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Influence of canyons on shelf currents: a theoretical study. Chen, Xiaoyang; Allen, Susan E. Aug 31, 1996

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. C8, PAGES 18,043-18,059,AUGUST 15, 1996  The influence of canyonson shelf currents: A theoretical study Xiaoyang Chen and SusanE. Allen Departmentof Earthand OceanSciences,Universityof BritishColumbia,Vancouver,Canada  Abstract. The influenceof submarine canyonson shelfcurrentsis studiedusingtheRossby adjustmentmethodfor a homogeneous, inviscidfluid on anf plane. The canyonin the model  is assumed to havevertical edges andconstant width.Thegeostrophic flowaround a canyon is foundto be dependentupontwo geometricparameters: the ratio of the depthof the canyon  to thedepthof theshelfandtheratioof thewidthOfthecanyon to theRossby radiusover the canyon. Moreover,a singleparameterdeterminesmostof the propertiesof the geostrophicstate. This parameteris called the canyonnumberand is a combinationof the  twobasicgeometric parameters. In thegeostrophiC .state aninfinitely longflat-bottom canyon Willactasa complete barrierto anapproaching shelfflow. Theapproaching flowis asymmetricallydivertedalong the canyon,and a net flux is generatedto the left of the flow  in thenorthernhemisphere. If thecanyoncutsa shelfbetweentheShelf,break andthecoast and connectsto a atrait(the geometryof Juande FucaCanyon)an inrcanyon(out-canyon)  currentwill be generated whenthe shelfbreakcurrentflowskeepingthe shelfat its left (fight)in thenorthern hemisphere. If thecanyonhasa stepped or slopedbottom,the  geostrophi c flowhasa singularity where thesteporslope meets theleftcanyon edge (lookingupcanyon)in the northernhemisphere.How can crossthe Canyonedgethroughthe  singularity, sothecanyonis no longera complete barrierto theapproaching shelfflow. In this case,as above,a net flux is generated to the left of the apprøaching shelfflow. 1. Introduction  Submarinecanyons are one of the main topographicfeatures of the coastal regions of the world's oceans. Observationshave shown that a canyonmay have very impor-  tanteffectsonthelocalcirculation. Manyobservational •Studies in the past focusedon tides or internalwavesand the ef-  canyonflows.CUrrents nearthe surfacearenot'strongly affectedby the submarine canyon;however,•currents at thebottomseemto alignwiththecanyon isoba•hs. :All of theseob-  servational studies support theexistence of coupling between shelf and canyon circulation.This coup!•ingexist• both in  shallowshelf-canyon systems (e.g.,Hudson ShelfValley [Mayer et al., 1982]) and in deep ones (e.g., .Juande Fuca  fectsof theserelatively highfrequency currents onsedimentCanyon [Freeland and Denman, 1982].). distribution andresuspension in canyons [Drakeet al., 1978; Severalefforts have been made to d6nstructtheoretical Gordon and Marshall, 1976; Hotchkiss and Wunsch, 1982;  Inntanet al., 1976;Keller et al., 1973;Keller and Shepard,  modelsto explainthedynamics of 'thecouplingbetweenshelf  andcanyon circulation. Freeland andDenman [1982] proposed  1978; Shepard et al., 1979]. The study of longer-timescale thatthe currentswithina narrowcanyonare.forcedby theun-  (and time mean)currentsin and aroundsubmarine canyons startedin the early 1980'swith Han et al. [1980] andFreeland and Denman [ 1982].  Han et al. [1980] found that the velocities at the bottom near the Hudson Shelf Valley were clearly aligned with the  balanced pressure gradient, whichis supplied by the geostrophicshelf flow just above the top of the canyon.In their theoretical model, the shelfzi:an•oninteraction allows waterto be raisedfrom depthsmuchgreaterthanthatnormally  expected from theclassical wind-driven upwellingmechanism. canyontopography. FreelahdandDenman[i982] andFreeland Their calculationscorrespondreasonablywell with the'trob-  etal.[1984] presented observations overthe continent/il shelf servations.However, their theory neglectedthe transwerSevenearVancOUver Islandthat showeda persistent deflectionof  the summer:coastalflow near a small submarinecanyonin this region. Hickey et al. [1986] describedmultiyear observations of currentsand suspended sedimentsin QuinaultCanyon.Their observations revealed a correlation between flow along the canyon axis and the along-shelfcirculation;the pressuregradient due to the geostrophicallybalancedcoastal flow forces upwelling or downwelling in the deep parts of the canyon.  Hunkins[1988] analyzeda setof approximately yearlongcurrent measurementsnear the head of and in Baltimore Canyon that show the existence of persistent upcanyon and down-  locity andhencetheCorioliseffectwithinthe canyon,limiting its validity to canyonsmuchnarrowerthan the Rossbyradius.  Thegeostrophic adjustment of a stratified coastal Current in the presenceof an infinitelylong, rectangular, flat-bottom  canyon is cons!.dered byKlinck[1988,1989].Hismodelincludesthefeedbackof theupwelleddensewateron thecrossshelfpressure gradient, whichwasnotconsidered in themodel of Freeland and Denman [1982]. In Klinck'smodel, the initial  flowontheshelfis assumed tobegeostrophic andbarotropic with trigonometricdependence in the along-canyon direction. For eachverticalmodethe decayscaleof the perturbation is determined by a scalethatis the shorterof theradiusof deforß  Copyfight1996by the AmericanGeophysical Union. Pagenumber96JC01149. 0148-0227/96/96JC-01149509.00  mation for that mode and the width of the coastal current. The  width of the canyon determinesthe strengthof the crosscanyonflow and thusthe strengthof the canyon'seffecton the 18,043  18,044  CHEN AND ALLEN: INFLUENCEOF CANYONS ON SHELFCURRENTS  shelf  Figure 1. The geometry of the stepped-bottomcanyon model. The canyonhas vertical walls and is infinitely long. The shelf is flat and infinitely wide with depth H 1. The canyonbottomis dividedby a stepinto two portions:a shallow upper canyonportion with depth H2 and a deeplower canyonportionwith depth H 3. Here x is in the cross-canyon direction,and y is alongthe centralaxis of the canyon. The stepis at y=d.  overlying coastal current, with the interaction becoming smalleras the canyonwidth becomessmallerthanthe current width or the radius of deformation.  Even in the case of  flow  over a narrow canyon,the isopycnalsat the top of the canyon are distorted, and there is also some residual circulation on the  shelf forced by the presenceof the canyon. It is tempting to seek a steadystate for the interactionbetween shelf and canyon circulation. One of the possibleapproachesis to use Rossbyadjustmentto derivethe steadystate solutionand avoid the relatively complicated,transientinitial value problem.One purposeof our studyis to applythis approach to investigate the basic dynamics in the interactive processof a shelfflow and a canyon.For a reviewof the open oceanadjustmentproblem,see Gill [1982, pp. 191-203].Gill et al. [1986] applied the Rossbyadjustmentmethodto study the topographicproblemand consideredthe problemof how a barotropicflow is modifiedwhen it passesover a stepliketopography,using linear analysisand numericaland laboratory  experiments.Extending the work of Gill et al. [1986], Allen [1988, 1996a] studiedRossbyadjustmentover a slope. Since the purpose of our research is to reveal the basic propertiesof the shelf-canyoninteractionand the effect of the canyon shapeon the circulation,as a first step this paper will considera homogeneous, inviscidfluid on anf plane.As will be demonstratedlater, a rotatingfluid, not initially in equilibrium, adjusts around a canyon to a final geostrophicstate throughlong modified doubleKelvin waveswhich transmitinformationalong the canyon.The geostrophicstatewill be derived by constructingthe solution from these long "canyon waves." The transient,propagatingwave solutionis investigated numerically by Chen [ 1996]. Three typesof canyongeometry,all with verticalwalls and constant width, will be considered.The simplest is an infinitely long flat-bottomcanyoncutting a flat shelf. Someeffects of topographywithin the canyonwill be investigatedusing an infinitely long canyonwith a step dividing it into two sections,one deep and one shallow but both deeperthan the shelf. The geometryof this type of canyonis shownin Figure 1. The third canyongeometryis designedto includethe main featuresof Juande Fuca Canyon. In this casethe canyoncuts througha flat shelf, the mouth of the canyonis at the shelf break, and the head of the canyonconnectswith a strait. The bottomof the canyonconsistsof four segments,with two flat regions and two slopesjoining them. The geometryof this type of canyonis shownin Figure 2. The coordinatesusedin this paper are oriented with x in the across-canyondirection andy along the central axis of the canyon. The governingequationswill be given in next section.The propertiesof the double Kelvin waves that exist in a flat-bottom canyon will be discussedin section 3. To provide the foundationfor this paper, the geostrophiccirculationover a flat-bottomcanyon will be describedin section4. The far-field solution will be found by constructinga solution of long canyon waves. This solution will be demonstrated,in the case of a specified initial conditionin which the fluid is at rest but in which there is a surfacediscontinuity,to be the geostrophic solution for a flat-bottom canyon in the far-field. An important parameter, or, will be defined in the processof solving  land  head slope coast  shelf  mouth slo  shelf break  deep ocean  Figure 2. The geometryof the Juande Fuca Canyonmodel.The bottomof the canyonconsistsof four segments:deepmouthpart, mouth slope,main body,and headslope;the edgesare vertical,and the width is constant. The shelf break,coast,and straitwalls are all vertical. The lengthof the straitis infinite. The shelf is infinitely long and is flat.  CHENAND ALLEN:INFLUENCEOFC,a2qYONS ON SHELFCURRENTS the problem.The flux and the full solutionwill be calculated underthe specialinitial condition. By applyingthe solutions obtained in section 4, discussion of the geostrophic state arounda stepped-bottom canyonwill be presentedin section  18,045  Thus geostrophic currents,us, must be parallelto the canyon walls, the canyon bottom step or slope, and the shelf break sinceat theselocationsVH • 0. The combination of requir-  ing no flow across the canyon edges and along the canyon over the canyon bottom slope implies that there is no flow tion arounda canyonwill be studiedin section6. A discussion over the canyon bottom slopes. Geostrophicflows cannot will be given in section7, and the conclusionswill be pre- cross these changes in depth (except at their intersections). sented in section 8. This requirement meansthat r/s is a constantalongthe edgeof eachpiecewisesegmentof the domain.We will usethis property later to determinethe boundaryconditionsfor (4).  5. Effects of the shelf break, coast, and strait on the circula-  2. Governing Equations  In a homogeneous, inviscidfluid on anf plane the governing equationsfor smalldisturbances are the linear shallowwa-  3. Canyon Waves  ter equations, (la)  Insteadof solvingthe completetime dependentproblem(5) in this section, we present an analysis of the properties of canyon waves and of the structure of long canyon waves. Since the purpose of this paper is to present the geostrophic  (lb)  circulation, i.e., the solution of (4), the transient solution of  (lc)  (3) will be deferredto a later paper. Consideran infinitely long canyonwith vertical walls and a constantdepth H2; the width of the canyonis a constant2L, andthe depthof the shelfis a constantH 1 ( H1 • H 2 ). Assume that the solutionof (5) takes a wavelike form  wheref is the Coriolis parameter,g is the accelerationdue to (possiblyreduced) gravity, H the undisturbeddepth of the fluid, and u=(u,v) the horizontalvelocity of the fluid. The coordinates  are defined in section 1.  The depthin all our canyonmodelsis constantexceptat the canyonedges,the shelf break, and the canyonslopes. In all  •lw(X,y,t)= E(x)exp[i(ky- (ot)],  where E(x) is a functionof x, (o>0 is the frequency,and k is the wave number in the along-canyondirection. Substituting (6) into the momentumequations(la) and (lb) gives  other regionsthe gradientof the topography,VH, is zero.  uw(x,y,t)=ig[f2_(o2 exp[i(ky(ot)],(7)  Therefore introduce an initial quantity H  QI(x,Y) ='•-•'I(Y)rll(Y),  (2)  vw(x,y,t) = f2_002exp[i(kyrot)], (8)  where•' = o•v! •x- o•u! o•yis therelativevorficityandthesubscriptI denotesa variableat the initial time t=0 and wherewe have assumed the initial conditions are independentof x. Manipulating (1) gives an equationfor r/alone in terms of  (6)  where E'(x) is the fu'st derivativeof E. Substituting(6) into (5) gives  this initial quantity  d 2Ea/2E=0, i=l,2, dx2  (9)  oti 2((o,k) =f2 - (o2+k2 gHii=1,2.  (1O)  where  7•-•--R2V 2+1rl(x,y,t)=Ql(X,y ), (3) where R=(gH)l/2/fisthebarotropic Rossby radius ofdeformation.  Becausewe are looking for waves trappedto the canyon,the  Solution of this initial value problem (3) can be found by addinga particularsolutionof it, which is the steadysolution r/s(x, y), given by  parameter a must bereal,i.e.,a• > 0.  R2V2rls(X,y) - rls(X,y)=Ql(X,y )  (4)  to the solution of the homogeneousequation, which is the transientwave solution rlw(x,y,t), given by  c)t. 2  +1rlw(X,y,t)=O, (5)  and with the initial condition rlw(X,y,O)=-Ql(x,y)= -rls(X,y). Equation(4) can be solvedonly afterthe valuesof r/s at the edges of the piecewise composeddomain have been determined; theseare the boundaryconditionsfor (4). A property of the boundaryconditionsfor (4) can be obtained by combining(1) and assuminga steadystate,  us .VH= 0.  The boundedsolutionof (9) has the form  Ai exp(a•x), x<-L,  E= A2exp(a2x)+B 2exp(-a2x), x<lL[, B• exp(-alX ),  (11)  x > L,  whereA•, A2, B2 andB3 are all nonzeroconstants. Substituting the form (11) into the equations for r/w anduw [(6) and(7)], therequirement of continuityof r/w and Huw at the canyon edges gives  exp(-a 1L)A1- exp(-a 2L)A2 - exp(a 2L)B2 = 0,  (12a)  H1(a'l(o-kf)exp(-o:1L)A 1- H2(o•2(o-kf)exp(-ot2L)A 2 +H2{a2(o +kf)exp(a2Z)B 2= O, exp(a2 L )A2 + exp(-a2 L )B2 - exp(-a• L )B3 =0,  (12b) (12c)  •: (•: •o- •) •xp(•:•)•: - •: (•: •o+•r)•xp(-•: •)•: +H1(a'l(o+kf)exp(-oqL)B 3= O,  (12d)  18,046  CHEN AND ALLEN: INFLI•NCE OF CANYONS ON SHELFCURRENTS  2. For long waves( k << 1/R 2, andso to<< f), (13) gives  which can be written in matrix form as  to  A•  Iy2-11  Co ='•'=cl(72+2ycoth•+l ' )it2 ' (14)  M B2=0.  where c1=(gH 1)112 isthelong-wave phase speed ontheshelf.  Fora nontrivial solution of (12),thedeterminant ]M]must be zero,whichyieldsthe dispersion relationfor canyonwaves  The parameterCo is the group speedand phasespeedof the long canyon waves. For the infinitely wide canyon limit  (fi-->oo), Co=(gH2)ll2-(gH1)ll2 which isidentical tothe doubleKelvin wavephasespeedfor a singlestepfoundby Gill et al. [1986].  Poincarewaves(first classwaves) are the only wavespossible in a barttropic, flat-bottomoceanon anf planefar from (13) lateral boundaries.These waves establishthe classicRossby adjustment [Gill, 1982], which will be the solution far from the canyon in our infinitely long canyon models. Note that whereot1 and tz2 are functionsof to andk as given by (10), the final, steady state will differ from the initial condition and coth( ) is the hyperboliccotangentfunction. The phaseand the groupspeedof canyonwavescan be ob- only in a narrow (one Rossbyradius wide) region aroundthe tained from the dispersionrelation (13). Both the phaseand original change in surface elevation. Farther from the initial the groupspeedare functionsof the wave number,so canyon disturbancethe propagatingPoincarewaveswill carryenergy waves are dispersive.Becausethe dispersionrelationis sym- but no surfaceheight changes. The presence of a change in depth allows second-class metric aboutthe to axis, the phase and energy (as well as information) of canyon waves propagate in both directions waves (potential vorticity waves [see Rhines, 1969]), of which canyonwavesare an example.Thesewavestravelalong along the canyon. The dispersion relation for canyon waves with •= a depth changeand can carry changesof surfaceelevationover 2L/R2=1 and72=H2/H 1=2, 3, 4, and5 is givenin infinitely long distances. Permanentchangesover long disFigure3, for k > 0. Obviously, for canyonwaves,the shorter tances, which determine the steady state, are controlledby the wave length is (comparedwith the Rossbyradiusover the long waves. Long Poincarewaves have zero groupvelocity, canyon), the smaller the group speedis (the derivativeof the but long double Kelvin waves have finite group velocity as dispersionrelation). The limiting casesare as follows: shownin (14) for the exampleof long canyonwaves. Canyon waves are dispersive in general; however, long 1. For shortwaves(k >> 1/R2), the groupspeedapproaches zero. canyon waves are nondispersive,and the group speed ap-  /n•j cøth(2Lø•2  D i specs  • on  Re ! a t: i on  for  Canyon  Waves  0.8  y2=4 Y•--3  y2=2  0.0 0  2  4  WRVE NUMBER  6  RLONG  CRNYON  lB  10  12  DIRECTION  Figure 3. Dispersion relation ofcanyon waves forthefiat-bottom canyon of fi = 1, y2= 2, 3,4, and5.The horizontal axis is the nondimensionalwave numberalong the canyon, kR2, and the vertical axis is the nondimensionalfrequency to/ f.  CHEN AND ALLEN: INFLUENCEOF CANYONSON SHELFCURRE•S  18,047  proachesa maximumvaluethatis determined by the geometric where A, B, C, and D are functions of two variables and are to parametersof the system.The group speedof shortcanyon be determined.An advantageof assumingthat (1) has a soluwavesapproaches zero. Thesepropertiesof canyonwaveswill tion of the form (19) is that be used in the next section to determine the far-field solution  A(y,0) = B(y, 0) = C(y,0) = D(y,0)= 0.  for a flat-bottom canyon.  Substituting(19) into (18a) gives  4. GeostrophicCirculation Over a Flat-Bottom  v(x,y,t)=-g Aexp[(x+L)/R 1],x<-L,  Canyon 4.1.  General  Far-field  Solution  The far-field solution for a flat-bottom canyon will be obtained in this section by constructingthe solution as long canyonwaves.The Poincarewavesare very much in presence but are not explicitly calculatedbecausethey are not important to the far-field  (20)  g 1{-Bexp[-(x+L)/ v(x,y,')  solution, as was noted in the last section.  Using potentialvorticity, the solutioncan be calculatedwithout a full calculation of the time variation. This method is the  standardtechniqueof Rossbyadjustment[Gill, 1982; Gill et  al., 1986]. In thefarfield(lyl>>•2)  th•surface isini-  (21a)  +Cexp[(x-L)/R2] }, x<l/•,  {2lb)  v(x, y,t)= g-D-D exp[-(xL)/R1],x>L.  (21c)  Then substituting(19) and (21) into (18b) gives  u(x,y,t)=-y rl•y- +  exp[(x+L)/R•], x<-L,  tially flat (i.e., r/l =const) assumethat the solutionof (1) takes the form  rl(x,y,t)=rll(y)-E(x)exp[i(ky-wt)],  (15)  whereE(x) hastheformof (11) with a 1= 1/R 1 and a 2 = 1/R 2 for long canyon waves. The corresponding u(x,y,t) and v(x,y,t) are given by (7) and (8), respectively.Now using (15), (7), and (8) to examinethe magnitudeof eachterm in the momentumequations,(la) and (lb), for long canyon waves  ( k << 1/ R2 and ro= cok), we have  f2  '  lYvl --glE'l,  igal __ gig' I, Ifulglcøg'f  where the subscriptt deno•s a derivativewith respectto •e and •e subscripty denotesa defivaavewi• res•ct to y. Using the bound•y conditionsthat the surface elevation and the cross-canyonflux •e continuousat the edgesof •e canyon, we have  (16a) (16b)  Atx=-L  ( (17a) (7b>  A = B + Cexp(•),  (22a)  1[-Bt +Ct exp(-•)]}, (22b)  (17c) H2 -'•2 {rlly-Byexp(-fl)-Cy 1[-Btexp(-•)+Ct]} Atx=L  Bexp(-•) + C = D,  (22c)  The first term of (la), i.e., (16a), is much smaller than the second and the third ones, i.e., (16b) and (16c), so the first term  is negligiblein (la); all termsof (lb), i.e., (17a), (17b) and (17c), are small but of the sameorder, so all termsmust be consideredin (lb). For a solutionof (1) having the form (15), the momentumequations(la) and(lb) reduceto  where•/, as definedin lastsection,is thewidthof thecanyon  -fv+gxx=0,  t•-•-+fu +g-•=0.  (22d)  made nondimensional by the Rossbyradiusover the canyon. If we assumethat the surfaceheightis rlL(y,t) at the canyon edge x=L and is rl_L(Y,t) at the canyonedge x=-L, we have r/_L = r/z- A and r/L = r/l - D. By combining with (22a) and (22c), (19) can alsobe expressedin anotherform, in termsof r/œand  (18b)  Using(11) for E(x), we write (15) in anotherform,  rl(x,y,t)=rli-A(y,t)exp[(x+L)/R•], x<-L, (19a) rl(x,y,t)= rll- B(y,t)exp[-(x +L)/ R2] +C(y,t)exp[(x-L)/R2], <lml, (19b) rl(x,y,t)=rll-D(y,t)exp[-(x-L)/R1], x> L, (19c)  rl(x,y,t)=rll+(rl_L-rll)exp[(x+ L)/R1], x<-L, (23a)  rl(x,y,t)= rll+•1{(rlL +rl-L-2r/l} cosh(L/R2) cosh(x/R 2)  18,048  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELF CURRE•S  dition will lead to a geostrophicstate [Gill, 1982, pp. 191-  sinh(x /R2) } x<ILl, (23b)  +(riL - ri-L) s'•'Z••22 )''  203]  ri(x,y,t) = ril+(rtL- rt•)exp[-(xL)!R1], x>L. (23c) This form of solution will be used later to analyze the geostrophiccirculationaroundthe other two canyonmodels. Now we havefour equations(22) for four unknowns A, B, C, and D. Forf>0 the procedureto solve the systemis given in the appendix. The functionsA andD are  (27a) ri(Y) =-ri0 sgn(y)[1-•xp(-lyl/•)], (27b)  v=0,  (27c)  whichrepresents a geostrophic jet with a widthof 2R1 anda coreat y=O. Given a flat shelf(no shelfbreak),far awayfromthecanyon  2W p(y-cot), (24a) (Ixl>> A(y,t)= ril(y)-[1-•]Wn (Y+Cot)Cr  + •),  adjustment will be thesameasthatin a  flat-bottom ocean, and the geostrophicstate is describedby  D(y,t)= ril(y)--•Wn(Y+Cot)1-•- p(Y-C0t ).(24b)(27).  Far awayfrom the coreof thisjet, initially,r/is -ri0  for y>> R1 and ri0 for y<<-Rl. This surfaceelevationinformationwill have been transmittedalong the canyonby the  In (24)  longcanyon waves provided thatt >>1/Ifl. It isrepresented  I7(coshfl-1)+ sinhfl ]112(25)  in (24) by Wp(y+cot)andWn(Y-Cot).Soforthespecific  =1- r(cosh/ +1)+ sinh'/ ,  initial condition(26), (24) reducesto (forf>O)  -or,y>0, D(y,t)= rio-(2or, y <O. or), y> 0,  where7 and fl weredefinedin section3. The ratio 7 is > 1  A(y,t)= rio(2-or),y<0,  for canyons.The canyonnumber cr• {0,1} (for fl • {0,o•}) is  (28a) (28b)  determinedby the geometryof the systemand is an important parameter in describing the geostrophic circulation of a canyon-shelf system. The wider the canyon is (hence the In the discussion above,(28) was obtainedby considering weaker the effects of one canyon edge on the circulationat the other edge), the smaller the canyonnumberis. It represents the physicalmeaningof Wp(Y+Cot ) and Wn(Y-Cot ). the interactivestrengthof one edgeof the canyonon the circu- However,(28) canalsobe obtainedby usingthepurelymathematicalmethodgiven below. lation inducedby the other edge.  Thequantities Wn(Y+Cot) andWp(Y-Cot)in (24) represent the information, which is the surface elevation in this  case, transmitted into the study region by the long canyon waves from the positive y direction and negativey direction, respectively.The canyonwave that is inducedby and intensified on the canyon edge at x=-L carries the information  W•(Y+Cot)andtravelsfromthepositive endof thecanyon into the studyregion, whereasthe canyonwave that is induced by and intensifiedon the canyonedge at x=L carriesthe information Wn(y-cot) andtravelsfromthenegativeendof the canyoninto the studyregion. Togetherthey determinethe final geostrophicstate. We will use the physical meaningsof  By usingcondition (20), (24) yieldsWn(Z)=W•(z)= ril(Z) wherez is a free variable. So (24) canbe writtenas  A(y,t)=ril(y )- 1-• ril(Y+Cot )- ril(Y-Cot), (29a)  D(y,t)= ri1(y)--•-ri1(y+cot )- 1-• ri1(y-cot). (29b) For the initial condition(26), in front of the wave front of the  longwave(lyl> cot),(29)gives A(y,t)=0andD(y,t)---O, sori remainsril(Y) andu=v=0;i.e., the statehasnotbeenadjusted;  behind thewavefrontof thelongwave(lYl<<cot,whichis equivalent to t>>/Ifl lyl>> > =co/Ifl),(29)  W•(y+ Cot)andWn(y- Cot)to studythegeostrophic circula- givesthe sameresultsas (28). The statehasbeenadjusted by  tion for all our canyonmodels. Iff<O, A will equal the right-handsideof (24b) andD will equal the fight-handside of (24a). Thereforeflow patternsin the southernhemisphereare the reverseof thosein the northem hemisphere.The following discussionwill be limited to the northernhemisphere(f>0). Substituting (24) into (22a) and (22c), B and C can be found. Then substitutingA, B, C, andD into (19) givesa solu-  tionof (1) expressed in terms of Wp(y+cot)andWn(Y-Cot). As will be demonstratedin an example in the next subsection, this solution is the far-field geostrophicsolution. 4.2.  Far-field  Solution  for  an  Initial  Surface  In the precedingdiscussionthe initial conditionswere quite general. To simplify further discussion,considerthe simple, initial  ri(x,y,t)= -ri0sgn(y) + ri0[sgn(y) +(1- cr)sga(x)]  I1>L,  (30)  cosh(x !R2)  ri(x,y,t) =-rl0 sgn(y)+ ri0sgn(y) cosh(fl /2) sinh (x_./..R_2..).]  +(l-a)sinh(fl/2) J'Ixl<, u(x,y,t)=O,  (30b) (30c)  v(x,y,t)=-I•ø1ø ]sgn(x)[sgn(y)+ (1-o')sgn(x)]  Discontinuity  classical  the long waves and  xxp[<L-Ixl>/g], Ixl>L,  (30d)  condition  ril (Y)=-ri0 sgn(y),  (26a)  ul =v• =0.  (26b)  For a flat-bottomopenoceanwith depthH1, this initial con-  v(x,y,t) =  sgn(y) cosh(• /2)  cosh(x /R2)]  +Oa)  'J'Ix[ <œ.  (30e)  CHEN AND ALLEN: INFLUENCEOF CANYONS ON SHELFCURRENTS 8  Equations (30) describea statethatis independent of t and the value of y and that is valid only underthe conditions  lyl>>  t>>1/Ifl,  18,049  it isthefarfieldgeostrophic so-  6  lution. If t is finite, the transienteffects due to shortercanyon  waveswill be present.It is obviousthat the effectsof the canyonon thegeostrophic shelfcirculation in thefar-fielddecayexponentially withthedistance fromthenearer edgeof the canyon. Thewidthof theaffected regionis approximately R1  ;  '",  ....... '..r ........... .-.-  ',,.>,,.;;;:; ....... i............  ============================ .............. ..:,. ......... :::::::::::::::::::::::::::  The solution(30), for two examples,is givenoutsidethe dashedlinesin Figure4. Thisfigureshowsthecontours of surwithin the dashedlines, where the flow turns near the canyon  ';:, •,  ................ ................. iil  on the shelf.  face elevation, which also form the streamlines. Details  ! •i•  .--.............. !.................... /iil  =, -2 I  edges,will becalculated in section 4.3. However, evenfor the far-field circulation,the picturethat emergesis mostintriguing. First, in the geostrophic statethe canyonactsas a com-  pletebarrierto the approaching jet, whichis completely deflectedalongthecanyon.Second,asa jet approaches a canyon in thenorthern hemisphere, mostof it is deflected to thefight.  -6  -8  i '"• i  -4  -2 CR055  0  i 2  i  i  4  6  8  CRNYON DIRECTION  25  The canyoninducesa flux alongit thatis of greatinterest. Integrating(27b) with respectto y for y•{-oo,•} givesthe flux approaching the canyon(in the +x direction)  F•=H• dy= 2gill//ø . f  29  15  (31)  Integrating (30d) and(30e) with respectto x for x givesthenetflux in the +y, along-canyon, direction  +H2  i  -6  X --  turned 180ø aroundthe Z axis.  Fy=H•  i  -8  Note also that the solutionis symmetricaboutthe origin. If the initial conditionis changedso that the geostrophic flow directionis reversed,the flow patternwill be thatof Figure4  z  +H• dx=Fx(r2-1)(1-(r).(32)  If f>0, thenFx>0, andthenFy>0. In thegeostrophic statefor a shelf-canyon system,thenetflux transported in the along-canyon directionis proportional to the flux approachingthecanyonandis to theleft of theapproaching flux;if 7  -1õ  -29  is a constant,the wider the canyon(the smaller0), the larger  isFy.The maximum transport isFx(72- 1)inthelimit ofinfinitelywidecanyon( • --• ooandhence(r --• 0).  -25  -25  -20  -15  -10 X --  4.3.  Full  Solution  for  an  Initial  CRESS CANYeN DIRECTIeN  Surface  Figure 4. Contoursof surfaceelevation// which also form the streamlinesfor the flat-bottomcanyon.Thick lines repreIn section4.2 the analyticfar-field solutionwas obtained. sentthe positionof the canyonedges.Dotted lines represent // values.Arrowsrepresent thedirection of theflow To completethe geostrophic solutionof (4), we will usethe negative far-field solutionas the boundaryconditionsto calculatethe in the northernhemisphere.The far-field solutionappliesoutsolutionin the regionswherethe streamlines turn,insidethe side the dashedlines. The rangeof valuescontouredis -//0 to •/0, andthe contourintervalis 0.2 t10. The positionof the dashedlines in Figure4. In theseregions,x andy are not very initial surfacediscontinuityis at y=O. The lengthunit is R2, largecomparedwith R1 and R2. 72= 2 and]•= (a)2 and(b)30. Discontinuity  For the initial condition (26), the surface discontinuity (equation(2)) becomes  Ql(x, y)=-rh(y). The solution of (4) should approach(30a) and (30b) as  The partialdifferentialequation(4) will be solvedf•rst insideandthenoutsidethe canyonbutonlyin theregionx <-L (from which the solutionin the region x > L can be easilyde-  rived owing to the symmetryof the solution). Writing the solutionof (4) as //2(x,y) inside the canyon //at thetwoedgesof thecanyonas -(1-ct)//0 at x=-L and  lyl •-Equations (30a)and(30b)givethefar-field values of  (4) with (.•,•)=(x/R2,y/R2) (1-•y)//0 at x= L, respectively. As was statedin section2, and nondimensionalizing for a homogeneous, inviscid,linearfluid no flow cancrossthe •2 =//2///0 gives  edgesof thecanyonin the steadystate,so // will be uniform alongeither edge of the canyon,i.e., tl_L =-(1-•y)//0 at •x=-L and//L =(1- •Y)//0at x= L for all valuesofyo  and  (•,9) //I(9) •2 ø•2•2(•'9) +o32•2 ø• 2 -•2(•,9)=.•o , I•l <• ß(33a)  18,050  CHEN AND ALLEN: INFLUENCEOF CANYONSON SHEI• CURRENIS The boundaryconditionsare  The boundaryconditionsare  +(1a),  •I (-•'->0,•) = -(1-  (33b)  =  •2('•,•"->+•) =+c0Sh(fi/2) -1]+(1-O)sinh(fi/2 ) cosh.• sinh.• (33c)  (37b) (37c)  •1('•,• '->oo)=-1 + aexp(•),  (37d)  •I (•,• -'>-=) = 1- (2- (•)exp(•),  (37e)  where•(•) is givenby (34). The solutionof (37) is  It is easy to demonstratethat a particularsolutionof (33a) is  •I (•,•) = •(•)+ o'exp(•)-exp(•)  •'(•)=-sgn(•)[1exp(-I•l) ].  (34)  (38)  Applyingthe techniqueusedby Gill et al. [1986], we put where  =  %  (35)  into (33). UsingFouriertransforms andexpressed in convolution form the solution  is  sinhi  •2(i,•)=•'(•)+(1-or) sinh(fi /2) (36) where  in which K1 is the modifiedBesselfunctionas definedbY AbramowitZand Stegun[1968].  oo cosh(•(P 2+1) 1/2)  Theshapes of G1(.•,½)versus • and•1(•--•) versus • are quitesimilarto thoseof G2(•,½) andE2(9-½), so (38) can alsobe rapidly evaluatednumerically. Combining(30a), and(30b),(36), and(38) givesthewhole picture of the solution in the steadystate as that shownin Figure 4. It can be seenthat as the canyonbecomeswider  Thetypicalshapes of G2(•,•) versus • (for the examples (fi -.->oo), theflowpattern neareitheredgeof thecanyon is inof i = 0.2, • = 2 andi = 0.5, fi = 2) andE2(Y- •) versus • distinguishable from thatfor a single-step topography derived (for the examplesof •= 0.5 and • =-1) are shownin Figure  5, fromwhichwe seethat G2(•,•) decreases veryquicklyas  I•l increases. So(36)canbeeasily evaluated numerically (itis sufficient totakelel<10inthenumerical evaluation). Following the same procedurebut writing the solutionof (4) as •1•(x,y) in the region x <-L and changingto nondi-  mensionalvariables (•,•)=((x+L)/Ri,y/R•) and •I --•1]•0, (4) becomes  by Gill et al. [1986]. This resultis expected,sincethe effects  of oneedgeof the canyoncannotbe felt at theotheredgeif the canyonbecomesinfinitely wide, For Rossbyadjustmentover a flat-bottomcanyon,the canyon acts as a complete barrier to the approaching geostrophic flow. The flow is diverted in both directions  alongthe canyon,with mostof theflow turningto thefightin the northernhemisphere.Within the canyon,a unidirectional current is generated.A shelf-canyonsystemgeneratesa net flux to the left of the shelfflow in the northernhemisphere,  •2 •2•1('•'•) • ø• 2 -•1(.•,•)=-•, •<0. (37a)  whichdecreases withincreasing canyonnumberfor a constant  depthratio of the system. The resultsmentionedabovewill be reversedin the southernhemisphere. 1.5  I  •G2,•=0.5  1.0  0.5  I  In the oceanthe depthof a canyonis neveruniform.As a first approximationto real situations,assumethat the canyonis composedof two portions:a shallow upper canyonportion with constantdepth H2 and a deep lower portionwith constantdepth H 3 (H 3 > H2). If the length scale of the region  0  -0.5-1.0  5. Geostrophic Solution Overa Stepped-Bottom Canyon  -  -  wherethedepthchanges from H2 to H3 is muchshorterthan the local Rossbyradius,the bottomof the canyoncan be rep0 2 4 resentedby a step as shownin Figure 1. The canyonis assumedto be infinitely long with verticalwalls, the stepis at Figure 5. Shapesof G2(•,•) (for •=0.2, fi=2, and y = d, the widthof the canyonis 2L, andthedepthof the shelf •=0.5, fi =2) andEi(•-•) (for •=-1 and0.5)versus•. is H1. The geostrophicsolutionaroundthis stepped-bottom -1.5  I  I  , I  I  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELF CURRE•  canyonwill be found by applyingthe resultsobtainedin the precedingsection. First, the far-field solutionwill be deter-  In the linear steadystatethe transportacrossa line between two pointsP and Q (say) in a regionof depthH is  mined.  5.1. .•.  18,051  gH  f (•lP-•l(2),  Analytical Far-Field GeostrophicSolution  totheright.Thus in ourcase, writing Z/]x•_• = z/ø•and •lx• = • aty=d,thefluxacross y=d+g, where g isin-  Define the following parameters:  finites•al  ,  is  •H  2L  ,  and •e flux acrossy = d- g is  [3 i=[3.•_1,  •[H1 <"-L3.S.)+ H3<.L._L3) +H,(.•-  ai(ri,0i)= •-[r,(cosh0 '+l}+sinh•i ' (39) •'i (coshfli -1) +sinhill ]1/2 lYi 2-11  •e •u•ement that•ese twofluxes•e •u• at y = d is  (7•2-722)•L+(y22-!)•_[2-(7•2-1)•_[•=0. (41a) In the regions far away from both the core of the geos•ophicflow •d •e step,•ogous m (•), for•0  cøi=Cl i7'i2+2•'icothill+1)•/2 ' e23= (1- 0'2)(1- o'3)(7'32 - 722),  8i= E2 + E3 ei+ •23 ,  (40)  •om consideration of •e regiony<d, and  • =•wv3(y+Co3t)+(1)w•(y-Co•t),(41d)  wherec1 was given in section3, and i=2, 3.  Assumethatthe regionwhere o•r/1//•y,• 0 is a small,lim-  •-L,=(!-•)Wv3(Y+Co3t)+•Wn3(Y-Co ). (41e)  ited region around y =0. Far enoughfrom the canyon bottom  step andtheregion where ar/l/ o9y • O,i.e.,lyl>>la+a2I, the circulation will' havethe samepatternasthatin thefar field for a flat-bottomcanyon.Thus the far-field solutionfor a stepped-  bottomcanyoncan be expressed in the form of (23) in terms  from consideration of •e regiony>d.  Combining(4!), the solutionexpressed in termsof Wv• (y + cmt) •d W,• (y - Co•t)c• befound:  of the unknown rl values at the canyon edges. In the  geostrophi cstate,.aswasdiscussed in section 2, values of surface elevation r/ a,tthe canyonedgesand canyonbottomstep must be constantso that there is no large-scaleflow acrossthe canyon edges or the canyon bottom step. Once these constant. s are found, the far-field geostrophicsolutionis given by (23). The geostrophicstate around a canyon is set up by canyon waves transmitting information in both directions along the canyon, as was discussedin the precedingsection. Assume that the information (surface elevation /•, in our case) trans-  "L3 =(1-•)[(282 WV2 )+(1-2 82 )Wn3 ]+•Wn3, (42c)  ß  whe• •i •d 8i (i = 2, 3) •e givenby (39) and(40),res•c•vely. Equations (42) •e •e generalgeos•ophic f•-field soluffon•ound a step.d-bottomc•yon for a nons•cific initial  condifon.  •en H2=H 5 (so eg=e 3, e23=0, and 82=83=0.5), •-L2 •d •-L3 •e identical. Toge•erwi• •L •ey •e consistentwith •e resultsfor a flat-boSomcanyon. W-b3(y+co3t) andWn3(Y-Co3t ), respectively, in theregion If •e inifi• condi•onis a s•face discontinuity (equations  mittedin the -y andthe+y directions is Wp2(y+co2t)and Wn2(y-co2t),  respectively, in the region y>d,  and  y< d. Thequantities Wn2(Y-Co2t ) andWp3(y+co3t) arethe similarto the last •ection,Wr2(y+c•t) and information carried awayfromthestep,whereas Wl•2(Y+Co2t ) (26)), Wn3(Y-Co3t ) in (42)•e -•0 •d •0, res•ctively. The cot-  andWn3(Y-Co3t) aretheinformation carried fromthepositive and negativeendsof the canyontowardthe step:.Write r/ atx=-L as r/_L2 for y>d and rl_L3for y<d, and rI atx=Las  r/L for y < d. The surfaceheightat the stepmustbe r/L becausethe double Kelvin wave propagatingalong the stepwill carrythe informationr/L awayfrom the x=L edge.Similarly, rI atx=L for y > d is also r/L. Since r/L is generallynot equal to /l-L2 or /l-L3, there must be a singularitywhere the step meetsthex=-L edge.Similarto the casediscussed by Gill et al. [1986] in which double Kelvin waves propagatealong a step toward a wall, there shouldbe flux acrossthe step(and in our case the x=-L edge), through the singularpoint, in order to conserve  mass.  resending solution is  nL=[(2-%)83-%82In0, n-L2=[283%-1]n0,  (43a) (43b)  n-j3=[1-28•(2- a3)]n0.  (43c)  Equations(42) or (43) with (23) give the far-field geos•ophicsolutionfor • •ound a stepped-bottom c•yon. •e flux approaching a step.d-bottomc•yon under•e inifi• condi•on(26) is Fx givenby (31), whereas•e flux in •e along-canyondkec•on is  Fy=2e382F•= 2e283F•.  (•)  18,052  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELF CURRENTS  If H 2 = H 3, (44) is identicalto (32), the valuefor the flat-bottom canyon. Again, there is a net flux generatedin the alongcanyon direction to the left of the approachingshelf flow in the northern hemisphere. •.2.  Numerical  Full  Geostrophic Solution  The geostrophicsolution near the canyonedgesand the step may be calculated from the far-field solution assuming geostrophy.However, as was discussedfor a similar case by Gill et al. [1986], in practice, geostrophywill not occur everywherefor a stepped-bottomcanyonas a resultof nonlinear or frictional effects and the advectionof potential vorticity throughthe singularity. The solutionachievedby solving(4) can be regardedonly as a first approximation. The complicated geometry of a stepped-bottomcanyon makes an analytic solution complicated. Consider instead a numericalsolutionof (4). After multiplyingby the grid space size, the f'mitedifferenceform of the elliptic type equation(4)  •2  -6  is  •i+l,j + •i-l,j + •i,j+l + •i,j-1 +ei,j•i,j = f i,j  (45)  X --  CR•SS CI:INYBN DIRECT I IN  wherei/j is the grid indexin x/y dimensionof a squaredomain.  Thecoefficients ei,j andfi,j canbecalculated fromthegrid spaceand the canyonsystemparametersY2 and Y3- If the  domainis largeenough, (27a)and(30a)and(3Oh)canbeused as the domainboundarycondition.In fact, the numericalsolution of (45) is done in pieceswith boundaryconditionsgiven by (43) along all depth discontinuities.The singularity also occurson a boundary.With the Chebyshevaccelerationand a reasonabletolerance,the solutionof (45) is calculatedby simultaneousoverrelaxation [Press et al., 1986, pp. 657-659]. The precisionof the solutionhas been checkedby comparing different spatial resolutions. Solutionsof (45) for someexamplesare given in Figure 6,  4  I g  which shows the contours of surface elevation, which also  form streamlines.Becausethe Rossbyradius over a steppedbottom canyon has different values for the deep and shallow portions,the Rossbyradiuson the sheif, R•, is chosenas the length scaleinsteadof the Rossbyradiusover the canyon. As is shownin the three panelsof Figure 6, a stepped-bottom canyonis not a completebarrier to an approachingshelf flow. The shelf flow can crossthe canyonedgein a small region where the junction occursbetweenthe stepin the canyon bottom and the left canyonedge (looking upcanyon)in the northern hemisphere;a stepped-bottomcanyon can also inducean along-canyonflux that is directedto the left of the approachingshelf flow in the northernhemisphere.Note that the solutionis no longer symmetricowing to the existenceof the step. Figures 6a and 6b, comparedwith Figure 4a, show  Figure 6. Contoursof surfaceelevation •, which also form the streamlines,for the stepped-bottomcanyon. Thick lines representthe position of the canyon edges and the canyon bottom step. Dotted lines represent negative • values. Arrows represent the direction of the flow in the northern hemisphere.The rangeof valuescontouredis - r/0 to r/0. The position of the initial surface discontinuity is at y=0. The  i" -6  -8  i  -8  I  -6  -4  -2  X --  •  0  i  i  I  2  4  6  -  8  CI:INY•N DIRECTION  25' ß ' ß ' ' ' ' ' ß t . , . 'i  ::::::::::::::::::::::::::::::::::::::::::::::::::::::::"..•  ....... ;;!l,,:,.. ................. ...'i.  :"'.... i . /•j• ii•.'. .'........... -1•  15  length unitis R1, y22= 2, •'32= 4, (a)Thestepis aty=4, •1 = 2, the grid pointsare 385 x 385, andthe contourinterval is 0.16 •0. (b) The stepis at y=-4, •1 = 2, the grid pointsare 385 x 385, and the contourintervalis 0.16 •/0. (c) The stepis at y=10, •1 = 30, the grid pointsare 401 x 401, andthe contour interval is 0.2  2g  25  -25  -20  -15  -lg X --  -5  0  5  10  CR•5S CI::INYONO i RECTi •N  15  28  25  CHENANDALLEN:INFLUENCE OFCANYONS ONSHELFCIJRRENTS that the flow on the fight-handside of the canyon(looking upcanyon) is almost unaffectedby the slope; however,on the left-hand side, much of the approachingflow that for a flatbottom canyon would turn right, is deflectedinto the canyon. In Figure 6c the canyonis wide enoughthat the two sidesare essentiallydecoupled. The solution in the canyon in the region of the junction betweenthe step and the canyonedge is similar to that observedby Gill et al. [1986] for a channelbut modified by the sourceof fluid from the shell  6. Geostrophic Circulation Over a SlopingBottom Canyon With a Coast and a Shelf Break 6.1. Values of the Surface Elevation Changes  at Depth  18,053  no variation in the along-shelfdirection, and conservationof potential vorticity. Boundary conditions include no flow through the coast, mass flux conservationover the sheff break  andconservation of mass. A full derivationis givenby Chen [1996, AppendixC]. The requiredsurfaceelevationvaluesare as follows: At the shelf break,  •I.,C = r•o [Aexp( DsB3 )+1]  (47a)  r/k= %[2Eexp(Dc )- 1]  (47b)  and at the coast  where  A={211-exp(2D c)l}{xp(os3osa)  x[(?'3 -1)exp(2OsB1 )+(?'3 +1)exp(2 Dc )]}-l,  This geometry(shownin Figure 2) has the grossfeaturesof Juan de Fuca Canyon including the shelf break, the coastline broken by the Strait of Juande Fuca and variationsof bottom depth. For convenience,the initial condition (26) is chosen with the fluid at rest and with a surfacediscontinuityalong a line in the across-canyon direction. It is assumedthatthe coastis at y = dc , the shelfbreakis at andDsm= dsa/ Ri , Dsm= dsB/ R3,andDe=dc / Ri. y = dsB,the lowerboundof thecanyonheadslopeis at y = dht  E={2exp(Dstt3 )It3cøsh(Dsztl >-sinh(Dsttl )]} x{2exp(Ds•3 )[73cosh(Dsm )-sinh(Dsm )]  x[( 73 -1)exp(2 Dsm )+(73 +l)exp(2 Dc )]}-1  Now considerthecasewith a canyonasshownin Figure7. while theupperboundof the slopeis at y = dhu,andthe lower At thecoastandat all changes in depth,thesurfaceheightwill boundof the canyonmouthslopeis at y = dmtwhile the upper be a constant as discussedin section 2. The waves, Kelvin boundof the slopeis at y = dmu. waves induced by the coast and double Kelvin waves induced The depthratiosin thismodel are definedas by the shelfbreak,areonesided;theypropagate with the shallowerwateror the coaston theright. Followingthedirection  ),j=Rj/R!=(hd/h1)1/2J=0,2,3,  (46)of propagationof thesewaves and of the doubleKelvin waves  whereRj is the barotropic Rossby radiuscorresponding to over the canyon slopes allows calculation of the surface depthhj. Specifically, h0 is thedepthof thestrait,h1 is the heightat all the depthchanges. depthon the shelf, h2 is the depthof the middlecanyonportion, and h3 is the depthof the deepcanyonportionand the deep ocean.  BecauseKelvin and doubleKelvin wavescan propagate only in the -x direction,the existenceof the canyonand the straitdoesnot affectthe adjustment processforx--> oo. Thus  To solve for the casewith a canyon,the surfaceelevationat in the steady state, the surface elevation at the coast the coast and shelf break in the casewithout a canyonare re- (denotedr/K) is given by (47b), and the surfaceelevationat quired. This solutioncan be found by assuminggeostrophy, the shelfbreak(denotedr/LC) is givenby (47a).  land  Y=dhu  land  head slope  y=dhi  y=dc  Ps shelf h1  TIK shelf  mouth slope  hi y--dmu y=dm•  x=-L x=L  Y=dsB  TI.L3  Figure 7. Top view of the Juande Fucamodelcanyon. The shadedregionsrepresentthe canyonbottom slopes.The dottedline represents the positionof theinitial surfacediscontinuity.The widthof the canyonas well asthe straitis 2L. The depthsin theinnerstrait,on the shelfoverthemiddlecanyon,andoverthe deep canyon(as well as in the deepocean)are h0, h1, h2 and h3, respectively. The surfaceelevationin the geostrophicstateat all depthchangesand boundariesis indicated. See text for othernotations.  18,054  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELF CURRENTS  At the part of the canyonedge at x = L betweenthe shelf break and the canyonmouth slope,which is contiguouswith the shelf break, the surface elevation must be the same value as  be found using conservationof mass and the properties of Kelvin  waves.  Solution  for riLS. Equating the flux acrossthe lines that at the shelfbreakfor x > L, riLC. y=dc-•, xe{-X,X}, andY=dc+•, xe{-L,L}, where The surfaceelevation at the canyonedge at x = L is transt; -->0 and X -->oo shownin Figure8a, gives mired by the canyonwavesalong the canyonedgein the +y direction and transmittedby the slope-inducedtopographic r•s = 2 ß (48) waves towards the canyon edge at x = -L. Regardlessof the shapeof the slope, the surfaceelevation over it is a constant, Solution for ri_LOo A strait acts similarlyto a canyon riLC, determinedby the canyonwaves. A constantsurfaceelwith the canyonwavesreplacedby Kelvin waves. Thus the evation over the slope implies that the fluid there, in the geostrophicstate, is stagnant(see Allen [1996a] for details). surfaceelevation at the two walls of the inner strait beyondthe canyon head slope is given by The canyonwavescarrythe surfaceheight riLC to the edgeof  )t2 2_1)riLC +rile  the  strait.  Beyond the canyonhead slope,in the geostrophicstate,the surfaceelevation is uniform along the strait walls. If the surface elevation along the strait wall at x = L in the lower strait is assumed to be ri/3, the Kelvin waves within the strait will transmitthis informationalong the strait wall at x = L toward the +y direction. Similar to the analysisfor the canyonmouth slope, the canyon head slope does not interfere with the transmissionof the information along the strait wall at x = L. The information ri• will be transmittedcontinuouslyalong the whole length of the strait wall at x = L and transmittedby the slope-inducedtopographicwavestowardthe straitwall at x  Note that riL• is equalto neither rizc nor rir. ThusPs is a singularpoint similar to that discussedin section5. At P s, where the coastmeets the canyonedge at x = L, the incoming Kelvin waves which carry the information rlff confront the incoming canyon waves which carry the information riLC. The outgoingKelvin waves from P s propagatealong the strait wall at x = L and transmitthe information riLXin the +y direc-  (49a)  "s(y+cøst)+TWps(Y-Cøst)' (49b) whereWns(Y+Cost ) is theinfomarion caffied bytheKelvin waves •om the f•thest end of the s•ait towed the canyon  headslope, Wps(y-Cost ) is theinfomation caffied bythe Kelvin  waves  moving  in  the  +y  direction,  and  as= 1-tanh(fi / 2).  El•inatingWps(y-Cost ) between (49a) and(49b) gives •-•0 =  (50)  2-as  where Wns=-ri0 for the choseninitial condition(26).  tion.  Along the strait wall at x = -L, the Kelvin waves that propagate in the -y directiontowardthe head slopemake the surface elevation a constantdenoted ri-ro. However, in the region where the strait wall at x =-L meets the canyon head slope, these incoming Kelvin waves confront the incoming slopeinducedtopographicwaves that carry the information ri•. The outgoing Kelvin waves that propagatetoward the mouth of the strait make the surfaceelevation along the strait wall at x = -L in the lower straita constantdenotedri-L2. The region where the canyonhead slopemeetsthe strait wall at x = -L is a singular line, an extensionof the singularpoint discussedin section  b)  T•L•  i!iY=dhu+ œ  landii•i.:....T....:.:',  5.  i.l,c  -  When the Kelvin wavesreach the junction of the coastand the canyon edge at x =-L, the task of transmittingthe inforland mation, ri-L2, is handedover to Kelvin wavesthat propagate in the-x direction along the coast and the canyonwaves that propagatetoward the canyonmouth slope. The region where the canyon edge at x =-L meets the canyon mouth slope is another singular line. The outgoing hl hl canyon waves that propagateaway from this region make the surfaceelevation along the canyonedge at x = -L in the deep .... ..•-y=d mu+ F_. canyonportiona constantdenotedri-L3' The double Kelvin waves that propagatein the -x direction along the shelf break for x <-L make the surfaceelevation at the sheffbreak ri-Z3ha The geostrophic solution of the steady state governing Figure 8. Close up views of the Juande Fucamodelcanyon equation(4) at all depthchangesand boundarieshasbeenana- near (a) the mouthof the strait,(b) the canyonheadslope,and lyzed qualitatively and is indicatedin Figure 7 for easy refer- (c) the canyon mouth slope. The surface elevation in the ence. Now the relations between these surface elevations must geostrophicstate at all depthchangesis indicated.  p Y=drnl' gq  ....  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELFC!JRRENTS  Solution for /•-L2' Equatingthe flux acrossthe lines y=dht-t; and y=dhu+t;, where t;--)0 and xt[-L,L}  a)  shownin Figure 8b, gives  -  /•-L2 =  +  "2  18,055  land land  head slope  "  where7 is definedby (46).  Solution for  //-L3' Considerthe rectangleshownin  Figure8c,whichincludes thecanyon mouthslope.Thefour  points of are A(-X, dml-e ), B(X, dml-e ), C(X,dmu q'the E),rectangle andO(-X,dmu q'E), in which e-->0 and X-->oo.Forcing themass fluxentering theboxtobezerofor  outh slope  the steadygeostrophic stategives  (?'32 -722)///_17+(722 - 1)r/-L2  =  .  Solutionsof the steadystate governingequation(4) have beenobtainedat all depthchanges(the canyonbottomslopes, the canyon edges, and the shelf break) and at all internal boundaries(the coast and the strait walls). Equation(4) can now be solved in each flat-bottom segmentfollowing the numericalprocedurediscussed in section5.2. 6.2  Numerical  Solution  for  Full  ,  -6  I  -6  ß  I  -4  ß  I  -2  X --  ß  I  0  2  ß  I  ,M,.-'  ß  4  6  CRESS CRNYBN OlEECTION  Domain  headslope  The steadystategoverningequation(4) was integratedin a squaredomainseveralRossbyradii in width. The valuesof the surfaceheight at all the depthchangeswere takenfrom the results of the previous section. The solutions at the open  land  land  ,-; ß !  !  t  !  !  l  ,  ß  ....  boundaries are as follows. ,  1. The solution at the right (southern) boundary is the geostrophicsolution over a single step parallel to a coast  ,  ,  I  f  withouta canyonas discussedin the precedingsection. The full solutionis given by Chen [ 1996].  ,,,,,-  2. The solution at the left (northern) boundary is the  geostrophicsolution over a single step parallel to a coast forced to have surface elevations at the coast and shelf break  given by •/-L2 and /•-L3, respectively.The full solutionis given by Chen [1996]. 3. The solution at the bottom (deep ocean) boundary is estimated from the surface elevation at the right and left bottom corners by linear interpolation. 4. The solutionat the top (strait) boundaryis given by the solution for geostrophicflow constrainedby the values of the  surfaceelevationat the straitwalls, r/_LOand r/Ls. The results of the numerical integration for two sizes of canyon are given. First considera canyonwith width of 1 Rossbyradius(the barotropicRossbyradiuson the shelf) and length of 4 Rossbyradii. The depthsare 100 m in the inner strait, 150 m on the shelf, 300 m in the middle canyon and 1800 m in the deepcanyonand over the deepocean. Relative to the line of the surfacediscontinuity(at y= 0), the coastis at 3 Rossbyradii, the shelf break is at -1 Rossbyradius, and the two canyon slopesare locatedbetween3.75 and 4.5 and between-0.6 and 0.2, again in Rossbyradii. The distribution of the surfaceelevation is shown in Figure 9a, and a close-up of the flow vectors in the region of the canyon is given in Figure 9b. Similar to the stepped-bottomcanyon,this geometryis not a completebarrier to the incomingshelf break current,part of which crosses the canyon walls at the canyon mouth slope. Some flow  is diverted  from  outside the shelf break into the  ....... ........  • mouth , slope  ,  ,  ,  B.428E.f2  HAXIHUH  X --  VECTOR  CRESS C.qNYeN O i RECT i 8N  Figure 9. Geostrophicstate around the large size Juan de Fuca model canyon.(a) Contoursof the surfaceelevation (b) Velocity field aroundthe canyon;note that this is an expanded view. The thick lines representthe positionsof the canyonedges,shelf break, coast, strait walls and boundaries of the canyonbottom slopes. Solid lines in Figure 9a representpositive rl, while dottedlines representnegativefl. The lengthscaleis R1. The rangeof r/contouredis from -r/0 to r/0, and the contourintervalis 0.14 r/0 where r/0 is half the height of the initial surfacediscontinuity,which was taken as 0.2 m in this example.  canyonthroughthe mouth. Some of the flow in the canyon "squeezes"out of the canyonthroughthe singularpoint where the coastmeetsthe right canyonedge. The secondcanyonsize chosencorresponds to the geometry of Juande FucaCanyon. The depthsarethe sameas in the previous case, but the canyon is 14 .kin wide (0.04 Rossby radii) and the shelf is 70 km wide (0.2 Rossbyradii). The cal-  18,056  CHEN AND ALLEN:INFLUENCEOF CANYONSON SHELFCURRENTS  Juan de Fuca Strait Vancouver Island  Olympic Peninsula  tial condition is changed so that the geostrophicflow direction is reversed, the flow patterns will be similar to those shownexceptwith all flow directionsreversed. 2. A canyoncan causethe inshoreexcursionof a shelf break current of either direction, but the net transportalong the canyonand the flow within the canyonis alwaysto the left of the shelf break current in the northernhemisphere.  de Fuca Cany( 7. Discussion  The analysisin this paperis basedon the linearshallowwater equationsfor the barotropiccase and thus representsthe strongesteffects of the topographyon the flow. Once stratification is considered, our results should be modified (see Allen  [1996b] for the narrow canyon case). The barotropicsolution can be directlyextendedto a specialbarocliniccase.If reduced deep ocean gravity is used and hencethe deformationscaleis the internal Rossbyradius,all resultsin this paper are retainedin the bottom layer of a systemwith a relatively deep overlyinglayer (one and a haft layer stratifiedmodel [Gill et al., 1986]). In this paper, the dispersionrelation for canyonwaves has X -- CR•SS CI:INY•N DIRECTION been calculated.The propertiesof long canyonwaves are disFigure 10. Geostrophicstate aroundthe model of Juande cussedand are essentialfor determiningthe geostrophicstate. Fucacanyon. Only the innergrid is shown. The thin linesare The wave solutionis not given in this paper,and thusthe adcontoursof the surfaceelevation•/, which are also the streamlines in the geostrophicstate. Thick linesrepresentthe posi- justmentproblem posedin section2 is only partially solved. tions of the canyonedges,shelf break,coast,straitwalls, and Even so, some basic properties of the effect of canyonson boundariesof the canyonbottomslopes. Arrowsrepresentthe shelf circulationhave been revealed. Some importantparamedirectionof flow duringa southerlycoastalflow. The rangeof tersdefinedin the processare ry and co , and they are expected q contouredis from 0.5 q0 to 0.98 q0, and the contourinter- also to be significantfor the wave solution. val is 0.01 •/0, where •/0 is half the heightof the initial surThe canyon number ry is an important parameterfor deface discontinuity. scribingthe geostrophicstate arounda canyon.The value of • is determinedby the geometryof the canyonsystem,i.e., the depth of the water layer on the shelf and over the canyon, culationwas done using a nestedgrid method. The first relax- the width of the canyon, and the Rossby radius over the ationwas donefor a 6 Rossbyradii by 6 Rossbyradii domain. canyon. For the bottom layer of the specialbaroclinicmodel, Then, relaxation was done on a smallergrid (shownin Figure the one and half layer model, three canyons(Juan de Fuca 10) using, as open boundaryconditions,valuesinterpolated Canyon, Astoria Canyon, and Moresby Trough) are chosento from the original large grid. Contoursof surfaceheight, illustrate the values of cr for real canyon systems(however, which are the streamlinesare shownin Figure 10. note that the upper layer is not relatively deepcomparedwith In the regionof Juande FucaCanyon,the observed circula- the bottom layer, so the calculations give only the approxition is determinedby a competitionamongseveraldifferent mate valuesof ry). For the Juan de Fuca Canyon systemthe physicalprocesses includingestuaryeffects,wind, and topo- depth of the bottom layer is assumedto be 50 m on the shelf graphiceffects. However,evenwith neglectof stratification, and 250 m over the canyon;the averagewidth of the canyonis somefeaturesof the flow aroundJuande FucaCanyonare simu- 7 km, while the local internal Rossbyradius is about 20 km latedby thissimplemodel. First,aswaspredicted by previous [Freeland and Denman, 1982]. Thus the calculatedo' is 0.684. models and in agreementwith this model, the southbound For Astoria Canyon the depth of the bottom layer is assumed shelf break current leads to an in-canyonflow as observedby to be 50 m on the shelf and 500 m over the canyon;the averFreeland and Denman [1982]. Second,at the mouthof Juande age width of the canyon is 7 km, while the local internal Fuca Strait, the flow around the corner of the Olympic Rossbyradiusis about30 km (B.M. Hickey, The responseof a Peninsulais strong and multidirectional[Thomsonetal., narrowcanyonto strongwind forcing,submittedto Journal of 1989]. Third, the modelpredictscyclonicflow to the northof Physical Oceanography1996, hereinafterreferredto as B.M. the canyon (most clearly seen in Figure 9a) in the region Hickey, submittedmanuscript,1996). Thus the calculatedry is wherethe Tully eddy is observed.The modeldoesnot give a 0.880. For Moresby Trough the depth of the bottomlayer is closededdy, which is to be expected;the presenceof nonlin- assumedto be 50 m on the shelf and 250 m over the canyon; earity and the surfaceoutflowfrom Juande FucaStraitmust the averagewidth of the canyon is 40 km, while the local inplay a role.Fourth,themodelshowsa shiftof thepositionof ternal Rossbyradiusis about 20 km [Crawfordetal, 1985]. the shelf break current as it flows over the canyonto further Thus the calculated ry is 0.171. Thus Astoria canyonis narinshore. Unfortunately,there are no detailedobservations of row with strong coupling between the two sides, Moresby the shelfcurrentoff the OlympicPeninsulato confirmor refute Troughis wide with weak coupling,andJuande Fucalies in between. this prediction. A simple classicalinitial conditionwas usedin mostof the Someotherpropertiesof flow aroundthisgeometryinclude discussion.As long as the forcing is small enoughthat the the following. 1. Unlike the infinitely long canyons,the streamlinesare not flow remains approximatelylinear, solutionsfor other initial symmetricowingto the existenceof the shelfbreak.If the ini- conditionscan be found by linear superposition.  CHENANDALLEN:INFLUENCE OFCANYONSONSHELFCURRENTS The topographyinvolved is simplified but containsthe basic featuresof a real canyon.Studyingthe circulationarounda flat-bottom canyon lays the foundationsfor analyzing that arounda more realistic, complicatedcanyon.The circulation pattern for an infinitely wide flat-bottom canyon is identical to that derived by Gill et al. [1986] for a single-steptopography, which has been demonstratedto be consistentwith numerical and laboratory experiments [Gill et al., 1986]. To study a canyonwith a steppedbottomis the first steptoward studyinga real, sloping-bottomcanyon.Then a geometrywas chosenthat has the major featuresof Juande Fuca Canyon,includingthe shelf break, coast,and Strait of Juande Fuca. Extensionof the resultsof this papermustbe madecarefully on account of the linear and inviscid assumptions.For instance,our resultsfor a very narrow, infinitely long flat-bottom canyon will lead to currentswithin the canyon too strong to have physicalmeaning.This deficiencyis probablydue to neglecting the effects of turbulent viscosity or advection. Inclusion of viscosity and nonlinearadvectionin a stratified fluid is expectedto improve our currenttheory. Our theoretical results are qualitatively consistent with someobservations.For example,the mean shelf break current along the west coast of VancouverIsland, where the Juande Fuca Canyon is located, is southeastwardin the summer.A mean in-canyoncurrent within the Juande Fuca Canyonwas  18,057  canyonedgesalong which they propagate,keepingthe deep water of the canyonto their left in the northernhemisphere. All but very longcanyonwavesaredispersive. 2. An importantparameterfor describingthe geostrophic state arounda canyonis the canyonnumber cy, which is deternfinedby the geometryof the canyonsystem(ty • {0,1}), for an infinitelynarrowcanyoncy-->1 and for an infinitely wide canyon cy---> 0. The canyonnumbermeasuresthe influenceof one canyonedgeon the other. 3. For an infinitely long flat-bottomcanyon,the canyonacts as a complete barrier to an approachingshelf flow in the geostrophicstate.The net transportalongthe canyonis to the left of the approachinggeostrophicflow in the northern hemisphere.  4. For an infinitely long stepped-bottom canyonthe canyon is not a completebarrier to an approachingshelf flow in the geostrophicstate. In the northernhemisphere,flow crossesat  the singularpoint where the left canyonedge (lookingup-. canyon) meets the canyon bottom step. The net transport alongthe canyonis to the left of the approaching geostrophic flow in the northernhemisphere. 5. For a canyonwith geometrysimilarto Juande Fuca,the canyon causesan inshore excursionof the geostrophicshelf break current, and flow also entersthe canyonacrossthe canyonwall at the mouth slope. Someflow exits the canyon observedin the summer [Freeland and Denman, 1982]. When a where the canyonand shelf intersect,but flow within the strait northwestwardshelf break current is forced by a winter storm is strongand an inflow. The net transportalongthe canyon in this region, an out-canyon current within the canyon was and the flow within the canyonare towardthe left of the shelf observed [Cannon, 1972]. However, the currentmagnitudes break currentin the northernhemisphere. predictedhere, approximatelyequal in the canyonto that over the shelf, are a factor of 3 too high comparedwith Cannon's Appendix: Solvingthe Systemof Partial [1972] observations. The overpredictionis expected;the apDifferential Equations proximation of homongeneousflow tends to overestimatethe As an examplewe presentthe solutionprocedurefor the effect of the topography. (22) forf> The theoreticalresultsare also verified by the observations systemof first-orderpartialdifferentialequations is similar.Substituting (22a) and(22c) aroundthe HudsonShelfValley [Mayer et al., 1982], wherethe 0. Forf < 0 theprocess strongestin-canyon and out-canyonflows are associatedwith into (22b) and (22d), the systemreducesto northeastwardand southwestwardwinds (and hence shelf currents), respectively. All these observationsare consistent with the predictionfrom our model with a shelf break.Using the conductivity-temperature-depthdata from specific La  cothfi +  Perouse cruises, Foreman [1992] simulated the summer geostrophiccurrent in the coastalarea southwestof Vancouver Island with a numerical  model. The model results show that  upwellingis associatedwith horizontalvelocity excursionsof the shelf break currentonto the shelf throughthe canyonsin this region. The dynamics for the horizontal movementof the currentsaroundthe canyonshave not beengiven. Our theoretical analysis provides a possible explanation of Foreman's numerical  results. The shelf break current has also been ob-  servedto turn shorewardand have a strongvelocitycomponent upcanyonover Astoria Canyonduring periodsof weakly vertically sheared southerly flow (B.M. Hickey, submitted manuscript, 1996).  At(y,t )-  'sinhfi Dt(y,t )  -( H2- H1)Ay(y,t)=-( H2--H1)lily,  (Ala)  1 At(y,t)+ 'sinhfi  Dt(y,t )  -  cothfi +  +(H2- H1)Dy(y,t)=(H2- H•)rlly.  (Alb)  Define  all = a22=  coth• +  ,  a12 =a21 =-  sinhi] ' -  8. Conclusions  In this paper the geostrophicadjustmentmethodhas been used to study the influence of submarine canyons on geostrophicshelf currents.The fluid is homogeneous, inviscid, andon anf plane. Someconclusions can be drawn. 1. The subinertial,doubleKelvin wavesover a canyondetermine the geostrophicsolution.These waves can propagatein eitherdirectionalongthe canyonbut are trappedto one of the  (A2a)  •,gj  hI = -h2 = -(H 2- H1) r/ly(y),  (A2b) (A2c)  (A2d)  andlet t = t(gt) and y = y(W), where W is a referencevariable, so A = A(W) and D = D(W). Equation(A1) thencollapses to  (blli-allP)Ay - a12pDy =hli-allA-a12b, (A3a) a•2pAy +(blli+a•p)Dy =h•i+a•2A+alllS.(A3b) wheretheoverdotdenotes thederivativewithrespect to •.  18,058  CHEN AND ALLEN: INFLUENCE OF CANYONS ON SHELF CURRENTS  Becauseon eachcharacteristicy + Cot= C1, the left-hand  The characteristicdirectionsare given by  blli-a11• -a12• [=0, a12• bali+alii21 j•= + coi  (A4a)  side of (A10) is a constant C3, and on each characteristic y-cot = C2, the left-hand side of (A12) is a constantC4, let these two constants  C3=-!H2-Hll(1-z1)Wp( ),  (A4b)  Co  where the positiveparameterco is definedas  C4=H2 - H1(1-z2 )Wn (y-cot), Co  gl/2  co-  be  /  whereWn andWp arearbitrary functions of onevariable and are relatedto the informationcarriedalongthe characteristics  or, in terms of 7 and • defined in section 3 and with  (see section 4.1). Then  C1=(gill) 1/2,  (Zla12 +all)A(y,t)+(Zlall +a12 )D(y,t)  C 0 C1 I•2--11 =  (72 +27•thfl +1) 1/2'  co  Equation (A5) is the group and phase speedof long canyon waves •at  was derived  =-(H2-H1){1-Zl)[wt,(Y+Cot)rlI(Y)] (A13a)  (M)  (X2a•2 +all)A(y,t)+(X2a•l +a12)D(y,t)  in section 3.  =(H2 - H1 )(•-;t2)[W• (y-cot)rh(y)].  From (A4b), we know that •e system(A3) is of hyperbo•c ty• and•e •o setsof ch•actefisticsare  (AS), (All), (A2a), and (A2b) into (A13), anddefininga parmeter found in the process  dy= codt, i.e., y-Cot = C2 where C1 and C2 •e constanU. From (A4a) the coefficientsof (A3a) and (A3b) must be in a constant•tio for a solution to exist. We defoe this ratio as Z, i.e.,  -a12•  hli-all•-a12•  o'=1-I al-••-2 all +/ al 1 - al 2  a12P- bill+all • =hli+a12•+all • =Z. (A6)  Substituting(A2d) into (A6), we obtain  cosh•5 +(H1)1/2 sinh•5 +(" 2)1/2 =1-[(H2 (H2)1/2 ' )l/2cosh]5+(H1)l  or, in termsof 7' and•6,  (Zal2+all) •(y,t)+(Zall +a12 ) dD(y,t)  tYiY']•) =1-['y(coshfl +1) + ' 7(coshfi1) +sinh•/ sinh]•] 1/2  dt  =o. On a ch•acmristic, y + Cot= C1 (i.e., dy= -Co• ), (A6) becomes  •1•--••1 +talc0 •,  Departmentof Fisheriesand Oceans/NSERCsubventionandan NSERC researchgrant.  (•1a12 +all)•(y,t)+(•la11 +a12 )dD(y,t)  Integrating (•),  o.  References Abramowitz,M., andI. A. Stegun(Eds.),Handbookof Mathematical  we have  Functions,1046 pp., Dover, Mineola, New York, 1968.  (•1a12 +all)A(y,t)+{•lall+a12 )D(y,t)  -0-Zl) c0-  =c3  (no)  where C3 is a const•t along a ch•acm•sfic y + Cot= C1. S•l•ly, on a ch•acte•sfic, y- Cot= C2 (i.e., dy= c0•), (A6) becomes  '  Z2 =  •1 - allC0  ,  (A11)  a12c0  •d  s•l•  W•,(y+ Cot)andWn(y - Cot). Acknowledgments.We are verygratefulfor thesuggestions from referees on ourfirstdraftandto R. Thomson for suggestions onX. C.'s thesiswhichwereincorporated here. Thisprojectwassupported by a  •d (A7) becomes  -  we obtain the solution of (A13) as (24) in the terms  (A8)  a12c0  c0  (^13b)  It is straightforward to solve this system. Substituting  • =-codt, i.e., y +Cot= C1,  •l[--all•_  Co  Cannon, G. A., Wind effects on currents observed in Juan de Fuca  SubmarineCanyon,J. Phys.Oceanogr.,2, 281-285,1972. Chen,X., Rossbyadjustment overcanyons, Ph.D.thesis,278pp.,Univ. of B.C., Vancouver, Canada, 1996.  Crawford,W. R., W. S. Huggert,M. J. Woodward,andP. E. Daniel, Summercirculationof the watersin Queen CharlotteSound,Atmos. Ocean, 23, 393-413, 1985.  to (A10), we have  (Z2al2 +all)A(y,t)+(Z2all +al2)D(y,t)  +(1-Z2) H2 - H1•I(Y)=C4 c0  Allen,S. E., Rossbyadjustment overa slope,Ph.D.thesis,206pp.,Univ. of Cambridge,England,1988. Allen,S. E., Rossbyadjustment overa slopein a homogeneous fluid,J. Phys.Oceanogr.,in press,i 996a. Allen, S. 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