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

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C1, PAGES 1285-1297, JANUARY 15, 2000 On subinertial flow in submarine canyons' Effect of geometry Susan E. Allen Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada Abstract. Shelf break canyons on the west coast of Canada and the United States have been observed to be regions of enhanced upwelling during southward currents compared to the surrounding shelf break. Most shelf break canyons from Oregon north cross only part of the continental shelf cutting from the shelf break toward the coast but end on the continental shelf well below the mixed layer. Juan de Fuca canyon, on the other hand, cuts the continental shelf from the slope to, and actually continues into, the Strait of Juan de Puca. This difference in geometry has a very strong effect on the subinertial flow around the canyon. Model canyon shapes, which include convergent bathymetric contours, axe constructed and motivated for Juan de Fuca canyon and a typical shelf break canyon. Geostrophic analytic solutions show that the in-cai yon flow in Juan de Fuca canyon is generated by first-order geostrophic dynamics, whereas in the majority of canyons, of which Astoria is an example, in-canyon flow is generated by higher-order effects. This difference is postulated to lead to the observed, very deep upwelling over Juan de Fuca canyon compared to more moderate, episodic upwelling over Astoria canyon. 1. Introduction Canyons are ubiquitous features of many continental shelves and, in particular, the western coast of Canada and the United States. Off Oregon, Washin.on, and British Columbia the canyons are mainly deep canyons in the sense that their rims are below the mixed layer. Most deep canyons are of finite length: that is, they cut from the slope onto the shelf and end on the shelf. A number of canyons, however, continue into straits or estuaries. This last set includes Juan de Fuca canyon off Vancouver Island, Bio Bio canyon off the coast of Chile, and the Congo canyon. The purpose of this paper is to investigate the differences in the dynamics between a canyon such as Juan de Fuca canyon and a finite length canyon, for example, Astoria canyon off the coast of the Colmnbia River. Juan de Fuca sub narine canyon runs frmn the shelf break off southern Vancouver Island into Juan de Fuca Strait. It has been associated with the large seasonal- scale upwelling over the nearby shelf [Freeland and Den- man, 1982]. More recent observations [Vindeirinho, 1998] show that temperature decreases at 75 m over a spur of Juan de Fuca canyon frmn early May 1993 to the end of July. Both of these observational studies show upwelling occurring on a seasonal timescale over Juan de Fuca canyon. Copyright 2000 by the American Geophysical Union. Paper number 1999JC900240. 0148-0227/00/1999JC900240509.00 Astoria sub narine canyon cuts the shelf break off the Colmnbia River between Oregon and Washington. It is 20 k n long but ends on the shelf 20 km from the coast in 100 m of water. Detailed current and conductivity- temperature-depth (CTD) measurements were made over and within Astoria canyon in 1983 [Hickey, 1997]. Temperatures within the canyon vary episodically as strong upwelling favorable winds occur. Unlike Juan de Fuca canyon, there is little or no seasonal-scale up- welling. A model for flow over Juan de Fuca canyon was pre- sented by Chen and Allen [1996]. They considered barotropic, linear flow and showed strong shoreward flow during periods of southward shelf currents. A layer-stratified numerical model gave similar results for the layer that lies at the level of the canyon rim [Chen, 1995]. Flow over a finite length canyon similar to As- toria canyon has been numerically modeled by Klinck [1996] and Allen [1996a]. Both studies considered an upwelling episode of short duration (10 days and I day, respectively). Strong upwelling was observed, particu- larly over the downstream side of the canyon near its head. Two models, one for Astoria canyon and one for Juan de Fuca canyon, will be introduced in section 2. The Juan de Fuca .cemyon model includes singular points, and the inclusion of these points in the geomet- ric model is justified in section 3. An investigation of the geostrophic flow around the canyon models is pre- sented in section 4. This investigation illustrates the fundamental difference in the dynamics between the two types of canyons. 1285 1286 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 2. Models The choice of geometrical model must satisfy two con- straints. It must be true enough to the real problem to contain most of the observed dynamics, but it must also be simple enough for, in this case, analytic treatment. For a finite length canyon consider the model drawn in Figure 1. It is a symmetric canyon of two depths em- bedded in a flat shelf. For a Juan de Fuca-type canyon consider the  nodel drawn in Figure 2. It is a symmetric canyon joined to a strait. The canyon has two depths, the strait has two depths, and the shelf is flat. Through- out this paper the x axis will be taken to lie parallel to the coast, and the y coordinate will be taken to decrease offshore, as ilhtstrated in Figure 2. The Coriolis param- eter will be assumed to be positive. The two edges of the canyon will be referred to as left and right as they appear in Figures 1-3, and 13-15 (the right side occurs at larger x than the left). The origin of the coordinates will be changed from section to section for convenience. The full governh g equations are the nonlinear shal- low water equations: Du -1 V2 02u D-  + 'f  : x u - -- V p + An u + Aw - z   (la) Po Ou Ov Ow + + - 0 Op = -Pg (lc) Oz Dp =0, (ld) D  where l is time, u- (u, v) is the horizontal velocity, w is the vertical velocity, p is the pressure, p is the density, po is a constant reference density, ] is the Coriolis pa- rameter, assumed to be positive and constant, g is the acceleration due to gravity, D/Di is the horizontal total derivative,  : is the vertical unit vector, An and Av are Figure 1. Plan view of the model topography for As- toria canyon. Figure 2. Plan view of the model topography for Juan de Fuca canyon. The directions of the x and y axes are given. The solid small circles mark the two sin- gular points in the closed strait case. The solid black square  narks an additional singular point in the open strait case. The open box shows the area considered in section 3.1. the horizontal and vertical eddy viscosities, respectively, and V is the horizontal gradient. The hydrostatic and Bousshmsq approximations have been made. Consider a simplified stratification, that of a series of homogeneous (p constant) layers. The flow is assumed to be linear (V/fL << 1, where V is a velocity scale and œ is a horizontal len h scale), steady (O/Or- 0), and inv cid. Inclusion of viscosity would cause the flow to slowly spin-down. It is  sumed that this process   much slower than the adjustment to geostrophy. Equa- tions (la)- (ld) s p fy to -1 fxu- Vp (2a) Po V-(Hu) - 0, (2b) where H is the undisturbed thickn s of the layer. These equations are degenerate for layers of constant thickess H; i.e., for those layers not in contact with the bottom and any deep layers over a flat bottom. As the topo aphies under consideration are piecewise flat, the solution for the steady state uses the conservation of potential vorticity equation (derived  om the t  dependent version of (2) [Gill, 1982])' 05 H Ox   - f (h- H) -0, (3) where h   the thickess of the layer. The distribution of potenti  yogicity is chosen to give a simple shelf/she  bre  current   the absence of a canyon. In layers h  contact with the topography, over the regions where the depth changes, (2) imp es that there . is no flow across the depth contours. This is the familiar tendency for geostrophic flow to follow bathymetry. As ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1287 geostrophic flow follows pressure contours, (2) implies that the pressure is constant over depth changes in the layer in contact with the topography. Therefore a complete steady state solution for geo- strophic flow over a canyon requires a determination of the pressure at the coast and at the other depth changes and then solution of (lc), (2), and (3) over the fiats. Although the required solution is steady, to find which steady solution is appropriate, one must consider the transient effects. In particular, the pressure over the depth changes is determined by the propagation of long topographic waves [Gill et al., 1986; Chen and Allen, 1996] and that along the coast is determined by Kelvin waves. As these waves keep the shallow water or land to their right, information propagates from the right to the left in Figures I and 2. Consider first the model of Astoria canyon. As the information (pressure) propagates along the shelf break and into the canyon, it encounters the change in depth within the canyon. Here the information splits and trav- els both ways: across the canyon at the depth change and around the canyon head. It meets on the other side of the canyon and continues along the shelf break. So, determination of the pressure at the depth changes is trivial as it is the same value as that at the shelf break. Consider now the model of Juan de Fuca canyon (Fig- ure 2). Information propagates again from the right along both the shelf break and the coast. Far to the right, the pressure at the shelf break Inust be differ- ent froIn that at the coast as there is a current on the shelf. The information traveling along the shelf break turns up into the canyon and splits at the canyon depth change. The information continuing up the right-hand canyon wall encounters the coast along which different information has been traveling. This convergence gives rise to a sh gular point. Note that such sh gular points can be avoided by slightly inodifying the topography as shown in Figure 3 and that real topography does not include singular points, although the contours Inay get very close. The Inodel we wish to consider is a layered, lh ear, inviscid, steady state flow over the two topographies shown in Figures I and 2. However, first, we Inust jus- tify the choice of using the singular points as shown in Figure 2 over the topography without singular points as shown in Figure 3. 3. Justification for Singular Points In this section we will consider a homogeneous layer in contact with the topography. For example, the mid- dle layer would be the layer of interest in a three-layer systmn with an upper layer representing the mixed layer (say 30 in deep), a middle layer extending down to up- per slope depths (say 250 m), and a deep layer below. Steady, low Rossby number flow in a homogeneous layer that is in contact with the bottom follows bathy- metric contours. However, bathymetric contours do not relnain a constant distance apart. For example, con- sider the 50 and 200 m contours north of Juan de Fuca canyon (Figure 4). In the region north of the canyon these contours are 60 km apart, while near the west end of Juan de Fuca Strait they close to 5 km. If the flow continues to follow the depth contours and remains geostrophic, the total flux between the contours remains constant. Hence the flow speed must accelerate by a factor of 12. Flow that is quite slow (low Rossby num- ber) where the isobaths are far apart will become much stronger where the isobaths converge. Given an acceler- ation of a factor of 12, it is quite possible that flow that has a low Rossby number north of the canyon has, at the west end of the strait, a moderate Rossby number and is no longer purely geostrophic. Essentially, this is the physical limitation. In prin- ciple, bathymetric contours can become infinitely close together, but it is unphysical to expect the flow to ac- celerate without bound. To investigate the behavior of flow as bathymetric contours converge, consider the ge- ometry of Figure 5 and consider a homogeneous layer in contact with the topography. Across the converging flow, Figure 3. Plan view of the modified topography model for Juan de Fuca canyon. Ou Ou Ou - 10p O-- +U xx+V - - fv ....  A,Vu+Av (4) y po Ox which is the x direction momentum equation. Assume that the flow is steady and that the flow across the depth contours, even as they converge, is much smaller than the flow along the contours. Thus lu[ << [v[, and (4) is dominated by the geostrophic balance. This gives Ap v - (s) pof Ax' where v is the velocity along the contours, Ap is the pressure change between the two contours, and Ax is the distance between the two contours. 1288 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 49'00'N 48'50'N 48'40'N 48'30'N 48'20'N 48'10'N 48'00'N 126'00'W 125'30'W 125'00'W 124'30'W 124'00'W Figure 4. Map of the Southern Vancouver Island Shelf showing the coastline and the 50 and 200 m depth contours. Along the converging flow, Ov Ov Ov -10p 02v Y Po Oy Away from the convergence, the flow is assumed to be geostrophic. However, as the contours converge, depth=h ] depth=h 2 Y Pl P2 Figure 5. Plan view of the convergence of two bathy-  netric contours with notation for section 3 marked. both the advection terms and the frictional terms in- crease. For a large difference in the initial pressure the nonlineax advection terms will dominate. For smaller pressure differences the horizontal or vertical viscos- ity will dominate. The relative importance is given by the Reynolds numbers, ReH -- VoAx/AH and Rev = VoH2/(AxAv). For typical shelf values, Vo = 0.3 m s -z the depth scale H = 100 m, and Ax = 10 kin. Ranges for AH are 10-1000 m   s -  and for Av are 0.001-0.1 m   s -  [Pedlosky, 1979]. The shelf is a rela- tively active area, so upper range values are probably  nore appropriate. This gives Res and Rev numbers of 300 to 3. We will assume here that the nonlinear advection terms dominate. Assrune that two contours converge over a length scale L much greater than the final distance between them, e. The velocity parallel to the contours, assum- ing the flow continues along the contours, is given by Ap _ (7) poe/' The velocity along the contours must accelerate as the contours converge. This acceleration is given by vOv/Oy and is provided by the pressure gradient in (6), thus v 2 Ap -- (8) L poL' Eliminating the velocity between (7) and (8) gives the  nini num distance ec that can separate the two contours while ensuring that the flow can re nain strictly along the contours: (_ o p) 1/21 = f (9) ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1289 Another way to quantify the convergence is given the final convergence e, we can compare it to the minimum convergence ec and define a convergence parameter G: or - / ) /  G- e VoAZ ' (11) The value of (7 detemnines the structure of the flow at the convergence. For large (7 the convergence is weak, and flow continues along the isobaths. For ranall (7 the convergence is strong and the flow crosses the isobaths. In section 3.1 the minionurn value of G(= 1.5) for which the flow can be considered weak (no isobath crossing) and the  naxi nmn value of G(- 0.3) for which the con- vergence can be treated as a singular point will be esti-  nated numerically. It is interesting to note that G is simply the square root of the reciprocal of the Rossby number evaluated at the convergence if the flow follows the isobaths. (This result follows from conservation of flux; the flow at the convergence is VoAx/e.) In stunmary, the  naximum acceleration possible for a given flow is li nited by available pressure difference (8), but for regions where the bathymetric contours sharply converge this acceleration may be insuificient to allow the flow to continue along the contours (7). As the flux  nust be accmmnodated, the flow will expand over a larger region. The flux between the two contours will reduce, which implies that the interface elevation along the contours is not conserved as the flow is principally geostrophic across the contours (frmn (4) with [u[ << To solve for flow in the region of sharp convergence of bathy netric contours, conservation of  nass must be used. This  nethod is exactly that used by Chen and Allen [1996] to solve for flow in the vicinity of the sin- gular points in the Jua  de Fuca model. Thus the use of singular points can be justified as a very good approx- i nation for broad low Rossby number flow that only beco nes strongly nonlinear in small regions. Figure 5 does not include any bending or curvature in the isobaths. However, at the head of canyons, isobaths turn 180 ø over the width of the canyon, W. Why can this effect be neglected? Consider two cases- one where there is curvature and a second where there is conver- gence. An esti nate of the effect of curvature is given by the Rossby number V/fW. An estimate of the effect of convergence is also given by the Rossby nmnber, in this case, V/f . However, whereas around the curve the velocity changes little so V   Vo, through the conver- gence the velocity increases to VoAx/e. Charts show that the width of a canyon is similar but larger than the isobath convergence seen at Juan de Fuca. That is, W _> e. Thus the Rossby number due to the conver- gence is larger than the Rossby nmnber due to curvature by at least Ax/e, which is order 10 for Juan de Fuca canyon. The effect of curvature is i nportant to canyons like Astoria, but it is a second-order effect. 3.1. Numerical Verification In this section a nmnerical solution of the nonlinear reduced gravity equations is used to illustrate the dy- na nics described above and to determine the critical values for (7. The use of a single active layer allows a large nmnber of different parameter sets to be simu- lated. Note that the lower layer is taken as the active layer. The model solves (4) and (6) with Av taken as zero, the pressure given as pot'h, and a conservation of volrune equation Oh + v. (nu) - o. The  nodel is a finite difference, leapfrog explicit method on an Arakawa C grid [Allen, 1996a]. It uses an enstro- phy- and energy-conserving formulation for the advec- tion and Coriolis terms [Arakawa and Lamb, 1981]. The topography investigated corresponds to the point in the Juan de Fuca canyon geo netry where a slope within the canyon  neets the left canyon wall as illustrated by the open box in Figure 2. To avoid boundary effects, the numerical boundaries have been greatly expanded com- pared to the box shown. The topography is illustrated in Figure 6. Parmneters used in the numerical solutions are given in Table 1. 401 120 m 40 m 200 m y=b X  5 . 60  o 160 by 20 Figure 6. Plan view of the bathymetry for the nu-  nerical si nulations showing the whole computational do nain with domain size in kilometers. The shallowest water (40  n) is to the left side; the deepest water (200  n) is at the bottmn right; and the depth at the upper right is 120  n. Depth contours, 20 m apart, are solid lines. The dashed horizontal line marks the potential vorticity discontinuity (y = b). The two arrows mark the position where cross sections were taken. The small- dashed square is the subdomain shown in Figure 8. 1290 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS Table 1. Para neters Used in the Numerical Simulations Parameter Value Grid size 2.5 km Time step 175-400 s f 1 X 10-4s -t Av 0 g  0.02 and 0.10 m s -2 Re 2000 Boundary condition- offshore constant gradient u, v,  1 Boundary condition- left constant gradient u, v, r/ Boundary condition- right constant gradient u, v, r/ Boundary condition- onshore 5 grid point sponge to 0 u, v, r/ The distribution of potential vorticity is chosen to give a si nple shelf/shelf break current in the absence of the canyon. For a southbound (in the positive x direction) current this i nplies higher potential vorticity near the coast, decreasing offshore:  ,_ { -fr Y<   0 y>b ' where P is the potential vorticity given by the term in square brackets in (3) and  /o is positive for a south- bound current. This distribution of potential vorticity gives a geostrophic jet the width of the Rossby radius, centred at y = b. Over the fiat shelf (on the left in Figure 6), u -   exp R ' (14) where R = (g'H) /=/f is the internal  ssby radius.  ther than sta ing the  nodel with a sharp poten- tial vorticity controt, the flow w  forced gradually. Spreading the forcing in time reduces the amplitude of the generated Poincar  waves. The forcing (tax adding of fluid to the father offshore section of the domain, y < b) w  ra nped up over half an inertial period and reduced down over half an ine ial period. There w  no forcing after the first inertial period. So, a er one iner- tial period the potential vorticity is a slightly smoothed version of (13). The line separating the unforced sec- tion of the do nain from the forced section   marked   a d hed line on Figure 6. If the flow h  a small Rossby number even through the convergence, the solution is linear. The two re- gions of topographic slopes carry information,   this c e the interface height, in the direction that keeps the shMlower water to the right. Thus information is propagating offshore (toward negative y) Mong the canyon wall and to the left along the slope. The inter- face height propagated along the canyon wall is that of the onshore section of the domain, whereas the interface height propagated along the slope is that of the offshore section. This configuration gives a convergence of dif- fering information. If the flow remains linear through the convergence, the value of the interface height along each topographic contour can be found by tracing the topographic contour back to its source on the onshore or right boundary of the domain. In particular, those contours originating at the onshore boundary have in- terface heights of zero. Figure 7a shows cross sections between the two arrows marked on Figure 6 of the in- terface height for a nearly linear case, (7: 2.5, at three different times. The sharp transition (with some over- shoot) between the topographic contours carrying r/= 0 and those (to the right) carryh g   = 0.125 m is clear. The variation between the different times shows that the flow is not completely steady. Topographic Rossby waves propagate along the canyon edge, causing oscil- lations in the depth at the canyon edge [Allen, 1996b]. This propagation can be seen by comparing Figures 8a and 8c, which show contours of the interface elevation at days 10 and 15 for the weakly forced case. The trough over the canyon edge at   y = 170 k n at day 10 has  noved to y: 120 km by day 15. Figures 8a and 8b show the interface elevation at day 10 for a weakly forced case, G = 2.5, and for a strongly forced case, G = 0.25, respectively. In Figure 8a the four surface height contours that enter the domain from the left side turn right and follow the canyon wall off- shore; they do not cross the canyon wall. The four surface height contours from the right-hand side simi- larly turn and follow the canyon wall. In contrast, in the more strongly nonlinear case, Figure 8b, two of the four contours do cross the canyon wall in the vicinity of the bathymetric convergence. As the flow is nearly geostrophic, this implies flow crossing the bathymetry here (Figure 9). Figure 7b shows the interface cross section for the strongly forced case. As flow has crossed the bathyme- try at the convergence, the minimum interface value at this cross section is greater than zero. Given a small G, the theoretical value for the minimum interface height can be calculated by conservation of mass. For this topography the value is r/o/2 or 6.25 m as shown in the appendix. If the flow follows the topography, the minimum value of the cross section is 0, whereas if flow acts as if there is a complete convergence, the minimum value is r/0/2. Thus we can use this parameter as a measure of the type ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1291 0.2 0.15 ß 0.05 _c -0.05 I I I I O0 150 200 250 300 350 X (km) 0 I I I I 1 O0 150 200 250 300 350 X (km) Figure 7. Cross sections of interface height for (a) G - 2.5 and (b) G - 0.25, respectively at three different times (solid, 10 days; dashed, 20 days; and dotted, 30 days). The position of the cross section is shown by arrows in Figure 6. The vertical axis is in meters; the horizontal axis is in kilometers. Note that the whole width of th  domain is not shown. of flow occurring. The minimum value of the interface elevation along this cross section (normalized by  0) was found as a function of time for a number of Cs. The value of r/0 was varied by 2 orders of magnitude giving a single order of magnitude variation in (7. As short topographic waves travel along the slope, t!xe  nini num interface value oscillates in time. Various different measures from this time series were calculated. All gave a si nilar transition between the linear result and the nonlinear result. Figure 10 shows the average value from days 5 to 30 of the minimum interface ele- vation along the cross section. The maximum value of G for which the flow follows the isobaths is m 1.5. The minimum value of G for which the flow acts like a sin- gular point is 0.3. At an intermediate point where G is m 0.5- 0.6, the behavior is halfway between. (Note that neither of the theoretical values were found for the asymptotes. This discrepancy is probably due to a number of factors, one is simply the estimation of the mini nmn interface height from a time-varying series.) Thus, for a velocity of 30 cm s -  between two con- tours initially 10 km apart the flow can continue to follow contours as long as they remain 8 km apart (G = 1.5). The totally converged solution becomes valid when the contours converge to within 2 kin (G = 0.3). 1292 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS A contour plot showing e as a fu, ction of Vo and Ax at G = 1.0 for a range of values relevant to the shelf is shown in Figure 11. For other values of G, such as the ,ni,fimum and  naximu,n values given above, multiply the e given in Figure 11 by the required G. 3.2. Field Evidence There are some field observations in support of G deter ni,fing the ability of geostrophic flow to follow 301 . 101 . 126. ß ß x 326. isobaths. Noble and Ramp [2000] used an array of six curre, t ,neter moorings to investigate the Califor- nia undercurrent in the Gulf of the Farallones off San Francisco. South of the moored array the shelf and slope form a broad, relatively shallow sloping continen- tal shelf. Within the mooring array the topography steepens dramatically (from 2ø-3 ø at the southern line to 9ø-10 ø at the norther,  line). At their souther,  line the distance between the 200 and 1000  n isobaths is 25 k,n. Their typical flow values are 25 cm s - . At the northern li, e the 200 and 1000 m isobaths are only 5 k,n apart. This gives a G of 0.6. This low value of G i,nplies that the flow will not be able to follow strictly the isobaths through this convergence. Noble and Ramp [2000] find that the undercurrent, clearly visible in the southern line, is not visible and assu,ned offshore of the northern line. U, der conditions where the undercurrent velocity drops to 8-12 cm s -h, so that G increases to 0.9 the volu,ne flux through the two lines becomes more simi- lar. Thus, under larger G the tendency to cross isobaths is s,naller. Considering just the upper part of the slope, the 200 ,n isobath and the 500 ,n isobath are 6 km apart at the southern line and only I km apart at the northern line. This topographic convergence gives G = 0.2 under strong flow and G = 0.4 under weak flow. At these low values of G the theory above predicts that the topogra- phy should act as a singular point. Indeed, Noble and Ramp [2000] find that the current meter over the upper slope at the northern line is always \"shadowed.\" 301 . 101 . 126. 1 X 326. 1 6. Figure 8. Plan view of the interface contours for (a) a weakly forced case (contours -0.025 to 0.125 ,n by 0.025 m), and (b) a strongly forced case (-12.5 to 15 m by 2.5 m) after 10 days and (c) the weakly forced case (contours -0.05 to 0.15 m by 0.025 m) after 15 days. Note that the whole domain is not shown. Positive contours, the zero contour, and negative contours are shown as solid lines, a dashed line, and dotted lines, respectively. ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1293 251. i , x 176. 176. 251. 0.42 I.E...  UAXI'- '  VECTOR Figure 9. Plan view of the velocity vectors for   - 0.25 flow at the bathymetric convergence. Flow is cross- ing the bathymetry along the canyon slope down the center of the domain shown. The velocity is in m s - . Flow is shown at day 10. Note that the domain shown is smaller than that for Figure 8. 4. Implications for Geostrophic Flow Now that the use of convergent bathymetric contours has been justified, we return to the geostrophic solution for flow over the two model geometries (Figures I and 2). Assrune a potential vorticity distribution given by (13) where y = b occurs on the shelf. Calculations of the interface heights at the coast and shelf break far to the right are given by Chen [1995] for the reduced gravity case. Rather than repeating the formulas here, we will use nu,nbers from an exa,nple. Assume that the active layer is 40 m deep over the shelf, 120 m deep over the shallow canyon section, and 200 m deep off- shore. The reduced gravity is 0.02 m s - , and the Cori- olis para,neter is 10 -4 s -  . This gives Rossby radii of R  - 8.9 k,n, R2 - 15.5 kin, and R3 - 20 km for the shelf, shallow canyon, and offshore, respectively. From Chen [1995] this gives the interface height at the shelf break far to the right as r/so - 0.90r/0 and that at the coast as r/½ - 0.17,/0. The flow vectors, in the absence of a canyon, for this geometry are given in Figure 12 (for,nula given by Chen [1995]). Consider first the model for Astoria canyon. The value of the interface height at the boundary between the shallow shelf and the canyon is determined by the value from the right at the shelf break r/ o. The value at the boundary between the two canyon depths is the sa,ne. Now that all the values at the canyon edges have been determined, the full solution can be found by nu- ,nerical solution of (lc), (2), and (3), which in this case reduces to R2V2r/- r/- -P/f. (15) The solution was found using relaxation (Figure 13). Parameters for the numerical relaxation are given in Table 2. For Juan de Fuca canyon the conditions far to the right are the same as those for Astoria. The value of the interface height on the right-hand side of the canyon and across the slope within the canyon is r/so. However, 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 ....... '   0.1 1 Figure 10. The average (days 5-30) of the mini,num interface height (normalized by r/0) across the canyon wall offshore of the bathymetric convergence. The chosen cross section is shown in Figure 6. The diamonds are for g  - 0.02 m s -  and r/o from 50 to 0.025 m. The crosses are for g  = 0.10 m s -  and r/o from 10 to 0.025 m. Lines simply join the points to aid the eye. 1294 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 50 45 40 ,......,35 20 5 Vo(m s -1 ) Figure 11. Contour plot showing the distance between two converging bathymetric contours that gives G = 1.0 as a function of initial velocity and initial distance be- tween the bathymetric contours. The region not con- toured h  the lower right corner below the dashed line represents the region with initial Rossby number above 1. For other values of G, multiply the value of e given by the required value of G. where the right side of the canyon meets the coast, there is a singular point. The value of the interface height for the right side of the strait can be found by conservation of  nass [Chen and Allen, 1996] and, for the values here, is 0.66  10- This value is carried across the slope in the strait. The value at the left-hand strait wall depends on the assumptions made about the strait. Chen and Allen [1996] assumed an infinitely long strait, i.e. no connec- tion between the two strait walls. This solution is given in Figure 14a. In Figure 14b the strait is considered to be closed. The real solution probably lies between these two and. for Juan de Fuca canyon, is likely determined by  nixing in the San Juan/Gulf Islands area. For the open strait case the upper left strait wall has an interface height of 0.27 r/0. A second singular point occurs at the strait slope. Again, using conservation of  nass. the value of the interface height for the lower left strait wall, along the left coast and for the upper left canyon, is 0.53 T10. The third singular point occurs where the canyon slope  neets the left canyon wall. The interface height for the lower left canyon wall and along the shelf break is 0.72 t10- The fluxes that these interface heights i nply are given in Figure 15a. For the closed strait the value of the interface height along the left side of the strait, the upper left canyon wall, and the left coast is 0.66 r/0. The value of the interface height for the lower left side of the canyon and the left shelf break is 0.78 r/0. The fluxes these interface heights imply are given in Figure 15b. In either the open or closed strait case a significant fraction of the shelf break current originally flowing off- shore of the shelf break ends up on the shelf. This flow enters the right shelf at the singular point between the coast and the canyon. In the Astoria canyon case, no flow  noves across the shelf break. Flow across bathy-  netric contours implies vertical motion (upwelling) in these models. Thus, for the Juan de Fuca model case, upwellinõ occurs within the canyon and over the right- hand canyon wall near the coast. For the Astoria canyon  nodel case, no upwelling occurs. -lO -15 -2o -25 -3o -35 4o 45 Coast  J elLbreak Figure 12. Flow vectors for geostrophic flow in the absence of a canyon given a shallow shelf, sharp shelf break, and deep ocean. Geometric parameters are given in Table 2. The dashed line  narks the shelf break. The dotted line at y - b marks the discontinuity in potential vorticity. ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1295 -40.0 -20 Figure 13. Plan view of the interface height for geostrophic flow over the Astoria canyon model. The solid lines  nark the topographic depth changes. Con- tours are 0.067-0.93 by 0.67 r o (contours 0.2, 0.4, 0.6, and 0.8  1o are shown dashed; the other contours are dotted). The domain size is in kilometers. The whole dmnain is not shown. Chen [1995] and Chen and Allen [1996] used the sixn- piest possible stratification, a single active layer. Exten- sion to a homogeneous layer in contact with the topog- raphy (with active layers above and below) is concep- tually trivial. Consider two cases: a single active layer case versus a multiple active layer case. In the multiple active layer case assume that the hmnogeneous layer in contact with the topography has the same undisturbed depth profile as the single active layer case. Further- more, assume that an initial potential vorticity distri- bution (varying through the active layers) is specified in such a way that in the absence of a canyon the flux over the shelf and the flux offshore of the shelf is the same in the two cases. Then the fluxes through the system Table 2. Para neters Used in the Geostrophic Solutions Parameter Value, km b 17.6 Canyon width 8 Shelf width 22 Canyon length, Astoria 14 Deep canyon length, Astoria 3 Deep canyon length, Juan de Fuca 8 Strait length modeled 25 Deep strait length 8 Domain size, Astoria 50 x 50 Grid size, Astoria 0.25 Domain size, Juan de Fuca 70 x 95 Grid size, Juan de Fuca 0.33 20.0 -30.0 X 30.0 20 -40.0 -30.0 X 30.0 Figure 14. Plan view of the interface height for geostrophic flow over Juan de Fuca canyon model (a) assuming that the strait is open, and (b) assuming that the strait is closed. The solid lines mark the topogra- phy depth changes. Contours are 0.067-0.93 by 0.067 (contours 0.2, 0.4, 0.6, and 0.8 r o are dashed; the other contours are dotted). Domain size is in k n. The whole dmnain is not shown. shown in Figure 15 are unchanged. This result can be justified as follows: (1) As described in sections 2 and 3 the pressure is used, instead of an interface height, as the governing variable for the multiple active layer case. (2) As the flow is geostrophic (except at the singular points), the flux between two lines determines the pres- sure difference between them. (3) Thus, as the fluxes in the absence of the canyon are the sazne in the two 1296 ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 3.1 Figure 15. Plan view showing the fluxes in x10   m   s -t for the geostrophic solution of Juan de Fuca canyon (a) assulning that the strait is open, and (b) assuming that the strait is closed. Shading corre- sponds to Figure 2. cases the pressure at the coast and shelf break can be taken to be the same (as the origin for pressure is arbi- trary, only pressure differences matter). (4) Pressures at depth changes beyond singular points are found by conservation of mass, and as the flux approaching the singular points in the two cases is the same, the pres- sures in the two cases will be the same. (5) Thus the pressure at all depth changes (coastline, canyon walls, canyon depth changes, and the shelf break) will be the same in the two cases. (6) As flux and pressure are di- rectly related, all the fluxes will be the same in the two cases. However, finding the full solutions (flow at each point) is not so trivial. Finding the potential vorticity distribution to give the fluxes or finding the fluxes from a given potential vorticity distribution is algebraically messy even in the three-layer case [Allen, 1996a]. Once the fluxes are found, finding the pressures at the vari- ous depth changes is straightforward. Then determin- ing the final solution requires relaxation over all layers of an equation complicated by the number of vertical  nodes now allowed. However, assuming that the first baroclinic inode would be expected to dominate and that the reduced gravity in the one active layer case is chosen so that the baroclinic Rossby radii  natch, changes due to the other modes should be s nall. The expected final result is that the solutions shown in Figures 13 and 14 would be very good approximations to the inultiple-layer case. In sum-  nary the single active layer gives the correct fluxes for a  nultiple-layer case, but details of the flow would be slightly different. 5. Conclusion A canyon such as Juan de Fuca, which has strong bathymetric convergences, can support geostrophic up- welling. However a canyon such as Astoria which does not have strong bathymetric convergences cannot sup- port geostrophic upwelling. Of course, upwelling is ob- served over Astoria canyon, but it is episodic and must be due to second-order dynamics such as time depen- dence or widespread nonlinearity. Thus, at Juan de Fuca we observe strong seasonal scale upwelling whereas at Astoria the upwelling is episodic. Appendix: Calculation of the Pressure After a Convergence for G- 0 Consider flow over the topography illustrated in Fig- ure 6. As G approaches 0, the convergence is so strong that there will be only one value of the pressure over the canyon wall offshore of the convergence. This value Oc  can be found by conservation of mass. The flux into the domain from the left is the integral in y of the velocity (14) multiplied by the depth, which gives  1oR f, where Rs is the Rossby radius in the shal- low shelf water. The flow out of the domain to the right is similarly rloR2mf, where Rm is the Rossby radius in the shallow part of the canyon. The flux out of the do nain, offshore along the wall of the canyon but over the shelf, is rlc Rs2.f. The flux into the domain, from the offshore, along the wall of the canyon, and over the canyon is rk,,R f, where Ra is the Rossby radius in the deep part of the canyon. Equating the incoming to the outgoing flux gives  %  H,, - H  ,lo = Hd - H  ' (A1) where H , H,,, and Hd are the undisturbed depths over the shelf, the shallow canyon, and the deep canyon, re- spectively. For the numerical values used in the simu- lation, Hd = 200 in, Hm = 120 m, and Hs = 40 m;  lc  = 0.5 1o. Acknowledgments. The author gratefully acknowl- edges the support of NSERC through the Research Grants and Strategic Grants programs and from NSERC and DFO Canada through the GLOBEC Canada project. Discussions ALLEN: GEOMETRIC EFFECTS ON UPWELLING OVER CANYONS 1297 with G. Holloway and comments on the maauscript by J. Klinck, C. Vindeirinho, R. Mirshak, and two anonymous reviewers were most helpful. References Allen, S. E., Topographically generated, subinertial flows within a finite length canyon, J. Phys. Oceanogr., 26, 1608-1632, 1996a. Allen, S. E., Rossby adjustment over a slope in a homoge- neous fluid, J. Phys. Oceanogr., 26, 1646-1654, 1996b. Arakawa, A., and V. R. Lamb, A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. Weather Rev., 109, 18-36, 1981. Chen, X., Rossby adjustment over canyons, Ph.D. thesis, 279 pp., Univ. of B.C., Vancouver, B.C., Canada, 1995. Chen, X., and S. E. Allen, Influence of canyons on shelf cur- rents: A theoretical study, J. Geophys. Res., 101, 18,043- 18,059, 1996. Freeland, H. J., aad K. L. Denman, A topographically con- trolled upwelling center off southern Vancouver Island, J. Mar. Res.,  0, 1069-1093, 1982. Gill, A. E., Atmosphere-Ocean Dynamics, 662 pp., Aca- demic, San Diego, Calif., 1982. Gill, A. E., M. K. Davey, E. R. Johnson, and P. F. Linden, Rossby adjustment over a step, J. Mar. Res.,  , 713-738, 1986. Hickey, B. M., The response of a steep-sided narrow canyon to strong wind forcing, J. Phys. Oceanogr., 27, 697-726, 1997. Klinck, J. M., Circulation near submaxine canyons: A mod- eling study, J. Geophys. Res., 101, 1211-1223, 1996. Noble, M., and S. Ramp, Subtidal currents over the cen- tral California slope: Evidence for spatial and temporal variations in the undercurrent and for local wind-driven currents over the outer slope, Deep Sea Res., Part II, in press. 2000. Pedlosky, J., Geophysical Fluid Dynamics, 624 pp., Spring- er-Verlag, New York, 1979. Vindeirinho, C.,  Vater properties, currents and zooplankton distribution over a submarine canyon under upwelling- favorable conditions, Master's thesis, 121 pp., Univ. of B.C., Vancouver, B.C., Canada, 1998. S. E. Allen, Department of Earth and Ocean Sciences, University of British Columbia, 6270 University Boulevard, Vancouver, B.C., Canada V6T 1Z4. (allen@ocgy. ubc.ca) (Received July 15, 1998; revised May 18, 1999; accepted July 20, 1999.) 


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