED

where D is the total domain. Both X and Y increaselinearly in time and proportionally to the diffusion co-efficient for the case of the diffusion of a point releasewhose solution was given in Eq. (61). This means thatthe quantities

2Xx 5 and (66)

QA9t

2Yy 5 (67)

2Qr A9t

should be equal to one for the numerical solution. Alsothe ratio p 5 (X/Y)/r2 should be constant and equal toone, as can be seen by the ratio x/y, which is one forthe analytical solution. The first two parameters measurethe effective diffusion in the two grid directions, where-as the last parameter measures the actual slope on whichthe method diffuses. If the parameter is lower, this meansthat the diffusion is too steep, whereas a higher valuemeans that the diffusion is too horizontal.

In Fig. 18, it can be seen that all the linear methodsproduce a correct average measure of horizontal dif-fusion. This is because the undershooting and over-shooting cancel out when performing the integral. Forthe CLASSIC scheme, the additional background dif-fusion clearly displays a 20% increase in effective dif-fusion, while AMPMIN has considerably reduced theeffective diffusion because of its systematic choice ofminimal amplitude fluxes (or zero fluxes when fluxeshave opposite signs). The flux-corrected method onlyslightly decreases the effective diffusion. In Fig. 19, itis demonstrated that the linear methods all produce acorrect average measure of diffusion in the vertical di-rection. The additional background diffusion does notshow up here for the CLASSIC scheme, since it wasadded only in the horizontal direction and the schemeis linear. The COMBI method shows a very strong in-crease in effective diffusion, which is consistent withthe interpretation of the stencil given beforehand (Fig.7). The AMPMIN version has again reduced effectivediffusion because of its AMPMIN function. The flux-


FIG. 18. Evolution of the measure of the horizontal diffusion. AMP-MIN clearly reduces diffusion while increasing the horizontal back-ground diffusion by 20% in CLASSIC can clearly be seen on thisintegral quantity. All linear schemes have the correct average hori-zontal diffusion while FLUXCORR slightly reduces it.

FIG. 20. Evolution of the measure of the direction of the diffusion.Consistently with the analysis of the horizontal and vertical diffusion,the average direction of diffusion is correct for the linear schemes,too flat for the CLASSIC scheme and slightly more upward orientedfor the FLUXCORR scheme. AMPMIN and COMBI modify the di-rection very strongly.

FIG. 21. Integral on h in the function of the grid point; the exactsolution is indistinguishable from the linear solutions. AMPMIN notonly shows the reduced diffusion as before but also the effect of theinconsistent flux calculations, which allow an unphysical peak in theintegrated field to remain. FLUXCORR has a slightly higher peakthan the exact solution, while the CLASSIC method reduces the peak.

FIG. 19. Evolution of the measure of the vertical diffusion. As forthe horizontal diffusion, AMPMIN reduces the diffusion, and thelinear schemes behave correctly. The FLUXCORR method slightlyincreases the vertical diffusion while the inconsistent COMBI methodshows the strong additional vertical diffusion consistent with the anal-ysis of the numerical stencil.

corrected method only slightly increases effective dif-fusion. Figure 20 shows that the linear methods all pro-duce a correct average measure of direction of diffusion.The additional background diffusion in CLASSIC leadsto a weakening of the effective slope, consistent withFigs. 18 and 19. The other methods increase the slope,with the worst effect in the COMBI and the AMPMINmethod.

Another integrated way is to look at the integral ofthe field in the h direction and show the distribution inthe function of j after 100 time steps. In principle, thisshould be exactly the same solution as obtained by apure horizontal diffusion. An inspection of Fig. 21

shows that the linear methods all produce a ‘‘correct’’answer, because the discrete summing of the linearschemes in the y direction leads to a discrete equationof the vertical average that is quite simply a horizontaldiffusion. For 100 small time steps, the numerical so-lution to this equation is indistinguishable from the exactsolution. The additional background diffusion in CLAS-SIC leads, as one should expect, to an overdiffusion.The flux-corrected method slightly underestimates thediffusion, while AMPMIN not only strongly underes-timates diffusion but also leads to a very particular andunrealistic shape of the diffused field. Though the in-tegral measure seems to indicate that the linear methods


FIG. 22. Density field used for the variable slope experiment.

FIG. 23. AMPMIN method for the variable slope experiment show-ing that the diffused field follows the density field by bending thepatch along the isolines.

FIG. 24. FLUXCORR method for the variable slope experiment.As for AMPMIN, the solution is following the density field.

are not so bad, the patterns show the problems they caninduce. These patterns also show that simple linear in-tegrals are not sufficient to characterize a field, sincethe wiggles’ contributions of the dispersion may cancelin the integrations (unless quadratic measures as in I2

are used).In order to verify that the methods also behave cor-

rectly with slopes that exceed the aspect ratio of thegrid, additional tests with |r| $ 1 were performed (butare not shown here). Both the AMPMIN and FLUX-CORR method behaved similarly to the case presentedabove, so that the adaptation from the original methodof Stelling and van Kester (1994) to a local stencil wassuccessful. After the constant slope case, we will finallytest the more promising methods in more general sit-uations.

2) VARIABLE SLOPE

In order to verify that the two monotonic nonlinearmethods also behave correctly in a situation where theslope is not constant, we performed a simulation inwhich the lines on which diffusion takes place arecurved as shown in Fig. 22

In this situation, both the FLUXCORR and the AMP-MIN method (Fig. 23 and 24) diffuse the signal on theselines by bending the patch along the isoline. Both meth-ods remain monotonic, except for very small negativevalues in the AMPMIN method, which are however inthe range of CPU rounding errors and several orders ofmagnitude lower than the undershooting induced by lin-ear methods.

b. Discussion

The flux-corrected method seems to have some sig-nificant diapycnal diffusion, but part of it is due to the


FIG. 25. Flux-corrected method after 1000 time steps, constant-slope case. The method keeps the solution in a rather narrow bandonly slightly larger than after 100 time steps.

FIG. 26. Linear scheme after 1000 steps, constant-slope case. Herethe solution has propagated into the overall domain.

FIG. 27. Flux-corrected method after 1000 time steps in a variableslope situation. As for the constant-slope case, the solution is main-tained in a narrower band.

fact that the nonlinear scheme needs several points tokeep the information contained in a narrow band. Typ-ically, after 100 iterations, the diapycnal signal is spreadover seven points containing a signal that is larger than1% of the central signal. When running the diffusion10 times longer (Fig. 25) this spreading is only on 10points, indicating that the initial diapycnal diffusion wasnecessary to create a large-scale signal. This is verydifferent from the linear methods: as seen in Fig. 26,the linear scheme has dispersed farther into the diapyc-nal direction and spreads over 21 points (again for a1% threshold), while the flux-corrected scheme some-how stabilizes. This indicates that for larger-scale sig-nals than the Dirac function, the flux-corrected methodwould exhibit less diapycnal mixing. This is also con-firmed for the variable slope case integrated over 1000time steps (see Fig. 27).

Presently, if we need to ensure a monotonic scheme,we suggest the use of the flux-corrected method. TheCOMBI method clearly exhibits too strong a diapycnaldiffusion, whereas the generalization of the the AMP-MIN method suggested by Stelling and van Kester(1994) leads to strange patterns in the diffused fields asstaircase patterns and non-Gaussian integrals. The in-consistent discretization in this case leads to a numericalsolution that is inconsistent with the mathematical prop-erties of diffusion. Since the AMPMIN method is alsonot easily implemented into vertically implicit schemes(the classical ocean model approach), the flux-correctedmethod seems to be the indicated choice. However,


when such a nonlinear scheme is used, we will face amajor problem when diffusing temperature and salinity.Since the nonlinear scheme may behave differently forthose two fields, there is no way to ensure that the com-bined contribution of each of these fields to densityfluxes along isopycnal surfaces is nil. This may be ac-ceptable, if the resulting modification in the pressurefield does not lead to an unstable coupling between mo-mentum and density. In the case of the classical linearscheme, the incomplete compensation was shown byGriffies et al. (1998) to be the major source of problemsin the isopycnal diffusion of the Modular Ocean Model.It is probable, though not certain, that the problem re-mains also for the nonlinear scheme, unless the insta-bility mechanism is suppressed by avoiding the intro-duction of new extrema in the T, S fields. If, despite themonotonicity, the presence of density fluxes on isopyc-nals still leads to instabilities, certain tricks may perhapsallow us to achieve this compensation even in nonlinearschemes. A simpler approach could thus be to use alinear scheme as in Griffies et al. (1998) (maybe witha more sophisticated averaging technique than the onedescribed in Fig. 9) for dynamically active tracers only,while dynamically passive tracers can be solved by us-ing the new nonlinear schemes. This means that errorsfor temperature and salinity fields are controlled so thatthey do not have an influence on isopycnals, the onlydynamically important feature. For dynamically passivetracers the use of our monotonic scheme (particularlyfor biological models or turbulent variables) thus elim-inates most problems associated with the rotated dif-fusion. For further improvements of our scheme, a wayto decrease the diapycnal diffusion in the flux-correctedmethod would be to use a truly 2D flux correctionscheme for nonadvective problems. This is, in our opin-ion, the direction for further research on discretization,rather than searching for larger stencil methods for ex-ample; a linear model is not expected to behave betterby using more points, since the risk of introducing othernegative coefficients increases.

6. Conclusions

In our opinion, the design of a discretized rotateddiffusion operator should have the following properties.

R Be conservative;R satisfy the monotonicity principle;R extend only over a nine-point stencil [for the (x, z)

case];R reduce to the classical horizontal stencil 1, 22, 1 when

no transformation is present;R be computationally efficient, since diffusion should

be small anyway, and a scheme should not be penal-ized by a second order term; and

R introduce the smallest possible diapycnal diffusion.

Consistency could be another basic requirement, butwe consider it less important than the other properties

because the isopycnal diffusion is a parameterizationthat should disappear when grid spacing decreases. Fur-thermore, inconsistent schemes are common and some-times behave even better than consistent schemes (e.g.,Cockburn et al. 1999). Nevertheless, if an inconsistentscheme is retained, one should carefully analyze its be-havior (see the AMPMIN and COMBI schemes).

We have shown that even when allowing for non-consistent schemes, unfortunately, no scheme analyzedhere satisfies all these requirements perfectly.

All linear (and inexpensive) schemes produce dia-pycnal dispersion and over- and undershootings. As alinear scheme that performs well in situations withstrong diapycnal mixing, we presented a slightly mod-ified linear method (LINEAR1) that may be sufficientin some cases where monotonicity is not a strong re-quirement. For linear methods, we showed that addinghorizontal background diffusion for purely numericalreasons is not recommended.

We presented a number of new methods to deal withthe monotonicity violation of linear schemes. Theseschemes are either nonlinear, inconsistent, or both non-linear and inconsistent. Only two schemes are candi-dates for implementation into general models: the AMP-MIN method keeps diapycnal mixing at a lower value,but leads to some strange patterns because of its incon-sistent discretization and diffusion, while the FLUX-CORR method has a satisfactory behavior in all thecases we tested, though one might still search for ascheme with even less diapycnal diffusion.

Presently, if we need to ensure a monotonic scheme,we suggest thus the use of the flux-corrected method atleast for dynamically passive tracers. For temperatureand salinity, it is not clear if the instability mechanismidentified in Griffies et al. (1998) is suppressed by themonotonic schemes. In case of the lack of compensationof temperature and salinity fluxes on isopycnals stillleading to instabilities, the use of a linear scheme as inGriffies et al. (1998) eliminates the problem for dynam-ically active tracers.

Though we investigated a large range of methods,other remedies could still be analyzed, but our feelingis that we already reached a level of complexity that issurprisingly high for a diffusive process introduced forparameterization reasons. This shows also that if theisopycnal diffusion is only introduced for numerical rea-sons, in order to eliminate grid noise, then the use ofthe rotated operator does not yield an efficient numericalfilter. In the case of a complex model in which somefiltering of the grid noise is required, one should use aspace-selective filter in the numerical grid or, even bet-ter, numerical schemes (generally for advection) that donot generate grid noise. Only if the rotated physicalparameterization supersedes the numerical filtering re-quirements should the rotation be discretized. In thiscase, we proposed monotonic schemes to deal with themonotonicty problem for the given coordinate system.Another way of decreasing the problem of the operators’


rotation is to design the coordinate system so that itfollows closely the direction of interest. A pure iso-pycnal layer model is an example of this, but somehybrid models can adapt their vertical grid positionscontinuously to density gradients in order to reduce rel-ative slopes.

Finally, we should mention that throughout the wholeanalysis, we supposed that the mixing paramaterizationis adequately performed by a Laplacian formulation,which is the standard version used in isopycnal diffusionparameterizations. This supposes relatively coarse-res-olution models, since for higher resolutions, more scale-selective biharmonic diffusions are generally retained.In this case however, even for unrotated operators, theclassical scheme is not monotonic, not to mention theproblems related to the rotation. One might ask whetherthis might be a source of problems in coupled biologicalmodels in the future, but since high-resolution modelsnot only use more scale-selective but also much lowervalues for the diffusion, this may be hidden by otherprocesses involved.

Acknowledgments. A model intercomparison wassupported by the European Union (MEDMEX ContractMAS2-CT94-0107 and MATER Contract MAS3-CT96-0051) and led to some questions addressed in the presentpaper. The development of the numerical code was partof the MEDNET project (MAS3-CT98-0189). E. De-leersnijder and P. P. Mathieu work in the framework ofContract GC/DD/09A with Belgium’s Federal Office forScientific, Technical and Cultural Affairs and Conven-tion N 92/02-208 d’Actions de Recherches Concerteeswith the Communaute Francaise de Belgique.

The National Fund for Scientific Research (Belgium)has given J. M. Beckers and E. Deleersnijder the op-portunity to be active in oceanographic research. P. P.Mathieu was a fellow of Fonds pour la Formation a laRecherche dans l’Industrie et l’Agriculture during thisresearch.

Anand Gnanadesikan and the anonymous reviewersmade some helpful comments on the numerical aspects.

APPENDIX

General Considerations on Consistency

In this appendix we analyze the consistency of someof the numerical schemes, showing that the FLUX-CORR scheme is consistent in contrast to the AMPMINscheme. In order to do so, we will develop the truncationerror of the diffusion operator in terms of the truncationerror of the fluxes.

A common way to design discretizations of a diver-gence such as term D,

]FD 5 2

]x

Dx DxF x 1 2 F x 21 2 1 22 2 2 3Dx ] F

5 2 1 , (A1)3Dx 24 ]x

is to calculate consistent discretizations of the flux F atthe interface of a finite volume and then to proceed toa volume integration of the divergence, resulting in thediscretized form

Dx Dx˜ ˜ ˜DDx 5 2F x 1 1 F x 2 , (A2)1 2 1 22 2

where ˜ stands for the discretized version of the ana-lytical expression. In this case, the truncation error ofthe scheme is D 2 D and includes truncation errors inthe fluxes and truncation errors due to the flux differ-encing. Assuming that we use a nth-order scheme forthe calculation of the fluxes, we have a truncation errorof the flux F 2 F defined by

Dx Dx DxnF x 1 5 F x 1 1 Dx a x 1 , (A3)1 2 1 2 1 22 2 2

where the function a depends on the mathematical for-mulation of the flux and its discretization. Typically, itconsists of some derivatives of the fields on which theflux operator acts.

We can then calculate the truncation error to be

Dx Dxa x 1 2 a x 21 2 1 22 22 3Dx ] F

n˜D 2 D 5 1 Dx .324 ]x Dx

(A4)

This shows that a high-order flux calculation does notnecessarily lead to a high-order scheme, since the fluxdifferencing performed here leads to a second-orderscheme. This phenomenon, known as degeneracy, isgenerally discussed when analyzing discretizations onnonuniform grids (e.g., Hoffman 1982) but also appliesto nonlinear conservation laws. When a first-orderscheme is used for flux calculation, even an inconsistentscheme may result if the function a is not differentiable.

This is the case for the AMPMIN scheme, where thefunction used to choose the flux involved at the interfaceis nondifferentiable (since different triads may be in-volved at the left and right interfaces, the truncationerror reads differently on both sides, which in generalcannot ensure that the differencing of a converges forDx → 0).

For linear schemes, function a is a differentiable func-tion depending on the slope r and a second derivativeof the field that is diffused.

In the case of the FLUXCORR method, the situationis more complicated, since the flux is evaluated by using


the equivalent diffusion coefficient modified by the fluxlimiter function.

First, we show that for Dx → 0 the parameter q usedin the flux limitation for the flux along the x axis tendsin a continuous way to 1:

C 2 Ci112m i2mq 5 , (A5)i11/2 C 2 Ci11 i

where m depends on the sign of the high-order localflux. One can develop the solution C in a Taylor seriesaround i, for example, by assuming that the solution issufficiently smooth (this might not be easily satisfied inregions near the thermocline, since the axes may crossthis discontinuity, but all truncation errors are affectedby this and our analysis here is therefore consistent withclassical analyses):

2]C ] C22 1 (1 2 2m)Dx 1 O(Dx )

2]x ]xq 5 , (A6)i11/2 2]C ] C

22 1 Dx 1 O(Dx )2]x ]x

where the derivatives are taken at i.This means that for very small grid sizes, no flux

limiting is used anymore, except at extrema, but theseoccupy a smaller fraction of the domain as resolutionincreases, since for q 5 1, all currently used flux limiterfunctions return the high-order flux, which in our casemeans that no additional diffusion is added.

Furthermore, for small grid sizes, the equivalent dif-fusion coefficient clearly converges toward its mathe-matical counterpart at least as fast as Dx.

This means that the error on the discrete fluxes is3Dx ] f

eF 2 F 5 A , (A7)324 ]x

which is differentiable and does lead to a consistentscheme.

Recent developments on consistency and conver-gence can also be found in Cockburn et al.’s (1999)recent paper. Beside general aspects of consistency andconvergence of nonlinear conservation laws, they showthat in some situations loss of consistency causes su-praconvergence (a cancellation of the loss of consisten-cy by an increased stability), provided that a conser-vative flux formulation with consistent fluxes is used insituations with a sufficiently smooth exact solution (inthis case the AMPMIN method could be tested again).

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