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Dynamics of advection-driven upwelling over a shelf break submarine canyon. Allen, Susan E.; Hickey, B. M. 2010

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Dynamics of advection‐driven upwelling over a shelfbreak submarine canyonS. E. Allen1and B. M. Hickey2Received 17 August 2009; revised 30 March 2010; accepted 16 April 2010; published 19 August 2010.[1] The response over a submarine canyon to a several day upwelling event can beseparated into three phases: an initial transient response; a later, much longer, “steady”advection‐driven response; and a final relaxation phase. For the advection‐driven phaseover realistically steep, deep, and narrow canyons with near‐uniform flow andstratification at rim depth, we have derived scale estimates for four key quantities.Observations from 5 real‐world canyon studies and 3 laboratory studies are used tovalidate the scaling and estimate the scalar constant for each scale. Based on 4 geometricparameters of the canyon, the background stratification, the Coriolis parameter, and theincoming current, we can estimate (1) the depth of upwelling in the canyon to within 15 m,(2) the deep vorticity to within 15%, and (3) the presence/absence of a rim depth eddycan be determined. Based on laboratory data, (4) the total upwelling flux can also beestimated. The scaling analysis shows the importance of a Rossby number based on theradius of curvature of isobaths at the upstream mouth of the canyon. This Rossby numberdetermines the ability of the flow to cross the canyon isobaths and generate thepressure gradient that drives upwelling in the canyon. Other important scales are aRossby number based on the length of the canyon which measures the ability of theflow to lift isopycnals and a Burger number based on the width of the canyon thatdetermines the likelihood of an eddy at rim depth. Generally, long canyons with sharplyturning upstream isobaths, strong incoming flow, small Coriolis parameter, and weakstratification have the strongest upwelling response.Citation: Allen, S. E., and B. M. Hickey (2010), Dynamics of advection‐driven upwelling over a shelf break submarine canyon,J. Geophys. Res., 115, C08018, doi:10.1029/2009JC005731.1. Introduction[2] Submarine canyons are ubiquitous and steep‐sidedfeatures(upto45°)thatfrequentlyindentthecontinentalshelfas much as 60 km [Hickey, 1995]. Off the coasts ofWashington State and British Columbia typical canyons areat least 600 m deep and on the order of 5–30 km wide. Thesefeatures are regions of enhanced upwelling [Freeland andDenman, 1982; Hickey, 1997; Vindeirinho, 1998] and areimportant for cross‐shelf‐break exchange including nutrientflux onto the shelf [Hickey and Banas, 2008]. They are bio-logically active areas with dense euphausiid and fish aggre-gations[Pereyraetal.,1969;Mackasetal.,1997;Allenetal.,2001] during summer upwelling favorable winds.[3] For eastern boundaries, upwelling along the coast andwithin the canyons occurs as a result of pulses of equator-ward currents due to local alongshore wind [Hickey, 1997] orequatorward currents generated by poleward propagatingshelf‐waves [Allen et al., 2001]. During an upwelling pulse,the response over the canyon can be separated into threephases: an initial transient phase (first inertial period), a nearsteady advection‐dominated phase and a relaxation phase(after the wind‐forcing dies). The initial transient phase isreasonably well explained by linear dynamics [Allen, 1996].In this paper a dynamical analysis of the second phasewill bepresented. This analysis allows results obtained by obser-vation and numerical simulation in the relatively few can-yons that have been studied to be extended to other canyons.[4] Canyons are three dimensional features with extremelysteep bottom slopes as wellas sharp changes inbottom slope.Incident flow is strong and the water column can be highlystratified. Because of this challenging environment, ob-servations within canyons sufficient to delineate the dynam-ics ineven the most rudimentary fashion have beenlimited toa handful of canyons [Allen and Durrieu de Madron, 2009].Moreover, the steep bottom slopes and strong stratificationhave proved to be a barrier to realistic numerical modelingstudies. For example, Allen et al. [2003] have demonstratedthat topography typical of many Pacific coast canyons resultsin numerical errors in vertical advection schemes for density.Attempts todate to model canyons using realistic topographyhave producedunrealistic flow anddensity fields where there1Department of Earth and Ocean Sciences, University of BritishColumbia, Vancouver, British Columbia, Canada.2School of Oceanography, University of Washington, Seattle,Washington, USA.Copyright 2010 by the American Geophysical Union.0148‐0227/10/2009JC005731JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, C08018, doi:10.1029/2009JC005731, 2010C08018 1of20is good observational data with which to compare [Klincket al., 1999] or have simply not had sufficient data to thor-oughly test the model [e.g., Skliris et al., 2001]. Even whenthe bottom boundary layer is well resolved, details of sepa-ration upstream of the canyon are not properly modeled[Dawe and Allen, 2010]. The goal of the present paper is touse available observations to extend the dynamical under-standing obtained to the multitude of canyons that lackobservationsaswellastoprovideadynamicalframework forfuture modeling efforts. Such a dynamical analysis has beenproduced for oscillatory flow over canyons [Boyer et al.,2004] although the specific length scales were not deter-mined. Kämpf [2007] has empirically derived a scaling forupwelling flux based on numerical results.[5] The dynamics of upwelling over a number of shelfbreak canyons have been examined using observations[Hickey et al., 1986; Hickey, 1997; Allen et al., 2001]numerical models [Klinck, 1988, 1989, 1996; Allen, 1996]and laboratory models [Allen et al., 2003; Pérenne et al.,2001; Mirshak and Allen, 2005]. Based on these studiesthe following picture of the second phase of upwelling overa canyon in a stratified environment emerges [Allen et al.,2001] (Figure 1):[6] 1. Near surface flow is only weakly affected by thecanyon and passes directly over the canyon. However, nearsurface isopycnals may be elevated as observed overBarkley Canyon [Allen et al., 2001].[7] 2. Flow just above the depth of the canyon rim (theedge where the near flat shelf meets the steep bathymetry)flows over the upstream rim of the canyon. As it crosses therim, it flows down into the canyon, stretching the fluidcolumns and generating cyclonic vorticity. The flow thenturns up‐canyon and flows across the canyon equatorward,shoreward and upward leaving the canyon shoreward of itsoriginal position. As the flow crosses the canyon it de-creases its depth and the stretching decreases. As the flowcrosses the downstream rim, fluid columns are compressedgenerating anti‐cyclonic vorticity. The stretching can bestrong enough to generate a closed cyclonic eddy at thisdepth (“rim depth eddy” in Figure 1).[8] 3. Flow over the slope at the depth of the rim of thecanyon and for some depth below (“upwelling current” inFigure 1) is advected into the canyon and upwells over thedownstream rim of the canyon near the head. This flow car-riesthedeepestwateradvectedontotheshelf.Vorticityinthisflowcanbegeneratedbybothflowseparation[Pérenneetal.,2001] and by stretching [Hickey, 1997]. Recent laboratory[Waterhouse et al., 2009] and observational studies [Flexaset al., 2008] suggest that the latter probably dominates.[9] 4. The upward vertical displacement of water parcelsdecreases with depth. Thus water deeper than that whichupwells onto the shelf is stretched within the canyon and hascyclonic vorticity (“deep flow” in Figure 1).[10] In order for upwelling to occur within a canyon,upwelling‐favorable currents must extend down to the shelfbreak depth. The formation of an undercurrent at shelf breakdepth (seasonal at Astoria Canyon) suppresses canyonupwelling [Hickey, 1997]. The results presented here are forcanyons experiencing upwelling‐favorable flow to shelfbreak depth and for canyons that behave like BarkleyCanyon (located off the West Coast of Vancouver Island atlatitude 48°30′N) and Astoria Canyon (located off the mouthof the Columbia River at latitude 46°15′N). In particular, thecanyon must have (1) a nearly uniform flow (little horizontalshear) approaching the canyon, (2) the canyon must notapproach the coast too closely, and (3) the canyon must bedeep, steep and narrow. The third constraint will be madeexplicit during the scaling analysis. In the scaling analysisFigure 1. Schematic of the advection‐driven flow over a submarine canyon.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080182of20we will assume that the cross‐shelf pressure gradient isnearly uniform along the length of the canyon; thisassumption requires that the approaching flow must be nearuniform across the shelf over the length of the canyon. If thecanyon approaches the coast closely, strong bathymetricconvergences occur and their affect on the flow must beconsidered [Allen, 2000; Waterhouse et al., 2009]. Thesefirst two provisos allow the inclusion of most canyons in theanalysis. However two major west coast canyons areexcluded: Monterey Bay Canyon because the regional flowis non‐uniform and it cuts the continental slope all the wayto the coast; and Juan de Fuca Canyon because it cuts thecontinental slope all the way to the coast into Juan de FucaStrait. In our comparison to the observations we include onecanyon, Redondo Canyon, which closely approaches thecoast. As expected, Redondo Canyon does not follow thedynamics and scaling as well as the other canyons. ForAstoria Canyon and other canyons influenced by a seasonalundercurrent at shelf break depth, observations are onlyconsidered during seasons with upwelling favorable flowextending down to shelf break depth.2. Scaling Analysis[11] In scale modeling of fluid flow a number of dimen-sionless numbers are selected that represent the essentialdynamics. The laboratory parameters are then selected sothat these numbers match between the real world and thelaboratory. Here we will perform a scaling analysis to allowthe observations from a few canyons (Astoria Canyon,Barkley Canyon, Carson Canyon, Quinault Canyon and acanyon off Tidra, North Africa) to be extended to other shelfbreak canyons.[12] The dynamic parameters of the canyon upwellingproblem are the incoming velocity, U, the Coriolis param-eter, f, the stratification characterized by the buoyancy fre-quency N and the coefficient of eddy viscosity n. We willconstrain the uniformity of U and N so each is characterizedby a single scale. In addition there are the geometric para-meters of the system, the depth at the shelf break, Hs, thelength of the canyon, L, the depth at the head of the canyon,Hh, and the width of the canyon at the shelf break Wsb.Three more geometric parameters define the shape of thecanyon: R the radius of curvature of the upstream isobaths,W the width of the canyon at mid‐length and Hcthe depthdrop across the canyon at the shelf break. Thus we have atotal of 11 parameters with two dimensions: length and time.According to the Buckingham PI theorem [Buckingham,1914] we will have 9 non‐dimensional groups.[13] The choice of non‐dimensional numbers to charac-terize upwelling through a canyon depends on the choice ofvertical scale. For processes within the canyon the verticalscale is determined by the flow, so an obvious choice for thevertical scale is DH= f‘/N for an appropriate horizontallength scale ‘, which we take here as L, the length of thecanyon. Given this choice for the vertical scaling, our anal-ysis will show that the non‐dimensional numbers that bestcharacterize upwelling through a canyon are two Rossbynumbers (U/fR and U/fL), a Burger number (NHs/fW) andan aspect ratio (L/Wsb). Each Rossby number is characterizedby the same velocity scale and Coriolis parameter; the lengthscale differs between them. Due to the three‐dimensionalityof flow over canyons it is necessary to be very specific as toexactly what length scales are chosen and which parts of theflow the various numbers characterize.[14] The other 5 non‐dimensional numbers can be takenas an Ekman number: Ek = n/fHs2, a vertical aspect ratio, Hs/L, a measure of the slope of the continental shelf (Hs− Hh)/Hs, and two additional Burger numbers Bc= NHc/(fW) andBs= NHs/(fL).[15] We will neglect friction, characterized by the Ekmannumber, Ek, in this analysis. Boundary layers will form overthe shelf, in the abyss, on the slope and within the canyon.With upwelling favorable flow, these layers will havetoward‐coast velocities [MacCready and Rhines, 1991].However, as the water is stratified, the Ekman layers on theslope and within the canyon will be quickly arrested[MacCready and Rhines, 1991]. In particular, consideringthe case of Astoria canyon, one can show that the time tocompletearrestoftheboundarylayerbetween150and400mdepth everywhere in the canyon is less than 1 day [Brink andLentz, 2010; Allen and Durrieu de Madron, 2009] becauseslopes are larger than 0.022 (assuming a drag coefficient of0.0029 following Brink and Lentz [2010]). Thus theseboundary layers do not carry flow across isobaths and withsufficient resolution one can show they occupy only a smallproportion of the volume [see Dawe and Allen, 2010,Figure 11]. Note, however, that the boundary layer on theshelf is not arrested and its impact on the shelf flow can beconsiderable [Jaramillo, 2005; Boyer et al., 2006]. Here wetake the shelf flow as a given, measured, value. Lastly,although the separation of the slope boundary layer isthought to bring vorticity into the canyon [e.g., She andKlinck, 2000], the amount of vorticity created by stretchingis sufficient to account for all the vorticity measured in thefield [Hickey, 1997; Allen et al., 2001]. In addition, thedependence of vorticity in the canyon with stratification isconsistent with stretching being the dominant mechanism ingenerating vorticity [Waterhouse et al., 2009]. Thus, for thecanyons considered here: steep‐sided canyons with realisticstratification, frictionisnotexpectedto playa significant rolein the dynamics within the canyon.[16] We will assume that the flow is hydrostatic so that thevertical aspect ratio Hs/L is very small and de‐couples thehorizontal and vertical scales. This is a standard approxi-mation in oceanography and has been recently verified forsteep canyons [Dawe and Allen, 2010].[17] Having eliminated the Ekman number and the verticalaspect ratio we are left with 7 non‐dimensional governingparameters. The data are insufficient to allow an empirical fittothese7parameters.Insteadweusethedynamicalequationsto constrain the expected response and then use the data forverification. The assumptions that the flow is hydrostatic andthat friction can be neglected, along with the Boussinesq,incompressible and non‐diffusive approximations are theonly a priori assumptions we will make before writing downour starting dynamic equations. However, through the anal-ysis we will make a number of restrictive assumptions on theflow being considered in order to simplify it. These aresummarized and labeled in Section 2.5. Through this processwewillfindthat2oftheremaining7parameters playnorole.Thecanyonsunderconsiderationwillbeassumeddeep(DeepCanyon Assumption, see section 2.5), so that the upwellingflow does not reach the bottom of the canyon and thus theALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080183of20canyon is deep enough that the total depth of the canyon Hcand its non‐dimensional form Bcplay no role. We will alsoassume that the shelf is shallow so that Bsis less than about 2(Shallow Shelf Assumption, see section 2.5). This restrictionwill be explained in detail in section 2.1.1; it effectivelyeliminates a role for Bs.[18] The upwelling considered in this paper is “steady”advection‐driven upwelling as opposed to the transientupwelling that occurs in the initial stages of an upwellingevent. The shallow water equations for steady flow of aninviscid stratified fluid are~uC1r~uC0f^k C2~u ¼C01C26orp ð1aÞrC1~u ¼ 0 ð1bÞ~uC1rC26 ¼ 0 ð1cÞ@p@z¼C0C26g ð1dÞwhere ~u is the horizontal velocity,^k is the vertical unitvector, p is the pressure, rois a constant reference density, ris the density and g is the gravitational acceleration. Thehydrostatic, Boussinesq, incompressible and non‐diffusiveapproximations have been made.[19] Taking the vertical component of the curl ofequations (1a) gives a vorticity equation [e.g., Holton, 1992]~uC1rC16 þ C16 þfðÞrhC1~uhþrhwC2@@z~uh¼ 0 ð2Þwhere ~uh, w are the horizontal components and verticalcomponent of the velocity, respectively, rhis the horizontaldivergence and z =^k ·r×~u is the vertical component of thevorticity. The vorticity equation (2) consists of three terms,an advection term, a stretching term and a twisting term. Thelatter term corresponds to the generation of vertical vorticityby “twisting” horizontal vorticity into the vertical. As theincoming flow is usually vertically sheared (stronger near thesurface), the incoming horizontal vorticity is directedonshore. However, the variations in vertical velocity occur asthe flow crosses the canyon and are thus mainly directedcross‐canyon. Thus the cross‐product will be small and wewill neglect this term. Substituting equations (1b) andequations (1c) one can derive a potential vorticity equation~uC1r f þC16ðÞ@C26@zþ@C26*@zC18C19C20C21¼ 0 ð3Þ[20] The description in the introduction illustrates that themostimportant characteristics oftheflowwithin andoverthecanyon include: (1) the depth of upwelling; (2) the upwellingflux; (3) the presence or absence of a rim depth eddy; and (4)the vorticity of the deep flow. These characteristics areconsidered below in terms of similarity theory. A notationlist is included for reference as needed.2.1. Deepest Water Upwelled Onto the Shelf[21] To determine the depth of the deepest water upwelledonto the shelf, we first estimate the strength of the flowcrossing the canyon (that is, the tendency of the flow tocross the canyon rather than follow the isobaths). Then wedetermine the acting pressure gradient, and finally calculatethe resulting deformation of the density field.2.1.1. Tendency of the Flow to Cross the Canyon[22] Upwelling flow through a canyon is driven by thecross‐shelf pressure gradient at the depth of the rim[Freeland and Denman, 1982; Klinck, 1989; Allen, 1996].The pressure field can be modified significantly by thepresence of the canyon. To quantify the magnitude of thepressure gradient in the presence of the canyon considerthe rim depth flow. We choose a curvilinear coordinatesystem that follows the shelf break isobath around the rim ofthe canyon. Far upstream of our (assumed isolated) canyon,the flow is rectilinear along the straight isobaths. Horizontalflow across the isobaths and vertical flow are zero. Flowalong the isobaths is geostrophic and the speed is uniform.We will assume this uniform flow continues along the iso-baths as it approaches the canyon and that it has a magnitudeof U.[23] The shelf break isobath at the upstream corner of thecanyon is approximated as an arc of radius R (Figure 2).Using polar coordinates around the projected center of thiscircle the vorticity z expands asC16 ¼@uC18@rC01r@ur@C18þuC18rð4Þwhich scales asO C16ðÞ¼URC0VRC25=2ð5Þwhere U is the along‐isobath velocity scale, V is the cross‐isobath velocity scale and p/2 ≈ 1 is the approximate angle ofrotation. The positive curvature (uC18/r) has been assumedlarger than the negative term ∂uC18/∂r because upstream theflow isuniform (Uniform Flow Assumption, seesection2.5).[24] The potential vorticity equation (3) is essentially abalance between advection of vorticity (the first term) andstretching (the second term). Consider following a stream-line from upstream, where the flow is assumed uniform, tothe canyon where it crosses or follows the topography.Upstream the vorticity is zero and the density perturbation iszero so the potential vorticity is f∂r*/∂z. Thus as the flowcrosses the topography the vorticity will be given byC16 ¼ f@C26*=@z@C26=@zþ@C26*=@zC01C18C19ð6ÞDiscretizing (6) one can writeC16 C25 fC261C0C26bzu1C0zubC18C19z1C0zbC261C0C26bC18C19C01C20C21ð7Þwhere we take rbas the isopycnal just above the bottomboundary layer and r1as a second isopycnal at some depthabove the topography where the flow is generally unaffectedby the topography. The depth z1and zbare the depth of r1ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080184of20and rb, respectively and z1uand zbuare the depths of theisopycnals upstream of the canyon. If the upper isopycnalr1is far enough above the topography that it is unaffectedz1= z1u. Writing the original distance between the iso-pycnals as h = z1u− zbuand the change in depth of rbasDh = zbu− zb, then (6) becomesC16 C25 fDhhð8Þwhere we have assumed Dh < h.[25] The change in depth Dh experienced by a column asit follows the streamline (off the lip of the canyon) is Dr∂h/∂r. And Dr is vDt and Dt is RDC18/u soC16 ¼ fDhh¼ fvuRDC18h@h@rð9ÞScaling this equation givesURC0VR¼ fVURh@h@rð10ÞorU C0VðÞ¼VRoð11ÞwhereRo¼UfRh=R@h@rð12Þ[26] The depth h is the vertical length scale over whichisopycnals above the canyon are affected by the canyon. Ifthe depth above the shelf break was infinite, this scale wouldbe given by the scale depth Dh. If the depth above the shelfbreak is very shallow, this scale would be given by the shelfbreak depth Hs. The ratio of the shelf break depth to thescale depth (Bs) varies from 0.5 to 1.8 for the canyonsconsidered here. Barkley Canyon with the 1997 stratifica-tion has the second highest value at 1.4. For this case wehave clear evidence that isopycnals were affected up to thebottom of the mixed‐layer at 10 m below the surface [Allenet al., 2001]. We have no detailed data for Tidra Canyonwith the highest ratio. Here we will assume that the strati-fication is weak enough, or alternatively, the shelf breakdepth is shallow enough, that isopycnals are affected closeto the ocean surface. This Shallow‐Shelf Approximationshould hold for Bs< 1.4 based on the Barkley Canyonobservations and it may hold for higher values. Thus wetake h = Hs.[27] The rate of change in depth of topography h/(∂h/∂r)=Hs/(∂h/∂r) is one over a length scale. We could carry thisscale through our calculations but it is generally similar tothe curvature R. For example, the ratio of the topographiclength scale to the curvature is about 0.9, 0.8, 0.7 and 1.0 forAstoria Canyon, Barkley Canyon, Carson Canyon and thelaboratory canyon used by [Allen et al., 2003], respectively.Thus we assume that the change in depth of the topographyscales similarly to the curvature and Ro= U/fR (RegularShape Assumption, see section 2.5).[28] A simple balance in (11) occurs for both weakincoming velocities RoC28 1 where V / RoU and for strongincoming velocities RoC29 1 where V ≈ U. To exactly esti-Figure 2. Plan view of the isobaths of Barkley Canyon showing the location of the shelf break, thelength L,thewidthW, the width at the shelf break Wsband the radius of curvature R for this canyonas an example. One over this radius multiplied by the incoming flow and divided by the Coriolis param-eter gives the Rossby number which determines the ability of the incoming flow to follow the topography.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080185of20mate the in‐between scaling for V it would be necessary toknow the relative size of the coefficients in front of theterms in (11). Without this information we can write thescaling with two unknown coefficients, both expected to beof order 1. So V = F(Ro)U whereF RoðÞ¼c1Roc2þRoð13Þ[29] Note that if the stratification is not weak or the shelfbreak depth is not shallow (Bs> 2), Ro= U/fR should bereplaced with Ro= U/(fRBs)=U/(NHs)(L/R) which is aFroude number rather than a Rossby number.2.1.2. Rim Depth Pressure Gradient[30] Now that the tendency for the flow to cross thecanyon has been determined, the pressure gradient along thecanyon can be quantified. Returning to equations (1a) andusing polar coordinatesur@uC18@rþuC18r@uC18@C18þuC18urrþfur¼C01C26or@p@C18ð14ÞThe first term is small compared to the third [see discussionbelow equation (5)]. By conservation of volume 1/r∂uC18/∂C18 =−∂w/∂z −∂ur/∂r scales as −V/R and so the second twoadvection terms scale similarly as UV/R but with oppositesigns. For strong flows their sum can be shown to be zero;for weak flows they are negligible compared to the Coriolisterm. Thus we neglect them and take the pressure gradientforce as scaling as fV= fUF and the pressure gradient alongthe upstream rim edge of the canyon as −rofUF. We notethat here we have implicitly assumed near uniform incomingflow over the shelf for the length of the canyon (UniformFlow Assumption, see section 2.5).[31] The relationship between the pressure gradient andthe Rossby number can be understood physically. For verywide canyons the cross‐shelf pressure gradient is modifiedover the canyon so that the flow can smoothly follow theisobaths. A measure of the ability of the flow to follow theisobaths is given by the Rossby number (Ro= U/fR) basedon the ratio of the required acceleration of the flow to turnonshore at the upstream side of the canyon to follow theisobaths (U2/R where R is the turning radius, Figure 2) tothe Coriolis force fU.[32] If the flow passes directly across the canyon (largeRo), the along‐canyon pressure gradient is identical to thatupstream of the canyon which, assuming geostrophy, isrofU. The actual pressure gradient along the canyon at therim level is expected to be O[rofUF(Ro)] where F is anappropriate function (as derived above). Physically one canargue that for very small Rossby numbers, the flow willfollow the isobaths and the pressure gradient along thecanyon will be negligible (F small). For very large Rossbynumbers the flow will cross directly over the canyon and thepressure gradient isexpected tobe close torofU(F order 1).[33] The pressure difference between the canyon mouthwhere the canyon opens to the deep ocean (Figure 2) and thecanyon head where the canyon ends on the shelf (Figure 2)is the pressure gradient multiplied by an appropriate length.Because advection onto the shelf occurs at depths near rimdepth, the length L of the canyon is taken as the length fromthe shelf break (canyon mouth) to the last isobath that issignificantly deflected by the canyon (the canyon head).2.1.3. Deformation of the Density Field[34] To determine the deformation of the density field weuse the natural coordinate system [e.g., Holton, 1992] wheres is along the horizontal direction of the flow and n is acrossit. Note that this choice implies that the flow in the directionn is zero. The equations (1) becomeu@u@s¼C01C26o@p@sð15aÞu2Rþfu ¼C01C26o@p@nð15bÞ@u@sþ@w@z¼ 0 ð15cÞu@C26@sþw@C26@z¼ 0 ð15dÞ@p@z¼C0C26g ð15eÞwhere R is the radius of curvature of the streamlines. Byconvention, R is taken to be positive when the curvature isto the left of the flow.[35] Consider the deepest streamline which crosses the rimof the canyon. It comes from a depth Hh+ Z at the mouth ofthe canyon and rises a distance Z to reach the depth of thehead of the canyon Hh. Along this streamline density isconserved (15d). Equation (15d) scales asU*GLC0WC26oN2g¼ 0 ð16Þbecause the dominant vertical gradient of density is theundisturbed gradient ∂r*/∂z. The Brunt‐Väisälä frequency isN where (N2= −g/ro∂r*/∂z), G is the scale of horizontaldensity perturbations, L is the length scale of the streamline,U*is the scale of the horizontal velocity of the upwellingstream and W is the vertical velocity scale.[36] By conservation of mass (15c) the horizontal andvertical velocity and linear scales are related according toU*LC0WZ¼ 0 ð17Þ[37] This streamline marks the depth where the totalpressure gradient is zero. The pressure change along thecanyon that drives upwelling was estimated above asrofUFL and this must balance the density gradient due tothe perturbation density. Scaling (15e) givesC26ofUFLZ¼Gg ð18ÞALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080186of20[38] Combining (16), (17) and (18) allows us to solve forthe three scalesZ ¼fULFN2C18C191=2¼ DHFRLðÞ1=2ð19ÞW¼ U*fUFN2LC18C191=2¼U*DHLFRLðÞ1=2ð20ÞG¼C262ofUFN2Lg2C18C191=2¼@C26*@zDHC18C19FRLðÞ1=2ð21Þwhere DH= fL/N and RL= U/(fL). The scale Z is the depthof upwelling, a quantity that will be compared with ob-servations in the subsequent section. Note that here we haveimplicitly assumed a near uniform N value over a depthrange Z below the depth of the head of the canyon (UniformStratification at Rim Depth Approximation, see section 2.5).[39] One can also derive the scale for Z from physicalarguments. Consider the water which passes through themouth of the canyon and is upwelled onto the shelf. Thedeepest isopycnal that is upwelled onto the shelf is that onewhich just touches the rim of the canyon at the canyon head(depth Hh). At the canyon mouth, this isopycnal has depthZ + Hh. At this depth there must be a balance in the canyonsuch that the pressure at the mouth of the canyon is the sameas that at the head (Figure 3). The tilted isopycnals provide abaroclinic pressure gradient to balance the rim depth pressuregradient so that below this isopycnal the flow does not gofrom the canyon up onto the shelf.[40] Assuming that the average tilt of the isopycnals is T =Z/L, the density difference along the canyon (length L)is∂r*/∂zT L. Thus the baroclinic pressure difference along thecanyon due to the tilted isopycnals (integrating over thedepth Z)isO(roN2Z2). Equating this baroclinic pressurechange and the pressure change at rim depth (fUroLF)gives equation (19).2.2. Upwelling Flux2.2.1. Speed of the Upwelling Stream[41] Quantitatively, to determine the upwelling fluxthrough the canyon, we scale (15b). This flow lies directlybelow the shelf‐depth flow crossing the canyon and it twistsmore up‐canyon as it goes deeper. However, most of theupwelling stream crosses the downstream canyon rim withonly a little flowing out the head of the canyon. The top ofthis flow is thus exposed to the same pressure gradientalong‐canyon as the shelf‐depth flow, and the lowest part ofthis flow is at Z and there is no pressure gradient. Thus theflow speed varies with depth but mid‐way through it will bedriven by a pressure gradient 1/2 the strength of the shelfbreak pressure gradient immediately above the upwellingstream which is fUF soU2*WsbþfU*¼12fUFð22Þwhere Wsbis the width of the canyon and the requiredturning radius for flow into the canyon. Writing Rw= U/(fWsb) we see that for small FRw/2 the balance is betweenthe last two terms. One would expect that the coefficients c1and c2in F would be of order 1 which will be verified laterusing the data. If we assume c1and c2are 1, for all canyonsFigure 3. Sketch to illustrate the baroclinic pressure gradient due to isopycnals (thin lines) tiltingtowards the head of the canyon. The sketch shows a cross‐section through the centreline of the canyon.The bottom‐topography away from the canyon is given by the bold dashed line. The height Z is deter-mined by balancing the pressure gradient at rim depth and the change in baroclinic pressure gradientdue to the tilted isopycnals. The depth Hh+ Z is the deepest water upwelled onto the shelf. The col-umn‐stretching deep in the canyon is given by the ratio Z/Dwwhere Dwis the vertical scale within thecanyon.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080187of20for which we have found data (Table 1), FRwis less than0.2 and so we will neglect this term (Weak to ModerateFlow Assumption, see section 2.5). Thus the upwellingstream velocity scales as UF with a coefficient expected inthe range of 1/2.[42] Physically, the upwelling stream velocity is propor-tional to the strength of the pressure gradient along thecanyon.[43] At this point it is useful to also consider the flowabove the rim of the canyon as in practice it is difficult todistinguish these two flows. In Section 2.1.1 we separatedthe shelf break depth flow into two components: U, the flowfollowing the isobaths and V, the flow crossing the isobaths.This latter flow is proportional to FU and will tend to shiftshoreward as it crosses the canyon.[44] The first component U which follows the isobathsshould return to its nominal depth on the downstream side ofthe canyon and thus should not contribute to the upwellingflux. However, if the shelf is flat there are no topographicgradients to drive this nearly geostrophic flow back towardthe shelf break. One can contrast the flow over a sloping‐shelf which has a strong off‐shore flow downstream of thecanyon [She and Klinck, 2000, Figure 6] to the flow over aflat shelf with flow from the canyon reaching the coast[Klinck, 1996, Figure 4a; Kämpf, 2007, Figure 5]. Thus for aflat shelf this component forms part of the shoreward fluxwhereas for a sloped shelf it does not.[45] We will assume the shelf is sloped. A measure of thestrength of the slope is (Hs− Hh)/Hsand this should besubstantial compared to the strength of the geostrophic flowon the shelf, given by say RL= U/fL. It is difficult to putlimits on this ratio M =(Hs− Hh)fL/(HsU) because obser-vational data does not generally include the full velocityfield. Our lowest non‐zero value is over Tidra Canyonwhere M = 0.4. The path of a single drifter above head‐depth shows flow directly along the canyon axis to the 50 misobath [Shaffer, 1976, Figure 44] which would suggest thatthis might be a flat‐shelf case. However, density sections upthe canyon and onto the shelf [Shaffer, 1976, Figure 20] arefar more similar to the sloping case [e.g., Dawe and Allen,2010, Figure 9] than they are to the flat‐bottom case [e.g.,Pérenne et al., 2001, Figure 15]. The second lowest non‐zero M value is 1.1 over Barkley Canyon. Here the ob-servations of salinity contours on a density surface showflow turning downstream at about head‐depth [Allen et al.,2001]. Thus our estimate is that a critical value for M isbelow 1 but might indeed be below 0.4 (Sloped‐shelfApproximation, see Section 2.5).2.2.2. Upwelling Flux[46] The flux up the canyon is the flux through the canyonmouth. The flow velocity scales as U*and the width is Wsbunless the canyon width is considerably wider than theRossby radius (Narrow Canyon Assumption, see section2.5). The deepest water upwelled is Hh+ Z. To calculatethe upwelling flux the vertical thickness of the upwelledstream must also be estimated.[47] At the upstream canyon rim, water from the shelf (therim depth flow) crosses the isobaths. It drops down into thecanyon and then separates from the topography. Below thisflow, the upwelling stream flows nearly parallel to the iso-baths as it has no shelf source. This flow is geostrophic tofirst order and so the pressure gradient along this line iszero. This requires a balance between the baroclinic pressuregradient due to the tilted isopycnals within the canyon andthe rim depth pressure gradient. As calculated above, at themouth, the depth at which this balance occurs is Z + Hh.Below this depth the water re‐circulates within the canyon.But this depth is also the depth of the bottom of the rimdepth flow crossing the canyon. Thus, against the upstreamrim of the canyon, the depth of the top of the upwellingstream and the depth of the bottom of the upwelling streamare the same (Figure 4). Thus on the upstream side of thecanyon the upwelling stream is pinched to zero verticalheight [see Hickey, 1997, Figure 3].[48] The rim depth flow crosses the canyon at an angle, isadvected slightly up‐canyon and crosses the downstreamrim. So at the downstream rim of the canyon, the bottom ofthe rim depth flow is rim depth. As the top of the upwellingstream is the bottom of the rim depth flow, at the down-stream rim, the top of the upwelling stream is rim depth. Atthe mouth of the canyon, the bottom of the upwelling streamis the depth of the deepest water upwelled, Z + Hh. Thus theupwelling stream forms a wedge with vertical thickness Z atthe downstream side and zero vertical thickness on theTable 1. Scales for Three Shelf Break Canyons on the West Coast of Canada and the United States, One on the East Coast ofNewfoundland, One on the Northwest Coast of Africa, and Three Model Canyons Used in Laboratory StudiesaNumber Astoria Barkley Quinault Carson Tidra Lab‐Allen Perenne Mirshak RedondoHs150 m 200 m 180 m 125 m 130 m 2.2 cm 2.5 cm 2.2 cm 80 mHh110 m 170 m 130 m 110 m 100 m 1. cm 2.5 cm 1.0 cm 40 mHc450 m 350 m 1200 m 375 m 470 m 3. cm 4. cm 3. cm 375 mU 0.20 ms−10.1 ms−10.2 ms−10.1 ms−10.15 ms−11.2 cm s−10.8 cm s−10.4–1.7 cm s−10.15 m s−1f 1.05 × 10−4s−11.08 × 10−4s−11.07 × 10−4s−11.03 × 10−4s−10.49 × 10−4s−10.52 s−10.5 s−10.4–0.7 s−10.81 × 10−4s−1L 21.8 km 6.4 km 16.6 km 16.5 km 5.1 km 8 cm 15 cm 8 cm 8.5 kmW 8.9 km 8.3 km 30.0 km 11.8 km 10.2 km 2.4 cm 8.6 cm 2.4 cm 7.0 kmR 4.5 km 5.0 km 44.0 km 18.4 km 2.7 km 1.4 cm 3.1 cm 4.0 cm 6.0 kmN 7.5 × 10−3s−15.0 × 10−3s−1(’97) 7.5 × 10−3s−19.7 × 10−3s−13.5 × 10−3s−12.2 s−12.4 s−12.2−4.4 s−19.5 × 10−3s−14.5 × 10−3s−1(’98)Wsb15.7 km 13 km 50 km 33.7 km 8.5 km 6.9 cm 20 cm 5.8 cm 8.5 kmDH305 m 138 m (’97) 237 m 175 m 71 m 1.9 cm 3.1 cm 1–2.5 cm 72 m154 m (’98)DW103 m 140 m (’97’) 334 m 174 m 52 m 0.8 cm 2.1 cm 0.3–0.9 cm 34 m156 m (’98)aA ninth shelf break canyon, which closely approaches the coast, Redondo Canyon, is included for contrast.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080188of20upstream side of the canyon. Hence for canyons narrowerthan the Rossby radius, the upwelling flux scales asF¼ U*WsbZ ¼ UWsbDHðÞF3=2R1=2Lð23Þwith an expected coefficient on the order of 1/4.2.3. Eddy Presence at Rim Depth[49] An eddy is generated at the depth of a canyon rim dueto stretching of the water column as it crosses the canyon[Hickey, 1997; Allen et al., 2001]. The rim depth flowcrosses the upstream canyon edge, feels the slope of thecanyon, and drops down to a depth Z. The thickness of thisflow is the full depth of the water column Hsbecause DHisgenerally larger than the depth of the water column. Thestretching of this flow across the canyon is given by Z/Hswhich generates a vorticity fZ/Hsassuming conservation ofpotential vorticity.[50] This cyclonic vorticity has a width given by the half‐width of the canyon W/2 and is embedded in a flow ofstrength U. If the flow velocity generated by the vorticity isequal to the background flow then the total flow will be zeroin one region. If the vorticity is stronger, closed streamlineswill form. The circulation 2p uR around a circle of radius Rdue to a patch of vorticity is equal to the area of the circlemultiplied by the vorticity. Thus the velocity due to thevorticity scales as fZW/Hsand an eddy forms if this velocityis large compared to the background velocity U; that is, ifE ¼fWUZHs¼1BuFRLC18C191=2ð24Þis large, where Bu= NHs/fWand the coefficient is expectedto be about 1/2.2.4. Deep Water Vorticity[51] Assuming conservation of potential vorticity, thestretching S of the water column multiplied by the Coriolisfrequency gives the deep cyclonic vorticity. The fluid ofinterest is that below Z to the deepest isopycnal that issignificantly elevated within the canyon. If this isopycnaloccurs at scale depth Dwbelow Z at the mouth of the can-yon, then the water column that has a length Dwat themouth of the canyon has a length Z + Dwnear the head ofthe canyon (Figure 3) and S = Z/Dw. For a geostrophicallybalanced flow we would expect a vertical scale depth givenby f‘/N where ‘ is an appropriate horizontal length scale. Weobserve that the velocity in the deep canyon is into thecanyon on the downstream side and out of the canyon on theupstream side.2.4.1. Scale Depth Deep in the Canyon[52] To quantify the scale depth Dw, we will scale theequations for the deep cyclonic circulation. Deeper than (Z +Hs), that is under the deepest isopycnal that upwells throughthe canyon, the pressure along the centre axis of the canyonis lower than at the side of the canyon, assuming the pres-sure gradient is in balance with the cyclonic circulation atdepth.[53] From (15b) and (15e) we can form the thermal windequationf@u@z¼gC26o@C26@nð25Þwhere we have assumed UD/fWsbis small and UDis thevelocity scale at this depth.[54] From conservation of potential vorticityf þuRC0@u@nC18C19@C26@z¼ f@C26*@zð26Þand so assuming ∂r/∂z ≈∂r*/∂z to first order then@u@nC0uR¼ f@C26@zC0@C26*@zC18C19@C26*@zC18C19C01ð27ÞCombining these equations gives@@n@u@nC0uRC18C19¼f2N2@2u@z2ð28ÞThe two terms on the left‐hand side have the same sign.This flow is the deep flow in/out on the downstream/upstream side of the canyon so the cross‐flow length scale isabout half Wsb. Scaling (28) we get that vertical length scaleof this flow is Dw= fWsb/2N.2.4.2. Deep Water Stretching[55] The deep stretching is of orderS ¼NZfWsb¼FRoðÞRL1=2LWsbð29Þwhere the coefficient is expected to be of order 2. Thevorticity deep in the canyon is given therefore by a com-bination of two Rossby numbers and an aspect ratio and isindependent of the stratification. This independence occursbecause both the drop into the canyon Z and the scale depthwithin the canyon Dwdecrease with stratification. Asstretching is a ratio between these depths, the vorticity isindependent of N. Note that for canyons deeper than thedepth of upwelling plus the scale depth (Dw+ Z < Hc), thetotal canyon depth is unimportant because the flow does notfeel the canyon bottom (Deep Canyon Assumption, seesection 2.5).Figure 4. Cross‐section through a canyon. The flow pat-tern is separated into three regions in the vertical: rim depthflow crossing the canyon, the upwelling stream flowing up‐canyon, and the deep flow recirculating within the canyon.The upwelling stream has a strong component into the pageand a weaker component across the canyon and up onto theshelf.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C080189of202.5. Restrictions on Cases Considered[56] During the scaling derived above we have made anumber of implicit and explicit assumptions that restrict thecanyons and flow under consideration. These assumptionsare summarized below:2.5.1. Uniform Flow[57] We have assumed the incoming flow is nearly uni-form along the length of the canyon and from the depth ofthe head of the canyon to the depth of the shelf break. Asmentioned in the introduction, this eliminates a number ofreal world canyons.2.5.2. Weak to Moderate Flow[58] We have assumed that FRw, which is approximatelya Rossby number squared, is small (<0.2). Larger flowspeed would require high order terms in the upwelling fluxcalculation.2.5.3. Uniform Stratification Near Rim Depth[59] Over a scale Z the stratification ∂r*/∂z or N has beenassumed nearly uniform. If the value of N was 20% lower ata depth Z we would expect an under‐estimate of theupwelling depth by about 10% as the back‐pressure pro-vided by the baroclinic pressure gradient would be bestestimated with the average N over the depth Z rather than thevalue at the depth of the shelf break. An error of 10% forthese scaling estimates is reasonable based on the order ofthe scatter in our empirical fits to observations. This con-straint is met by most canyons as typically the pycnocline ismuch shallower than shelf break depth. For Barkley Canyonthe variation in N is about 12%, for Astoria it is larger at18% mainly due to the larger value of Z.2.5.4. Shallow Shelf Break Depth[60] We have assumed that the isopycnals over the canyonare affected to near the surface of the ocean so that theeffective depth over the canyon is set by the shelf breakdepth. This assumes that Bsis not too large (<2).2.5.5. Sloped Continental Shelf[61] We have assumed that the continental shelf is slopedso that onshore flow is inhibited on the shelf due to its slope.This assumes that M is large: greater than 1 is a conservativelimit. If the continental shelf is flat or only weakly sloped,our analysis will under‐estimate the total flux onshore as itwill not include a branch of the incoming flow that followsthe canyon rim to the head of the canyon and is not turnedback to the shelf break by the shelf‐topographic slope.2.5.6. Steep Canyon Walls[62] We have assumed friction is not important as thebottom boundary layer will have arrested within the canyonand over the continental slope. For a given canyon, thisconstraint can be evaluated using the Brink and Lentz [2010]criteria.2.5.7. Deep Canyon[63] For the upwelling flux scaling we have assumed thedepth of the canyon is much greater than Z. For the deepvorticity scaling we have made the stronger assumption thatthe scale depth, Dwis of order of, or smaller than, the depthof the canyon.2.5.8. Narrow Canyon[64] We have assumed, in the upwelling flux scaling, thatthe canyon width is narrower than about 2 Rossby radii, a.For wider canyons, 2a should replace Wsbas the width scalefor the flux.2.5.9. Regular Shape at Upstream Corner[65] We have assumed that the topographic length scalenear the upstream corner of the canyon, Hs(∂h/∂r)−1,issimilar to the radius of curvature of the isobaths, R. Thisratio is very close to one for all the canyons considered herebut may eliminate certain types of canyons.3. Scaling Applications[66] In the following sections, the scaling analysis isapplied to eight different canyons, five real world and threelaboratory models. For each canyon, the maximum depth ofupwelling, presence of a rim depth eddy and the deep can-yon vorticity are estimated from available data. Results arecompared with estimates derived from the scaling argumentsand the fit between the observed and scaled results is used toestimate the scalar coefficients for the scaling factors forupwelling depth, eddy presence and deep vorticity. Forcontrast we also give results for Redondo Canyon whichclosely approaches coast and therefore can be expected tohave somewhat different dynamics [Allen, 2000]. We usethe information from the flux estimates of a single labora-tory canyon study to compare with upwelling flux [Mirshakand Allen, 2005]. Geometric and flow characteristics foreach canyon are given in Table 1. Geometric parameters arederived from bathymetric maps; flow velocity is for nearrim depth and is estimated from nearby current meters; andthe Brunt‐Väisälä frequency is also for near rim depth and isestimatedfromnearbyCTDcasts.Non‐dimensionalnumbersfor the nine canyons are given in Table 2.[67] Four of the canyons are located on the U.S. orCanadian west coast. One canyon is on the northwest coastof Africa and one is on the east coast of Newfoundland. Thefirst five are subject to intermittent upwelling conditionswhereas Carson Canyon off the coast of Newfoundland onlyoccasionally receives upwelling‐favorable flow. The shelfwidth varies from about 30 km to 60 km. Redondo Canyonextends closest to the coast (2 km). Widths vary from about8 km (Barkley) to 30 km (Quinault). A variety of shapes areTable 2. Dimensionless Parameters for Six Shelf Break Canyons and Three Laboratory Model CanyonsaNumber Astoria Barkley Quinault Carson Tidra Lab‐Allen Perenne Mirshak RedondoRo0.42 0.19 0.04 0.05 1.1 1.6 0.52 0.22–0.93 0.31F 0.32 0.17 0.05 0.06 0.56 0.65 0.36 0.20–0.51 0.26RL0.09 0.14 0.11 0.06 0.60 0.29 0.11 0.11–0.47 0.22Rw0.12 0.07 0.04 0.03 0.36 0.33 0.08 0.15–0.64 0.22Bu1.2 1.1 (’97) 0.42 1.0 0.91 3.9 0.72 2.9–7.6 1.31.0 (’98)aA ninth shelf break canyon, Redondo Canyon, which closely approaches the coast, is included for contrast.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801810 of 20represented, from the squat Barkley canyon to a long, thinlaboratory canyon.3.1. Available Data[68] Astoria canyon is a narrow, deep canyon (9 km by600 m) situated off the U.S. west coast just offshore of theColumbia River (Figure 5). The walls of Astoria canyon aresteep, approaching 45° in some locations. The canyon isrelatively symmetrical in shape and is aligned such that thecanyon axis is roughly perpendicular to the direction of thelocal isobaths. Since the shelf circulation is quasi‐geostrophic,the regional flow follows the northwest to southeast directionof the local isobaths to intersect the canyon axis at roughlyright angles [Hickey, 1989]. An 18 element moored velocity/temperature array deployed for several months during theearly upwelling season as well as CTD/transmissometersurveys during one strong upwelling event have been used toprovide valuable “ground truth” data for model studies ofsubmarine canyons [e.g., Klinck et al., 1999]. The data usedin this study were all obtained prior to the development of apoleward undercurrent at the shelf break depth [Hickey,1997]. The data has proved especially valuable becausemoorings were placed at closely spaced (≈2–5 km) intervalsboth along and across the canyon (Figure 5). The data wereused to provide time‐ and space‐dependent estimates ofverticalvelocityandofrelativeandstretchingvorticity.Theseestimates were compared with results from available modelsas well as with along‐shelf wind and velocity incident on thecanyon to provide a detailed description of time variableupwelling within a canyon [Hickey, 1997].[69] Barkley Canyon is situated off the southern end ofVancouver Island (Figure 5). The shelf in this region is wide(60 km) and shelf break depth deep (200 m). In this region,shelf waves propagating from the south can have a strongereffect on long‐shore currents than the local wind [Hickey etal., 1991]. The California Undercurrent is somewhat deeper(core at 400–500 m depth) off Vancouver Island than it isfurther south and it does not penetrate onto the shelf nearBarkley Canyon [Krassovski, 2008]. Barkley Canyon isshort (17 km) for its width (8 km) (Figure 6). Incident flowis approximately perpendicular to the canyon axis. In 1997four current meter moorings were deployed in the canyon(Figure 6) and a detailed CTD survey of the region wasperformed [Vindeirinho, 1998; Allen et al., 2001]. The CTDsurvey extended both upstream and downstream from thecanyon which allowed the use of diagnostic model todetermine the mean currents around the canyon [Allen et al.,2001]. These currents clearly show a strong, closed eddy atrim depth (Figure 6). The study was repeated in 1998.Conditions differed from 1997; a summer storm of maxi-Figure 5. Bathymetry of Astoria Canyon. Inset shows the locations of Astoria Canyon, BarkleyCanyon, and Quinault Canyon. Mean currents above and below the rim are shown as solid and dottedarrows, respectively. Note the tendency for the near‐surface flow above 100 m to pass unaffected overthe canyon, the rim depth (150‐m) flow to go up the canyon and the deep flow to form a possibly closedcyclonic gyre. The two CTD stations used for estimating the depth of upwelling are marked with trian-gles.Thesolidstraightlinemarksadetailedhydrographicsurveylinealongwhichstretchingwascalculated.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801811 of 20mum wind speed 14 m s−1on Jul 14 reversed upwellingcurrents from Jul 11–21, several days before the CTD sur-vey over the canyon which was completed on Jul 25. Thelack of a synoptic CTD survey meant the diagnostic modelcould not be used for 1998.[70] Quinault submarine canyon is a broad, relatively deepcanyon (≈30 km by 1200 m) located off the Washingtoncoast 60 km north of Astoria canyon (Figure 7). Quinault issubject to roughly the same environmental conditions asAstoria canyon because both wind stress and ambient cur-rents are large scale in this region [Hickey, 1989]. QuinaultCanyon was studied with current meters and sediment trapsprimarily placed along the axis and with CTD surveys alongand across the canyon [Hickey, 1989].[71] A scaled physical model of a shelf/slope geometryincluding a canyon in a stratified density field was con-structed in the laboratory (Figure 8) [Allen et al., 2003]. TheLab‐Allen canyon shelf break depth is 2.2 cm and the depthat the headof the canyon is 1 cm. The canyon is 7 cm wide,8 cm long and strongly tapered towards the head. The flowwasforcedbychangingtherotationrateofthetank.Theflowwas observed using neutrally buoyant particles with theposition of the particles noted at intervals and velocities cal-culated from position differences. Other than the initial den-sity distribution (determined by the filling parameters) thedensity distribution was unknown so that all parameters wereestimated from the measured velocity field. Particle depthwas estimated in two ways. First the particles were illumi-nated by colored light using different colors above and belowthe shelf break depth. A more precise measure of the particledepth was obtained from a numerical model (S‐CoordinateRutgersUniversityModelVersion3.1(SCRUM)[Hedström,1997]) configured to accurately reproduce the laboratoryexperiment.[72] A second, similar but not identical canyon (Lab‐Mirshak), was used in the same shelf/slope topography toestimate the net effect of a canyon on spin‐down [Mirshakand Allen, 2005]. The change in spin‐down can be writtenas an extra drag caused by the canyon and assuming geo-strophic flux one can estimate an on‐shore flux through thecanyon ofYo¼FDC26FLð30Þwhere FDis the drag force [Mirshak and Allen, 2005].Results from these experiments will be used to verify theflux scaling.[73] Carson canyon is a wide canyon (34 km) of moderatelength (16 km) on the east coast of Newfoundland. Typicalcurrents are downwelling favorable but in June 1981 theeffects of a current reversal were documented by threecurrent meters moored over the canyon [Kinsella et al.,1987].[74] A detailed CTD survey of a canyon off Tidra, North‐west Africa shows the effect of strong upwelling flows[Shaffer, 1976]. Tidra is a small canyon (width 10 km,length 5 km) with a sharp upstream corner (R = 2.7 km) atlow latitude (20°N) so moderate flows (0.15 m s−1) lead tohigh Rossby numbers.[75] A laboratory study of impulsively started flow over acanyon was performed at the Arizona State Universityrotating table facility [Pérenne et al., 2001]. Their canyon iswider and shorter than the Lab‐Allen canyon used by Allenet al. [2003]. It is also less tapered with width at half‐lengthalmost 50% of the width at the shelf break. Velocity datawas collected at three depths: above, at and below rim depth.This canyon will be referred to as “Perenne‐Lab”.[76] Redondo submarine canyon is a narrow canyon withsteeply sloping sides (≈2 km wide by 500 m deep) thatprojects into the Santa Monica Bay shelf (Figure 9). Itextends to within 2 km of the coast. The flow is predomi-nantly equatorward over the canyon in the upwelling season[Hickey et al., 2003].3.2. Comparisons of Scaling to Observations3.2.1. Upwelling Flux[77] As the experiments of Mirshak and Allen [2005]provide the largest number of data points and thus providethe best estimate of the function F, we will begin bycomparing the observed upwelling flux to the scaling esti-mate. We fit the non‐dimensional observed upwelling fluxFoUWsbDh¼FDC26fLUWsbDhð31Þto the non‐dimensional scaled fluxFUWsbDh¼F3=2R1=2L¼c1Roc2þRoC18C193=2R1=2Lð32ÞAs we have no direct estimates of F, it is not possible todetermine c1; it always appears multiplied by an unknownconstant (in this case the scale for the flux). Hence we willassume c1= 1 and absorb it into the coefficients for theparameters that can be estimated.Figure 6. CTD stations (pluses) and moorings (asterisks)around Barkley Canyon in July 1997. The thick solid lineis a closed streamline at 150 m depth from a diagnosticmodel inversion of the density field. This streamline showsthe boundary of a closed eddy about 50‐m above rim depth(200‐m). The moorings were in the same position in 1998.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801812 of 20[78] Using the Levenberg‐Marquandt algorithm fornon‐linear least‐squares optimization, the observed non‐dimensional flux (31) was fit toc3c1:52Ro1þRo=c2C18C193=2R1=2Lð33Þto give c2= 0.9 and c3= 1.0 (Figure 10) with an root meansquare error of 26%. The value of c2is reasonably wellconstrainedbythedata;avalueof0.8leadstoa33%erroranda value of 0.95 leads to a 35% error.3.2.2. Depth of Upwelling[79] The depth of the deepest upwelling was estimated foreach canyon in different ways, depending on the dataavailable for each canyon. Barkley Canyon[80] To estimate the deepest depth of upwelling overBarkley Canyon in 1997 the densest water observed over theshelf (stations BCD2 (150 m depth) and BCC2 (172 mdepth)) was compared to a CTD cast at the mouth of thecanyon (BCC4 (772 m depth) (Figure 6)). In 1997, thedensest water on the shelf, 26.64 st, was observed inthe bottom boundary layer at BCC2 right at the head of thecanyon. This water occurred at a depth of 226.5 m at theFigure 7. (top) Bathymetry of Quinault Canyon. Solid dots mark the location of CTD stations. Arrowsgive the geostrophic velocity (0/500 db) during an upwelling event. Data have been extrapolated intoregions shallower than the reference level using the method of Montgomery [1941]. Geostrophic currentsperpendicular to the CTD transects show the tendency for the flow to follow local isobaths. (bottom)Geostrophic velocity (0/1000 db) across Quinault Canyon during an upwelling event. Off shelf flow isshaded. The data are consistent with the presence of a rim eddy over the canyon and cyclonic vorticity deepin the canyon.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801813 of 20canyon mouth and thus the deepest water upwelled onto theshelf came from deeper than 226.5 m. Similarly the deepestdepth of upwelling was found from CTD sections for AstoriaCanyon, Quinault Canyon, Barkley Canyon in 1998, CarsonCanyon and the canyon off Tidra. Astoria Canyon[81] The Astoria Canyon calculation is based on a stationat the mouth and a station at the head taken during anupwelling event in May 1983 (Figure 5). Quinault Canyon[82] The Quinault Canyon calculation was based on asurvey in May, 1979 that fortuitously took place during anupwelling event. Carson Canyon[83] For Carson Canyon the depth of upwelling wasestimated from the observed rise of the 1°C isotherm[Kinsella et al., 1987, Figure 5]. Tidra Canyon[84] For the canyon off Tidra it was estimated from thedepth of the 26.7–26.8 staveraged from stations 123, 124and 140 versus mouth stations 132 and 125 [Shaffer, 1976,Figures 14, 17, and 20].Figure 8. Bathymetry of the Lab‐Allen canyon. All dimensions are in cm. The tank is cylindrical (radius50 cm) with the coast and shelf running around the outside. The deep ocean is toward the center of thetank. The diamonds mark the sequential position (0.5 s apart) of tracers in the flow. The last position ismarked by a square. All the tracers shown were at 1.5 cm to 1.6 cm depth (above the shelf break depth of2.2 cm). The tracers show coastward flow over the downstream edge of the canyon consistent withupwelling. The tracers do not show any evidence of an eddy over the canyon.Figure 9. Bathymetry of Redondo Canyon. Solid and dot-ted arrows mark the currents below and above the rim of thecanyon, respectively on May 3, 1988 during a strongupwelling event. Upper water column flow (46, 50, and100 m) is directed across the canyon with little apparentinfluence by the canyon. A cyclonic circulation patternoccurs at and below rim depth (100 m; 130 and 150 m).ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801814 of Lab‐Allen Canyon[85] For the Lab‐Allen canyon [Allen et al., 2003] thedepth of upwelling was estimated from the depth of thedeepest particle that was observed to upwell onto the shelf(1.9 cm). Extrapolating to the mouth of the canyon thisparticle would have originated from a depth of 2.0 cm. Perenne‐Lab Canyon[86] For the Perenne‐Lab canyon the depth of upwellingcould not be determined beyond that it was less than 5 cmdepth based on flow patterns at that depth [Pérenne et al.,2001, Figure 10a]. Redondo Canyon[87] For Redondo Canyon the depth of deepest upwellingwas derived from time series of temperature within and overthe canyon. Specific upwelling events have been identifiedin previous research [Hickey et al., 2003]. Results[88] The observed deepest depth of upwelling is in rea-sonable agreement with predictions from the scaling anal-ysis (Figure 11). The farthest departure from the line isRedondo Canyon where we expect different dynamics tohold due to its very close (≈2 km) approach to the coast. Thelinear fit of the observed to the predicted values (neglectingRedondo Canyon) givesZDH¼ 1:3 FRLðÞ1=2þ0:06 ð34Þwith an average fractional error of 21%. The intercept is tobe expected; even in the absence of a canyon, upwelling isoccurring along the coast and this effect is not removed fromthe observations.[89] The value measured from particle tracking (Lab‐Allen Canyon) appears to be lower than the values measuredfrom CTD profiles. The particle tracking method provides alower limit as the whole water column is not sampled. Theupwelling depth estimated from the current meter array atRedondo Canyon is higher than values for the other canyons.Water from 80 m is upwelled to 40 m in Redondo Canyonconsistent with the stronger upwelling expected in canyonswhich closely approach the coast [Allen, 2000].3.2.3. Deep Vorticity[90] Depending on what data was available for each can-yon, deep vorticity within the canyons was estimated eitherfrom current meter records or by estimating stretching fromdensity records. Barkley Canyon[91] The deep vorticity over Barkley Canyon in 1997 wasestimated from CTD profiles at station BCC4 at the mouth ofthe canyon and at station BCC3 (630 m depth) in the centreof the canyon. The stretching was calculated for isopycnals0.1 stapart and averaged over the stretching region (215 mto 345 m). For 1998, the vorticity was based on the averageof two stations in the canyon versus the average of a stationupstream and downstream of the canyon. Astoria Canyon[92] For Astoria Canyon, deep stretching vorticity wasestimated using upstream hydrographic data compared to in‐canyon data to estimate changes in layer depth (Figure 12).Calculation of relative vorticity using the shear betweencurrent meters at similar depths gave a much smaller valueof 0.2f rather than about 1.0f. However, these current meterswere at 230 m and thus perhaps were too shallow to samplethe strongest stretching seen at 300 m depth. Quinault Canyon[93] On the other hand for Quinault Canyon, deep vor-ticity was estimated using along‐canyon geostrophic flowon the two sides of the canyon during an upwelling event inMay 1979 as well as the data shown in Figure 7 from anupwelling event in October 1981 [Hickey, 1989]. Lab‐Allen Canyon[94] For the Lab‐Allen Canyon, the vorticity deep in thecanyon was estimated from the shear between two deepparticles, one on each side of the canyon.Figure 10. Flux through the canyon, observed versus thepredicted scaling. The straight line (slope 1.0) is the best fit.Figure 11. Depth of upwelling, observed versus the pre-dicted scaling. The values are labeled: A‐Astoria, B‐Barkley(data from years 1997 and 1998), C‐Carson, L‐Lab‐Allen,R‐Redondo, Q‐Quinault, and T‐Tidra Canyon. Note thestronger upwelling observed in Redondo Canyon whichclosely approachesthe coast and thus has different dynamics.The straight line (slope 1.3) is the best fit for the othercanyons.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801815 of Perenne‐Lab Canyon[95] For the Perenne‐Lab canyon it was estimated fromflow patterns at 5 cm depth [Pérenne et al., 2001, Figure 10]. Redondo Canyon[96] For Redondo Canyon the deep vorticity was esti-mated from current meters at 130 and 150 m on oppositesides of the canyon (Figure 9). Results[97] The estimated deep vorticity shows a linear rela-tionship with the observed deep vorticity for the four can-yons excluding Redondo Canyon (Figure 13). The linear fitof the observed to the predicted values (neglecting RedondoCanyon) givesS ¼ 2:5 FRLðÞ1=2LWsbþ0:3 ð35Þwith an average error of 16%. The intercept is to beexpected; even in the absence of a canyon, upwelling andthus stretching is occurring along the coast and this effect isnotremoved from theobservations. Redondo Canyonhastheweakest deep vorticity, again consistent with its distinctlydifferent dynamics.3.2.4. Eddy Presence3.2.4.1. Barkley Canyon[98] In 1997, the presence of a rim eddy over BarkleyCanyon at 100 to 150 m depth is apparent in the currentsderived from a diagnostic model [Allen et al., 2001]. Theeddy has a radius of 5 km and peak velocities of 30 cm s−1with a vorticity of 0.3 f (Figure 6). The eddy was observedin 1998 using current meter observations. Average currentsfor Jul 25 show flow generally southward over the canyon at150 m depth but the four current meters (Figure 6) clearlyshow a cyclonic eddy at 250 m depth. Astoria Canyon[99] For Astoria canyon, the presence of a rim eddy dur-ing upwelling was deduced from available current meterdata near the rim (Figure 5). Quinault Canyon[100] Over Quinault canyon, the presence of a rim eddywas deduced from geostrophic velocities across the canyonduring an upwelling event in October 1981 (Figure 7). Carson Canyon[101] An eddy is evident around Carson canyon fromcurrent meter observations [Kinsella et al., 1987, Figure 8]. Tidra Canyon[102] A 100‐m deep drifter over the canyon off Tidrashows a rim depth eddy [Shaffer, 1976, Figure 44]. Lab‐Allen Canyon[103] The particle trajectories over the Lab‐Allen Canyonclearly show that there is no rim‐level eddy (Figure 8). Perenne‐Lab Canyon[104] However, the broader Perenne‐Lab canyon doeshave an eddy [Pérenne et al., 2001, Figure 7]. Redondo Canyon[105] The presence of a rim eddy over Redondo Canyonwas deduced from moored array data on two sides of thecanyon (Figure 9). Results[106] The value of the eddy number for all 8 canyons isgiven in Table 3. All real canyons and the Perenne‐Labcanyon had a rim depth eddy whereas the Lab‐Allen canyondid not. Redondo Canyon, which closely approaches thecoast, appears to fit the theory though this is probably for-tuitous. The observations imply that the critical value of theFigure 12. Stretching vorticity across Line 1 over AstoriaCanyon (Figure 5) relative to a station upstream. Units aref. Shading marks cyclonic vorticity. Note the strong cyclonicvorticity deep in the canyon. Even stronger cyclonic vor-ticity is seen at rim depth in the vicinity of the observed rimdepth eddy. From Hickey [1997].Figure 13. Deep vorticity, observed versus the predictedscaling. Labeling as in Figure 11. Note that the weakestvorticity is observed in Redondo Canyon illustrating thedifferent dynamics over this canyon. The straight line (slope2.5) is the best fit for the other canyons.Table 3. Eddy Likelihood From Scaling Versus the Observationof the Presence or Absence of a Rim Depth EddyaCanyon E Rim EddyAstoria 1.5 yesQuinault 1.4 yesPerenne 1.2 yesBarkley ’98 1.0 yesTidra 0.93 yesBarkley ’97 0.92 yesCarson 0.92 yesRedondo 0.76 yesLab‐Allen 0.33 noaHere E is eddy likelihood from scaling.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801816 of 20eddy number for the presence of an eddy is greater than 0.33and less than 0.92. So an eddy will be present if E ≥ 0.9.[107] The parameter E is primarily determined by theinverse of the Burger number fW/NHs. Laboratory experi-ments have often been configured to match a field BurgerNumber based on the length of the canyon or the shelf breakwidth. As laboratory canyons are often strongly tapered, theappropriate Burger number Buis thus too high and rim deptheddies are not generated. The Pérenne experiments are anexception, with a much less tapered canyon and a smallBurger number [Pérenne et al., 2001].4. Discussion[108] The dynamics of upwelling over submarine canyonsare nonlinear and complex. Even small topographic varia-tions can cause significant changes in velocity and densityfields [Klinck et al., 1999]. However the study presentedhere shows that the basic response of the stratified fluidwithin and above the canyon can be estimated from bulkgeometric parameters. The scaling analysis was performedassuming near‐steady advection‐driven upwelling driven bythe flow that is nearly uniform over the length of the canyon.The canyon is assumed not to closely approach a coast (thedistance between the coast and the head of the canyon isgreater than 0.5(UX/f)1/2where X is the distance from thecoast to the Hhisobath away from the canyon [Allen, 2000]).These assumptions eliminate the initial transient upwellingwhich occurs as an equatorward wind event starts, and alsoeliminate from consideration canyons such as Juan de FucaCanyon (approaches coast), Monterey Canyon (regionalflow is non‐uniform) and Redondo Canyon (approachescoast). However the analysis is applicable to the majority ofcanyons along the shelf break of the West Coast of Canadaand the United States and in other regions where upwellingoccurs and where and/or when the undercurrent is notpresent or lies below shelf break depth.[109] As we proceeded through the scaling analysis wemade further simplifying assumptions (Section 2.5). Theserestrict the analysis to steep‐sided, narrow, deep canyonscutting into sloped continental shelves that are not overlydeep. The canyons must be steep enough that the Ekmanlayers arrest within a day [Brink and Lentz, 2010]. We haveassumed the canyons are less than two Rossby radii wideand deep enough that the upwelling stream does not reachthe bottom. The continental shelf is assumed to have a slopeand be shallow enough that the shelf Burger Number basedon the length of the canyon is less than 2. All theseassumptions are met by the real canyons presented here.[110] The scaling analysis was tested for four measurablequantities against observations collected in the field and datacollected from three laboratory canyons. The reasonableagreement between the observed values and the predictedvalues gives us confidence in the scaling analysis. A leastsquares fit was used to set scalar coefficients for the scaledparameters.[111] While deriving the scaling analysis, an estimate forthe size of the scalar coefficients was made. The coefficientfor the deepest depth of upwelling, 1.3, is close to 1, asexpected; the coefficient for the vorticity, 2.5, is close to 2,as expected; the critical number for the presence of an eddy,between 0.33 and 0.92, is close to 1/2 and the coefficient c2in F 0.9 is close to 1 as expected. The only deviationbetween our initial expectations and the observations occurswith the coefficient for the upwelling flux, that we expectedto close to 1/4 and is actually 1. The difference is partiallydue to the difference in what was estimated and what wasmeasured. The estimate was for the upwelling stream alone.However, the measurement will include all water turnedthrough the canyon (including the rim depth stream) to givea total onshore flux. The fit here implies that the rim depthstream has 3 times the volume of the upwelling stream. Notethat this does not include the extra onshore flux that occursif the continental shelf is flat, because the Lab‐Mirshakcanyon shelf is sloped.[112] For most real‐world canyons, the Rossby number Rois small (<0.5) and F [defined in (13)] can be approximatedby 1.1Ro. This approximation allows simplification of thescaling (Table 4). In particular, the depth of upwelling canbe expressed as Z = 1.4(U/N)×(L/R)1/2. Thus the depth ofupwelling is linearly proportional to the strength of the flowacross the canyon and inversely proportional to the strengthof the stratification. Longer canyons and canyons with sharpinflow regions will have stronger upwelling.[113] Making the same approximation for upwelling fluxgivesF¼ 1:2U3WsbL1=2NfR3=2C18C19ð36ÞThus upwelling flux is proportional to the cube of thestrength of the flow across the canyon. The flux is strongerfor wider or longer canyons and for canyons with sharpinflow regions and the flux is inversely proportional to thestrength of the stratification. This steady upwelling flux forAstoria Canyon is equivalent to a wind‐driven Ekman massflux due to a 16 m s−1wind or about twice the flux thatwould be associated with the actual observed peak Bakunwind over the canyon during the 1983 measurements.[114] Similarly, the vorticity in the deep water can beexpressed as Sf = 2.6UR−1/2L1/2Wsb−1. Thus the vorticitydeep in the canyon increases with the strength of flow acrossthe canyon and is independent of the stratification. Vorticityis stronger for longer canyons and weaker for wider canyonsor for canyons with smoother inflow regions.Table 4. A Synopsis of the Simplified Scaling for Four Signifi-cant Components of the Upwelling Process for Small RoaFeature Symbol ScalingEffectsUNfLR WsbWDepth of upwelling Z 1.4UNLRC18C191=2⇑ + – ↑↓ ––Upwelling flux F 1.2U3WsbL1=2NfR3=2⇑⇑ ++↑ + ⇑ –Deep vorticity Sf 2.6UL1=2WsbR1=2⇑ ––↑↓ + –Eddy presence E 1.0fWL1=2NHsR1=2– + ⇑↑↓ – ⇑aUpward/downward pointing arrows imply the component increases/decreases with the parameter. Strength of variation increases from ↑ to ⇑to ⇑⇑.ALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801817 of 20[115] The condition for an eddy at rim depth is E large andfor small Rossby number E ≈ fWL1/2/(NHsR1/2). The like-lihood of an eddy increases with the ability of the fluidcolumn to generate vorticity by stretching fW/NHsand withthe ratio of the canyon length to radius of curvature. All thereal canyons studied here had eddy numbers greater than 0.7and rim depth eddies were observed over each of thesecanyons. The generic canyons of some previous modelingstudies [e.g., Allen et al., 2003] are generally too triangular(narrow near the head W C28 NHs/f) to have rim eddies.Realistic canyons taper much less towards the head than hasoften been assumed in modeling studies. The canyon used inthe Pérenne et al. [2001] study has a more realistic hori-zontal shape.[116] One of the most important geometric parameters isthe shape of the shelf break upstream of the mouth of thecanyon. The radius curvature of this region, R, determinesthe tendency of the flow to cross the canyon bathymetry.Many canyons (Astoria, Figure 5; Barkley, Figure 6) appearto have fans in this area which increase the radius of cur-vature. These fans may be due to winter sediment fluxesfrom the canyon being transported poleward by the winterpoleward currents. These fans greatly increase R and thusdecrease upwelling within the canyon. Recent results for abuoyancy current traveling over a canyon found a similarimportance of the upstream radius of curvature as to whetherthe current crossed the canyon (separated) or flowed into thecanyon [Sutherland and Cenedese, 2009].[117] The upwelling flux data used in this paper weremeasured in a laboratory study by Mirshak and Allen[2005]. Those authors used an empirical scaling to deter-mine the dependence of the flux on powers of N, U and f:flux varied as U2(U/f)nN−1, where n = 0.66 ± 0.35. Lengthscales were not varied in their study. Here we find the samerelationship with respect to N and between U and f. How-ever, our relationship for flux as a function of Ro (≈U/f)isnot a simple power relationship because of the shape of theF function (13). When plotted over the measured range ofRo and compared to the power law relationship including itserror the new relationship falls within the bounds of theMirshak and Allen [2005] relationship (not shown). Thus,over the range of Ro used (0.2–1.0) the two relationships arenot significantly different. However, the new dynamicallybased relationship predicts more moderate upwelling at highRossby number: this result is more realistic than the powerlaw relationship of Mirshak and Allen [2005] which fails asRo goes to infinity.[118] Mirshak and Allen [2005] did not vary the shape ofthe canyon and thus the dependence on canyon length scaleswas assumed and not determined. They extrapolated theirlaboratory results to Astoria Canyon based on the similaritybetween the shapes of the canyons. However, the ratiobetween the radius of curvature and the length of AstoriaCanyon is twice as large in the laboratory canyon. This leadthem to underestimate of the flux through Astoria Canyoncompared to the more thorough analysis given here.[119] Our parameterizations for other canyon variablessuch as depth of upwelling and vorticity provide estimatesconsistent with real world observations, suggesting that ourgeneral approach is correct. We have no field observationsfor flux; however, our results can be compared to previousscaling analyses. For oscillatory flow over a canyon:dependencies for the net flux through the canyon on strati-fication, flow speed and Coriolis parameter are similar toours; the flux increases with increasing flow speed, whereasincreasing stratification and Coriolis parameter decrease theflux [Boyer et al., 2004; Haidvogel, 2005]. Length‐scaledependence has not been analyzed for oscillatory flow[Boyer et al., 2006]. However a scaling analysis for steadyflow (similar to our analysis) that was based purely onnumerical results [Kämpf, 2007] differs substantially fromour results. Dependence on stratification is consistent withour results. However, the Kämpf [2007] dependence on theCoriolis parameter is in the opposite direction from ours:ours decreases upwelling flux as f increases, whereas hisincreases the flux. Applying the final formula by Kämpf[2007] to laboratory canyons gives erroneously large re-sults, implying that its dependence on f cannot be extrapo-lated to the high values found in the laboratory. Thedependence on incoming flow is more similar in the twoanalyses: however, our dependence is stronger, being pro-portional to the cube of the incoming flow rather than thesquare. The length‐scale dependence of the flux for theKämpf study is simply the depth of the canyon whereas oursis its length and width. The canyon depth does not appear inour analysis because we are assuming much deeper canyons.Our shallowest canyon, Barkley Canyon, has a shelf breakto center depth of over 350 m. Kämpf [2007] is consideringcanyons from 45–230 m deep. Another type of comparisonis to apply the estimates of Kämpf [2007] and our estimatesto the real canyons for which there is data. Comparing thevalues of flux through the canyons, our fluxes are substan-tially (1.6 to 29 times) smaller than his.[120] The difference between the estimates appears to bethe slope of the continental shelf. The central case canyonfrom Kämpf [2007] is within the parameter range studiedhere for the other restrictions (Section 2.5). It does indeedhave uniform moderate flow and uniform stratification nearrim depth, and it is narrow and regular. It is deep enoughthat the upwelling flux should be unaffected by the canyondepth (which would not be true for the shallower canyonsconsidered by Kämpf [2007]). The numerical canyon is notas steep as the canyons considered here and given the samedrag coefficient as used by Brink and Lentz [2010] wouldnot have arrested bottom boundary layers for the first 3.5–3.9 days. However, the linear friction coefficient used (7 ×10−4ms−1) in the numerical model gives a higher dragwhich should lead to arrested boundary layers within a day.Our conjecture is that the crucial difference between thenumerical canyons of Kämpf [2007] and the canyons con-sidered here is that Kämpf [2007] uses a flat continentalshelf. Thus in addition to the flux estimated here, the fluxover his canyons includes a branch of the incoming flowthat follows the canyon rim to the head of the canyon and isnot turned back to the shelf break by the shelf‐topographicslope. Note that the final formula presented by Kämpf[2007] cannot be extrapolated to laboratory settings.[121] Few submarine canyons have been studied with thedetail of Astoria or Barkley Canyon. The scaling presentedhere for upwelling over coastal submarine canyons allowsestimation of the effects of upwelling over the manyunstudied canyons with minimal measurements. Besidesdetailed bathymetry (to estimate the geometric parametersR, L, Wsband W), a local CTD cast (to estimate N near rimALLEN AND HICKEY: DYNAMICS OF UPWELLING OVER A CANYON C08018C0801818 of 20depth) and local current meter data upstream of the canyon(to estimate the incoming, near shelf bottom velocity Uduring an upwelling event) are required to perform thesuggested scaling. Given these parameters, the depth ofupwelling in the canyon can be estimated to within 15 m,the deep vorticity can be estimated to within 15% and thepresence or absence of a rim depth eddy can be determined.Based on the laboratory data, total upwelling flux can alsobe estimated.[122] The scale analysis developed herein demonstratesthe important point that in spite of very steep topographyand variable spatial structures, some aspects of canyondynamics in upwelling environments obey relatively robustand simple relationships. This information can be used toadvantage in regions where detailed canyon data are lack-ing. In particular, one of the reasons canyons are of greatinterest is that they are known to enhance cross‐shelfexchange of properties such as carbon and nutrients.Therelationships presented here can be used to estimate suchfluxes within a given region that includes canyons with areasonable degree of certainty. In addition, we expect thatthe relationships for depth of upwelling, deep vorticity andeddy presence, which are strongly constrained by data froma number of canyons, will provide critical benchmarks forfuture modeling in canyon regions.Notationa Rossby radius NHs/f.Bcnon‐dimensional canyon depth NHc/(fL).Bsnon‐dimensional shelf break depth NHs/(fL).BuBurger number NHs/(fW).DHdepth scale fL/N.Dwscale depth deep in the canyon fWsb/(2N).E eddy number fWZ/UHs.Ek Ekman number n/(fHs2)f Coriolis parameter.F a function of Rodefined in (13).g gravitational acceleration.h fluid column length.Hcdepth change across the canyon mouthHhdepth at the head of the canyon.Hsshelf break depth.^kvertical unit vector.‘ unspecified horizontal length scale.L length of the canyon.M ND shelf‐slope (Hs− Hh)fL/(HsU)N Brunt‐Väisälä frequency near rim depth.(r, C18) polar coordinates around the projected center ofthe upstream isobaths.R radius of curvature of streamlines (naturalcoordinates).R radius of curvature of shelf break isobath ups-tream of canyon.p pressure.RLRossby number U/fL.RoRossby number U/fR.RwRossby number U/fWsb.(s, n) natural coordinates, s along flow, n across flow.S stretching deep in the canyon.T isopycnal tilt along the canyon.~u =(~uh, w) flow velocity: horizontal and verticalcomponents.U strength of velocity upstream of the canyon.UDvelocity scale deep in the canyon.U*strength of upwelling stream.V strength of flow across the canyon at rim depth.W width of canyon at half‐length.Wsbwidth of canyon at shelf break.u horizontal velocity in natural coordinates.(ur, uC18) polar, horizontal, velocity components.z vertical distance, positive upwards.Z vertical depth change of deepest isopycnalupwelled onto shelf.G strength of horizontal density perturbations.F flux of water upwelled onto shelf.r density.roconstant reference density.r*horizontally uniform, undisturbed density.n viscosity or eddy viscosityW strength of vertical velocity.z vertical component of relative vorticity.[123] Acknowledgments. This work was supported by NationalScience and Engineering Strategic and Discovery Grants to S. Allenand by grants to B. Hickey from the U.S. National Science Foundation(OCE‐9618186, OCE‐0234587, and OCE‐0239089) and from the Centerfor Sponsored Coastal Ocean Research of the National Ocean and Atmo-spheric Administration (NA17OP2789). This is contribution 50, 25, and324 of the RISE, ECOHAB PNW, and ECOHAB programs, respectively.The statements, findings, conclusions, and recommendations are those ofthe authors and do not necessarily reflect the views of the NSF, NOAA,ortheDepartment ofCommerce.Theauthors thankN.Banas,M.Dinniman,J. Klinck, R. Mirshak, and A. Waterhouse for comments on earlier versionsof the manuscript and an anonymous referee whose insightful comments ledto the analysis of the restrictions on the cases considered.ReferencesAllen, S. E. (1996), Topographically generated, subinertial flows within afinite length canyon, J. Phys. Oceanogr., 26, 1608–1632.Allen, S. E. 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