A sigma-coordinate primitive equation model for studying

Deep-Sea Research I 45 (1998) 543—572

A sigma-coordinate primitive equation model forstudying the circulation in the South Atlantic.

Part I: Model configuration with error estimates

Bernard Barnier, Patrick Marchesiello,Anne Pimenta De Miranda, Jean-Marc Molines,

Macky Coulibaly

Laboratoire des Ecoulements Ge&ophysiques et Industriels, Institut de Me&canique de Grenoble, BP 53, 38041Grenoble Cedex, France

Received 27 November 1995; received in revised form 27 March 1997; accepted 29 July 1997

Abstract

This paper describes the configuration of a topography-following (sigma) coordinate, numer-ical ocean model for studying the circulation in the South Atlantic. An analysis is performed (i)to ensure that the model configuration does not introduce a numerical bias in the modelsolution and (ii) to give estimates of numerical errors. The model is the Semi-spectral PrimitiveEquation Model (SPEM) from Rutgers University (Haidvogel et al., 1991). Two importantissues relating to the sigma-coordinate are investigated: the pressure gradient calculation andthe diffusion of tracers. Errors in the pressure gradient calculation are investigated by simula-ting an ocean at rest, and the choice is made to reduce errors by smoothing the bathymetry.A smoothing criterion is derived that permits a limitation of the errors in the pressure gradientcalculation to an acceptable level (i.e. maximum errors on velocities below a millimeter persecond). It is applied to define the model bottom topography. Errors in the tracer fields, inducedby a diffusion scheme operating along constant sigma surfaces, generates large unrealisticvelocities (of the order of 10 cm/s). A rotation of the diffusion tensor into geopotentialcoordinates is proposed. Tests show that errors are then reduced to an insignificant level. Therotation of the diffusion tensor is therefore retained. The numerical treatment of the openboundaries and the flux conditions that yields the most realistic circulation is also described.Open boundary conditions are based on radiation conditions and relaxation to climatology.They appear to be numerically robust, and to be able to bring into the South Atlantic basin thenecessary information from the outer oceans. A configuration of the SPEM model to study thelarge scale circulation in the South Atlantic is then obtained. Errors due to model configurationare shown to be small compared to the signal one wants to simulate, and their spatial pattern isknown, which will facilitate the interpretation of the model simulations presented in followingpapers. ( 1998 Elsevier Science Ltd. All rights reserved.

0967-0637/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 9 6 7 - 0 6 3 7 ( 9 7 ) 0 0 0 8 6 - 1

1. Introduction

The South Atlantic Ocean has a wide opening to the Indian Ocean to the east and,to the west, receives a large inflow from the Pacific Ocean through Drake Passage. Tothe north, it communicates over an extensive area with the North Atlantic, where themajor sources of cold deep waters for the world ocean are located. Consequently, theSouth Atlantic is a region where many water masses coexist (Peterson and Witworth(1989) defined as many as eight different water masses in this area), and the dynamicsof this ocean basin play an important role in determining the characteristics of thewater masses and the meridional heat and fresh water fluxes of the upper flow of theAtlantic overturning cell.

The recent emphasis on the role of the ocean in influencing climate and theidentification of the Atlantic overturning cell as an important element of the worldclimate (Toggweiler, 1994) have focused more attention on the South Atlantic withinthe World Ocean Circulation Experiment (WOCE), as demonstrated at the 1994Bremen Colloquium (Berger and Wefer, 1996).

The first important coordinated field program carried out in the South Atlantic wasthe South Atlantic Ventilation Experiment (SAVE) in the late 80s. The WOCEhydrographic program is now sampling the mass field of the South Atlantic at anunprecedented level. Intensive experiments (using hydrography, moorings, surfacedrifters and sub-surface floats), in special study areas such as Confluence (Provost etal., 1995), the Brazil Basin (Hogg et al., 1996) and the Cape Basin, are beginning toprovide additional information on many physical processes relevant to the generalcirculation. The dense sampling and the quality requirements of WOCE measure-ments will result in a tremendous increase in our knowledge of the properties of thewater masses and the ocean circulation in this basin. However, field experiments donot have the synoptic character required for the calculation of “turbulent” fluxes inthe ocean (by “turbulent” here we mean subgrid-scale with regard to the space—timesampling of hydrography). Consequently, the WOCE program also includes impor-tant modeling objectives, these being to develop and test ocean models that will beuseful for predicting climate change and to analyze the WOCE field data.

A modeling strategy aimed at understanding the ocean circulation in the SouthAtlantic will need to consider carefully the two most important geographical featuresof this ocean basin: the bottom topography and the openings towards the otheroceans (Fig. 1).

The treatment of the bottom topography in basin-scale ocean circulation models iscurrently a matter of concern. Although parameterization of certain topographiceffects on large-scale circulation (such as topographic stress) is being investigated (Ebyand Holloway, 1994), an important issue that we believe is still being overlooked innumerical modeling is the choice of a vertical coordinate. There are three differenttypes of vertical coordinates currently used in oceanography: the geopotential orz-coordinate used in the Geophysical Fluid Dynamics Laboratory (GFDL) oceanmodel (Bryan, 1969; Cox, 1984), the isopycnic or p-coordinate of the Miami IsopycnalCoordinate Ocean Model (MICOM) (Bleck and Smith, 1986), and the topography-following or sigma-coordinate used in the Princeton ocean model (POM) (Blumberg

544 B. Barnier et al. / Deep-Sea Research I 45 (1998) 543—572

Fig. 1. A schematic representation of the bathymetry of the South Atlantic ocean basin (from Peterson andStramma, 1991). The bold straight lines are the model open boundaries.

and Mellor, 1987) and in the Semi-spectral Primitive Equation Model (SPEM)(Haidvogel et al., 1991). One expects significant differences to show up in model resultsdepending on the choice of vertical coordinate, because of different numerical treat-ments of the primitive equations and the exact resolution employed. Results are nowavailable from these different models, but it is only very recently that the modelingcommunity has begun to investigate the impact on model solutions of one coordinatesystem compared to another.

The second feature (the openings of the South Atlantic basin) is also important,since modeling the ocean circulation is this area involved dealing with complicatedinter-basin exchanges. Barnier et al. (1996) have discussed the various numericalapproaches to the problem. One of these approaches consists of modeling the worldocean and studying the South Atlantic as a sub-domain. The numerical problem ofopen boundary conditions is thus avoided, since inflow and outflow conditions at thelimit of the South Atlantic are then determined by overall model behavior. However,this is not fully satisfactory, and there are still large differences between the varioushydrographic and model estimates of the fluxes at the limits of the South Atlantic.

B. Barnier et al. / Deep-Sea Research I 45 (1998) 543—572 545

Moreover global models are not yet capable of simulating transport realistically in theAntarctic Circumpolar Current (Barnier et al., 1996).

Another approach is to limit the model domain to the South Atlantic basin. In thiscase, the simulated circulation of the South Atlantic will largely depend upon thefluxes prescribed at the boundaries. This approach requires an a priori knowledge ofincoming mass, heat and salt fluxes, which is derived from climatologies of hydro-graphic data. It has the advantage of enabling a large number of experiments to beconducted (since the modeled domain is reduced), and it is thus possible to performmany sensitivity studies. The difficulty is to provide a solution to the numericalproblem of the open boundaries.

The major scientific objective of the present study is to understand, throughnumerical model simulations, how the fluxes at the limits of the South Atlantic basincontribute in determining the large-scale circulation and the associated transport ofproperties (volume, heat and salt) in this part of the world ocean. The modelingstrategy is as follows: The model domain is restricted to the South Atlantic basin(including the Weddell Sea) so that a large number of experiments can be run.A topography-following vertical-coordinate (sigma-coordinate) is considered. Thistype of coordinate, which is widely used in meteorology, has not been used routinelyin oceanography for general circulation problems. Most applications have been forlakes, estuaries, coastal oceanography (Ezer and Mellor, 1992), and process studies(Beckmann and Haidvogel, 1993). Basin-scale studies are still in the preliminary phase(Ezer and Mellor, 1994; Barnier et al., 1996).

The sigma-coordinate was first introduced by Phillips (1957) in a meteorologicalmodel. However, hypotheses that are valid in large-scale atmospheric models (likea small aspect ratio between the topographic height and the thickness of the fluid) arenot necessarily valid in ocean basins, where major topographic features (seamount,mid-ocean ridge and continental shelf) are of the same order as the ocean depth.Therefore, the lower limit of the fluid can be close to the surface, resulting in iso-sigmaand density surfaces that cross at wide angles. Under these conditions, the pressuregradient calculation and the parameterization of horizontal diffusion have to becarefully considered in order to avoid first-order circulations induced by numericalerrors and non-physical mixing of water masses.

The present paper (Part I) describes the efforts that have been made to reacha model configuration that we consider adapted to studying the circulation of theSouth Atlantic. It also describes the treatment of the open boundaries and the fluxconditions giving the most realistic model circulation.

In Part II, which follows, Marchesiello et al. (1998) present the results of a series ofexperiments investigating the South Atlantic circulation from the model response tovarious surface forcings. A new explanation is proposed for the seasonal variability ofthe meridional heat transport at 30°S.

The paper is divided into six sections. Section 2 features a short presentation of thenumerical model, and this is followed in Section 3 by a description of the SouthAtlantic model configuration. The process of smoothing the bottom topography iscarefully described and error estimates related to pressure gradient calculations areprovided to demonstrate that they have reached an acceptable level. The treatment of


the open boundaries is described in Section 4 and the parameterization of diffusion isinvestigated in Section 5. The paper concludes with a discussion of the modelconfiguration in relation to the scientific objectives of the project.

2. The numerical model

2.1. ¹he original SPEM model

The model is a sigma-coordinate primitive equation model (SPEM) developed byHaidvogel et al. (1991). Initially, SPEM is a Semi-spectral Primitive Equation Modelusing a bottom-topography following coordinate (sigma) in the vertical.

The sigma-coordinate transformation is given by

p"2z

h#1 (1)

where h(x, y) is the local depth and sigma ranges from !1(p(1. Orthogonalcurvilinear coordinates are used in the horizontal. The prognostic variables are thebaroclinic horizontal velocities (u, v), the barotropic streamfunction t (the Bous-sinesq, hydrostatic and rigid lid approximations are made), the salinity S, and thepotential temperature ¹. Density, pressure and vertical velocity are calculated diag-nostically. The numerical solution is based on a spectral method in the vertical(Chebyshev polynomials) and finite differences on a C-grid in the horizontal. Thevertical grid is not staggered, so all variables are calculated at the same sigma-level.Model equations and numerical solution techniques can be found in the SPEM User’sManual (Hedstrom, 1994). Errors arising from pressure gradient calculations areanalysed in Beckmann and Haidvogel (1993) for the case of a steep isolated seamount.They agree with the results of Haney (1991) showing that the error increases withsteeper topography and for a stratification in which the high vertical modes have largeamplitudes. A corrected algorithm, which approximates the pressure gradient on thehorizontal and uses the vertical Chebyshev polynomials to extrapolate into thebottom, is shown to reduce the error, but its application appears to be severelyrestricted by numerical stability.

2.2. Model changes for basin-scale applications: finite differencing in the vertical

The application of the model to realistic basin-scale simulations imposes severalconstraints. One such constraint, of general importance to most basin-scaleapplications of a sigma-coordinate model, is the representation of the dynamicallyimportant water masses, which requires a large number of vertical levels. Sucha constraint would be of general interest for most basin-scale applications of a sigma-coordinate model. For the South Atlantic Ocean, a minimum of 20 levels seems to beneeded to account for 5—7 water masses (Semtner and Chervin, 1992). Consequently,the vertical levels can be very close to each other in shallow areas (for a total depth of500 m, the distance between two consecutive levels is 2 m near the surface; Table 1). In


this situation, the explicit treatment of vertical diffusion imposes an unacceptablyshort time step. To turn that problem around, the technique of finite differences on thevertical was implemented to enable an implicit treatment of vertical diffusion. Animplicit scheme with the spectral expansion in Chebyshev polynomials requires theinversion of a full matrix at each time step, but the associated computational cost makesthis solution unaffordable. Therefore, the polynomial expansion has been dropped.Another reason for dropping the polynomials in a realistic configuration is that spectralmethods are generally not suitable for representing the discontinuity of the main andseasonal thermoclines unless a very large number of modes is used. The finite differencescheme used in this study is a classical second-order scheme based on Simpson’s rule(Baranger, 1977; Fortin, 1995). The latest version of the code (SPEM5), developedafter this study, now uses finite differences on a staggered grid in the vertical.

3. The South Atlantic model configuration

3.1. Model geometry

The model domain extends meridionally from 16°S to the Antarctic continent(76°S), and longitudinally from 68°W to 20°E. For the first application of the model toa basin-scale model study, we do not intend to resolve mesoscale eddies. Theelementary horizontal grid is square (Fig. 2), with a coarse resolution of 1.375° inlongitude (j) and 1.375°]cos / in latitude (/). The resolution (107 km at 45°S) istherefore roughly six times the local internal radius of deformation, and is in the rangeof the resolutions recommended by WOCE for coarse resolution experiments(WOCE-NEG, 1994). The continents and the Falkland Islands are treated witha masking technique based on the capacitance matrix (Wilkin et al., 1995), a methodsuccessfully employed for this type of problem by Blayo and Le Provost (1993) forquasi-geostrophic layer models. Twenty sigma levels are used in the vertical, witha higher resolution near the surface (Table 1). The vertical distribution of the sigmalevels follows a monotonic third-order polynomial. The model cost is similar to that ofother primitive equation models, and one year of integration of the South Atlanticmodel takes about 1.5 h on a vector super-computer.

3.2. Initialisation and forcing used for testing the model

To estimate model errors, we choose two different initial states. The first hasa density field that is homogeneous in the horizontal and is referred to as the “restingstratification” state. The second, referred to as the “realistic” state, has a density fieldderived from the annual mean temperature and salinity obtained from the climatologyof Levitus (1982).

The idealized “resting stratification case” (Table 2) has a temperature field thatvaries only with depth. The salinity is a constant, and the ocean is at rest (zerovelocities). The equation of state is linear. Consequently, the density field has nohorizontal pressure gradient, and if no forcing is applied the ocean should stay at rest.


Fig. 2. The isotropic horizontal grid of the model. The resolution is 1.375° in longitude (j) and1.375°]cos / in latitude (/).

For the experiments performed with the resting stratification configuration, noforcing is applied, the domain is closed (no open boundary condition is applied), andthe bottom topography is that shown in Fig. 3, described in Section 3.3.

The “realistic case” (Table 2) has temperature and salinity fields derived from theannual mean climatology of Levitus (1982). The equation of state is the UNESCOequation of state derived by Jackett and McDougall (1994), which computes in situdensity as a function of potential temperature, salinity and pressure. The fluid isinitially at rest (no motion). The wind forcing used is the annual wind stress climatol-ogy of Hellerman and Rosenstein (1983), applied as a body force over the first 50 m.Temperature and salinity are relaxed to the climatology of Levitus (1982) at allvertical levels contained within the first 50 m, with a relaxation period of 30 daysscaled by the real depth defined by the corresponding sigma levels to the depth of thebody force. This was done to assure homogeneous surface forcing independently ofthe distribution of the surface sigma levels. Inside the domain, a relaxation toclimatology is also applied to the temperature and salinity fields with a time constantof 5 years. In experiments performed with the realistic stratification, the open bound-ary conditions are operating, and the bottom topography is that shown in Fig. 3.

The above forcing was found easy to implement for the numerical tests presentedhere. However, in Part II, which discusses the seasonal ocean circulation in the SouthAtlantic, no relaxation is applied in the interior, and a different surface forcing is used(Marchesiello et al., 1998).


Table 1Configuration of the South Atlantic model. Depth z(k) of the 20 vertical levels in sigma p(k), for variousvalues of the total ocean depth, h

Vertical level number k Sigma level p(k)

Depth z(k) (m)

h"5500 m h"3000 m h"500 m

1 1.000 0 0 02 0.993 !20 !11 !23 0.984 !43 !24 !44 0.973 !74 !40 !75 0.957 !118 !64 !116 0.935 !179 !98 !167 0.905 !262 !143 !248 0.865 !372 !203 !349 0.814 !513 !280 !47

10 0.749 !691 !377 !6311 0.669 !909 !496 !8312 0.573 !1174 !640 !10713 0.459 !1488 !812 !13514 0.324 !1858 !1014 !16915 0.168 !2288 !1248 !20816 !0.012 !2782 !1518 !25317 !0.217 !3346 !1825 !30418 !0.449 !3984 !2173 !36219 !0.709 !4700 !2564 !42720 !1.000 !5500 !3000 !500

Table 2Characteristics of the initial states used for the numerical experiments

Initial state Resting stratification Realistic

Equation of state o"1028.2!0.14¹ UnescoTemperature (°C) ¹(z)"20 ) ez@1000 ¹(x,y,z) from Levitus (1982)Salinity (psu) S"35 S(x,y,z) from Levitus (1982)Wind forcing None Hellerman and Rosenstein (1983)

annual wind stressThermohaline forcing None Relaxation to Levitus (1982) ¹,SOpen boundaries Closed OpenBottom topography As in Fig. 3 As in Fig. 3

3.3. Bottom topography

The determination of bottom topography is the most crucial element in a sigma-coordinate model. The discrete form of the pressure gradient terms may produce largesystematic errors over steep topography (Messinger, 1982). For fixed horizontal and


Fig. 3. The smoothed bottom topography in the model: maximum depth 5500 m, minimum depth 500 m,contour interval 500 m.

vertical resolutions, a reduction of the errors to an acceptable level requires a signifi-cant smoothing of the bathymetry. The smoothed topography used in this study isshown in Fig. 3. It retains the major topographic features of the South Atlantic shownin Fig. 1 and has been determined by smoothing the ETOPO5 bathymetry (NOAA,1988). Smoothing is carried out according to a criterion involving the bottom slopeand the resolution of the horizontal and vertical grids in order to reduce errors in thepressure gradient (PG) calculation and to avoid hydrostatic inconsistency (Messinger,1982; Haney, 1991).

The pointwise smoothing procedure is as follows: A dimensionless local slopeparameter r

h(i,j) is defined at every horizontal grid-point (i,j) as the maximum relative

variation of the ocean depth h over a grid element:

rh(i, j)"

*x,y

h

hM x,y"

maxMMDh(i#1, j )!h(i, j)D, Dh(i, j#1)!h(i, j )DN12minMh(i#1, j)#h(i, j ),h(i, j#1)#h(i, j )N

(2)

According to Beckmann and Haidvogel (1993), the local pressure gradient errorsare limited to a few percent with values of r

h(i,j) less than 0.2, if the variations in the

density field are concentrated within the lowest vertical modes. This is expected to bethe case in coarse resolution simulations of the South Atlantic circulation. Conse-


quently, the model bottom topography is first smoothed until values of rh(i, j ) are

always less than 0.15.The second source of error in PG calculations is hydrostatic inconsistency. Messin-

ger (1982) proposed a criterion stating that the numerical scheme becomes consistentif the maximum relative increase in bottom topography is less than the distancebetween the two consecutive sigma levels. What is usually missing in the literature isthat the criterion depends on the vertical integration scheme. The higher the order ofthe scheme, the less restrictive the criterion is. Despite that, authors usually suggestusing the standard criterion adapted to the first-order scheme on a vertical staggeredgrid (Beckmann and Haidvogel, 1993). We have proposed a similar criterion here, totake into account the finite-difference scheme used in the present version of the model.The bottom topography, already smoothed to limit local PG errors with r

h(0.15, is

again smoothed until the following criterion is respected everywhere in the model forevery vertical level k:

K*x,y

h

hM x,yp(k!1)!1

p(k#1)!p(k!1)K(1 (3)

The smoothing algorithm is applied everywhere, but it uses a selective system ofweights, the values of which are set locally by a “distance” to the criterion. The result isthat the topography is not modified by the smoothing at points where the criterion issatisfied. For the horizontal and vertical resolutions defined in the present study,Eq. (3) is more severe than Eq. (2), and a violation of hydrostatic inconsistency isshown to have a dramatic effect on the model solution. This is illustrated in Fig. 4,which compares the barotropic streamfunction obtained after one year of integrationin an experiment where Eq. (3) is respected with that obtained from an experimentwhere it is not. Both experiments are conducted with identical initialisation andforcing fields as described in the “realistic case” in Section 3.2, and both use thesmoothed bottom topography of Fig. 3. The only difference between the experimentsis in the distance separating the two deepest sigma levels: in one experiment thisdistance is 800 m (the value given in Table 2), and Eq. (3) is respected everywhere,whereas it is reduced to 600 m in the other experiment, so that there are a few areas,where the topographic slopes are the steepest, where Eq. (3) is not respected. However,we insist that the present test only illustrates, but does not quantify, the impact of thehydrostatic inconsistency on model results, since other effects due to the influence ofthe vertical spacing on PG calculation could also contribute.

It appears clear in Fig. 4 that the violation of the hydrostatic consistency criterionhas a drastic impact on the model solution in the key region of the Confluence area. Inthe case where the consistency is respected, the convergence of the Brazil andMalvinas currents is reasonably localized near 40°S, whereas in the case it is not, theBrazil current flows down to the Falkland Islands at 50°S and then turns northwardalong with the Malvinas current. Noticeable differences are also observed over theWalvis Ridge and in the Antarctic Circumpolar Current at 30°W, 52°S, near thetopographic feature that represents the South Sandwich Islands.


Fig. 4. The impact of the hydrostatic inconsistency. The model barotropic streamfunction after 1 year ofintegration in a diagnostic mode (a) in the case where the criterion for hydrostatic inconsistency is respected,and (b) in the case where it is not. The contour interval is 10 Sv.


Fig. 5. The velocity field at 260 m after 30 days of integration in the resting stratification case. The arrowbelow the plot indicates a velocity of 1.5]10~3 m/s.

Estimates of the pressure gradient errors related to the smoothed bottom topogra-phy shown in Fig. 3 can be computed from experiments starting from a “flat”density field, as in Beckmann and Haidvogel (1993). In such experiments, the initialstate is that of the resting stratification case as described in Section 3.2. The modelis integrated with no forcing applied, and the resulting velocity field (Fig. 5) givesan estimate of the maximum pressure gradient errors. In the present case, the averageerror in the velocity field is less than a half-millimeter per second, while the maximumerror reaches a millimeter per second at very few locations. From these tests, itmay be concluded that the smoothed topography satisfies our criterion of non-significant error in the pressure gradient calculations, and it is thus used in Part II(Marchesiello et al., 1998) to investigate the South Atlantic circulation with theSPEM model.

Achieving such small numerical errors has a cost: the ocean physics associated withsteep topography is neglected. However, coarse resolution experiments are known toproduce weakly-advective and highly-diffusive flows, and this reduces the impact offlow—topography interactions on the model solution. As discussed in Part II(Marchesiello et al., 1998), we believe that despite the smoothing of the bathymetry,the sigma-coordinate model does not lose more of the physics than z-coordinatemodels.


4. The open boundary conditions

Although the South Atlantic ocean is bounded by three continents, its boundariesexhibit wide openings to other oceans (Fig. 1). These are treated in the model throughthe use of open boundary conditions. The model domain therefore presents three openboundaries where model variables need to be calculated: to the west (68°W) at DrakePassage, to the north (16°S) between Brazil and Angola, and to the east (20°E)between South Africa and Antarctica. Only the southern limit of the domain is totallybounded by land (the Antarctic continent). According to Red and Cooper (1986), anopen boundary is a computational border where the aim of the calculations is to allowthe perturbations generated inside the computational domain to leave it withoutdeterioration of the inner model solution. However, an open boundary may also haveto let information from the outer oceans enter the model, and should support inflowand outflow conditions.

The problem of open boundaries in primitive equation ocean modelling has beentackled by Stevens (1990), who proposed and tested numerical schemes for the GFDLmodel. These conditions were successfully used in the Fine Resolution AntarcticModel (FRAM) experiment (Stevens, 1991). However, the SPEM code, written on anArakawa C-grid, is significantly different from the GFDL code, written on anArakawa B-grid, and the numerical solutions proposed by Stevens (1990) could not beapplied. A specific open boundary condition had to be defined for SPEM, and this ispresented below.

4.1. ¹he radiation condition

Open boundaries are treated here with a method combining radiation conditionsand relaxation to a climatology, so information from the outer oceans can influencethe model solution. The physical assumptions are therefore slightly differentfrom those of Stevens (1990), who used radiation conditions for tracer fieldsonly and used linearized momentum equations to calculate velocities at boundarypoints.

The wave equation used in SPEM to derive the radiation condition is based uponthe work of Raymond and Kuo (1984), who considered phase velocities both normaland tangential to the boundary. However, a relaxation term similar to the one used byBlumberg and Kantha (1985) was added to the right-hand side of the wave equationto modify the condition of Sommerfeld (1949). The resulting radiation condition,written for a prognostic model variable /, is:

L/

Lt#(c

x#u

a)L/

Lx#c

y

L/

Ly"

1

q(/!/

clim) (4)

where x and y stand for directions respectively normal and tangential to the bound-ary. /

climis a climatological estimate of the variable / at the boundary, and q is a time

relaxation constant. Following Stevens (1990), an advection velocity term uais added


to the phase speed cx

in the direction normal to the boundary. The phase speedscx

and cyare calculated from the / field surrounding the boundary point as follows:

cx"

L/!L//Lt

Lx (L//Lx)2#(L//Ly)2

and

cy"

L/!L//Lt

Ly (L//Lx)2#(L//Ly)2or c

y"0. (5)

The radiation condition is applied for every prognostic variable in the model (thebarotropic streamfunction t, the horizontal normal velocity u and the tangentialhorizontal velocity v, the potential temperature ¹ and the salinity S). This condition,applied with u

a"0 and c

yO0, has been shown to behave rather well for quasi-

geostrophic layer models (Blayo, 1992).The possibility of setting the tangential phase speed c

yto zero is retained here. This

means that the propagation of the signal in the direction parallel to the boundary isneglected in radiation Eq. (4), but the expression for c

xgiven in Eq. (5) is not changed.

It was observed that in the case where cyO0, instabilities could grow at boundary

points adjacent to land points. For the present application, greater numerical stabilitywas found when the condition was applied with c

y"0 for all variables. This choice

was thus made for the study.

4.2. ¹he numerical scheme

Wave Eq. (4) is integrated using an explicit leap-frog numerical scheme derivedfrom the work of Orlanski (1976). This scheme is the most consistent with the C-gridand the explicit numerical schemes used in the SPEM numerical code to solve theprimitive equations. It is different from the scheme of Stevens (1990), who used anupstream scheme on a B-grid. With the above choice for c

y(c

y"0), the corresponding

numerical expression for the value of / at the boundary, at time step n#1, is:

/n`1b

"

1!r!R

1#r/n~1b

#

2r

1#r/n

b~1#

R

1#r/clim

b(6)

with r"m

bm

b~1

/n~2b~1

!/nb~1

/nb~1

#/n~2b~1

!2/n~1b~2

#mbua*t and R"

2*t

q, (7)

where m is the scale factor introduced by the curvilinear coordinates, *t is the modeltime step, q is the relaxation time scale, and u

a"un

bthe velocity component normal to

the boundary. Superscript n indicates the time, subscriptb

indicates the boundarypoint, and subscripts

b~1and

b~2indicate grid points before the boundary, as shown

in Fig. 6.However, the numerical scheme does not alone constitute a boundary condition.

The conditions of application of the scheme are also important and differ depending


Fig. 6. Sketch of the grid at the open boundary. Subscripts b, b!1 and b!2 indicate the boundary andthe inner points, respectively.

on whether one deals with inflow or outflow conditions, the two situations beingdetermined from the sign of the normal phase/advection speed c

x#u

a.

The boundary condition (Eqs. (6) and (7)) described above appears to be numer-ically the most stable, and also gives the most physical results.

4.3. Application of the open boundary condition in an outflow situation

In the case of an outflow, (i.e. when cx#u

ais directed outwards from the domain),

the values of the model prognostic variables at boundary points are calculated byapplication of Eq. (6) and Eq. (7) with the following adaptations:

1. The relaxation time scale in Eq. (7) is q"5 years;2. For the barotropic streamfunction t, and the normal and tangential compo-

nents of the velocity fields u and v,ua"0 in Eq. (7);

3. For the tracer fields ¹ and S:ua"un

bin Eq. (7), the flow component normal to the boundary at time n, at the

boundary point;r is calculated with Eq. (7) with /"v, the tangential velocity field, and not with/"¹ or /"S.

In many experiments, it was noted that the calculation of r from the tangentialvelocity field, instead of from the tracer fields, significantly improved the stability ofthe numerical scheme and the calculation of the tracer fields at the boundary,although at present it is not possible to provide a clear explanation for this.

4.4. Application of the open boundary condition in an inflow situation

In the case of an inflow, (i.e. when cx#u

ais directed into the domain), the values of

the variables at boundary points are calculated as follows:

1. For the t, u, and v fields, by application of Eq. (6) and Eq. (7) with:a relaxation time scale in Eq. (7) set to q"15 days;ua"0 in Eq. (7);


2. For the tracer fields ¹ and S:The radiation condition (Eq. (6)) is not applied, and the values of ¹ and S arespecified from their climatological estimates.

Application of this latter condition, as opposed to application of the radiationcondition with a 15-day relaxation, revealed no significant difference in the behaviorof the tracer fields. The above condition was therefore retained for simplicity. This isnot true for the dynamical variables, which evolve faster over time and for whichradiation has an essential role to play, even if relaxation is dominant.

4.5. Determination of the climatological values at the boundary

In the way it has been defined, the open boundary condition can be considered asa forcing at the lateral limits of the domain, and it is clear that the volume transportand the fluxes of heat and salt at the open boundaries will have a considerable impacton model results. In outflow situations, these fluxes are mainly determined by themodel solution inside the domain, since the relaxation to climatology at the boundaryis weak (5-year time constant).

The situation is different with inflows, as it is always the case at Drake Passage.Relaxation to climatological fields at the boundary is strong (15 days or shorter timescale). The climatological values of the model prognostic variables at the boundarywill thus have a significant impact on the model solution, and they must therefore becarefully determined.

The climatological values of the temperature and salinity fields, ¹clim

and Sclim

areestimated from the climatology of Levitus (1982). The baroclinic velocities u

climand

vclim

are obtained by integration of the thermal wind equation.The determination of the climatological barotropic streamfunction t

climis crucial

since it fixes the total volume transport at the boundary, and also largely determinesthe values of the heat and salt transport through the boundary.

At Drake Passage, the climatological streamfunction is defined with two objectives:to obtain a total volume transport of 130 Sv, which is close to the estimate of Reid(1989), and to obtain a heat transport of 320°C ) Sv, close to the values proposed byGeorgi and Toole (1982) and Rintoul (1991). The condition relating to volumetransport is met by setting t

clim"0 Sv on the Antarctic continent, and

tclim

"!130 Sv on the American continent. The condition relating to the heat flux isobtained by imposing a variation of t

climwith latitude across the passage, which yields

a barotropic velocity with the profile shown in Fig. 7a, with a larger flow in thenorthern part of the passage, in agreement with recent WOCE observations (King andAlderson, 1994). A Gaussian or uniform profile results in a weak transport of heat(200°C )Sv).

In the ACC at 20°E, the barotropic streamfunction is determined so as to producethe velocity profile shown in Fig. 7b, with a 5 Sv inflow for the Agulhas Current andalong the Antarctic, and a total volume transport of 130 Sv. At the northern boundaryalong 10°S, the total meridional volume transport is zero.


Fig. 7. Schematic representation of the barotropic velocity profile (a) at Drake Passage, for a total volumetransport of 130 Sv and a temperature flux of 320°C ) Sv; (b) between Africa and Antarctica, for a 5 Sv inflowat the tip of Africa and into the Weddell sea, and a 140 Sv outflow for the ACC.

4.6. Results

The behavior of the open boundary conditions in a realistic simulation of the SouthAtlantic circulation was found to be quite satisfactory. For example, the streamfunc-tion shown in Fig. 4 does not show any distortion of the streamlines in the vicinity ofthe boundaries, and a meridional section of the salinity field at 30°W (Fig. 8) revealsno impact of the open boundary at the northern limit of the domain.

The ability of the model to dynamically adjust inflows and outflows is remarkable.Fig. 9ab, shows the velocity field at two different depths after 10 years of model


Fig. 8. Meridional section of the salinity at 30°W after 10 years of model integration in a realistic case. Thenorthern limit is an open boundary. The contour interval is 0.1.

integration in a diagnostic mode. These should be compared with the velocity fieldsproduced by the world ocean simulation of England and Garcon (1994), also shown inFig. 9. The purpose here is to compare inflows and outflows at the limits of the SouthAtlantic basin in both models.

The model is able to produce flow patterns very similar to the solutions provided bythe world ocean model. Near the surface (Fig. 9a), the open boundary conditionallows the model to reproduce the outflow of the Benguela Current and the inflow ofthe Brazil Current at the northern boundary. At the eastern boundary (20°E), theinflow of the Agulhas current is small (5 Sv in Fig. 7), and flows around the tip ofAfrica. In the world ocean model, a strong Agulhas Current retroflects at about 20°E,(the location of the open boundary in our model), and only a small part enters theSouth Atlantic basin. Beyond the region of retroflection, the flow is weak andcomparable in both models. The ACC flows out between 40 and 60°S, and there isa small inflow along Antarctica.

At depth (Fig. 9b), the open boundary model is able to generate a significant deepwestern boundary current along South America, and the outflow of the ACC splitsinto two branches at 45°S and 60°S.

At Drake Passage, the inflow is rapidly confined to the northern part of the strait.Although the constraint of the open boundary is greatest at Drake Passage, the open


Fig. 9. SPEM velocity fields after 10 years of model integration in a realistic case at depths of (a) 70 m, and (b)2250 m. Open boundary conditions are used between continents. Velocity field in the South Atlantic fromthe world ocean simulation experiment of England and Garcon (1994), at depths of (c) 70 m, and (d) 2250 m.


Fig. 9 (continued)


boundary condition lets the model generate a small outward flow at a depth of2250 m. The realism of this feature is not discussed here, but it shows that boundaryconditions leave model dynamics with sufficient opportunity to adjust.

Similar comparisons at depths of 280 m and 900 m (not shown) also showremarkable similarities in the flow patterns at the model limits. In particular,the open boundary conditions are shown to reproduce the western boundary currentsrather well at the northern boundary. The model solution shows inflowing currentsat surface and deep levels, and an outflowing current at intermediate levels.These features appear in the global model of England and Garcon (1994) as wellas in Semtner and Chervin (1992). The open boundary conditions describedin detail here have demonstrated their stability, and their ability to providethe information needed by the model to study the circulation in the SouthAtlantic.

Until now, a lack of confidence in open boundaries has meant that most studies ofthe South Atlantic have been undertaken with global models. The present results,however, open new opportunities for sensitivity studies of the ocean circulation in thesouthern hemisphere.

5. Diffusion of heat and salt

5.1. ¹he horizontal diffusion operator

The effects of eddy mixing in coarse resolution experiments are often said to bebetter parameterized by a diffusion along isopycnal or along neutral surfaces. How-ever, for the first application of the model to a basin-scale simulation, we adopta rather conservative approach and use horizontal diffusion, but the model will verycertainly evolve towards more sophisticated parameterizations.

Diffusion in ocean models using a sigma-coordinate is commonly handled bya Laplacian operator acting along surfaces of constant-sigma. This may produceunwanted vertical diffusion in regions where the bottom slope is steep. To obtaina clear understanding of the problem, a simple analytical development is useful. Ina two-dimensional cartesian (x,z) frame, the horizontal diffusion term for the potentialtemperature ¹ can be written in a conservative form as:

*¹"

1

h

LLxAKH

hL¹LxB (8)

where x derivatives are taken along constant-z surfaces, and KH

is an eddy-viscositycoefficient. In the (x,p) coordinate system, the horizontal derivative derived fromEq. (1) is:

L¹Lx K

z

"

L¹Lx Kp#

L¹Lp

LpLx

"

L¹Lx Kp!

p!1

h

Lh

Lx

L¹Lp

, (9)


which yields the following formulation for the diffusion term (Eq. (8)):

*¹"

1

h CLLxAKH

hL¹LxB!

LLx A(p!1)

Lh

Lx

L¹LpB!

LLp A(p!1)

Lh

Lx

L¹LpB

#

LLpAKH

(p!1)2

h ALh

LxB2L¹LpBD

(*¹1) (*¹

2) (*¹

3)

(*¹4)

(10)

On the right-hand side of Eq. (10), all x-derivatives are now taken along constant-sigma surfaces. The horizontal diffusion operator appears as the sum of four terms.Most atmosphere and ocean models retain only the first term, (*¹

1), known as the

iso-sigma diffusion term. The main reason for neglecting the other terms is that thevariation in the bottom topography over an elementary grid space is small. Mellorand Blumberg (1985) suggest the iso-sigma formulation as being the most correct inthe case of an Ekman bottom boundary layer. Jamart et al. (1986) investigated thisproblem and found, with scaling arguments, that the 4th term, (*¹

4), could in some

applications be one or several orders of magnitude larger than the vertical diffusionparameterized in the model. In this case the iso-sigma formulation artificially inducesa significant vertical diffusion.

This issue is of primary importance in our realistic simulations of the South Atlanticcirculation. Because the vertical resolution is rather coarse (20 sigma-levels, seeSection 3), the depth increment of the vertical grid can be several hundred meters nearthe bottom (see Table 1). In the case of large topographic slopes, the diffusion of saltand temperature may generate strong geostrophic currents.

In our efforts to obtain a model configuration having the smallest numerical bias,we therefore tested and compared the iso-sigma diffusion scheme (scheme 1), whichconsiders only the first term of Eq. (4) and is widely used in ocean simulations, and thegeopotential diffusion scheme (scheme 2) which retains all terms of Eq. (4) andoperates along horizontal surfaces.

5.2. Errors induced by the artificial vertical diffusion of heat and salt

Resting stratification experiments were performed to investigate the behavior of twodifferent numerical schemes for the horizontal diffusion of heat and salt.

On the one hand, when scheme 1 is used, diffusion occurs along sigma surfaces,which produces unwanted vertical diffusion in regions where the bottom topographyis steep, as seen in the vertical section of the temperature field shown in Fig. 10a.Associated density currents are strong even near the surface (10 cm/s at 250 m depth).The associated circulation shows large-scale barotropic flows of several tens of Sv(Fig. 11a).


Fig. 10. Vertical section of the potential temperature at 30°W after 30 days of model integration in theresting stratification case, (a) with iso-sigma diffusion (scheme 1), and (b) geopotential diffusion (scheme 2).The contour interval is 1°C.


Fig. 11. The barotropic streamfunction after 30 days of model integration in the resting stratification case,(a) with iso-sigma diffusion (scheme 1), and (b) geopotential diffusion (scheme 2). The contour interval is 5 Sv.


Fig. 12. The barotropic streamfunction in the realistic case after 1 year of model integration, (a) withiso-sigma diffusion (scheme 1), and (b) geopotential diffusion (scheme 2). The contour interval is 10 Sv.


On the other hand, scheme 2 diffuses tracers strictly horizontally and keeps theisotherm completely flat (Fig. 10b). In this case, the small residual circulation ob-served (Fig. 11b) is driven by the pressure gradient errors.

The same type of comparison is made with the results from realistic case experi-ments. The differences in the model solutions are not as great as in the restingstratification case, which confirms that this latter type of experiment is extreme in theevaluation of errors, as was noticed for the pressure gradient calculation (Section 4).

When scheme 1 is used (Fig. 12a), the circulation clearly shows a strong correlationwith the prominent topographic features like the Rio Grande rise, the Walvis ridge,and the continental shelves of South America and Africa. The Malvinas currentovershoots as far as 35°S. These features disappear when the geopotential diffusion(scheme 2) is used (Fig. 12b) and the resulting circulation is clearly more realistic.

These results confirm that the iso-sigma diffusion of tracers has an unrealisticimpact on the model circulation, and scheme 2 has therefore been retained for thepresent study. In the simulations presented in this study, the diffusion of momentum isiso-sigma. The effect on the model solution is considerably smaller because velocitiesdo not directly enter the calculation of the pressure gradient, as do temperature andsalinity. Nevertheless, iso-sigma diffusion of momentum is expected to produce a smallsmearing of the velocity field in regions of large topographic slopes. In versions of thenumerical model released after this work, the geopotential diffusion of momentum hasbeen included.

6. Discussion

The application of a numerical model to a general circulation problem begins withthe configuration of the model. Before any physical meaning can be drawn from theresults of model simulations, we have to make sure that the model configuration doesnot introduce non-physical bias in the model solution, and it is important to havesome idea of the numerical errors.

Such an analysis was performed with the SPEM model. Two major issues relatingto the sigma-coordinate were investigated: the pressure gradient calculation and thediffusion of heat and salt.

The errors in the temperature and salinity fields induced by an iso-sigma diffusionscheme were found to be large enough to generate unrealistic velocities, which were asgreat as those of the large-scale circulation the model was aiming to simulate. Therotation of the diffusion tensor into the geopotential coordinate was shown toimprove model results considerably. Higher resolutions should not reduce this effectof sigma diffusion, and the rotation of the diffusion tensor is certainly necessary formodelling studies dealing with large ocean basins. Our conclusion is different fromthat of Mellor and Blumberg (1985), who put forward the iso-sigma diffusion schemeas the most physical formulation, arguing that near the bottom, turbulent diffusion isconstrained by the topography. This is probably true in the benthic boundary layer,but above it the slope of the surface vertical coordinate must have no impact upon theprincipal axes of diffusion. The results obtained by Mellor and Blumberg (1985) with


a two-dimensional, high vertical resolution model are certainly not applicable toa fully three-dimensional ocean model where the vertical resolution does not resolvethe bottom boundary layer.

Our conclusion is that diffusion along sigma surfaces is not appropriate for generalcirculation problems, since the vertical resolution is usually coarse near the bottomfor most applications (see Table 1). The rotation of the diffusion tensor into thegeopotential coordinates (as used in the present study), and, probably even more so,into the isopycnal coordinate, must be considered.

Pressure gradient errors are difficult to estimate. Resting stratification experiments,as they are performed in this study, give an estimate of the maximum error, butmust be considered with care because they are difficult to interpret. For example,if a linear stratification is used instead of an exponential as in Table 2, thepressure gradient error is reduced to zero, and if the full equation of state is usedinstead of the linear equation of state, pressure gradient errors are more than twice asgreat but exhibit the exact same spatial pattern. However, this type of experiment isnecessary because it indicates the spatial correlation that is expected in the error, andcan provide clues for smoothing the bottom topography, as was the case in the presentstudy.

We made the choice to reduce the pressure gradient error by smoothing thebathymetry. A smoothing criterion was derived and applied, which produced thetopography shown in Fig. 3. Another solution for reducing the PG error, based on thework of Gary (1973), was tested with no success. Gary (1973) proposed subtractinga mean density profile oN (z), whose contribution to horizontal differencing is zero, fromthe instantaneous density o(x,y,z,t) profile. The resulting perturbation density fieldo@"o(x, y, z, t)!oN (z) was used in the model for the pressure gradient calculation, andif o@ is small enough, the pressure gradient errors can be smaller. This solution (Gary,1973) did not improve the solution in our study because the density field is far frombeing homogeneous over the South Atlantic and o@ would not become small every-where. This solution was therefore not applied.

Consequently, the pressure gradient calculations are performed with the standardalgorithm, the smooth bottom topography being used to keep the errors belowa millimeter per second.

Open boundary conditions based on radiation condition in the normal and tangen-tial direction, combined with a relaxation to climatology, have been described andtested. They appear to be numerically robust, and to be capable of introducing thenecessary information from the outer oceans into the South Atlantic basin. As will beseen in Part II, long-term integrations of the model have made it possible toinvestigate the impact of the open boundary on the model circulation. These openboundary conditions have also been tested in a channel configuration to radiate outeddies and waves (Nguyen and Verron, 1996).

To conclude, a configuration of the SPEM model with which to study the large-scale circulation in the South Atlantic has been obtained. It accommodatesthe limitations of the sigma-coordinates and the openings of the South Atlanticto the other oceans. Error estimates related to model configuration are small com-pared with the signal to be simulated, and we have a reasonable knowledge of their


spatial pattern, which will facilitate interpretation of the model simulations presentedin Part II.

Acknowledgements

The authors are supported by the Centre National de la Recherche Scientifique.This research was funded by the Institut National des Sciences de l’Univers andIFREMER (Contrat IFREMER 94-1-430096) through the Programme Nationald’Etude de la Dynamique du Climat as a contribution to WOCE. Support forcomputations was provided by the Institut du Developpement et des Ressources enInformatique Scientifique. The authors are also grateful to (in alphabetical order) A.Beckmann, D. Haidvogel, K. Hedstrom, A. M. Treguier and J. Wilkin for frequentdiscussion and interaction on the SPEM model. Finally, we would like to thank T.Nguyen for his generous and invaluable help with modelling and computing tech-niques, and three referees whose thorough work contributed to a real improvement ofthe manuscript.

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