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On subinertial flow in submarine canyons: Effect of geometry. Allen, Susan E. Jan 31, 2000

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105,NO. C1, PAGES 1285-1297,JANUARY 15, 2000  On subinertial flow in submarine canyons' Effect of geometry Susan E. Allen Department of Earth and Ocean Sciences,University of British Columbia, Vancouver,British Columbia, Canada  Abstract. Shelf break canyonson the west coast of Canada and the United States have been observedto be regionsof enhancedupwellingduring southwardcurrents comparedto the surroundingshelf break. Most shelf break canyonsfrom Oregon north crossonly part of the continentalshelf cutting from the shelf break toward the coast but end on the continental shelf well below the mixed layer. Juan de Fuca canyon,on the other hand, cuts the continentalshelffrom the slopeto, and actually continuesinto, the Strait of Juan de Puca. This differencein geometryhas a very strongeffecton the subinertialflow around the canyon. Model canyonshapes,which include convergentbathymetric contours,axe constructedand motivated for Juan de Fuca canyon and a typical shelf break canyon. Geostrophic analytic solutions show that the in-cai•yonflow in Juan de Fuca canyonis generatedby first-order geostrophicdynamics,whereasin the majority of canyons,of which Astoria is an example, in-canyonflow is generatedby higher-ordereffects. This differenceis postulated to lead to the observed,very deep upwellingover Juan de Fuca canyon comparedto more moderate,episodicupwellingover Astoria canyon. 1.  Introduction  Astoria sub•narinecanyoncuts the shelf break off the ColmnbiaRiver betweenOregonand Washington. It is  Canyonsare ubiquitousfeaturesof manycontinental 20 k•n long but ends on the shelf 20 km from the coast shelvesand, in particular, the westerncoastof Canada in 100 m of water. Detailed current and conductivityand the UnitedStates. Off Oregon,Washin.on, and temperature-depth (CTD) measurementswere made  BritishColumbiathe canyons are mainlydeepcanyons overand within Astoriacanyonin 1983[Hickey,1997]. in the sensethat their rimsare belowthe mixedlayer. Temperatures within the canyon vary episodically as Most deepcanyonsare of finite length:that is, they strong upwelling favorable winds occur. Unlike Juan upcut from the slopeonto the shelf and end on the shelf. de Fuca canyon,there is little or no seasonal-scale A numberof canyons,however,continueinto straitsor welling. A modelfor flow over Juan de Fuca canyonwas preestuaries.This lastsetincludesJuandeFucacanyonoff Vancouver Island,BioBiocanyonoffthecoastof Chile, sentedby Chen and Allen [1996]. They considered and the Congocanyon.The purposeof thispaperis to barotropic, linear flow and showedstrong shoreward investigatethe differencesin the dynamicsbetweena flow during periods of southward shelf currents. A canyonsuchasJuandeFucacanyonanda finitelength layer-stratifiednumericalmodelgavesimilar resultsfor canyon,for example,Astoriacanyonoffthe coastof the the layerthat liesat the levelof the canyonrim [Chen, Colmnbia River. 1995]. Flow overa finite length canyonsimilar to AsJuan de Fuca sub•narinecanyonrunsfrmn the shelf toria canyonhas been numerically modeled by Klinck [1996]and Allen [1996a]. Both studiesconsidered an break off southern Vancouver Island into Juan de Fuca Strait. It hasbeenassociated with the largeseasonal- upwellingepisodeof shortduration (10 daysand I day, scaleupwellingoverthe nearbyshelf[Freeland andDen- respectively).Strongupwellingwas observed,particuman, 1982]. More recentobservations [Vindeirinho, larly over the downstreamside of the canyonnear its 1998] showthat temperaturedecreases at 75 m over head. Two models, one for Astoria canyon and one for a spur of Juan de Fuca canyonfrmn early May 1993 to the end of July. Both of theseobservationalstudies Juan de Fuca canyon, will be introduced in section model includessingular show upwellingoccurringon a seasonaltimescaleover 2. The Juan de Fuca.cemyon points,and the inclusionof thesepointsin the geometJuan de Fuca canyon. ric modelis justifiedin section3. An investigationof the geostrophic flow aroundthe canyonmodelsis preCopyright 2000bytheAmerican Geophysical Union. sentedin section4. This investigationillustratesthe Papernumber1999JC900240. fundamentaldifferencein the dynamicsbetweenthe two 0148-0227/00/1999JC900240509.00  types of canyons. 1285  1286 2.  ALLEN:GEOMETRICEFFECTSON UPWELLINGOVERCANYONS  Models  The choiceof geometricalmodelmust satisfytwo constraints. It must be true enoughto the real problem to contain most of the observeddynamics,but it must also be simpleenoughfor, in this case,analytic treatment. For a finite length canyonconsiderthe model drawn in Figure 1. It is a symmetriccanyonof two depthsembedded in a flat shelf. For a Juan de Fuca-type canyon considerthe •nodeldrawn in Figure 2. It is a symmetric canyonjoined to a strait. The canyon has two depths, the strait hastwo depths,and the shelfis flat. Throughout this paper the x axis will be taken to lie parallel to the coast, and the y coordinate will be taken to decrease offshore,as ilhtstrated in Figure 2. The Coriolis parameter will be assumedto be positive. The two edgesof Figure 2. Plan view of the model topographyfor Juan the canyonwill be referredto as left and right as they de Fuca canyon. The directions of the x and y axes appear in Figures 1-3, and 13-15 (the right side occurs are given. The solid small circles mark the two singular points in the closedstrait case. The solid black at largerx than the left). The origin of the coordinates square•narks an additional singular point in the open will be changedfrom sectionto sectionfor convenience. strait case. The open box showsthe area consideredin  The full governh•gequationsare the nonlinearshal-  section  3.1.  low water equations:  Du -1Vp+AnV2u+Aw•-•z 02u D-•+'f•:xu- -• (la) Po Ou  Ov  Ow  + + -0 Op= -Pg Oz Dp D•  =0,  (lc) (ld)  wherel is time, u- (u, v) is the horizontalvelocity,w is the vertical velocity,p is the pressure,p is the density,  the horizontaland vertical eddy viscosities,respectively, and V is the horizontal gradient. The hydrostatic and Bousshmsqapproximations have been made. Considera simplifiedstratification, that of a seriesof  homogeneous (p constant)layers. The flow is assumed to be linear (V/fL << 1, where V is a velocityscale and œ is a horizontallen•h scale),steady(O/Or- 0), and inv•cid. Inclusion of viscositywould causethe flow to slowly spin-down. It is •sumed that this process• much slowerthan the adjustment to geostrophy.Equations (la)- (ld) s•p•fy to  pois a constantreference density,] is the Coriolisparameter,assumedto be positiveand constant,g is the acceleration dueto gravity,D/Di is the horizontaltotal  derivative, •: isthevertical unitvector, An andAv are  fxuV-(Hu)  -1 Po  Vp  - 0,  (2a) (2b)  where H is the undisturbed thickn•s of the layer. These equationsare degeneratefor layers of constant thickess H; i.e., for those layers not in contact with the bottom and any deep layers over a flat bottom. As the topo•aphies under considerationare piecewiseflat, the solutionfor the steady state usesthe conservation of potential vorticity equation (derived•om the t•  dependentversionof (2) [Gill, 1982])'  05H Ox •  - f (h-H) -0,  (3)  whereh • the thickess of the layer. The distribution of  potenti• yogicity is chosento give a simpleshelf/she• bre• current • the absenceof a canyon.  In layersh• contact with the topography,over the regionswherethe depthchanges,(2) imp•esthat there is no flowacrossthe depth contours.This is the familiar tendencyfor geostrophicflow to followbathymetry. As .  Figure 1. Plan view of the model topographyfor Astoria canyon.  ALLEN: GEOMETRIC  EFFECTS  ON UPWELLING  geostrophicflow followspressurecontours,(2) implies that the pressureis constantover depth changesin the layer in contact with the topography. Therefore a complete steady state solution for geostrophic flow over a canyon requires a determination of the pressureat the coastand at the other depth changes  and then solutionof (lc), (2), and (3) over the fiats. Although the requiredsolutionis steady,to find which steady solution is appropriate, one must consider the  OVER CANYONS  1287  The Inodel we wish to consideris a layered,lh•ear, inviscid,steadystate flow over the two topographies shownin FiguresI and 2. However,first, we Inustjustify the choiceof usingthe singularpointsas shownin Figure 2 over the topographywithout singularpoints as shown in Figure 3.  3. Justification for Singular Points  In this sectionwe will considera homogeneouslayer transient effects. In particular, the pressureover the depth changesis determinedby the propagationof long in contact with the topography. For example, the midtopographicwaves[Gill et al., 1986; Chenand Allen, dle layer would be the layer of interest in a three-layer 1996]andthat alongthe coastis determined by Kelvin systmnwith an upper layer representingthe mixed layer waves. As these waveskeep the shallow water or land (say30 in deep),a middlelayer extendingdownto upto their right, informationpropagatesfrom the right to per slopedepths(say 250 m), and a deeplayer below. Steady, low Rossbynumber flow in a homogeneous the left in Figures I and 2. Considerfirst the model of Astoria canyon. As the layer that is in contact with the bottom follows bathyinformation(pressure)propagatesalongthe shelfbreak metric contours. However, bathymetric contours do not and into the canyon,it encountersthe changein depth relnain a constant distance apart. For example, conwithin the canyon. Here the information splitsand trav- sider the 50 and 200 m contours north of Juan de Fuca els both ways: acrossthe canyonat the depth change canyon (Figure 4). In the region north of the canyon and around the canyon head. It meets on the other side these contours are 60 km apart, while near the west of the canyonand continuesalong the shelfbreak. So, end of Juan de Fuca Strait they closeto 5 km. If the determination of the pressureat the depth changesis flow continuesto follow the depth contoursand remains trivial as it is the same value as that at the shelf break. geostrophic,the total flux betweenthe contoursremains constant. Hence the flow speed must accelerate by a Considernowthe modelof JuandeFucacanyon(Figfactor of 12. Flow that is quite slow (low Rossbynumure 2). Information propagatesagain from the right ber) where the isobathsare far apart will becomemuch along both the shelf break and the coast. Far to the stronger where the isobathsconverge.Given an accelerright, the pressureat the shelf break Inust be differation of a factor of 12, it is quite possiblethat flow that ent froIn that at the coast as there is a current on the shelf. The information traveling along the shelf break has a low Rossby number north of the canyon has, at turns up into the canyonand splits at the canyondepth the west end of the strait, a moderate Rossby number change. The informationcontinuingup the right-hand and is no longer purely geostrophic. Essentially, this is the physical limitation. In princanyon wall encountersthe coastalong which different ciple, bathymetric contourscan becomeinfinitely close informationhas beentraveling. This convergence gives together, but it is unphysicalto expectthe flow to acrise to a sh•gularpoint. Note that suchsh•gularpoints celerate without bound. To investigatethe behavior of can be avoidedby slightlyinodifyingthe topographyas flow as bathymetric contoursconverge,considerthe geshown in Figure 3 and that real topographydoes not ometry of Figure 5 and considera homogeneous layer include singularpoints, althoughthe contoursInay get in contact with the topography. very close. Acrossthe convergingflow,  Ou  Ou  Ou  - 10p  O--•+U•xx+V•y - fv ....poOx•A,Vu+Av  (4)  which is the x direction momentum equation. Assume that the flow is steady and that the flow across the depth contours,even as they converge,is much smaller  than the flow alongthe contours.Thus lu[ << [v[, and (4) is dominatedby the geostrophic balance.This gives v-  Ap  pofAx'  (s)  where v is the velocity along the contours,Ap is the Figure 3. Plan viewof the modifiedtopographymodel pressurechangebetweenthe two contours,and Ax is the distance between the two contours. for Juan de Fuca canyon.  1288  ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS  49'00'N  48'50'N  48'40'N  48'30'N  48'20'N  48'10'N  48'00'N  126'00'W  125'30'W  125'00'W  124'30'W  124'00'W  Figure 4. Map of the SouthernVancouverIslandShelfshowingthe coastlineand the 50 and 200 m depth contours.  both the advection  Alongthe converging flow,  Ov  Ov  Ov  -10p  Y  Po Oy  02v  Awayfromthe convergence, theflowis assumed to  terms and the frictional  terms  in-  crease.For a large differencein the initial pressurethe nonlineax  advection  terms  will dominate.  For smaller  pressuredifferencesthe horizontal or vertical viscosity will dominate. The relative importanceis given by  be geostrophic. However, as the contours converge,the Reynoldsnumbers,ReH -- VoAx/AH and Rev =  VoH2/(AxAv). For typical shelf values,Vo = 0.3 m s-z the depth scaleH = 100 m, and Ax = 10 kin. Rangesfor AH are 10-1000m• s-• andfor Av are  0.001-0.1m• s-• [Pedlosky, 1979].The shelfis a rela-  depth=h ] depth=h 2  tively active area, so upper range valuesare probably •nore appropriate. This givesRes and Rev numbers of 300 to 3. advection  We will assume here that  terms  the nonlinear  dominate.  Assrunethat two contoursconvergeover a length scale L much greater than the final distance between them, e. The velocity parallel to the contours, assuming the flow continuesalong the contours,is given by  _  Ap  poe/'  (7)  The velocity along the contoursmust accelerateas the contoursconverge. This accelerationis given by  vOv/Oyandisprovided by thepressure gradientin (6),  thus  Y  Pl  P2  v2 Ap  -L  poL'  (8)  Eliminatingthe velocitybetween(7) and (8) givesthe •nini•num distanceecthat can separatethe two contours  whileensuringthat the flowcan re•nainstrictly along the contours:  Figure 5. Plan view of the convergence of two bathy•netric contours with notation for section 3 marked.  =  f (_•o p)1/21  (9)  ALLEN:  GEOMETRIC  EFFECTS  ON UPWELLING  Another way to quantify the convergenceis given the final convergencee, we can compareit to the minimum convergence ecand definea convergence parameterG:  CANYONS  1289  by at least Ax/e, which is order 10 for Juan de Fuca canyon. The effectof curvatureis i•nportant to canyons like Astoria, but it is a second-ordereffect. 3.1.  -  OVER  Numerical  Verification  In this section a nmnerical  solution of the nonlinear  reduced gravity equations is used to illustrate the dy-  or  / )•/•  G- e VoAZ '  The value of (7 detemnines the structure  na•nics described  above and to determine  the critical  for (7. The use of a singleactive layer allows (11)values a large nmnber of different parameter sets to be simu-  of the flow at  lated. Note that the lower layer is taken as the active  the convergence.For large (7 the convergence is weak, layer. The model solves(4) and (6) with Av taken as and flow continuesalongthe isobaths.For ranall(7 the zero, the pressuregivenas pot'h, and a conservationof convergence is strongand the flow crossesthe isobaths. volrune equation In section3.1 the minionurn valueof G(= 1.5) for which Oh  the flowcan be considered weak(no isobathcrossing)  + v. (nu)- o.  and the •naxi•nmnvalueof G(- 0.3) for whichthe convergencecan be treated as a singularpoint will be esti-  The •nodelis a finite difference,leapfrogexplicit method on an ArakawaC grid [Allen, 1996a].It usesan enstroIt is interestingto note that G is simply the square phy- and energy-conserving formulation for the advecroot of the reciprocalof the Rossbynumber evaluatedat tion and Coriolisterms[Arakawaand Lamb,1981]. The the convergence if the flow followsthe isobaths.(This topographyinvestigatedcorresponds to the point in the result follows from conservationof flux; the flow at the Juan de Fuca canyongeo•netrywherea slopewithin the convergence is VoAx/e.) canyon•neetsthe left canyonwall as illustrated by the In stunmary,the •naximum accelerationpossiblefor a open box in Figure 2. To avoid boundaryeffects,the givenflowis li•nitedby availablepressuredifference(8), numericalboundarieshavebeengreatly expandedcombut for regionswhere the bathymetric contourssharply pared to the box shown. The topographyis illustrated convergethis accelerationmay be insuificientto allow in Figure 6. Parmnetersusedin the numericalsolutions the flow to continuealongthe contours(7). As the flux are given in Table 1. •nust be accmmnodated,the flow will expand over a larger region. The flux betweenthe two contourswill 401 reduce,which impliesthat the interfaceelevationalong •nated numerically.  the contoursis not conservedas the flow is principally  geostrophicacrossthe contours(frmn (4) with [u[ << 120 m  To solvefor flow in the regionof sharp convergence of bathy•netric contours,conservationof •nassmust be used. This •nethod is exactly that used by Chen and  Allen [1996]to solvefor flow in the vicinity of the singular pointsin the Jua• de Fuca model. Thus the useof singularpoints can be justified as a very good approxi•nation for broad low Rossbynumber flow that only beco•nesstronglynonlinearin small regions. Figure 5 doesnot includeany bendingor curvaturein the isobaths. However,at the head of canyons,isobaths turn 180ø over the width of the canyon, W. Why can this effectbe neglected?Considertwo cases-one where there  is curvature  and a second where there is conver-  gence.An esti•nateof the effectof curvatureis givenby the Rossbynumber V/fW. An estimateof the effect of convergenceis also given by the Rossbynmnber, in this case,V/f•. However,whereasaroundthe curvethe velocity changeslittle so V • Vo, through the convergencethe velocityincreasesto VoAx/e. Charts show that the width of a canyonis similar but larger than the isobath convergenceseen at Juan de Fuca. That is, W _>e. Thus the Rossbynumber due to the convergenceis larger than the Rossbynmnberdue to curvature  y=b 40 m  200 m  X  60 •o 160 by 20  •5•.  Figure 6. Plan view of the bathymetryfor the nu•nerical si•nulationsshowingthe whole computational do•nain with domain size in kilometers. The shallowest  water(40 •n) is to the left side;the deepest water(200 •n) is at the bottmnright;and the depthat the upper right is 120 •n. Depth contours,20 m apart, are solid lines. The dashed horizontal line marks the potential  vorticity discontinuity(y = b). The two arrowsmark the positionwherecrosssections weretaken. The smalldashedsquareis the subdomainshownin Figure 8.  1290  ALLEN: GEOMETRIC  Table  EFFECTS  ON UPWELLING  OVER CANYONS  1. Para•neters Used in the Numerical Simulations Parameter Grid size  Value 2.5 km  Time step  175-400 s  f  1 X 10-4s-t  Av  0  g•  0.02and0.10m s-2  Re  2000  Boundary condition- offshore constant gradient u, v, •1 Boundary condition- left constant gradient u, v, r/ Boundary condition- right constantgradient u, v, r/ Boundary condition- onshore 5 grid point spongeto 0 u, v, r/  each topographiccontourcan be found by tracing the topographiccontour back to its sourceon the onshore or right boundary of the domain. In particular, those contoursoriginating at the onshoreboundary have interface heightsof zero. Figure 7a showscrosssections betweenthe two arrowsmarked on Figure 6 of the interface height for a nearly linear case,(7: 2.5, at three 0 y>b ' differenttimes. The sharptransition (with someovershoot) betweenthe topographiccontourscarryingr/= 0 where P is the potential vorticity given by the term in and those(to the right) carryh•g• = 0.125 m is clear. square brackets in (3) and •/o is positive for a south- The variation between the different times shows that bound current. This distribution of potential vorticity the flow is not completelysteady. TopographicRossby givesa geostrophicjet the width of the Rossbyradius, wavespropagate along the canyon edge, causingoscilcentred at y = b. Over the fiat shelf (on the left in lationsin the depthat the canyonedge[Allen, 1996b]. Figure 6), This propagationcan be seenby comparingFigures8a  The distribution of potential vorticity is chosento give a si•npleshelf/shelfbreak current in the absence of the canyon. For a southbound(in the positive x direction)currentthis i•nplieshigherpotential vorticity near the coast, decreasingoffshore:  •,_{ -fr Y<•  and 8c, which show contours of the interface elevation at  u- • exp R  '  (14)days10 and 15 for the weaklyforcedcase.The trough  over the canyon edgeat • y = 170 k•n at day 10 has •noved to y: 120 km by day 15. whereR = (g'H)•/=/f is theinternal•ssby radius. Figures 8a and 8b showthe interfaceelevationat day •ther than sta•ing the •nodel with a sharp poten10 for a weakly forcedcase,G = 2.5, and for a strongly tial vorticity controt, the flow w• forced gradually. forced case, G = 0.25, respectively.In Figure 8a the Spreadingthe forcing in time reducesthe amplitude of four surface height contoursthat enter the domain from the generatedPoincar• waves. The forcing (taxadding the left side turn right and follow the canyonwall offof fluid to the father offshore section of the domain, shore; they do not cross the canyon wall. The four y < b) w• ra•nped up over half an inertial period and surface height contours from the right-hand side simireduceddown overhalf an ine•ial period. There w• no  forcingafter the first inertial period. So, a•er oneiner- larly turn and follow the canyonwall. In contrast, in tial period the potential vorticity is a slightly smoothed the more strongly nonlinearcase,Figure 8b, two of the versionof (13). The line separatingthe unforcedsec- four contoursdo crossthe canyonwall in the vicinity tion of the do•nain from the forced section • marked • of the bathymetric convergence.As the flow is nearly geostrophic, this impliesflow crossingthe bathymetry a d•hed line on Figure 6. here (Figure 9). If the flow h• a small Rossbynumber even through Figure 7b showsthe interface crosssection for the the convergence,the solution is linear. The two restrongly forced case. As flow has crossedthe bathymegions of topographicslopescarry information, • this try at the convergence, the minimum interface value at c•e the interface height, in the direction that keeps this cross section is greater than zero. Given a small G, the shMlower water to the right. Thus information the theoretical value for the minimum interfaceheight is propagatingoffshore(toward negativey) Mong the can be calculated by conservation of mass. For this canyonwall and to the left along the slope. The interface heightpropagatedalongthe canyonwall is that of topographythe valueis r/o/2 or 6.25 m as shownin the the onshoresectionof the domain, whereasthe interface  appendix.  If the flow followsthe topography,the minimum value heightpropagatedalongthe slopeis that of the offshore of the crosssection is 0, whereasif flow acts as if there section. This configurationgivesa convergenceof differing information. If the flow remains linear through is a completeconvergence, the minimumvalueis r/0/2. the convergence, the valueof the interfaceheight along Thus we can usethis parameteras a measureof the type  ALLEN:  GEOMETRIC  EFFECTS  ON UPWELLING  OVER CANYONS  1291  0.2  0.15  ß  0.05  _c  -0.05  O0  I  I  I  I  150  200  250  300  350  I 250  I 300  350  X (km)  0 1O0  I 150  I 200  X (km)  Figure 7. Crosssections of interfaceheightfor (a) G - 2.5 and (b) G - 0.25, respectively at threedifferenttimes(solid,10 days;dashed,20 days;anddotted,30•days).The positionof the crosssectionis shownby arrowsin Figure 6. The vertical axis is in meters;the horizontal axis is in kilometers.  Note that the whole width of th• domain is not shown.  of flow occurring. The minimum value of the interface minimum value of G for which the flow acts like a sinelevationalongthis crosssection(normalizedby •0) was gular point is 0.3. At an intermediatepoint where G is found as a function of time for a number of Cs. The m 0.5- 0.6, the behavioris halfwaybetween. (Note valueof r/0 wasvariedby 2 ordersof magnitudegiving that neither of the theoretical values were found for a singleorder of magnitudevariation in (7. the asymptotes. This discrepancyis probably due to As short topographicwavestravel along the slope, a numberof factors,oneis simplythe estimationof the t!xe •nini•num interface value oscillates in time. Various different measures from this time series were calculated.  mini•nmninterfaceheightfrom a time-varyingseries.) Thus, for a velocityof 30 cm s-• betweentwo con-  All gave a si•nilar transition between the linear result  tours initially 10 km apart the flow can continue to  and the nonlinearresult. Figure 10 showsthe average follow contoursas long as they remain 8 km apart value from days 5 to 30 of the minimum interface ele- (G = 1.5). The totally convergedsolutionbecomes vation along the crosssection. The maximum value of valid whenthe contoursconverge to within 2 kin (G = G for which the flow follows the isobaths is m 1.5. The 0.3).  1292  ALLEN:  GEOMETRIC  EFFECTS  ON UPWELLING  A contourplot showinge as a fu,•ctionof Voand Ax at G = 1.0 for a rangeof valuesrelevantto the shelfis shownin Figure 11. For other valuesof G, suchas the ,ni,fimumand •naximu,nvaluesgivenabove,multiply the e givenin Figure11 by the requiredG. 3.2.  Field  OVER CANYONS  isobaths. Noble and Ramp [2000]usedan array of six curre,•t,neter mooringsto investigatethe California undercurrent in the Gulf of the Farallones off San  Francisco. South of the mooredarray the shelf and slopeforma broad,relativelyshallowslopingcontinental shelf. Within the mooringarray the topography  Evidence  steepensdramatically (from 2ø-3ø at the southernline There are somefield observations in supportof G to 9ø-10ø at the norther,•line). At their souther,•line deter•ni,fingthe ability of geostrophic flow to follow the distance between the 200 and 1000 •n isobaths is 25 k,n. Their typical flow valuesare 25 cm s-•.  At the  northernli,•e the 200 and 1000m isobathsare only 5 k,n apart. This gives a G of 0.6. This low value of G  301 .  i,npliesthat the flow will not be able to followstrictly the isobathsthroughthis convergence. NobleandRamp [2000]find that the undercurrent,clearlyvisiblein the southern line, is not visible and assu,nedoffshoreof the northern  line.  U,•der conditions where the undercurrentvelocity dropsto 8-12 cm s-h, so that G increases to 0.9 the volu,ne flux through the two lines becomesmore similar. Thus, underlarger G the tendencyto crossisobaths  ß  is s,naller.  ß  101 . 126.  x  326.  Consideringjust the upperpart of the slope,the 200 ,n isobath and the 500 ,n isobath are 6 km apart at the southern line and only I km apart at the northern line. This topographicconvergence givesG = 0.2 under strongflow and G = 0.4 under weakflow. At theselow valuesof G the theoryabovepredictsthat the topography shouldact as a singularpoint. Indeed, Noble and Ramp [2000]find that the currentmeteroverthe upper slopeat the northern line is always "shadowed."  301 .  101  . 126.  1  X  326.  1•6.  Figure 8. Planviewof the interface contours for (a) a weaklyforcedcase(contours -0.025to 0.125,n by 0.025m), and(b) a strongly forcedcase(-12.5to 15 m by 2.5 m) after10 daysand (c) the weaklyforcedcase(contours -0.05to 0.15m by 0.025m) after15 days.Notethat the wholedomainisnotshown.Positive contours, thezerocontour, andnegative contours areshown  as solidlines,a dashedline, and dottedlines,respectively.  ALLEN:  GEOMETRIC  EFFECTS  ON UPWELLING  251.  OVER CANYONS  1293  2). Assrunea potential vorticity distributiongivenby (13) where y = b occurson the shelf. Calculationsof the interface heights at the coast and shelf break far  to the right are givenby Chen[1995]for the reduced gravity case. Rather than repeatingthe formulashere, we will usenu,nbersfrom an exa,nple. Assumethat the active layer is 40 m deep over the shelf, 120 m deep over the shallowcanyonsection,and 200 m deep off-  shore.The reducedgravityis 0.02m s-•, andthe Coriolispara,neter is 10-4 s-• . ThisgivesRossby radiiof R• - 8.9 k,n, R2 - 15.5 kin, and R3 - 20 km for the  shelf,shallowcanyon,and offshore,respectively.From Chen[1995]this givesthe interfaceheightat the shelf break far to the right as r/so- 0.90r/0and that at the coast as r/½- 0.17,/0. The flow vectors,in the absence of a canyon,for this geometryare givenin Figure 12 (for,nulagivenby Chen[1995]). Consider first the model for Astoria canyon. The 176. value of the interfaceheight at the boundarybetween 176. 251. x the shallow shelf and the canyonis determinedby the 0.42 I.E...• UAXI'-•'• VECTOR value from the right at the shelfbreak r/•o. The value at the boundary betweenthe two canyondepthsis the Figure 9. Plan view of the velocity vectors for • sa,ne. Now that all the valuesat the canyonedgeshave 0.25 flowat the bathymetricconvergence. Flow is crossbeen determined, the full solutioncan be found by nuing the bathymetry along the canyonslope down the centerof the domainshown.The velocityis in m s-•. ,nericalsolutionof (lc), (2), and (3), whichin this case i  ,  Flow is shown at day 10. Note that the domain shown is smallerthan that for Figure 8.  reduces to  R2V2r/- r/- -P/f.  (15)  The solutionwas found using relaxation (Figure 13). Parametersfor the numerical relaxation are given in  4. Implications  for Geostrophic Flow  Table 2.  For Juan de Fuca canyon the conditions far to the Now that the useof convergentbathymetriccontours right are the same as those for Astoria. The value of hasbeenjustified, we return to the geostrophic solution the interfaceheighton the right-handsideof the canyon for flow over the two modelgeometries(FiguresI and and acrossthe slope within the canyonis r/so.However, 0.5  0.4  0.3  0.2  0.1  0  -0.1  -0.2  ....... 0.1  ' • 1  Figure 10. The average(days5-30) of the mini,numinterfaceheight(normalized by r/0)across the canyonwall offshoreof the bathymetricconvergence. The chosencrosssectionis shownin  Figure6. The diamonds arefor g• - 0.02m s-• andr/ofrom50 to 0.025m. The crosses are for g• = 0.10m s-• andr/ofrom10to 0.025m. Linessimplyjoin the pointsto aidthe eye.  1294  ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS  50  strait. The valueat the left-hand strait wall dependson the assumptionsmade about the strait. Chen and Allen  45  [1996]assumedan infinitelylongstrait, i.e. no connection betweenthe two strait walls. This solutionis given in Figure 14a. In Figure 14b the strait is consideredto  40  ,......,35 20  5  Vo(m s-1) Figure 11. Contourplotshowing thedistance between two converging bathymetriccontoursthat givesG = 1.0 as a functionof initial velocityand initial distancebetweenthe bathymetriccontours.The regionnot contoured h• the lower right cornerbelowthe dashedline representsthe regionwith initial Rossbynumberabove 1. For othervaluesof G, multiplythe valueof e given by the required value of G.  wherethe right sideof the canyonmeetsthe coast,there is a singularpoint. The valueof the interfaceheightfor the right sideof the strait can be found by conservation of •nass[ChenandAllen,1996]and,for the valueshere, is 0.66 •10-This valueis carried acrossthe slopein the  be closed. The real solution probably lies betweenthese two and. for Juan de Fuca canyon,is likely determined by •nixingin the San Juan/Gulf Islandsarea. For the open strait casethe upper left strait wall has an interfaceheightof 0.27 r/0. A secondsingularpoint occurs at the strait slope. Again, using conservation of •nass.the value of the interface height for the lower left strait wall, along the left coast and for the upper left canyon,is 0.53 T10.The third singularpoint occurs where the canyonslope•neets the left canyon wall. The interfaceheightfor the lowerleft canyonwall and along the shelfbreak is 0.72 t10-The fluxesthat theseinterface heightsi•nply are given in Figure 15a. For the closedstrait the valueof the interfaceheight along the left side of the strait, the upper left canyon wall, and the left coast is 0.66 r/0. The value of the interfaceheightfor the lowerleft sideof the canyonand the left shelfbreak is 0.78 r/0. The fluxestheseinterface heightsimply are givenin Figure 15b. In either the open or closedstrait casea significant fractionof the shelfbreak currentoriginallyflowingoffshoreof the shelf break ends up on the shelf. This flow enters the right shelfat the singular point betweenthe coast and the canyon. In the Astoria canyon case, no flow •novesacrossthe shelf break. Flow acrossbathy•netric contoursimpliesvertical motion (upwelling)in these models. Thus, for the Juan de Fuca model case, upwellinõoccurswithin the canyonand over the righthand canyonwall near the coast. For the Astoria canyon •nodel case,no upwellingoccurs.  Coast  -lO  -15  -2o •J•elLbreak -25  -3o  -35  4o  45  Figure 12. Flow vectorsfor geostrophic flow in the absenceof a canyongivena shallowshelf, sharpshelfbreak, and deepocean.Geometricparametersare givenin Table 2. The dashedline •narksthe shelfbreak. The dottedline at y - b marksthe discontinuity in potentialvorticity.  ALLEN: GEOMETRIC  EFFECTS ON UPWELLING  OVER CANYONS  1295  20.0  -40.0 -20  -30.0  Figure 13. Plan view of the interface height for geostrophicflow over the Astoria canyonmodel. The solid lines •nark the topographicdepth changes.Contours are 0.067-0.93by 0.67 r•o(contours0.2, 0.4, 0.6, and 0.8 •1oare shown dashed; the other contoursare dotted). The domain sizeis in kilometers. The whole dmnain  X  30.0  20  is not shown.  Chen[1995]and ChenandAllen [1996]usedthe sixnpiestpossiblestratification,a singleactivelayer. Extension to a homogeneous layer in contact with the topography (with active layersaboveand below) is conceptually trivial. Considertwo cases:a singleactive layer caseversusa multiple active layer case. In the multiple active layer caseassumethat the hmnogeneous layer in contactwith the topographyhas the sameundisturbed depth profile as the singleactive layer case. Further-40.0 -30.0 more, assumethat an initial potential vorticity distribution(varyingthroughthe activelayers)is specified in sucha way that in the absenceof a canyonthe flux over Figure 14.  X  30.0  Plan view of the interfaceheight for geostrophic flow overJuande Fucacanyonmodel(a) assuming that the strait is open,and (b) assuming that in the two cases. Then the fluxes through the system the strait is closed.The solidlinesmark the topographydepthchanges.Contoursare0.067-0.93by 0.067 (contours 0.2, 0.4, 0.6, and 0.8 r•oare dashed;the other Table 2. Para•netersUsed in the Geostrophic  the shelf and the flux offshore of the shelf is the same  contoursare dotted). Domain size is in k•n. The whole  Solutions  dmnain is not shown.  Parameter  Value, km  b  17.6  Canyon width  8  Shelf width  22  Canyon length, Astoria Deep canyon length, Astoria Deep canyon length, Juan de Fuca Strait length modeled Deep strait length Domain size, Astoria Grid size, Astoria Domain size, Juan de Fuca Grid size, Juan de Fuca  14 3 8 25 8 50 x 50 0.25 70 x 95 0.33  shownin Figure 15 are unchanged.This result can be justified as follows:(1) As describedin sections2 and 3 the pressureis used,insteadof an interfaceheight, as the governingvariablefor the multiple activelayer case. (2) As the flow is geostrophic(exceptat the singular points),the flux betweentwo linesdeterminesthe pressure differencebetweenthem. (3) Thus, as the fluxes in the absenceof the canyon are the saznein the two  1296  ALLEN: GEOMETRIC  EFFECTS  ON UPWELLING  3.1  OVER CANYONS  However, assumingthat the first baroclinic inode would be expected to dominate and that the reduced  gravity in the oneactivelayer caseis chosenso that the baroclinicRossbyradii •natch,changesdue to the other modesshouldbe s•nall. The expectedfinal result is that the solutionsshownin Figures 13 and 14 would be very goodapproximationsto the inultiple-layercase.In sum•nary the singleactivelayer givesthe correctfluxesfor a •nultiple-layercase,but details of the flow would be slightly different. 5.  Conclusion  A canyon such as Juan de Fuca, which has strong bathymetric convergences, can support geostrophicupwelling. Howevera canyonsuchas Astoria which does not have strong bathymetricconvergences cannot support geostrophic upwelling.Of course,upwellingis observedover Astoria canyon, but it is episodicand must be due to second-orderdynamics such as time dependence or widespreadnonlinearity. Thus, at Juan de Fuca we observestrongseasonalscaleupwellingwhereas at Astoria the upwellingis episodic.  Appendix: Calculation of the Pressure After a Convergence for G- 0 Considerflow over the topographyillustrated in Figure 6. As G approaches0, the convergenceis so strong that there will be only one value of the pressureover Figure 15. Plan view showingthe fluxes in the canyonwall offshoreof the convergence.This value x10• m• s-t for the geostrophic solutionof Juande Oc• can be found by conservationof mass. The flux into the domainfrom the left is the integral Fucacanyon(a) assulning that the strait is open,and  (b) assuming that the strait is closed.Shadingcorre- in y of the velocity(14) multipliedby the depth, which gives•1oR•f,whereRs isthe Rossby radiusin the shal-  spondsto Figure 2.  low shelfwater. The flow out of the domain to the right casesthe pressureat the coast and shelf break can be taken to be the same (as the origin for pressureis arbitrary, only pressuredifferencesmatter). (4) Pressures at depth changesbeyondsingularpoints are found by conservationof mass,and as the flux approachingthe singular points in the two casesis the same, the pressuresin the two caseswill be the same. (5) Thus the pressureat all depth changes(coastline,canyonwalls, canyondepth changes,and the shelf break) will be the samein the two cases.(6) As flux and pressureare directly related, all the fluxeswill be the samein the two  cases.However,findingthe full solutions(flow at each point) is not so trivial. Finding the potential vorticity distributionto givethe fluxesor findingthe fluxesfrom a given potential vorticity distribution is algebraically messyevenin the three-layercase[Allen,1996a].Once the fluxesare found, findingthe pressuresat the various depth changesis straightforward. Then determining the final solutionrequiresrelaxation over all layers of an equation complicated by the number of vertical •nodes now allowed.  is similarlyrloR2mf, whereRm is the Rossbyradiusin the shallow part of the canyon. The flux out of the do•nain, offshorealong the wall of the canyonbut over  the shelf,is rlc•Rs2.f.The flux intothe domain,from the offshore,alongthe wall of the canyon,and over the  canyon isrk,,R•f, whereRa istheRossby radiusin the deep part of the canyon. Equating the incomingto the outgoingflux gives •%•  H,, - H•  ,lo= Hd- H•'  (A1)  where H•, H,,, and Hd are the undisturbed depths over the shelf, the shallowcanyon, and the deep canyon, respectively. For the numerical values used in the simulation, Hd = 200 in, Hm = 120 m, and Hs = 40 m; •lc• = 0.5•1o. Acknowledgments. The author gratefully acknowledgesthe support of NSERC throughthe ResearchGrants and StrategicGrants programsand from NSERC and DFO Canadathroughthe GLOBEC Canada project. Discussions  ALLEN: GEOMETRIC  EFFECTS  ON UPWELLING  with G. Holloway and commentson the maauscript by J. Klinck, C. Vindeirinho, R. Mirshak, and two anonymous reviewerswere most helpful.  OVER CANYONS  1297  Gill, A. E., M. K. Davey, E. R. Johnson, and P. F. Linden, Rossbyadjustment over a step, J. Mar. Res., •, 713-738, 1986.  Hickey, B. M., The responseof a steep-sidednarrow canyon to strong wind forcing, J. Phys. Oceanogr.,27, 697-726, 1997.  Klinck, J. M., Circulation near submaxinecanyons:A modReferences eling study, J. Geophys.Res., 101, 1211-1223, 1996. Allen, S. E., Topographicallygenerated,subinertialflows Noble, M., and S. Ramp, Subtidal currents over the central California slope: Evidence for spatial and temporal within a finite length canyon, J. Phys. Oceanogr.,26, 1608-1632, 1996a.  variations  in the undercurrent  and for local wind-driven  currents over the outer slope, Deep Sea Res., Part II, in Allen, S. E., Rossbyadjustmentover a slopein a homogepress.2000. neousfluid, J. Phys. Oceanogr.,26, 1646-1654, 1996b. Arakawa, A., and V. R. Lamb, A potential enstrophyand Pedlosky,J., GeophysicalFluid Dynamics,624 pp., Spring-  er-Verlag, New York, 1979. Vindeirinho, C., •Vater properties, currents and zooplankton distribution over a submarine canyon under upwellingChen, X., Rossbyadjustmentover canyons,Ph.D. thesis, favorable conditions, Master's thesis, 121 pp., Univ. of 279 pp., Univ. of B.C., Vancouver,B.C., Canada, 1995. B.C., Vancouver, B.C., Canada, 1998. Chen,X., and S. E. Allen, Influenceof canyons on shelfcurrents: A theoreticalstudy, J. Geophys.Res., 101, 18,04318,059, 1996. S. E. Allen, Department of Earth and Ocean Sciences, Freeland, H. J., aad K. L. Denman, A topographicallycon- University of British Columbia, 6270 University Boulevard,  energyconserving scheme fortheshallowwaterequations, Mon. Weather Rev., 109, 18-36, 1981.  trolledupwellingcenteroff southernVancouverIsland, J.  Vancouver,B.C., Canada V6T 1Z4. (allen@ocgy. ubc.ca)  Mar. Res., •0, 1069-1093, 1982.  Gill, A. E., Atmosphere-Ocean Dynamics,662 pp., Academic, San Diego, Calif., 1982.  (ReceivedJuly 15, 1998; revisedMay 18, 1999; acceptedJuly 20, 1999.)  


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