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How does the El Niño-generated coastal current propagate past the Mendocino escarpment? Allen, Susan E.; Hsieh, William W. May 29, 1997

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JOURNAL OF GEOPHYSICAL  RESEARCH, VOL. 102, NO. Cll, PAGES 24,977-24,985, NOVEMBER  15, 1997  How does the E1 Nifio-generated coastal current propagate past the Mendocino escarpment? Susan E. Allen and William  W. Hsieh  Department of Earth and Ocean Sciences,University of British Columbia, Vancouver, Canada  Abstract. During an E1 Nifio, an internal coastal Kelvin wave bore propagates poleward along the west coastof North America, leaving behind a steady anomalous coastaljet. A nonlinear, two-layer, f plane, shallow-watermodel is used to determine the amplitude change of the steady coastal current over an escarpment. As the E1 Nifio-generated warm coastal current passesthe Mendocino escarpment off northern California, its amplitude is marginally enhanced. In contrast, for a cold coastal current, the amplitude will be reducednorth of the Mendocino escarpment. When the Kelvin wave bore travels over a depth increase,the amplitude change  is predictedto be muchlargerthan overa depth decrease(as in the caseof the Mendocinoescarpment).This modelis alsoapplicableto bottom water flowing equatorwardalong a westernboundary. In this case,much larger amplitude changes are found.  This region of small spatial gradients correspondsto the steady warm coastal current which is left behind after E1 Nifio, a prominent atmosphere-oceaninteraction the passageof the initial bore. Offshore the expected centered around the tropical Pacific, is characterized shape is the exponential decreaseover the Rossby raby the appearance of anomalouslywarm waters in the dius, typical of Kelvin waves. This model is consiseastern and central tropical Pacific on an interannual tent with a nonlinear wave balance alongshoreand a timescale[Philander,1990]. ThesetropicalE1Nifio ef- geostrophicbalance crossshore. fects spread to the extratropical northern hemisphere The northward traveling warm coastal current assoby two main mechanisms:(1) The warm water prop- ciated with the Kelvin wave encountersthe steep Menagates poleward along the west coast of North AmerIntroduction  docinoescarpmentoff California(about 41øN), where ica by coastallytrappedKelvin waves[Pares-Sierraand the ocean depthsabruptly changesfrom about 4000 to O'Brien, 1989],and (2) the atmosphere responds to the 3000 m further north. There has been a series of inheating in the central equatorial Pacific by developinga triguing theoretical studies on Kelvin wave transmisseriesof alternating high- and low-pressurecellsin the sion or the adjustment to steady flow over escarpments northern hemisphereknown as the atmospherictelecon- and ridges[Johnson,1985; Gill et al., 1986;Allen, 1988; nection[Wallaceand Gutzler,1981]. Killworth, 1989a, b; Johnsonand Davey, 1990; Johnson The coastal signal is clearly seen in both the tem- 1990; Willmort and Grimshaw, 1991; Johnson, 1993; perature and the sea level data [Enfield and Allen, Willmort and Johnson,1995;Allen, 1996].Thesestud1980]and can be explaineddynamicallyas a first-mode ies and the numericalstudy of Wajsowicz[1991]have baroclinic Kelvin wave. The strength of the anoma- shown that low-frequencyflow over a sharp depth delies, for example, a changein the surfacetemperature crease in the alongstream direction generatesdouble of approximately 5øC and associated alongshore curKelvin waves [Rhines, 1969; Longuet-Higgins,1968a, rents of 40 cm/s in October 1982 off Oregon [Huyer b; Willmort, 1984]that propagatealongthe depth deand Smith, 1985], indicatesa stronglynonlinearphe- crease. nomenon. With the observed sharp temperature rise An incomingKelvin bore, in the caseof a sharpdepth occurringover days, the signalcan be dynamicallymod- decreasein the northern hemisphere, will generate an eled as an internal Kelvin wave bore. In the alongshore offshorepropagating,longdoubleKelvin modeas shown direction the signal has a sharp drop followedby a long  flat regionof depressed pycnoclineheight (Figure 1).  in Figure2a, with possiblesignificantenergylossfor the incident  Kelvin  bore.  Since the warm coastal current  was observedduring an E1 Nifio north of the MendoCopyright1997by the AmericanGeophysical Union. Paper number 97JC01583.  0148-0227/97/97JC-01583509.00  cino escarpment[Huyer and Smith, 1985],the coastal current is clearly capable of propagating past the escarpment. Our first objective is to theoretically estimate how the steady coastal current formed after the 24,977  24,.978  ALLEN AND HSIEH:ESCARPMENTEFFECT ON EL NIfO COASTALCURRENT North Sea surface  .••  the nonlinear advection terms, important for the large observedpycnoclinedisplacements.  Pycnocline Flow over a Steep Depth Decrease  Bore  Figure 1. Sketch of the alongshorestructure of the coastal Kelvin  wave bore.  The problem of long, nonlinear Kelvin waves was considered by Bennett [1973]. The equationsindicate that such wavesmust steepen, and he postulatesthat they would eventuallyform a bore. This result is consistent with experiments on similar nonrotating flows  passage of the Kelvinwaveboreis affectedby the Mendocinoescarpment.No attempt is madeto analyzethe suchas thoseby Woodand Simpson[1984]. For long balinitial interaction between the bore and the escarpment. waves,Bennett [1973]showsthat the cross-shore ance is geostrophic, and providing potential vorticity is Furthermore,the studiesconcerning escarpments listconserved,the cross-shoreshape of the long-shorevelocity field and the sea surfaceelevation is the classic Kelvin wave exponential. As the wavelengthbecomes alongshore direction(Figure2b)istotallydifferent from  ed above showedclear asymmetry; the situation for a Kelvin waveencountering a steepdepthincreasein the  the steepdecrease shownin Figure2a. In Figure2b the muchlarger than the (bareclinic)Rossbyradius,the longdoubleKelvinwavecreatesa singularity in thelin- cross-shorevelocity introduced by the nonlinear terms ear solutionat the coastline(becausethe wavepropa- vanishes. Considera Kelvin wave bore (Figure 1) propagatgatesshoreward) [Johnson, 1985],sofrictionalandnoning northwardpast an escarpmentinto shallowerwater linear effectsbecomeimportant [Allen, 1988]. Hence (Figure 2a) in a two-layerstratifiedfluid in the norththe main energylossfor the incidentKelvin bore in ern hemisphere (Figure 3). After the passageof the Figure2b is fromfrictionratherthanfromthe offshore bore, the flow tends toward a steadystate. As there is propagating longdoubleKelvinmodein Figure2a. Our little alongshore variation and no cross-shore flow, all secondobjectiveis to study the differencebetweenan the advection terms in the momentum equations of the alongshore depthincrease andan alongshore depthdeshallow water model are 0. In this steady state region creasein influencingthe steadycoastalcurrentleft after south of the escarpment,assumingpotential vorticity The largebodyof research on scatteringoverridges is conserved,the disturbancehas an offshoreinterface and escarpmentscannot directly answerthese ques- displacementwith the classicKelvin waveshape,  the passageof the Kelvin bore.  tions. Most of the previousresearchhas considered the linear homogeneous problem[Johnson,1985; Gill et al., 1986; Johnson,1990; Willmortand Grimshaw,  wherey increases offshoreand the bareclinicRossbyra-  1991]. Stratifiedflowswereconsidered in the linear diusis R - {g'hH/[(H+ h)f•']}•/•',withH theupper regime[Killworth,1989a,b; Johnson,1993; Willmort andJohnson,1995].Only Allen[1988]considered nonlinearflow, but all her theoreticalresultswerefor homogeneous flow. Wajsowicz[1991]usedthe BryanCox modelincludingthe advectiontermsbut considered caseswell within the linear regime.As the Kelvin wavegeneratedby an E1Nifio canhavea largepycnoclinedisplacements compared to the averagepycnocline  Shallow  depth[ttuyerandSmith,1985],weconsider a nonlinear bore. In the linear limit, our resultsagreewith thoseof Wajsowicz[1991].  We will proceed in the spiritof Killworth[1989a,b], who consideredKelvin wave transmissionover a ridge,  and use a "black box" approach. The details in the  Deep  Shallow  b) interactionregionnearthe escarpment will not be con- a) sidered. Our problemis somewhatsimplerthan the (a) case1, the incoming ridgecase.We eitherhaveflowovera depthdecreaseFigure 2. Sketchshowing Kelvin wavebore, the outgoingbore and the offshore  (stepup,case1) for whichfrictional effects should re- propagating, longdoubleKelvinmodeand(b) case2, just the bores,asno offshore propagating longdouble  main small and a momentum balance can be used, or  we haveflowovera depthincrease (stepdown,case2) Kelvin mode is excited. The shadedregion represents  for which the flow is confined near the coast and a mass the "black box" around the escarpmentwhere evanes-  balancecan be used. The ridge, of course,includesa centwaves,nonlinearandfrictionaleffectsoccur.Our  isforthenear-steady currents formedafterthe stepup and a step down,hencecombining the com- analysis of the Kelvin wavebore. plexities.The simplergeometry allowstheinclusion of passage  ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTALCURRENT  24,979  Figure 3. Verticalsectionshowingthe variablesin the two-layermodel.  layer depth, h the lower layer depth, •/• the interface where r/c and r/t are the baroclinic and barotropic comdisplacementat the coast,g' the reducedgravity and f ponents, respectively, and a is the barotropic Rossby the Coriolis parameter. The shape is consistent with radius,with a2 - g(H + h2)/?. The othervariables Bennett's[1973] resultsfor a nonlinearKelvin wave. are given by The assumptionof potential vorticity conservationrests -g' h2 on assumingweakdissipationand that the changeswere  g(n+n•.) TM •P(-•/n•)  not due to the advection of a new water mass but rather  H+h2  a deepeningof the already existingone. For a pure baroclinic signal the surfacedisplacement  + n• ,•,•p (-•/•),  (•)  relatedto (1) is g'h  ( - -,• g(•+•) •xp (-•/n),  g'H  (•.)  • •  (- y/ S• )  +.("+ •) r•,exp ah•.f (-y/a,),  with the lower-layer,alongshorevelocity,  (7)  g'H  u- Ri(H+n)'rl•' exp (-y/R),  (3)  U  __  and the upper-layer,alongshorevelocity,  U - -hu/H,  (4)  (8)  to order9'/9, assumed small[LeBlondandMysak,1979, to order 9'/g [LeBlondand Mysak, 1979, sections16 sections16 and 24]. For now,ignorethe regionnearthe and 24]. escarpment,shown shadedin Figure 2a, and consider the flow further north. If a distanceof 5 Rossbyradii is far enoughfrom the escarpmentthat no lower-layer fluid from south of the escarpmenthas been advected that far, then the potential vorticity of the fluid will  Now considerthe flow in the interaction region near the escarpment.In this regionthe initial bore generates transientwaves(evanescent Poincar•and topographic  waves)anda longdoubleKelvinwavealongthe escarpment [Johnson,1990] which settlesdown to a steady  not havechanged(wheresmalldissipation hasbeenas- current flowingoffshore.Along the coastthe initial bore sumedawayfrom the escarpment).Thereforewe can generatestransient Kelvin wavesand a long Kelvin wave assume a cross-shoreshape similar to the disturbance south of the escarpment but with a smaller baroclinic  which settles down to a steady coastal current. Instead  of considering the detailsof the complicated transient  RossbyradiusR2 due to the shallowerdepth. Inter- and steady flow, we will restrict attention to the moaction with the escarpmentalso excites a barotropic mentum balancealong the wall. Kelvin mode and a long doubleKelvin modeout along The first momentum equation for the upper layer is the escarpment.So northward of the escarpment, ou ou ou •(  ot+vy•-• + v• - Iv - -g• +•,,, (•)  24,980  ALLEN AND HSIEH: ESCARPMENTEFFECT ON EL NI•IO COASTALCURRENT  and for the lower layer is  Ou ot + Ou + Ou .fv---g -  •?c- •7,,, 1- 2hh2(H•?,,,) +0 •  +Ft' (10)  r/c rho 1  (16a)  saa h 2(1-rho/H )]'  (16b)  where F representsthe frictional effects. Considera few inertial periods after the head of the bore has passed, with the flow having reached an almost steady state. wi•h Ah = h- h• and assuming H/h • I1 - n/HI Along the wall v, V _--0, as there can be no flow through that is, •ocannot approach H. Note that for small the wall. upper layer depth, even for relatively large pycnocline Thus along the wall, displacements,the changeover the escarpmentis small.  ou  oi_  ou  U•-• + gOa• - at + F=,  Ou+ g•-• O(+ g,Oa• O,•_ u•-• - Ou Ot+Ft.  The last term on the right-handsideof (16b) with the smallfactorH/h2 removed(i.e., multipliedby h2/H)  (11) is contouredas a function of the other two parameters Ah/h and •oH  (12)  in Figure4.  Due to nonlineartry,the solutionfor a depressedpy-  cnocline(negative•o is verydifferentfrom that of an Theseequationscan be integratedacrossthe escarpcase,there elevated pycnocline is a small-amplitude (positive•0). gain In the as can negative be seen •o mentfrom a position-X0 several(say5) Rossbyradii upstreamto a positionXz several(say5) Rossbyradii from (16a), whereasin the positive•ocase thereis a downstream. As the left-hand sides are exact differsmall-amplitude loss. The differenceis due to the dy-  entials,if we assumethe right-handsidesto be 0, the integralscan be evaluatedwithout knowingthe details of the flowin the regionof the escarpment. The integral of the left-handsideof (11) is  namicsat the depthchange.As the depthdecreases (h becomes smaller),the magnitudeof the upperlayerflow U decreases (from(3) and (4)). With a negativepycnoclinedisplacement •o (3) and (4) imply U is positive, that is, northward flow. Thus the depth decreaseleads to a smaller positive U, that is, flow deceleration. The pressureforce required for the flow decelerationimplies  Xl  xo  (•3)  2  -Xo  whichcan be expandedby substitutingthe expressions  (2), (4), (6) and (8). The integralof the left-handside of (12) is  f_x• u Xo( OuO(g,O•?) 22  higherpressureand sealevel(but lowerpycnocline) to the north. Hence the original negative pycnoclinedis-  x• -Xo , (14)  !i  which can be expandedby substitutingthe expressions  (1), (2), (3), (5), (6)and (7). The integration givestwo equations for themagnitudes r/t (whichis small)andr/•.  .......  ,  •\  ......"'.050.  ,,  -,  hh2 H2  rk.  ,  '100•  _  -----_..  -..... .050  ,. .005  ..................... .005 .......................... .005 ....................  0.0  ...................... .005 ................................... .005 ..........  (H-h2)(H-h) 2)1/2]  +'  '  -...  '"  H- h:• -1+ 1+ h•H  •.100_ -..  \  Solvingfor r/• gives  •7•-  .200_ •400  ,. •  ,,  (15)  ,,' - .005  /  -.o5o  to order(g,/g)Z/2andis accurate to theorderof  i f_•c• 0u Y + As Ft must removeenergy,(15) is an overestimate of the magnitude of •. The valuesof Ft shouldbe of the sameorder in the regionof the escarpmentasawayfrom  -.0o5-.o•o -0.9  •  ,  o.o  Ah  0.4  h  Figure 4. Contourplot of the amplitudeloss((•, n•)/n•.), scaledup by h2/H for case1, flow over a it, so the loss due to this term is similar to the losses steepdepthdecrease,assumingH/h2 smalland H/A <<  seenall along the coast. A simple solution is obtained in the limit of small upper layer depth comparedto the lowerlayer depth:  I1 - rl,,,/HI• (equation (16b)).Solidcontours arein incrementsof 0.1; dotted contoursare as labeled. Negative valuesof amplitudelossimply an amplitudegain.  ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NIfO COASTAL CURRENT 0.3  !  ,  ]  ,  i  i  •  i  ,  i  ,  ,  ,  ,  ,  0.25  0.2  0.15  80100  i  i  i  i  •  i  i  1000  30 ,,,I '  10  ........ ß  Exact  the downstream  coastal  current  and from the offshore  current along the escarpment.  The exactsolution(15) andthe approximate solution (16b) are comparedby plottingthe amplitudegain (in percent)as a functionof the upperlayer thicknessH,  Upper layer thickness H (m)  ',  five interface elevationchange. Thus negativeV• values are reinforcedand positive V• valuesare decreased.In conclusion,the warm E1 Nifio-generated coastal current should have a slight amplitude gain after passing the Mendocino escarpment. A routine, but somewhat tedious energy calculation, confirmsthat the small-amplitude gain doesnot imply an energy gain. Owing to the smaller Rossbyradius north of the escarpment,the energy decreasesregardlessof the sign of V•. At steady state, in a frictionless model, the energy balanceis suchthat the inflow energy from the coastal current equalsthe outflow energy from  ..... Approx. solution 300  24,981  while holdingconstantV• at -75 m, h at 4000 m, and h2  solution  at 3000m (Figure5a). The amplitudegainisverysmall (< 0.3%)as the upperlayerthickness is variedfrom 76  - .... Approx. solution  to 1000 m. For the caseof V• - +75 m, the amplitude  lossincreases asH decreases (Figure5b). Generally,the approximatesolution(dashedcurve)agreeswell with the exact solution(solidcurve),but as H -+ V•, the  _.o 3 .-E  1. '  •  •..,•.•,.,..•_  0.:3 •  __., ___•_•_•_  Flow over a Steep Depth Increase  b)  I  I  ]  0.1 ' 80100  approximate solution approachesinfinity, though the exact solution stays finite.  I  I  300  I  I  I  i  I  I  1000  The previousanalysiscannot hold for flow down an escarpment as it implies an increasein energy. Thus Upper layer thickness H (m) the friction term F must not be small. In particular, F;gure 5. Comparisonbetweenthe approximatesolu- besidesthe wavesgeneratedin case 1, there will be tion (dashedcurve)and the full solution(solidcurve) a swift boundarycurrentgovernedby nonlinear,fricfor case1, flow over a steep depth decreasewhere the in-  cidentamplitude r/,0is(a) -75rnand(b) +75 m. Kelvin tional and inertial effectscloseto the coast[Johnson,  waveamplitudegain/loss(in percent)is plottedversus !985; Allen, 1988; Wajsowicz,1991]. However,in this upper-layerthicknessH in casea/caseb. Other param- caseflow is not divertedout alongthe escarpment,and eters are given in the text. volumeflux must be conservedin the regionof the escarpment[Allen, 1996]. The shortdoubleKelvin waves that propagateout along the escarpmentdo not carry any volumeflux [Johnson,1993]. placementwill be slightly intensifiedby the depth deThe incomingvolumeflux in the upper layer is  crease to the north.  In the case of a positive pycnoclinedisplacement U is negative. The depth decrease,which reducesthe magnitudeof U, leadsto a lessnegativeU to the north. (17) Hence the pressureforce must push southward to give where. and U are givenby (1) and (4), respectively. an accelerated,more negativeU to the south, implying, The incoming lower-layerflux is again,higherpressureand sealevel (but lowerpycnocline) to the north. Hencethe originalpositivepycnoclinedisplacementwill be reducedby the depth decrease  •o øø g'hH g'h •. (H- •7)Udy - -f(H +h)•7• +2f(H +h)•7,,,,  to the north.  These results can also be seen from the conservation  j•o •(h+v)udy g'hH - f(H +h)•  g' H + 2f(H +h)•'• (18)  •o order(g'/g)X/•andwhere. andu aregivenby (1)  equation(13), wherethe approximatebalanceof the and (3), respectively. Assuming •ha• potential vorfici•y advection is confined within a region of say 5 Rossbyradii of •he esdepth decrease.As the flow is principally baroclinic, a carpment, outside •ha• region •he outgoingvolume flux negativesurfaceelevationchangecorresponds to a posi- in •he upper layer is two terms on the right-hand side implies that the sea  levelmustincreaseto the north as U 2 is reducedby the  24,982  ALLEN AND HSIEH' ESCARPMENT EFFECT ON EL NI•TO COASTAL CURRENT 0.7  g(H + h9.)Hrlt g'h9. Hrl• h•.f f(H + h•.)  +2f(H+h=)'  ,  ,  ,  ,  (19)  to orderg'/g with r/and V giYenby (5) and(8), respectively. The outõoinõlower-layerflux is  oo+ (20)  ß0000 .0000 .20 .20•00  •o order(g'/g)X/•',with.and u givenby (5) and(?), respectively.  The lower-layerincomingvolumeflux (18) mustequal the outgoingvolumeflux (20) and similarlyin the upper layer (equations(17) and (19)). Usingthe same  -0.9-' -i  o.o Ah notation as in the previoussection,equatingthe fluxes h and solvingfor r/• betweenthe two resultingequations Figure 6. Contourplot of the amplitudeloss((rh. gives  r/•)/•/,•), scaledup by h,./H for case2, flowovera steep depthincrease,assuming H/h2 smalland H/h << [1rl•/H[ •' (equation(23b)). Contoursare in increments of 0.1.  (21)  m and rho= -75 m (Figure7a) and ,•,o= +75 m (Figure 7b). The amplitudelossis generallymuchhigher  In the linear limit, rho• 0, (21) gives  than for case 1. Also in contrast to case 1, the ampli-  tude losseventuallyrisesas H increases:  + + H) H))  shows  that the amplitude lossincreaseslinearly with H when  ,  rl•/H <<0, in contrast to case1 (16b),wheretheampli-  in agreement with Wajsowicz [1991], showing thatthere is finite attenuation even as a linear disturbance crosses  a steepescarpment to deeperwater. In the linearlimit,  (15) showsthat thereis no attenuation asa lineardisturbancecrosses a steepescarpmentto shallowerwater,  tude lossapproachesa constantwhen rh0/H << 0. The approximatesolution(dashedcurve)generallyoverestimates the amplitude loss. For positive r/,•, as H -• r/w,againthe approximatesolutionapproaches infinity, whilethe exactsolutionstaysfinite (Figure7b).  againin agreement with Wajsowicz [1991]. Assuming smallupper-layer depth,that is, H/h •( 1 Applications andassuming H/h <( 11- •7,,,/HI •', then Let us estimate the effects of the Mendocino escarpment on anomalouscoastalcurrentsduring an E1 Nifio. With h = 4000 m, h•. = 3000 m, H = 150 m, and  r• • n• I h2 h (1 (1-rho/H) HAh - •./2H) ]'  rh0= -75 m, the gainin the amplitude(15) is 0.21%. (23a) If the pycnoclinewereraisedby 75 m insteadof being depressed during an E1Nifio, the amplitudelosswould  (23b) propagatinginto deeperwater (case2), that is, from have been 0.59%. In contrast, if the disturbance were  loss(21) wouldhave with Ah = h•.-h. In (23b),for smallupper-layer depth, 3000to 4000m, the amplitude the decreasein amplitudeis proportionalto the ratio of  been1.0%and1.7%forrh0= -75 m andr/w= +75 m,  the upper-layerdepthto the lower-layer depth(in the sameway as for case1) and is thereforeitselfsmall. Unlike case1, amplitudelossoccursfor both positive and negativerho The relativeamplitudeloss,with the smallfactorH/h2 removed(i.e.,multiplied by h:•/H), is plottedasa functionof Ah/h andrhoH in Figure6. The amplitudeloss,as a functionof the upperlayer  respectively, muchlargerthan the corresponding values in casei. The very smallvaluesfor all the amplitude changesare due to the shallowupper layer compared to the lower layer, leading to a relatively large distance between the interface and the uneven bottom.  If we examine the effectsof a smaller amplitude disturbance, say r/• = -10 m, the amplitude gain is thicknessH, is illustrated for h - 3000 m, h•. - 4000 0.039% in case1 versusan amplitudelossof 1.16% in  ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTALCURRENT '  '  i  :  Exact  m, the amplitudeloss(about40%) is muchhigherthan for case1. As •/w decreases, the percentageamplitude  solution  --x--approx.  lO  24,983  lossfor case2 increases slightly,in contrastto case1, wherethe percentagelossdropssteeply.Theseresults comparewell with the numericalmodelresultsof Wajsotoicz[1991]. Unlikefor the E1Nifio-generatedcoastalcurrent,an escarpmentresultsin a substantialamplitudelossfor a  solution  bottom-water current, as the interfaceis located much closerto the unevenbottom. Figure 9 illustrateshow the amplitudelossis affectedby varyingthe distance betweenthe interfaceandtheoceanbottom,whilekeepo.1  80 100  ing the meanoceandepthconstantfor (a) r/w- -100 1000 m and (b) r/w - 9-100m. In general,as the interface  3•0  is placedfarther from the oceanbottom, the amplitude loss(or gain)decreases. A minorexceptionoccurs whenthe interfaceis locatedveryfar from the bottom, sothat the upperlayerdepthH is smallenoughto start approachingthe interfacedisplacementr/w,therebyin-  Upper layer thickness H (m) 3  .  ß  ß  I  :  .....  Exact  solution  approx.  solution  creasing the amplitudeloss(case1, Figures9b and5). Conclusions  A nonlinear,two-layer,f plane shallow-watermodel was used in this study. In the real world the baroclinic =.  3  Rossbyradiuschangessubstantiallywith latitude, and togetherwith the presenceof continentalshelftopography leadsto substantialchanges in the coastaltrapped wavesolutionwith latitude [Brink, 1982]. Theseand  b) ß  1  ß  ,  I  i  i  i  i  3•0  80100  Upper layer thickness  i  1000 H (m)  the beta effect were not included in our model.  Fromour theory,the amplitudechanges in the region of the Mendocinoescarpmentsufferedby the steady  Figure ?. Comparisonbetweenthe approximatesolu- coastalcurrentafter the passageof the Kelvin wavebore tion (dashedcurve)and the full solution(solidcurve) duringan E1 Nifio wasfoundto be surprisinglysmall. for case 2, flow over a steep depth increase where the  incidentamplituder/• is (a)-75 m and (b) +75 m. Amplitudeloss(in percent)is plottedversusupper-layer thicknessH. Other parametersare given in the text.  lOO  !  i  i  i  i  i  i  i i  i  i  case2; for r/w - +10 m, the reductionin amplitude is 0.045% in case1 and 1.23% in case2, that is, the effect is now much larõer in case2 than in case 1 by a factor of almost 30. This chanõeis a reflectionof the fact that unlike case1 the percentaõeamplitude lossfor case2 is  nonzeroevenwhenr/w-->0 (22). The  exact  solutions  found  for  cases 1 and  2 are  also valid for bottom water flowinõ over escarpments. Bottom waters formed at hiõh latitudes flow equatorward as westernboundary currents[Storereeland  Arons,1960].The presence of escarpments andcanyons should greatly enhance the spreading of the bottom water away from the western boundaries, and in the Southern Ocean, away from Antarctica. In Figure 8 the amplitude loss for case 1 is plotted as a function of the incident interfaceamplitude r/w, with H - 3500  :  0.1 I 0  '  ' 30  "  Case  1  - -. -- Case  2  ' • ' ''! 100  '  ' 300  Incident amplitudes! w(m)  Figure 8. Amplitudeloss(in percent)in a bottomwater coastal current flowing over a steep depth de-  crease(case1, solid curve)and flowingover a steep depthincrease(case2, dashedcurve)as a functionof  its incidentamplitude. The upper-layerdepthH is 3500 m, h = 500 m, and h2 = 1000 m. For case2 (dashed m, and the averagelowerdepth,(h 9-h2)/2, is only 750 curve),with H - 3500 m, h - 1000m, and h2 -- 500 m. The escarpmentis 500 m high.  24,984  ALLENAND HSIEH:ESCARPMENT EFFECTON EL NI•IO COASTALCURRENT  •  60  :  ',,  Case  plitude approachedO, againin contrastto case1. Anotherinterestingcontrastis that in case2, both positive andnegativepycnocline displacements werereducedin magnitudeafter passingthe escarpment to the north, whereasin case1 the negativepycnoclinedisplacements (warmcurrents)wereslightlyintensified, whilethe positive pycnocline displacements (coldcurrents)werere-  1  --.--Case 2  40  20  duced.  Acknowledgments. We wish to thank Joe Tam for assistancewith the contour plots. This researchwas supported by grants from the Natural Sciencesand Engineering Re-  -20 .  'a) -40 ''''!  search Council  ....  0  • ....  • ....  1000  Distance  between  • ....  ! ....  2000  interface  I ....  '-.  4000  bottom  -'  •' ' -.  Allen, S. E., Rossby adjustment over a slope, Ph.D. thesis, 206 pp., Cambridge Univ., Cambridge, England, 1988. Allen, S. E., Rossby adjustment over a slope in a homogeneous fluid, J. Phys. Oceanogr.,•6, 1646-1654, 1996. Bennett, J. R., A theory of large-amplitude Kelvin waves, J. Phys. Oceanogr., 3, 57-60, 1973. Brink, K. H., A comparison of long coastal trapped wave theory with observationsoff Peru, J. Phys. Oceanogr.,  Case 1  '-......  '0  897-913, 1982. Enfield, D. B., and J. S. Allen, On the structure and dynamics of monthly mean sea level anomalies along the Pacific coast of North and South America, J. Phys. Oceanogr., /0, 557-578, 1980. Gill, A. E., M. K. Davey, E. R. Johnson, and P. F. Linden, Rossbyadjustment over a step, J. Mar. Res., 44, 713-738,  "  I  References  (m)  - -. - - Case 2  • 10  E  • ....  3000  and ocean  of Canada.  •  <  1986.  b)  , , , , ! t • • , ! ....  0.1  0  1000  Distance  between  I , ,,  , I , , , , ! ....  2000  interface  i ....  I , , , ,  3000  and ocean  4000  bottom  (m)  F]õure 9. The amplitudeloss(in percent)plottedasa function  of the distance between the interface and the  Huyer, A., and R. L. Smith, The signature of E1 Nifio off Oregon 1982-1983, J. Geoph•ls.Res., 90, 7133-7142, 1985. Johnson, E. R., Topographic waves and the evolution of coastal currents, J. Fluid Mech., 160, 499-509, 1985. Johnson, E. R., The low-frequency scattering of Kelvin waves by stepped topography, J. Fluid Mech., •15, 2344, 1990.  averageoceanbottom,for case1 (steepdepthdecrease) Johnson, E. R., The low-frequency scattering of Kelvin m and (b) •7,• - +100 m. The depth changeat the  waves by continuous topography, J. Fluid Mech., 173-201, 1993.  escarpmentis 500 m, the mean oceandepth is 4000 m, Johnson, E. R., and M. K. Davey, Free-surface adjustment and the interface displacementis 100 m at the coastal and topographic wavesin coastalcurrents, J. Fluid Mech., wall. Negativevaluesof amplitudelossimply an ampli•19, 273-289, 1990. tude gain. Killworth, P. D., How much of a baroclinic coastal Kelvin wave gets over a ridge?, J. Phys. Oceanogr.,19, 321-341, 1989a.  Killworth, P. D., Transmission of a two-layer coastal Kelvin wave over a ridge., J. Phys. Oceanogr., 19, 1131-1148,  Even more surprisingly,there was a very slight ampli1989b. tude gain, instead of a loss! The small changeresulted LeBlond, P. H., and L. A. Mysak, Waves in the Ocean,602 from the large spatial separationbetweenthe interface pp., Elsevier Sci., New York, 1979. and the uneven ocean bottom. In contrast, when ap- Longuet-Higgins, M. S., On the trapping of waves along a discontinuity of depth in a rotating ocean, J. Fluid Mech., plying our theory to bottom water traveling overan es31,417-434, 1968a. carpment, the amplitude changewas substantial,as the  Longuet-Higgins, M. S., Double Kelvin waveswith continuous depth profiles, J. Fluid Mech., 34, 49-80, 1968b. For Kelvin wave bores descendingan escarpment Pares-Sierra, A., and J. J. O'Brien, The seasonal and the (case2), the amplitudelosses for the steadyflowwere interannual variability of the California current system: A numerical model, J. Geophys.Res., 9•, 3159-3180, 1989. generally much higher than for bores ascendingan esPhilander, S., El Nifio, La Ni•a, and the Southern Oscillacarpment(case1). In case2, no longoffshoredouble tion, 293 pp., Academic, San Diego, Calif., 1990. Kelvin mode was generatedas in case 1, but frictional Rhines, P. B., Slow oscillationsin an ocean of varying depth, effects became very important, hence the much larger I, Abrupt topography, J. Fluid Mech., 37, 161-189, 1969. amplitude loss. In case2 the percentageamplitude loss Stommel, H., and A. B. Arons, On the abyssalcirculation of the world ocean, II, An idealized model of the circulation did not approach 0 even when the incident wave aminterface  was located much closer to the ocean bottom.  ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•TO COASTAL CURRENT pattern and amplitude in oceanic basins, Deep Sea Res., 6, 217-233, 1960.  24,985  presence of a step escarpment, J. Mar. Res., 53, 49-77, 1995.  Wajsowicz,R. C., On stratified flow over a ridge intersecting Wood, I. R., and J. E. Simpson,Jumps in layered miscible coastlines, J. Phys. Oceanogr., 21, 1407-1437, 1991. fluids, J. Fluid Mech., 140, 329-342, 1984. Wallace, J., and D. Gutzler, Telecormectionsin the geopotential height fields during the northern hemispherewinS. E. Allen and W. W. Hsieh, Oceanography, Deter, Mon. Weather Rev., 109, 784-812, 1981. partment of Earth and Ocean Sciences, University of Willmort, A. J., Forced double Kelvin waves in a stratified British Columbia, 6270 University Boulevard, Vancou-  ocean, J. Mar. Res., •42, 319-358, 1984. Willmort, A. J., and R. H. J. Grimshaw, The evolution of coastal currents over a wedge-shapedescarpment, Geophys. Astrophys. Fluid Dyn., 57, 19-48, 1991. Willmort, A. J., and E. R. Johnson, On geostrophic adjustment of a two-layer, uniformly rotating fluid in the  ver, B.C., Canada, V6T 1Z4. (e-maih allen@eos.ubc.ca; hsieh@eos.ubc.ca) (ReceivedFebruary16, 1996; revisedApril 18, 1997; acceptedMay 29, 1997.)  


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