Sensitivity of an idealised subtropical gyre to the eastern boundary conditions *

The eastern region of the North Atlantic Subtropical Gyre (NASG) shows an anticyclonic loop formed by the Azores Current (AC), the Canary Current (CC), and the North Equatorial Current (NEC). The AC progressively turns to the south and southwest into the CC, before leaving the region towards the west as the NEC (Fig. 1a, reproduced from Stramma, 1984). The total amount of water that transits the region is about 12 Sv (Stramma, 1984; Siedler and Onken, 1996). North of 30oN and off the African continental platform there is permanent onshore geostrophic transport of about 1-2 Sv. This water flux necessarily recirculates south close to the coast until it rejoins the interior deep ocean between 20 and 25oN (Stramma, 1984; Stramma and Siedler, 1988), in what we may call the eastern branch of the CC. This eastern branch is so narrow that it does not show up in Stramma (1984) and Stramma and Siedler (1988) maps, which where obtained from deep ocean data. Arhan et al. (1994), however, have inferred an eastward transport of about 11 Sv into the eastern boundary layer and at densities lower than 27.25 sigma-theta, which appears to feed southward alongshore currents. One could be tempted to associate the permanent eastern branch with the baroclinic coastal upwelling jet that develops in any upwelling region (Pelegrí et al., 1997). This may be partially true but a difficulty with this interpretation is that north of 30oN upwelling does not take place during winter. This suggests that the mechanism of coupling between the interior and coastal oceans is not simply forced by coastal upwelling but is a complex process requiring careful analysis. Data collected during the CANIGO project have improved the knowledge of the Canary Current area. Some of these measurements have been made very near to the continental platform and SCI. MAR., 65 (Suppl. 1): 187-194 SCIENTIA MARINA 2001


INTRODUCTION
The eastern region of the North Atlantic Subtropical Gyre (NASG) shows an anticyclonic loop formed by the Azores Current (AC), the Canary Current (CC), and the North Equatorial Current (NEC).The AC progressively turns to the south and southwest into the CC, before leaving the region towards the west as the NEC (Fig. la, reproduced from Strarnma, 1984).The total arnount of water that transits the region is about 12 Sv (Stramma, 1984;Siedler and Onken, 1996).North of 30°N and off the African continental platform there is permaficnt üfishürc.geos@ophic wmspoñ of -2 S".This water flux necessarily recirculates south close to the coast until it rejoins the interior deep ocean between 20 and 25"N (Stramma, 1984; Stramma and  Siedler, 1988), in what we rnay cal1 the eastern branch of the CC.This eastem branch is so narrow *Received Febniary 11,2000. Accepted August 28,2000.that it does not show up in Stramma (1984) and S m m a and Siedler (1988) maps, which where obtained from deep ocean data.Arhan et al. (1994), however, have inferred an eastward transport of about 11 Sv into the eastem boundary layer and al densities lower than 27.25 sigma-theta, which appears to feed southward alongshore currents.
One could be tempted to associate the permanent eastem branch with the baroclinic coastal upwelling jet that develops in any upwelIing region (Pelegrí et al., 1997).This may be partially true but a difficulty with this interpretation is that north of 309N upwelling does not take place during winter.This suggesis the mecnanisrn of coupiiñg lwíween the interior and coastal oceans is not simply forced by coastal upwelling but is a complex process requiring careful analysis.
Data coliected during the CANIGO project have improved the knowledge of the Canary Current area.Some of these measurements have been made very near to the continental platfonn and confirm that the recirculation of the eastern branch of the CC takes place in a very narrow region (Pelep' e? al., 1999).They have shown that this water flux sometimes continues southward between the Canary IsIands and the African coast, but at orher times it appears to rejoin the interior ocean at Cape Ghir before flowing south through the Canary 4rchipelago (Fig. lb).
The airn of this study is to examine how the oasíd ocean may &ve Ule easrem branch of íiie t.For this purpose we use a very simple one-layer ilasigeostrophic model driven by idealised zonal mds, and m -the eastem boundary conditions simulate the effect of the coastal ocean.The main tint is that the water in the coastal region moves uthwards, breaking lines of constant planetary 3rticity along a meridional band of constant potenal vorticity.We will see that this gives rise to anti-~clonic vorticity at the boundary, which may be terpreted a s being generated by the coastal jet.In order to solve the vorticity and continuity equations we need to specify boundary conditions for both the vorticity and the stream function.The standard boundary conditions are no-slip at the meridional boundaries, slip at the zonal boundaIies, and no-normal flux at al1 boundaries (Case 1, Table 1); a slightly different possibility is to use a slip condition at the eastem boundary (Case 2, Table 1).These boundary conditions are specified as follows (Roache, 1982): where the subscnpt b indicates the grid point at the boundary and n is the direction normal to the wall.The finite difference expression for the no-slip condition (first order forrn) where An is the distance, normal to the wall, fiom -One alternative to the above standard boundary conditions is as follows.First, we may replace the no-normal flux condition by a condition that allows a coastward flux across the eastem boundary, that is, where the subscript e indicates the grid point at the eastem bGUn&q.T&r;g iilto accguiii ú\2i w y sí-  constant q 2 0 m N slip 10-20% constant q 3 2 4 % no-slip 27-32% wnstant q 20-27ON

Case
no-slip 10-20"N constant q 3 2 m N slip 27-32W constant q 20-27% slip 10-20"N constant q 32-40"N no-slip 27-32% constant q 20-27% no-slip 10-20% constant o 3 2 4 % slip 2?-32% constant q 20-27% slip 10-20% em vounaary corresponds to the í5Wparaíiei, the non-zero flux condition will allow the outflow of interior water into the near-coastal region and its reappearance further south (see Fig. la).Second, we may modify the slip/no-slip conditions through the specification of a meridional band of constant potential vorticity, which would dlow southward flux, by requiring: As the basin has constant depth, this is eqiiivdent to Cei-.fc = const.On a mid-latitude pplane, the Coriolis parameter is expressed as: f = fo+Jg .Taking this into account, the relative vorticity along the eastem boundq takes the form: where yc is the separation latitude and y, is the northem limit of the basin.This is the general form of this condition, considenng the no-slip condition at the eastern boundary between y = O and y = yc.Notice that yE = O when the standard condition of no-normal flux is considered at latitudes between y = yc and y = y, .We may notice that this condition involves the generation of anticyclonic vorticity at the eastem boundary, as if it were induced by the horizontal shear produced by the intense coastal jet.any combination between them and the original ones, are theoretically reasonable.Our purpose here is to examine how they modify the numerical results and which modification bears the closest resemblance to the actual strearnlines in the region.
The numerical integration of the equations is done using the finite-difference rnethod.The Leapfrog scheme was used to solve the vorticity equation and a successive over-relaxation (SOR) approach was used for the Poisson equation (Roache, 1982).The Jacobian in the vorticity equation was written following Arakawa7s scheme (Arakawa, 1966).We used a time lagged diffusion to reduce numerical instabilities (Roache, 1982), and an Asselin time filter (Asselin, 1972) was included to avoid instabilities induced by the Leapfrog method.A constant mesh spacing in the x (longitude) and y (latitude) directions was used, the spatial :es=!i?U=i, ~"L?E; &p+= Q.50.

TEST CASES
The standard conditions, which fuliy ignore the interaction of the coastd upwelling region with the interior flow, correspond to Cases 1 and 2 of our simulaíions (Table 1).On the other hand, the influence of the coastal ocean on the interior gyre may be assessed through a proper specifcation of the eastem boundary conditions as discussed above.A summary of the eastem boundary conditions used for the main test cases is presented in Table 1.This is not an exhaustive list of al1 possible simulations but illustrates the main cases we have considered.
Except for the standard cases (1 and 2), the eastem boundary was divided into two or four subregions.The physical justification behind the division into two subregions, one between 10 and 20" and the other between 20 and 40°N, corresponds to the main circulation pattem of the eastem subtropical gyre (Fig. la).This pattem shows the AC and CC impinging eastward north of the Canary Islands and separating near Cape Blanc (near 20°N) as a quasipermanent giant f k n e n t (Gabnc et al., 1993), leaving behind &e so-cailed shadow zone (Luyten et al., 1983;Kawase and Sarmiento, 1985;Stramma, 1984;Schmitz and McCartney, 1993).
n All resuits discussed in the next section corre- what could be interpreted as the flow separation near Cape Ghir (at about 3S0N), its reincorporation to the coastal ocean south of the Canary Islands, and its defminve separation near Cape Blanc. Figure 3 illustrates the potentiai vorticity dismbution near the eastern boundary that corresponds to Cases 1,5 and 9. h all three cases the vorticity distribution is dominated by the planetvy vorticity, but Case 5 rhnws the existente of m e meridional hand of constant potential vorticity at the eastem boundary, while Case 9 includes two bands.An interesting point is that the effect of these conditions on the vor- zontal extent as compared with the modification experimented by the flow pattern.In some instantes it may appear as if this small region is of the sarne slze as the g i d used for tiie numericai caicuiations (i.e.0.5").Using better numerical resolution, how- ever, does not change the character of the solution and takes much more computational time.
The details of the results illusuated in Figures 2 and 3 depend on the wind forcing and on the mendional limits of the eastem boundary regions.The wind forcing is quite critica1 in controlling the amount of recirculating water in the whole subtropical gyre and particularly in the eastem branch of the eastern boundary cunent.In this paper we have used an idealised wind stress corresponding to a sinusoidal meridional wind variation with an amplitude of about 6 m S-'.This appears to be a reasonable approximation, the exact wind distnbution not being ocean in the eastem boundary circulation.We made numerical runs for severa1 different wind stresses, resulting in different arnounts of recirculating water but similar qualitative patterns.Regarding the meridional limits of the eastem boundary subregions, we performed a number of experiments which illustrate that these lirnits are important in controlling the small-scale pattems near the eastern boundary, but do not produce any major change in the large-scale patterns.Some of these modifications will be illustrated below.
Table 1 indicates the main simulations we have used to examine how important the precise eastern boundary conditions may be, and Figure 4 shows the stream functions for a selection of these numericai runs (with each flow line corresponding to 1 Sv).   4 indicates that an ?crease in the meridional extent of this subregion liows an increase in the arnount of water that recirulates through the coastal ocean, and that this takes dace in a rather smooth (and realistic) fashion.

'ONCLUSIONS
Despite &e skipG&y oí Ui et rndei, we iina thaí ie chosen specification of the eastem boundary conxions provides qualitative, and to a certain extent iantitative, features of the eastem boundary crirrent gime which show similarities with the observams.The most coherent nurnerical results correond to the case in whích both non-zero normal flux 3 constant potential vorticity are specified simulta-neously, they are rather robust and they do not depend on the choice of slip or no-slip cónditions.These boundary conditions sirnulate the most coherent physical situation, in which southward water flow in the coastal ocean (implied by the constant poten-tia1 vorticity condition) is fed srnoothiy through water drainage of the interior ocean.At a certain latitude the flow separates from the coast and recirculates, either locally (e-g.at 32ON in Case 9) or defínitively (e-g.at 20°N in CaseS), into the deep ocean.

FIG
FIG I. -(a) Mean mnua! circulaticm sheamlines fm the uppr 800 m according to Stramma (19M); each flow h e represents 1 Sv.(b) Schematic mean water transport (in Sv) as obtained from two years of quanerly XBT data along rhe nansecrs shown.
nis condition: together with the possibility of nonzro normal flux, produces rather realistic simulaions of the easternmost branch of the CC.THE ONE-LAYER QUASIGEOSTROPHIC MODEL A sLTIPle quásigeosú-oP~,jc o rlei ayer rn"&i is used to represent the circulation pattem of the NASG.The ocean basin is idealised as a rectangular basin with a flat bottorn and constant depth D on a mid-latitude P-plane.It lies in the 77-15OW longitude and 10-40°N latitude bands.We have assumed a hornogeneous and incompressible fluid subject to a zonal, steady, anticyclonic single-gyre wind stress pattern.The model was formulated using the quasigeostrophic barotropic vorticity equation (Pedlosky, 1979): and the Poisson equation: where ( is the relative vorticity and y the strezrc function, J(w, c) is the Jacobian, D is the water de ?= (5 g , O the surface wind stress.~y h edensity of water, 6, = (24 / fJR the Elanar.thickness, r = f, 6, / 2 0 the bottom friction r cient, and A, the horizontal diffusion coefficier,-The wind stress and its curl have the form The non-dirnensionalisation was performed by scaling: (x, y) = L(x', y'), (u, v) = U(u', v'), = (U/L))r' , rx = z , T ' , and t = (1/& L)t', where the pnmes indicate non-dimensional variables (Pedlosky, 1979).Thus, the non-dimensional vorticity equation becomes: where the non-dimensional pararneters are the Rossby number E = U/(&L2), the non-dimensional bottom friction coefficient , U = r/(P&), the wind forcing coefficient a = (ron)/(pd,LPDU), and the horizontal Ekrnan number E= (6,lL)j; the Munk boundary layer scale is deñned as S , E (A, / PJD. x i d point (b-1) to grid point (b).

nm
3 U U l C ~I V l l l V U P l l l y CWIIGJYW11U3 LV Ll1C YLIJIJPnent upwelling region off northwest Africa, between Cape Blanc and Cape Bojador.In this subregion we may expect water leaving the interior ocean and feedínp a southward fiowing coastal jet, providing a sensible physical justification for a constant poten-tia1 vorticity band.The 27-32"N subregion roughly a N represents the boundary between Cape Bojador and Cape Ghir, characterised by the presence of the o Canary Islands very close to the upwelling region and on the path of the southward CC.In this subre-E pion upweiiing is usually relatively weaic and we a could imagine that the Canary Islands may cause a perturbation to the CC.The 32-40°N subregion 3 roughly corresponds to the boundary region between -- S Cape Ghir and the Strait of Gibraltar, where signifi-E cant onshore geostrophic flux has been observed Figure2presents the strearnlines of the whole subtropical gyre for Cases 1, 5 and 9, with each flow line corresponding to 2.5 Sv.These figures
FIG. 5. -Inte3pted volurne transport, each flow line represents 1 Sv.Case 3.1 corresponds to Case 3 with the northem subregion ranging from 30 to W N .Cace 3.2 corresponds to Case 3 with the northem subregion ranging from 25 to 40°N.Case 5.1 corresponds to Case 5 with the northem subregion ranging from 30 to W N .Case 5.2 corresponds to Case 5 with the norrhem subregion ranging from 25 to 4O"N (ns: no-slip: q = c: constant potencial vorticity; y = O: zero normal flux; y * 0: non-zero nomal flux).

Figure 5
Figure 5 illustrates, for Cases 3 and 5 , how the rastem boundary circulation pattem is modified when the meridional limits of the eastem boundary subregions are changed.The top figures conespond tc Case 3 but with the upper subregion spanning from 30 to 40°N (Case 3.1) and from 25 to 40°N (Case 3.2).A comparison with Case 3 in Figure 4 ndicates that an increase in the meridional extent of Tic subregion allows more water to recirculate xough the coastai region.In aIl cases, however, this unredistically ejected out of the interior ocean in very narrow area close to the northem lirnit of the sasin.The bottom figures correspond to Case 5 but ~i t h the upper subregion spanning from 30 to 40 "N Case 5.1) and frorn 25 to 40°N (Case 5.2).A commison with Case 5 in Figure4indicates that an ?crease in the meridional extent of this subregion liows an increase in the arnount of water that recirulates through the coastal ocean, and that this takes dace in a rather smooth (and realistic) fashion.