Meridional Transport across a Zonal Channel: Topographic ...klinck/Reprints/PDF/maccreadyJPO2001.pdfmeridional topographic ridge in the lower layer. The model is forced only by a steady

JUNE 2001 1427M A C C R E A D Y A N D R H I N E S

q 2001 American Meteorological Society

Meridional Transport across a Zonal Channel: Topographic Localization

PARKER MACCREADY AND PETER B. RHINES

School of Oceanography, University of Washington, Seattle, Washington

(Manuscript received 29 September 1999, in final form 5 September 2000)

ABSTRACT

Experiments are performed using a two-layer isopycnic numerical model in a zonal channel with a largemeridional topographic ridge in the lower layer. The model is forced only by a steady meridional volumetransport in the upper layer, and develops a current structure similar to the Antarctic Circumpolar Current.Meridional volume flux across time-mean geostrophic streamlines is found to be due to a combination of thegeostrophic eddy bolus flux and the lateral Reynolds stress. The proportion of each depends on the strength ofthe forcing. The Reynolds stress increases with the forcing, while the bolus flux is relatively constant. Topographylocalizes the eddy fluxes at and downstream of the topography, where eddy energies are greatest. The strengthof the zonal transport is governed by the onset of baroclinic instability and so is relatively insensitive to thestrength of the meridional transport.

1. Introduction

This work is a numerical study of meridional transportacross a zonal channel in the absence of direct wind orbottom boundary layer ageostrophic transports. Al-though highly idealized, it is motivated by observed andmodeled properties of the Antarctic Circumpolar Cur-rent (ACC). The ACC has a large potential density rangethat does not contact the surface or bottom topographyon a circumpolar path. Killworth and Nanneh (1994,their Fig. 3) find this lack of contact to apply in theFRAM model in the density range so 5 27–28.4 kgm23 (with only about 2% outcropping up to 26 kg m23)over the latitude band 658–568S. Ivchenko et al. (1996,p. 766), looking on a time-mean barotropic streampathin the middle of the FRAM ACC also find a broad rangeof nonoutcropping or nongrounding density surfaces,between so 5 27.35–27.82 kg m23 (;500 to 2000 m).The meridional overturning circulation of the SouthernOcean, sketched in Fig. 1, is forced by the creation ofbottom and surface/intermediate water masses, whichare exported equatorward in deep boundary currents andthe wind-driven Ekman layer. The poleward return flow,nominally in the North Atlantic Deep Water (NADW)potential density range, brings in relatively warm, saltywater in the Atlantic sector, and oxygen poor ‘‘old’’water in the Pacific sector (Toggweiler and Samuels1992). The return flow is largely in the density rangelacking significant surface or bottom contact in a broadcircumpolar region, and is the focus of this study.

Corresponding author address: Parker MacCready, University ofWashington, Oceanography, Box 357940, Seattle, WA 98195-7940.E-mail: [email protected]

The dynamics of the ACC have been the subject ofa great deal of research in the last decade, particularlynumerical modeling. The Fine Resolution AntarcticModel (FRAM: FRAM Group 1991) was the first eddy-resolving model of the Southern Ocean to use realisticforcing. Results from FRAM (Killworth and Nanneh1994; Ivchenko et al. 1996) largely confirm the con-jecture of Munk and Palmen (1951) that the wind stressdriving the ACC is balanced by bottom form stress ontopographic ridges. Olbers (1998) gives a concise re-view of these dynamical results. FRAM’s meridionalcirculation was weaker than is thought realistic, due toour poor knowledge of wintertime forcing and convec-tion (Saunders and Thompson 1993). Nevertheless, themeridional circulation that developed, when analyzedin potential density coordinates by Doos and Webb(1994), is seen to have a middepth poleward flow. Kill-worth and Nanneh (1994) find that the vertical diver-gence of form stress balances the time-mean meridionalcirculation. Marshall et al. (1993) point out that transienteddies must be fundamental to the form stress in anisopycnal analysis since by orienting the horizontal pathof integration to be along a time-mean geostrophicstreamline the mean geostrophic mass flux (and formstress) must vanish. Gille (1997b) and Best et al. (1999)analyze Southern Ocean numerical model balancesalong mean streampaths. Ivchenko et al. (1996) and Gil-le (1997a) analyze momentum and potential vorticitybudgets in coordinate systems that follow both isopyc-nals and mean streampaths. These studies generally con-firm Marshall et al.’s view of the importance of transienteddies. However many questions remain, particularlyconcerning the geographic structure of the eddy effects.

1428 VOLUME 31J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

FIG. 1. Diagram of the presumed meridional circulation of the Southern Ocean. Surface wateris driven equatorward by the zonal wind stress, then subducted at latitudes north of Drake Passageto form Antarctic Intermediate Water (AAIW). Antarctic Bottom Water (AABW) is formed bycooling in the Weddell Sea and entrainment of overlying Circumpolar Deep Water. North AtlanticDeep Water (NADW), and other waters in the same potential density class, are assumed in thispaper to flow poleward to balance the equatorward flow above and below.

Ivchenko et al. (1996) show plots of crosspath velocityfrom FRAM, and find that it tends to occur in eddy-rich regions downstream of topography. But they findno clear sign or pattern to the fluxes. By carrying outthe analysis in a simpler model we are able here todelineate some of these patterns. One important issueis the exact definition of the streampath: often the bar-otropic streamfunction is used, but on any depth or layerthe baroclinic structure may cause the local geostrophicvelocity to veer off the barotropic streampath. We carryout our meridional transport analysis in the same layerwhere the geostrophic streampath is defined and soavoid this veering contribution, which may easily over-whelm the eddy mass transport of interest.

In this paper we explore volume transport balancesbetween two isopycnals, particularly across curvingtime-mean geostrophic streamlines. We use an idealizednumerical model configuration to analyze the eddy flux-es, the localization of the meridional flux by topography,and the sensitivity of the circulation to different forcingstrengths. In section 2 the numerical model setup andbasic output are presented. In section 3 the mass fluxacross both zonal and time-mean geostrophic stream-lines is analyzed. Section 4 presents results from ex-periments with different forcing. Section 5 is a discus-sion of the implications of these results.

2. Numerical model setup

We use the Hallberg Isopycnic Model (Hallberg andRhines 1996). The model solves the shallow-water equa-tions in a series of layers of different density, solvingfor layer thickness and horizontal velocity at each time

step. The vertical discretization is the time-varying layerinterfaces. We use two layers (Fig. 2), the minimum toallow baroclinic instability and a forced layer free fromtopographic intersections. We force the upper layer bysteadily adding volume to it on a strip near the northernchannel wall and removing volume on a strip near thesouthern wall. Thus, the upper layer is analogous to theNADW potential density class in the Southern Ocean,and we are ignoring the wind-forced upper layer. Theinitial upper and lower layer thicknesses are 1000 and3000 m, respectively, and deviate from this by 6250m, so there is no outcropping or grounding of the in-terface. The bottom topography is a Gaussian ridge,1500 m in height, with a width of 148 (;893 km, e-folding to e-folding). Treguier and Panetta (1994) per-formed a number of numerical experiments using a qua-sigeostrophic model, with 2–3 layers, and a variety ofbottom topographies. They find that even fairly smalltopographic heights (5%–10% of the total depth) mayexert a strong net form drag on the flow and that thestress increases markedly as the topographic length scalebecomes large compared with the internal Rossby ra-dius. Our experiments have relatively large topography,which causes blocked f /H contours in the lower layerand also effectively blocks the development of muchzonal transport in that layer. There is a linear bottomfriction with stress approximating that due to a dragcoefficient of 2 3 1023. The sidewalls are free slip. Theinternal density jump gives a 50-km internal Rossbyradius, while the (spherical) grid spacing is 1⁄8 in latitudeand ⅓8 in longitude, yielding a nominal grid spacing of18.6 km, all at the central latitude of 2558. The channelextent is 208 in latitude and 808 in longitude (zonally


FIG. 2. Sketch of the domain for the numerical simulations. The model has two layers with originalthickness 1000 and 3000 m. The bottom topography is a single Gaussian ridge extending the entire meridionalwidth of the channel. The flow is forced by continuously adding water to the upper layer in a 18 band 18off the northern wall (black band), and removing the same amount of water from the upper layer in a similarband near the southern wall (white band).

reentrant). The run is designed to resolve the most un-stable baroclinic eddies, the wake from the ridge, andthe jets associated with the current. The main run wediscuss is forced with a poleward volume flux in theupper layer of 2.2 Sv (Sv [ 106 m3 s21). This wouldbe equivalent to 10 Sv if the channel, and the forcing,were of 3608 longitudinal extent. The model is spun upfrom rest and takes about 5000 days to equilibrate (Fig.3). We analyze output, saved every 10 days, from day5000 to 10 000.

The model spinup is quite similar to that describedby Straub (1993). The first effect of the forcing is toimmediately set up a 2.2 Sv poleward transport in thelower layer (Fig. 3d). The meridional transports in Fig.3 are calculated at midlatitude. The lower layer transportis required to allow the eventual north-to-south shoalingof the interface. The rapid north–south connection inthe lower layer is caused by flow along the ridge on f /H contours. The Coriolis force acting on the forcedsouthward flow in the upper layer near the forcing stripsaccelerates a zonal flow to the east. As the upper layerthickens at the north and thins at the south, the mag-nitude of the meridional potential vorticity (PV) gradientdecreases in the lower layer. As the PV gradient goesto zero in the lower layer (Fig. 4), baroclinic instabilitydevelops, initially near the walls at about day 2000. Thisis also the time at which the poleward meridional trans-port begins to be felt in the upper layer. The baroclinic

eddies have two well-known effects (reviewed in Rhines1994): they cause an interfacial form stress that accel-erates the lower layer and redistribute momentum lat-erally, concentrating it here into one strong and twoweak sinuous jets. These are evident in the mean upper-layer surface height (Fig. 5), which follows contours ofthe time-mean geostrophic streamfunction. The tenden-cy to form multiple jets, sometimes called ‘‘zonation,’’in regions where the forcing scale is many Rossby radiiwide, is described in Panetta (1993) and Treguier andPanetta (1994). The jets exist in both layers downstreamof the ridge due to the eddy form stress coupling. Theundulations in the lee appear to be barotropic short Ross-by waves, with zonal wavelength approximately 640km. The time-mean energy fields are plotted in Fig. 6.The KE of the mean flow, and eddy KE and PE, areconcentrated within the jets, particularly downstream ofthe ridge where the mean flow is strongest.

3. Meridional transport across closed paths

Within a layer the momentum equations may be writ-ten exactly as

u 1 k 3 u(z 1 f ) 5 2=B 2 D. (3.1)

Here u 5 (u, y) is the horizontal velocity, z 5 y x 2 uy

is the relative vorticity, f is the Coriolis frequency, andk the vertical unit vector. Subscripts x, y, and t denote


FIG. 3. Zonal and meridional volume transport vs time for both layers of the larger numericalsimulation. The run equilibrates in about 5000 days.

FIG. 4. Time-mean (days 5000–10 000) potential vorticity (1027 m21

s21) in upper (a) and lower (b) layers of the numerical simulation.

partial derivatives. The Bernoulli function is B 5 p/ro

1 (u · u)/2, where p is the pressure anomaly and ro isa reference density. The dissipative terms used in themodel are a biharmonic lateral friction and interfacialand bottom friction; however only bottom friction isimportant to the balances that we present, the otherterms being negligible. Adopting subscripts 1 and 2 for

the upper and lower layers respectively, the bottom fric-tion is given by

FyxD 5 iD 1 jD 5 u , (3.2)2 2 22 h2

where F 5 1024 m s21. Multiplying (3.1) by the layerthickness h and rearranging to solve for the volumetransport,

2hp1 1y 2 yhu 5 2 huz 2 hu 2 hy 2 hD (3.3)y t1 2f r 2o

1 hp 1x 2 xhy 5 2 hyz 1 hu 1 hu 1 hD . (3.4)x t1 2f r 2o

The first term on the right is the transport due to thecorrelation of layer thickness and geostrophic velocity,the next three are the nonlinear and time-dependentterms, and last is the frictional contribution. The analysisof Marshall et al. (1993) highlights the utility of lookingat the balance along time-mean geostrophic streamlines,because upon such a path the large-scale meandering ofthe current is removed. Recasting our balance upon anarbitrary curvilinear path with alongpath and crosspathvelocities U 5 (U, V) and coordinates (s, n), the cross-path transport may be written as

1 hp 1s 2 shV 5 2 hVz 1 hu 1 hU 1 hD . (3.5)s t1 2f r 2o

Separating all dependent variables into time-mean (ov-


FIG. 5. Time-mean surface height contours (m) with vectors showing the lower-layer transport.

erbar) and eddy (prime) parts and taking the time av-erage of (3.5) we find

1 h p h9p9 1s s 2 shV 5 1 2 hVz 1 hu 1 hU 1 hD .s t1 2f r r 2o o

(3.6)

We have only explicitly separated out the eddy and meancontributions in the pressure term because it is only therethat the choice of path will cause a significant simpli-fication. Along a path with 5 const the first term onpthe right drops out, and only the eddy pressure termcontributes. Physically this is the geostrophic eddy ‘‘bo-lus’’ flux (Rhines and Holland 1979) due to the corre-lation of thickness anomalies and crosspath geostrophicvelocity. Evaluating (3.6) in upper and lower layers, weuse p1/ro 5 gh1 and p2/ro 5 gh1 1 g9h2, where g isgravity, g9 the reduced gravity of the interface, and h1,2,3

are the elevations (positive up) of the free surface, in-terface, and bottom topography, respectively. Note thath1 5 h1 2 h2, h2 5 h2 2 h3, and we will take pathssuch that 1s 5 0. In upper and lower layers then, (3.6)hmay be written (focusing only on the pressure term) as

1h V 5 [g(h9 2 h9)h9 1 other terms] (3.7)1 1 1 2 1sf

1h V 5 [g9(h 2 h )h 1 gh9h9 1 g9h9h92 2 2 3 2s 2 1s 2 2sf

1 other terms]. (3.8)

Integrating around a closed path after a statisticallysteady balance has been reached and assuming no sig-nificant correlation of terms with f (as is the case inour simulations), we find

forced volume transport

5 h V dsR 1 1

1ù 2g h9h9 ds 1 other terms ds (3.9)R 1 2s R[ ]f

0 5 h V dsR 2 2

1ù g9 h h ds 1 g h9h9 dsR 2 3s R 2 1s[f

(i) (ii)

1 other terms ds . (3.10)R ]The eddy bolus flux term is written as a form stress(pressure times interface slope) in (3.9) after the pathintegration, and typically acts as a drag on the upper-layer flow. In the lower layer the same bolus flux term[(ii) in (3.10)] appears with opposite sign and generallyaccelerates the flow in that layer. Thus, even though theeddy pressure terms are different in the two layers in(3.7) and (3.8), they almost exactly mirror each otherin the path integrals (3.9) and (3.10) due to other termsdropping out. This effect is quite clear in the analysisof the numerical results below.

The lower-layer time-mean geostrophic flux term[(3.10) term (i)] is a form stress (not necessarily a drag)on the bottom topography. Obviously there is no eddyform stress on the bottom because the topography isstationary.

One may instead choose a path on which the lower-layer pressure is constant, in which case the bottom formstress, term (i) in (3.10), would vanish and bottom fric-


FIG. 6. Mean KE of the mean (a) and eddy (b) fields, and mean PE (c) of the eddy field, forboth layers of the numerical simulation. The energies are given in J m22 and the basin averagesare shown in the panels. The eddy energies are greatest in the jets in the lee of the ridge. Thebasin average APE of the mean field is 462 450 J m22.

tion would be the only term available to slow the lowerlayer. In fact, such a statement holds for any closedgeostrophic streampath, whether in the ACC or a sub-tropical gyre. We choose to consider only the case wherethe upper-layer pressure is constant on integration paths,because that is the layer in which we diagnose mech-anisms of crosspath volume transport.

While the formulation of the numerical model makesit simple to carry out an integration within a givenisopycnal layer, it is more difficult to evaluate termsupon curving horizontal paths. This is particularly trueas one tries to look at small flows across the muchlarger mean geostrophic current. Small errors in in-terpolation or path orientation combine and may lead

to large spurious crosspath fluxes from the mean geo-strophic flow. In practice we will evaluate all the termson the rhs of the Cartesian equations (3.3) and (3.4)and then interpolate each term to points along a givenpath with 1 5 const. We will assume that the upper-hlayer mean pressure term is zero on this path. The otherterms (the ones which are not negligibly small) aregenerally oriented enough across the path so that theircrosspath contributions may be reliably calculated. Wewill also present results from standard zonal integra-tions, in which case the mean pressure terms are, ofcourse, retained.

The statistically steady zonal and meridional (‘‘Car-tesian’’) transport equations in each layer are


1h u 5 2gh h 2 gh9h9 2 h u z 1 negligible terms (3.11)1 1 1 1y 1 1y 1 1 1[ ]f

geo1x bolus1x rv1x

1h y 5 gh h 1 gh9h9 2 h y z 1 negligible terms (3.12)1 1 1 1x 1 1x 1 1 1[ ]f

geo1y bolus1y rv1y

1h u 5 (2gh h 2 g9h h ) 1 (2gh9h9 2 g9h9h9 ) 2 h u z 2 Fy 1 negl. terms (3.13)2 2 2 1y 2 2y 2 1y 2 2y 2 2 2 2[ ]f

geo2x bolus2x rv2x fric2x

1h y 5 (gh h 1 g9h h ) 1 (gh9h9 1 g9h9h9 ) 2 h y z 1 Fu 1 negl. terms . (3.14)2 2 2 1x 2 2x 2 1x 2 2x 2 2 2 2[ ]f

geo2y bolus2y rv2y fric2y

Short names for the various terms are given in (3.11–3.14) such as ‘‘bolus1y’’; each includes the 1/ f termand has units m2 s21, or volume transport per unit hor-izontal distance. For the path analysis the analogousterms will have names like ‘‘bolus1n’’ for the eddy geo-strophic bolus transport across the path.

The upper-layer Cartesian ‘‘bolus’’ and ‘‘rv’’ terms,typically two orders of magnitude smaller than the‘‘geo’’ terms, are plotted in Figs. 7 and 8. This scalingis consistent with the preponderance of APE in the time-mean fields (Fig. 6). The bolus flux is strongest in thelee of the ridge (Fig. 7), and is often directed almostnormal to the mean geostrophic streamlines. The rv flux(Fig. 8), often referred to as the Reynolds stress (Mar-shall et al. 1999), is larger than the bolus flux by abouta factor of 3, but is mostly directed along mean geo-strophic streampaths, particularly at and in the lee ofthe ridge. Interpolating these fields to the streampathsand evaluating the crosspath components we find (Fig.9) that the normal bolus flux is more often directedpoleward, consistent with the sign of the volume forc-ing. The normal bolus flux is greatest on the first twomeridional excursions of the current at and downstreamof the ridge. The normal component of the ‘‘rv’’ term(Fig. 10) is also strongest on the ridge and in the lee.Marshall et al. (1999), and previous work reviewedtherein, find that the bolus flux term is the dominantcontribution to meridional transport when one considerslarge-scale averages. What is notable in Figs. 9 and 10is that the flux terms normal to the streamline are lo-calized in space by the topography. Such localizationhas been seen in the analysis of atmospheric eddy fluxes(Lau and Wallace 1979) and in the ocean (Cronin andWatts 1996). The volume flux is clearly greater wherethe eddy energies are greater (Fig. 6), although there isnot an exact correlation feature by feature.

Ivchenko et al. (1996) examined the velocity acrossthe time-mean barotropic streamfunction in FRAM.

They plot locations where this velocity is significantand find that it is strongest just downstream of topog-raphy, but without a consistent pattern in sign. Our anal-ysis clarifies this process by using the mean geostrophicstreampath in the isopycnal layer where the flux is an-alyzed, so no standing-eddy flow is aliased into thecrosspath flux terms. Gille (1997a) analyzed the PVbudget in the Semtner–Chervin model, looking on anisopycnal surface along a path of constant Montgomerypotential. This is essentially identical to our streampathframe of reference. Her major finding was that PV wasnot conserved along the path, varying by ;25%, par-ticularly in a large jump downstream of Drake Passage.In our model the mean PV is much more nearly constantalong mean geostrophic streamlines. It is possible thatthe dynamics of the ACC in the region of Drake Passage,the Scotia Arc, and southern Argentina are similar to adeep western boundary current. This would suggest amore direct connection to dissipation in the bottomboundary layer. Our numerical model is designed to bemore like the rest of the ACC, which does not passthrough such constrictions.

One might guess that the bolus flux in various lo-cations could be due to specific eddy processes such asthin eddies moving equatorward in one location andthick eddies moving poleward in another location. How-ever, looking at scatterplots (not shown) of instanta-neous eddy thickness versus normal geostrophic eddyvelocity there were no clear patterns with location.

Considering terms in the crosspath transport budgeton a single contour, we may compare upper and lowerlayer budgets. In the upper layer the bolus and ‘‘rv’’terms are strongly correlated in space in the undulationsin the lee of the ridge. Thus, the two terms generallyreinforce each other at a given location along a stream-line. This contrasts markedly with the classical resultfrom zonal integrals of such flows (Shepherd 1983; Pa-netta 1993; Treguier and Panetta 1994) in which the two


FIG. 7. Time-mean surface height contours (m) with vectors showing the upper-layer geostrophic eddy bolus flux.

FIG. 8. Time-mean surface height contours (m) with vectors showing the upper-layer nonlinear ‘‘rv’’ flux (vectorscale the same as in Fig. 7).

terms tend to have opposite shapes as a function oflatitude. In the lower layer the bolus flux term generallymirrors that in the upper layer. The lower-layer ‘‘rv’’term is much smaller than its upper-layer counterpart.Bottom friction is important only in the lee of the ridgewhere the deep jet is strongest. By far the largest termin the lower-layer crosspath flux budget is the meangeostrophic term ‘‘geo2n.’’ This term is ( ,g9/ f )h h2 2s

which upon path integration (neglecting changes in f )is identical with the bottom form stress, term (i) in(3.10). When the path integral of the ‘‘geo2n’’ term isnonzero, it is equivalent to having a net mean form stresson the bottom topography.

Path integrals of the various terms are evaluated forall circumpolar contours in Fig. 11. In the upper layerh1

(Fig. 11a) both the bolus and the rv terms slow the jets,in about equal measure. This is different from the pattern

seen on zonal integrals in previous channel simulations.There the form stress term tends to slow the jet, whilethe ‘‘rv’’ term tends to accelerate and sharpen it. Thissuggests that the large-scale meandering of the current,aliased by zonal averaging, may give rise to the accel-erating tendency of the ‘‘rv’’ term in zonal averages.This pattern has been seen in zonal integrals in Treguierand Panetta (1994) (although in their case the eddy formstress was balanced by zonal wind stress instead of me-ridional transport) and in Marshall et al. (1999). In thelower layer (Fig. 11b) the balance is mainly betweenthe bolus term, accelerating the layer, and bottom formstress ‘‘geo2n’’ which slows it, as in other simulationsof the ACC (Ivchenko et al. 1996). The upper and lowerlayer integrated bolus terms are essentially identical inmagnitude and opposite in sign, as suggested by (3.9)and (3.10). The lower-layer terms should sum to zero,


FIG. 9. Time-mean surface height contours (m) with vectors showing just the normal component of the upper-layergeostrophic eddy bolus flux.

FIG. 10. Time-mean surface height contours (m) with vectors showing just the normal component of the upper-layernonlinear ‘‘rv’’ flux (vector scale the same as in Fig. 9).

nominally. Errors are due to low-frequency variabilityof the flow, subsampling of the results in time, and thedifficulty of accurately calculating the largest term,‘‘geo2n.’’

Comparing the path integrals with a more standardzonal integral (Fig. 12) the main difference is that theupper-layer mean geostrophic flux term ‘‘geoly’’ comesto dominate the bolus and ‘‘rv’’ terms. The growing im-portance of the ‘‘geoly’’ term is the expression of formstress on the time-mean interface undulations associatedwith the standing wave structure of the jets. The ‘‘rv’’term shows the classical tendency of accelerating the jetjust south of 608S. This is one effect of including thestanding wave into the zonal integral. In terms of physicalprocesses the path integral provides a much clearer pic-ture, as advocated by Marshall et al. (1993).

4. Sensitivity to different forcing strengths

Marshall et al. (1999) perform numerical experimentssimilar to those presented here, although without bottomtopography. They typically use three layers in a channel,forced by maintaining specified north–south PV gra-dients in the upper layers. This results in a meridionaltransport, as in our simulations, although the strengthof the transport is not specified a priori. They find thatthe meridional transport in a layer is approximately lin-early proportional to the mean north–south PV gradientin that layer. To test this in our case, where bottomtopography is significant, we ran four numerical exper-iments using our channel model in a smaller domain,108 in latitude and 208 in longitude, with the same me-ridional ridge as in the larger experiment. These runs,


FIG. 11. Path integrals of the cross-path transport terms. (a) Upper layer, (b) lower layer. Theupper and lower bolus terms almost exactly balance each other. The upper-layer forced transportis 22.2 Sv.

FIG. 12. Zonal integrals of the meridional transport terms. (a) Upper layer, (b) lower layer. Thecontribution from the standing eddies ‘‘geoly’’ in the upper layer now becomes the dominant term.


FIG. 13. Path integrals of the cross-path transport terms from a small-domain numericalsimulation, forced by 20.28 Sv meridional transport. (a) Upper layer, (b) lower layer.

especially those with stronger forcing, develop a singlejet, locked to the ridge as before. We vary the forcingstrength, using 21.11, 20.56, 20.28, and 10.28 Sv(which would be 220, 210, 25, and 15 Sv if extendedto a full 3608 domain; negative forcing means polewardtransport). Due to the smaller domain the flows equil-ibrate rather quickly, and all have equilibrated by day5000. The run with positive forcing is quite differentfrom the others. It does not develop a concentrated jetand has no coherent upper-layer PV structure or lockingto the topography, but is included here for contrast withthe more standard eastward flowing runs.

The integrated time-mean cross-path transport anal-ysis for two runs is shown in Figs. 13 and 14, for 20.28and 21.11 Sv forcing, respectively. The primary de-pendence on forcing strength is that the run with stron-ger forcing develops a jet with larger north–south ex-cursions. The jet in the 20.28 Sv run (Fig. 13) is pri-marily slowed by the bolus flux, which accounts formost of the poleward mass transport. The ‘‘rv’’ fluxtends to accelerate the jet. This is consistent with theclassic pattern seen in zonal integrals (e.g., Treguier andPanetta 1994). The situation is reversed however, in the21.11 Sv case (Fig. 14), with the strongest slowing ofthe jet and most of the poleward flux being due to the‘‘rv’’ term. The small-domain 20.56 Sv run (not shown)and the large-domain 22.2 Sv run (Fig. 11) fall midwaybetween these two cases, with bolus and ‘‘rv’’ fluxesbeing of equal importance. The path-integrated bolusflux is relatively insensitive to the strength of forcing,

and instead it is the ‘‘rv’’ term that changes most toabsorb the changes in net meridional transport.

The relation between the time- and zonally averagedmass transport and meridional PV gradient, outside ofthe forcing strips, is plotted for all four runs in Fig. 15,using the same scaling as in Marshall et al. (1999). Inall our runs the transport is equilibrated with the forcingat all latitudes, and there is a spread of PV gradientbecause the jet structure tends to separate out the flowinto a front with strong gradient surrounded by plateausof weaker gradient. Nevertheless, looking at the meanvalues for each run, the pattern that emerges for the runswith negative forcing is that the PV gradient is approx-imately constant, and independent of forcing strength.The reason for this (Straub 1993) may be argued asfollows. The PV gradient in the upper layer of our runsbuilds up until the PV gradient in the lower layer van-ishes, at which point the eddy activity of baroclinicinstability becomes a very efficient method of trans-porting mass poleward, and further north–south tiltingof the interface is inhibited. Runs with stronger forcinghave stronger eddy energy fields and, hence, the eddyflux terms are able to carry the extra transport for thesame PV gradient. What makes our runs different fromthose of Marshall et al. (1999) is that in ours the eddystrength is set by the meridional transport in the layerthat is forcing the overall flow, whereas in their exper-iments the overall flow structure was set by an imposedPV gradient in the upper layer, but the mass flux relationto PV gradient was evaluated in the intermediate layer.


FIG. 14. Path integrals of the crosspath transport terms from a small-domain numericalsimulation, forced by 21.11 Sv meridional transport. (a) Upper layer, (b) lower layer.

FIG. 15. Time-mean zonally averaged meridional volume transportfor the four small-domain numerical simulations (upper layer), plottedvs normalized time-mean, zonal-mean meridional PV gradient. Pointsplotted for a run of a given strength cluster near a given value onthe y axis, and the mean for a given run is plotted as a large filledcircle. All runs with negative forcing develop about the same PVgradient. The run with positive forcing develops only a small PVgradient (the larger gradient being in the lower layer).

Their intermediate layer was forced with a smaller PVgradient and thus did not control the eddy strength ofthe flow. Our experiments support their general conclu-sion that thickness flux may be down the PV gradient,but the value of the flux apparently is also controlledby the eddy strength, and not just the size of the PV

gradient. Our experiments do, however, support the con-cept that the strength of the bolus flux may be controlledby the PV gradient. Also, as Marshall et al. (1999) note,the downgradient flux concept may not work at all insome regions. This is clear in the lower layer of the10.28 Sv experiment (not shown). There the magnitudeof the zonal-mean meridional gradient of PV is rela-tively large, but it has zero mean meridional volumetransport. This highlights the fact that the eddy fluxesfrom baroclinic instability are really global properties,relating to the vanishing of the PV gradient somewherein the domain, and are not controlled by a single layer.

Stone (1978) proposed the concept that the mean me-ridional density gradient in the atmosphere would begoverned by the onset of baroclinic instability. Applyingthis idea to the Southern Ocean, Straub (1993) developsa prediction for the zonal transport in a two-layer flowwhen baroclinic instability has limited the meridionalinterface tilt. Assuming that the lower-layer zonal trans-port is blocked by topography he finds

2g9H H bDy1 2upper layer zonal transport 5 , (4.1)2f o

where H1 and H2 are resting layer thicknesses in a chan-nel of meridional width Dy, f o is f at midchannel, andb is the meridional gradient of f . For our small-domainruns (using Dy 5 68, the width of the unforced region)(4.1) gives about 50 Sv, somewhat larger than the threeruns with poleward forcing, which develop transportsbetween 36 and 41 Sv. Expression (4.1) predicts 177Sv for the larger-domain run, which has about 130 Sv.


We conclude that the ‘‘baroclinic instability governor’’on the interface tilt is a zeroth-order description of thebasic dynamics.

5. Summary and discussion

We have presented results from numerical simulationsof two-layer channel flow over topography, forced bya meridional mass transport in the upper layer. Analyz-ing the cross-channel transport mechanism on time-mean geostrophic streampaths we find that both eddyform stress and lateral Reynolds stress may retard thejets. The relative importance of the geostrophic eddybolus flux versus the Reynolds stress as the meridionaltransport mechanism was found to depend on forcingstrength. The Reynolds stress grew with the forcing,while the bolus flux was relatively constant. Focusingon the transport mechanisms as a function of positionalong a given path, we find that both transport mech-anisms were larger on and in the lee of the topographicridge, where eddy energies were also highest. The twomechanisms tend to be correlated along the path, insteadof being anticorrelated as seen in the path or zonal in-tegrals. The reason for this correlation is not clear.

A series of smaller numerical simulations were doneto test the sensitivity of the flow to different forcingstrengths. It was found that the total zonal transport (andwith it the time-mean meridional potential vorticity gra-dient) was markedly insensitive to the strength of me-ridional forcing. The PV gradients instead appeared tobe controlled by the onset of baroclinic instability.While the meridional mass transport in the forced layerwas always down the PV gradient, it was not linearlyrelated to that gradient, instead increasing with the eddy(and forcing) strength, while the PV gradient remainedrelatively constant.

While these experiments are highly idealized, theymay have important implications for the dynamics ofthe Antarctic Circumpolar Current. Regions of strongeddy activity have been observed (Ivchenko et al. 1997;Gille 1997a,b; Best et al. 1999) downstream of topo-graphic obstacles and channel constrictions. Our resultssuggest that these may be the main geographic locationsfor poleward transport of deep waters in the NADWpotential density class. In addition, the meridional tiltof isopycnals across the ACC may be very insensitiveto the strength of wind or buoyancy forcing, being gov-erned by the onset of baroclinic instability instead ofthe forcing strength. This suggests that the role of theSouthern Ocean as a connection point between surfaceand abyssal oceans may persist over many extremes ofclimatic forcing.

Acknowledgments. This work was generously sup-ported by the National Science Foundation under GrantsOCE-9301819 and OCE-9529813. We are indebted toLuAnne Thompson and three anonymous reviewers forcomments, to Robert Hallberg for the use of his nu-

merical model, and to David Darr for programming as-sistance.

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