Impact of the continental shelf slope on upwelling through submarine canyons

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: OCEANS, VOL. 118, , doi: /jgrc.20401, 2013 Impact of the continental shelf slope on upwelling through submarine canyons T. M. Howatt 1 and S. E. Allen 1 Received 15 March 2013; revised 13 September 2013; accepted 17 September 2013; published 24 October [1] Submarine canyons that cut into the continental shelf are regions of enhanced upwelling. The depth of upwelling and flux through the canyons determines their role in exchange between the shelf and the open ocean. Scaling analyses that relate these quantities to the strength of the flow, stratification, Coriolis parameter, and topographic shape parameters allow their estimation in the absence of a full numerical simulation or a detailed field study. Here we add the effect of the continental shelf slope to the scaling of the depth of upwelling, upwelling flux, and deep water stretching. The scaling is then tested using a three-dimensional primitive equation model over 18 distinct geometries. The impact of the continental shelf is significant for real canyons with changes in the depth of upwelling of up to 11% and of the flux of upwelling of up to 70%. The numerical simulations clearly show three types of canyon upwelling, a symmetric time-dependent flux, the dominant advectiondriven flux, and a new flux that appears to be related to internal waves. They also suggest that the canyon width is more important than the upstream canyon shape in determining the strength of the flow across the canyon. Citation: Howatt, T. M., and S. E. Allen (2013), Impact of the continental shelf slope on upwelling through submarine canyons, J. Geophys. Res. Oceans, 118, , doi: /jgrc Introduction [2] Submarine canyons are often regions of enhanced biological productivity [e.g., Hickey and Banas, 2008]. This enhancement can be related to aggregation [Pereyra et al., 1969; Allen et al., 2001] or to enhanced nutrient sources due to enhanced mixing [Kunze et al., 2002] or upwelling [Bosley et al., 2004]. Submarine canyons are important conduits between the coastal ocean and the open ocean both in bringing ocean nutrients onto the shelf and in transporting dense coastal water and sediments into the ocean [Allen and Durrieu de Madron, 2009]. They are also difficult regions to observe due to operational difficulties over the steep topography and to the often intense fishing presence. In large numerical models it is difficult to resolve submarine canyons sufficiently. Thus, it is useful to have a simple way to estimate the efficiency of canyon exchange empirically. In particular, a dynamical scaling analysis for advection-driven upwelling gives the depth of upwelling, upwelling flux, and deep vorticity as a function of the canyon geometry, stratification, and incoming flow strength [Allen and Hickey, 2010, AH subsequently]. 1 Department of Earth, Ocean, and Atmospheric Sciences, University of British Columbia, Vancouver, British Columbia, Canada. Corresponding author: S. E. Allen, Department of Earth, Ocean, and Atmospheric Sciences, University of British Columbia, Main Mall, Vancouver, BC V6T 1Z4, Canada. (sallen@eos.ubc.ca) American Geophysical Union. All Rights Reserved /13/ /jgrc [3] Advection-driven upwelling over submarine canyons is driven by the cross-shore pressure gradient in the canyon that cannot be balanced by alongshore flow due to blocking by the topography [Freeland and Denman, 1982]. Thus, upwelling requires that the shelf flow be in the direction opposite to the propagation of Kelvin and shelf waves. As the shelf flow approaches the canyon it tends to turn offshore [Hickey, 1997; Allen et al., 2001; Dawe and Allen, 2010] (purple line in Figure 1). As it crosses the canyon wall, its water columns drop into the canyon, the fluid is stretched and strong cyclonic vorticity turns the flow shoreward so that it crosses the canyon at an angle [AH]. As the flow reaches the downstream rim of the canyon, near the canyon head (purple line in Figure 1), its columns are compressed again, the vorticity is anticyclonic and the flow turns offshore [Allen, 1996; Dawe and Allen, 2010]. The deeper flow, initially along the continental slope, turns into the canyon, flows along the canyon against the downstream canyon wall and upwells (under the shelf flow) at the canyon head and along the downstream rim near the head [AH] (blue line in Figure 1). This flow is thick on the downstream side and pinched to nearly zero thickness on the upstream side [Hickey, 1997; Allen and Hickey, 2010; Dawe and Allen, 2010]. Below the slope water that is upwelled, flow in the canyon is toward the head on the downstream side and returns toward the ocean on the upstream side of the canyon leading to deep cyclonic vorticity [Hickey, 1997; Allen and Hickey, 2010] (orange line in Figure 1). [4] Using dynamical considerations, field data and laboratory data, a scaling analysis for upwelling over steep, 5814

2 Figure 1. Schematic of the flow around a submarine canyon. The surface flow, shelf flow, and the deep water flow are shown, as well as the flow for two types of upwelling discussed in this paper: timedependent upwelling (green) and advection-driven upwelling (blue, purple, and orange). The canyon rim is also shown (dashed line) and is defined as the position of the maximum vertical topographic curvature at the edge where the continental shelf starts to dip down into the canyon. deep canyons provided estimates of the deepest water upwelled onto the shelf, the flux of the upwelled water, the deep vorticity in the canyon, and the presence or absence of a cyclonic eddy just above the canyon rim depth [AH]. Due to the difficulty of measurements in submarine canyons [Hickey, 1997] the number of canyons with observations is limited. Building multiple sizes and shapes of canyons for laboratory studies is prohibitive. Thus, a systematic study of this scaling analysis can only be done numerically. [5] The accurate numerical simulation of stratified flow over steep topography is difficult due to pressure gradient errors [Haney, 1991], vertical advection errors [Allen et al., 2003], and flow separations errors [Dawe and Allen, 2010]. A detailed investigation of upwelling over a canyon using the MITgcm [Marshall et al., 1997] showed convergence at small enough grid size, but a systematic difference between the numerical simulation and the laboratory experiment it was simulating [Dawe and Allen, 2010]. Most of the processes over the canyon were simulated well including the depth of upwelling. However, the generation of vorticity at the upstream side of the canyon was too small in the model. This means the model does not reproduce the cyclonic eddies observed above some field canyons. Accepting this discrepancy, we systematically investigated the depth of upwelling, the upwelling flux, and the deep water stretching in the model by varying the geometry of the canyon and the slope of shelf, the stratification, the strength of forcing, and the Coriolis parameter. [6] Numerous numerical modeling studies of flow over canyons have been done since the initial study of the strong barotropic response within a canyon [Klinck, 1989], including investigations of specific canyons [e.g., Skliris et al., 2004] and the impact of canyons in coastal regions [K ampf, 2010]. Canyons have been used as test beds to study accuracy of numerical models [e.g., Perenne et al., 2001] and the impact of boundary condition choices [Dinniman and Klinck, 2002]. Previous systematic numerical studies of the impacts of geometry, stratification, or flow on upwelling over canyons include K ampf [2007] and a study of the importance of the width [Hyun, 2004]. Some numerical modeling studies have used flat shelves [e.g., K ampf, 2007] whereas others have used sloped shelves [e.g., Dinniman and Klinck, 2002]. The importance of the shelf slope has been recently shown; it reduces the amplitude and horizontal extent of the topographic Rossby waves downstream of a canyon or headland [K ampf, 2012]. Here we present a systematic investigation of the impact of the slope. With the inclusion of the shelf slope this study quantifies a number of upwelling characteristics, namely the depth of upwelling, upwelling flux, and deep water stretching. [7] In section 2, we present the numerical model and the method we used to analyze the results. In the next section, we briefly review the scaling analysis of [AH] and broaden it to include the effect of the continental shelf slope. This is followed by the results, a discussion of the implications, and conclusions. 2. Methodology 2.1. The Model [8] The Massachusetts Institute of Technology general circulation model (MITgcm) [Marshall et al., 1997] is a nonhydrostatic primitive equation model configurable to model the ocean, atmosphere, climate, and laboratory water tanks. The scenario modeled here consists of a canyon in a laboratory tank as per Dawe and Allen [2010]. A canyon cuts into the continental shelf with the center of the tank 5815

3 Figure 2. A contour plot of the Basic slope topography in a tank. The topographic scale is in centimeters. The thick black contours show the shelf-break line, the shelf-break isobath around the canyon and the center cylinder. consisting of a flat abyss (Figure 2). The previous study by Dawe and Allen [2010] resulted in a careful and positive evaluation of the MITgcm numerical model for upwelling over a laboratory size canyon; therefore, a laboratory canyon configuration rather than a real-world canyon configuration was used in this study. The dynamics and the nondimensional numbers, with the exception of the frictional processes, are the same on both the laboratory and real-world scales. In the real world, turbulence is the major frictional process, whereas in the laboratory, molecular viscosity dominates. However, the difference in frictional processes can be neglected as friction has been shown multiple times to be unimportant in the canyon upwelling process [e.g., Allen and Durrieu de Madron, 2009; K ampf, 2012]. An approximate horizontal scale for the tank compared to the real world is 1 cm:1 km, where a typical model canyon in this study has a width of 10 cm and a typical real canyon width would be on the order of 10 km. The vertical scale is exaggerated in the laboratory tank with an approximate scale of 1 cm:100 m. [9] The model was run using cylindrical coordinates with a central cylinder added to the topography to maintain numerical stability [Dawe and Allen, 2010]. The model used a viscosity of 10 6 m 2 s 1 and a diffusion coefficient of m 2 s 1, both are molecular values for water. The use of molecular values is appropriate as there is no turbulence in the laboratory tank. The vertical resolution of the model is 0.05 cm. In the horizontal the resolution in r is 0.15 cm and the resolution in varies so that there is a finer grid spacing of 0.25 over the canyon and a coarser grid spacing away from the canyon, reaching a maximum of 30. The model was run assuming that the system was nonhydrostatic to allow for internal waves, but the results were very similar to the hydrostatic case [Dawe and Allen, 2010]. To mimic a change in rotation rate of the tank, the flow was forced with a body force acting clockwise [Dawe and Allen, 2010]. No-slip boundary conditions along the topography and no friction against the central cylinder were used. [10] To test the scaling many runs with some variations are required. Each run begins with stationary flow with linear stratification in the vertical direction which is uniform in the horizontal direction. The body force is then applied which causes a current to flow over the canyon and create an upwelling event. The water is forced over 26.7 s. A complete run is 30 s. The forcing was stopped before the end of the run to allow the advection-driven upwelling to be dominant over the time-dependent upwelling. [11] Fifty-six different runs were done as part of this study. We use a naming scheme instead of a simple number to refer to the runs. The name refers to the difference from the Basic case (Table 1). The different cases included in the experiment consisted of varying the stratification in the tank by changing the Brunt-V ais al a frequency, N (High and Low N); the rotation rate of the tank by varying the Coriolis parameter, f (High and Low f); the forcing velocity of the water which is represented by the velocity upstream of the canyon, U (High, Medium High, Medium Low, Low and Extra Low U); the geometry of the canyon (Figure 3) such as the canyon length, L (Big and Small L), the radius of curvature of the canyon, R (Sharp and Smooth Entry), and widths, W at half of the length of the canyon (Big and Small W), and W sb at the shelf-break (Big and Small W sb ), all canyon widths (Big and Small Width), canyon width and length (Big and Small); and also by varying the slope of the continental shelf (Double Slope, Big Slope, Small Slope, Half Slope, Flat Slope). [12] The values of the Coriolis parameter and Brunt- V ais al a frequency took only three values as they were set at the beginning of the run (Table 1). Other parameters had a range of values as modifying one topographic parameter often impacted others and U is a response of the model, not a set value (Table 2). The continental shelf slopes ranged from flat to (Figure 4); note that due to the vertical Table 1. Variations of Slope, N, f, and U Parameters Variable Name Value s Flat Slope Half Slope Small Slope Basic Slope Big Slope Double Slope N (s 1 ) Low N 1.1 Basic N 2.2 High N 4.4 f (s 1 ) Low f 0.42 Basic f 0.52 Highf 0.62 U (cm s 1 ) Extra Low U Low U Medium Low U 1.01 Basic U Medium High U 1.92 High U

4 Figure 3. The geometry of the canyon for the Basic case is shown. The thick black contours show the shelf-break line and the shelf-break isobath around the canyon. W sb is the shelf-break width, W is the width of the canyon, L is the length of the canyon from the shelf-break width to the head of the canyon, R is the radius of curvature of the canyon, and R W is the width of the canyon across the 2 cm shelfbreak depth isobath. exaggeration in the tank, these would correspond to realworld values 10 smaller Canyon Geometries and Other Parameters [13] Numerical topographies were generated consisting of a submarine canyon cutting into a shelf/slope topography. There are key measurements which describe the geometry of the canyon: the radius of curvature of the canyon, R; canyon length, L; canyon width, W; and canyon width at shelf-break, W sb [AH]. To mimic the way these measurements are made for field experiments these measurements were taken from a topographic contour plot (Figure 3). The continental shelf slope is another key parameter. The standard error or uncertainty for each parameter was estimated to aid in evaluation of numerical agreement with scaling theory. [14] The length of the canyon, L, is defined as the length from the shelf-break (black contour in Figure 3) to the head of the canyon, where the head of the canyon is the last isobath that is impacted by the canyon in the discretized z coordinates. The uncertainty in L is estimated by taking half of the distance between isobaths which are one grid step above and one grid step below that of the head depth found in the z coordinates. The uncertainty in the coordinates z, r, and is taken to be half of the grid spacing. [15] The width of the canyon, W, is measured at half of the length of the canyon and between the upstream and downstream points of the canyon where the topography first starts to deepen into the canyon. The uncertainty in W was estimated by taking half the difference in this width and that taken one point deeper into the canyon on either side. [16] The next two scales are less easy to define. The canyon width at the shelf-break, W sb, is measured between the corners where the 2 cm shelf-break contour of the canyon would meet the 2 cm contour of the topography if there was no canyon. As the corner does not come to an exact point, there is some leeway in the measurement. This leeway was taken as the error in the measurement of W sb. [17] The radius of curvature of the canyon, R, is the hardest scale to uniquely identify. Visual determination was deemed unreliable. Instead a circle was fit to the topographic points that lie along the 2 cm contour near the upstream corner of the canyon using the Levenberg- Marquardt algorithm. The error in R was estimated by the difference in the solutions which used a different number of points in calculating R. The difference between the solutions was approximately 0.2 cm. In the Flat Slope case where the whole shelf had a depth of 2 cm a slightly lower (2.025 cm) contour was used which produced a better contour. [18] An alternative to the radius of curvature used in the Rossby number Ro is the width of the canyon at the canyon rim, R W, which is measured at half of the length of the canyon and bounded by the shelf-break depth contour on both the upstream and downstream sides (black contour in Figure 3). The error in R W was estimated by taking half the difference in the widths for the maximum and minimum allowable canyon lengths, determined from the error in L. [19] The continental shelf slope, s, is found by taking the topographic slope from the coast, or wall of the tank in this model, to the shelf-break away from the canyon. The uncertainty in the shelf slope was estimated by using the uncertainty in z at both the coast and shelf-break. [20] The velocity of the water upstream of the canyon is also important in governing the strength of an upwelling event. This velocity along the isobaths, U, was found by averaging the velocity profiles upstream of the canyon between and cm depth at the 1.76, 1.88, and 2.0 cm isobaths for the Basic case and at the same spatial locations for the other cases. The uncertainty in U was estimated by taking half of the largest difference of the velocity at the three locations upstream of the canyon and within the same depth range; this value was found to be 0.02 cm s 1 for the Basic case and is used for all other cases. [21] The velocity in the downstream corner of the canyon was used in the Froude number definition because the Froude number is used to examine flow in the downstream corner rather than upstream where U is measured. This velocity, U c, was found by averaging a profile in the downstream corner of the canyon between and cm depth. The uncertainty in U was also used for the uncertainty in U c Analysis Determining Upwelling Depth [22] The vertical movement of water is tracked by the salinity of the water. Over the duration of an upwelling event deeper water will rise to shallower depths. Neglecting diffusion, the salinity of the water at a given depth will indicate from which depth the water originated. [23] The depth of upwelling in this study is defined as the depth change of the densest water upwelled onto the shelf. As this water occurs near the head of the canyon and upwelling is weak near the mouth, it is approximately given by Z, the change in depth along the canyon of the isopycnal that just touches the bottom at the head of the canyon (Figure 5). To determine the depth of upwelling from the model results the maximum salinity at head depth over 5817

5 Table 2. Values for the Different Cases With Corresponding Error a Case L (cm) W (cm) W sb (cm) R ðcm Þ R W ðcm Þ s U (cm s 1 ) Basic Basic High U Basic Low U Basic Medium High U Basic Medium Low U Basic Low N Basic High N Basic Low f Basic High f Small Big Big Width Small Width Big L Small L Big W sb Small W sb Big W Small W Sharp Entry Smooth Entry Double Slope Double Slope High U Double Slope Low U Double Slope Low N Double Slope High N Double Slope Low f Double Slope High f Half Slope Half Slope High U Half Slope Low U Half Slope Extra Low U Half Slope Low N Half Slope High N Half Slope Low f Half Slope High f Flat Slope Flat Slope High U Flat Slope Low U Flat Slope Extra Low U Flat Slope Low N Flat Slope High N Flat Slope Low f Flat Slope High f Big Slope Big Slope High U Big Slope Low U Big Slope Low N Big Slope High N Big Slope Low f Big Slope High f Small Slope Small Slope High U Small Slope Low U Small Slope Low N Small Slope High N Small Slope Low f Small Slope High f a Blank entries have the same values as the entry above. the whole tank was determined. The depth at which this salinity occurred at the beginning of the model run, compared to its final depth (head depth) gives the depth of upwelling. The uncertainty in each of the minimum and maximum depths used to calculate the depth of upwelling was estimated to be half the grid spacing in z. 5818

6 Figure 4. Cross sections of the topography from the center to the wall of the tank as shown in Figure 2. These cross sections are taken away from the canyon and illustrate the different slopes used in the various canyon cases. Values corresponding to the names for the different slopes are given in Table 1 and range from to Volume of Water Upwelled onto the Continental Shelf [24] The volume of water upwelled onto the shelf per radian along the coast was determined by summing, per grid cell, the water that is above the shelf-break depth and was originally deeper than the shelf-break depth as determined by its salinity. Three sources of upwelling can be Figure 6. (top) A plan view of the height of the upwelled water above the shelf (in cm) for the Flat Slope High U case. Black lines show the tank edge, the shelf-break, the shelf-break isobath around the canyon, and the center cylinder. (bottom) A plot of the model computed volume of water upwelled per radian which is separated into three sources of upwelling as indicated by different Gaussian curves. The advection-driven upwelling is isolated and used in flux calculations. Figure 5. A vertical cross section along the central axis of a canyon. The dot-dash line shows the topography of the shelf/slope away from the canyon. The top plot is an expansion of the bottom plot. The head of the canyon is where the canyon bathymetry meets the shelf/slope bathymetry away from the canyon and the mouth of the canyon is at the shelf-break location away from the canyon. On the bottom plot: the depth of the head of the canyon, H h ; the shelf-break depth, H s ; and the canyon length, L. Best seen on the top plot: the bold blue isopycnal is the deepest isopycnal to reach the depth of the head of the canyon and the depth it rises is Z, the depth of upwelling. The distance above the head depth that the shallowest tilted isopycnal is found is Z b. The flow is coming out of the page and the case shown is the Double Slope. observed (Figure 6). Since the focus is the dominant, advection-driven upwelling, the volume of water upwelled due to advection was separated from that due to time dependence (symmetric over the canyon) and that due to internal wave uplift (only occurred in some cases) by fitting Gaussian curves to the different peaks. A sum of Gaussian curves is fit to the measured volume values as a function of using the Levenberg-Marquardt algorithm for curve fitting. [25] The volume of water upwelled was found at 5 s intervals and from that the flux was calculated as: ¼ V 30s V 25s 30s 25s where is the flux, V 25s and V 30s are the volumes of water upwelled onto the shelf at 25 and 30 s, respectively, and are found from integrating the Gaussian representing the advection-driven upwelling. [26] The uncertainty for the upwelling flux is estimated by calculating the flux between 29 and 30 s and taking the difference from the flux value used over s. ð1þ 5819

7 Vorticity [27] Vorticity, in cylindrical coordinates is: ¼ 1 where is the vertical component of vorticity, u is the velocity in the direction of negative, and v is the velocity in the r direction. [28] In the tank, the azimuthal velocity has a large shear due to the forcing of the flow increasing with radius. To isolate just the vorticity generated by the canyon, only the variation in the radial velocity was used. So the deep vorticity due to canyon was estimated as: v iþ1 v i r jþ 1 2 d where i, j, k are indices which indicate components in, r, and z, respectively, d is the grid spacing in in radians, and r is on a staggered grid compared to the velocity measurements. The deep vorticity was measured at three quarters length from the head of the canyon and taken to be the maximum value within the canyon, below the shelf-flow Figure 7. (top) Total vorticity in the canyon at three quarters length of the canyon. (bottom) Vorticity due to variations in the radial velocity generated by the canyon at three quarters length of the canyon (3). The vorticity in both plots is at 30 s and for the Flat Slope High U case. Units for both plots are s 1. The dashed contours indicate negative values. ð2þ ð3þ near the rim (purple line in Figure 1) and not including the boundary layers along the canyon walls (Figure 7). [29] The uncertainty in vorticity was estimated by finding the maximum vorticity in the center at half length from the head of the canyon and taking the difference from that found at three quarters length from the head of the canyon. 3. Scaling Analysis [30] The dynamic parameters of advection-driven canyon upwelling include the incoming velocity, U; the Coriolis parameter, f; and the Brunt-V ais al a buoyancy frequency, N, which characterizes the stratification [AH]. Friction is neglected here. Geometric parameters include the shelf-break depth, H s ; the depth at the head of the canyon, H h ; the canyon length, L; the canyon width at half length, W; the canyon width at the shelf-break, W sb ; and the radius of curvature of the upstream isobaths at the shelf-break depth, R [AH]. The canyon is considered deep enough that its depth does not limit upwelling. The two dimensions in this upwelling problem are length and time. With these nine parameters and two dimensions there will be seven nondimensional groups according to the Buckingham PI theorem [Buckingham, 1914]. [31] The four dominant nondimensional groups already described by [AH] consist of an aspect ratio L=W sb,two Rossby numbers Ro ¼ U=f R and R L ¼ U=fL, andaburger number B u ¼ NH s =fw. The three additional numbers were a vertical aspect ratio, H s =L, one additional Burger number, and a measure of continental slope, s, or scaled as the topographic Burger number B T ¼ sn=f.herewewillshowthe importance of the shelf-slope B T on the depth and volume of upwelled water. In addition, the numerical results suggest that R W ¼ U=f R W characterizes the incoming shelf-flow turning over the canyon better than Ro ¼ U =f R. [32] Some assumptions have been made which restrict this scaling analysis to certain canyons [AH]. These assumptions are: (1) nearly uniform incoming flow along the length of the canyon so that the pressure gradient along the canyon is nearly uniform, (2) weak to moderate flow, specifically that F R W < 0:2 wheref¼ro= ð0:9 þ RoÞ, (3) uniform stratification over the depth of upwelling, (4) a shallow shelf-break depth, specifically that NH s = ðflþ < 2, (5) that the canyon is deep, specifically that 2H c N= ðfw sb Þ > 1, where H c is the maximum depth of the canyon across the mouth, (6) that the canyon is narrow, specifically that it is less than 2 Rossby radii (NH s /f) wide, (7) that the canyon has steep walls and the water column is stratified so that the bottom boundary layer is arrested [Brink and Lentz, 2010], and (8) that the canyon has a regular shape at the upstream corner so that the topographic length H s ð@h=@rþ 1,whereh is the depth, is close to the radius of the curvature of the isobaths R Depth of Upwelling [33] As upwelling occurs, the isopycnals within the canyon are tilted, particularly near the head of the canyon. Assuming there is little or no upwelling at the mouth of the canyon, the depth of upwelling is approximately Z, the change in depth along the canyon of the isopycnal that just touches the bottom at the canyon head (Figure 5). The pressure due to these tilted isopycnals within the canyon inhibits further upwelling. This pressure can be determined from the 5820

8 hydrostatic balance as o N 2 Z 2 with N 2 ¼ g= =@z where is the horizontally uniform, undisturbed density [AH]. However, the pressure due to tilted isopycnals above the head depth, Z b, also restricts flow moving further up the shelf (Figure 5) and should also be included in the scaling. At depth Z b above the canyon head there is no density difference between the head and mouth of the canyon by definition. However, at the head depth there is a substantial density difference due to these tilted isopycnals. Neglecting the smaller isopycnal deflections at the mouth of the canyon, this density difference can be estimated as D ¼ ð@=@zþz ¼ ð o =gþn 2 Z.SoD, the difference in density between the head and the mouth, can be linearly interpolated as: D ¼ on 2 Z 1 Z ð4þ g Z b where Z is the depth above the head depth. [34] Integrating the hydrostatic pressure due to this density difference above the canyon gives: Dp Zb ¼ 1 2 on 2 ZZ b ð5þ [35] Thus, the total back pressure due to both tilted isopycnals within and above the canyon can be expressed as: Dp total ¼ o N 2 Z 2 1 þ Z b Z where is an unknown coefficient of order one. This pressure must balance the available along canyon pressure change. In the absence of a canyon but for a section of shelf of length of the canyon, the available pressure change would be o ful [AH]. The pressure change would also have this strength if the flow passed directly over the canyon. However if flow is weak and follows the isobaths, there is no pressure difference along the canyon. Thus, the along canyon pressure difference can be written o fulf where F ðroþ is the function that determines the strength of cross-isobath flow compared to the along-isobath flow and is 1 for high Rossby number and zero for low Rossby number. A suitable function is F¼Ro= ð0:9 þ RoÞ [AH]. Balancing the available along canyon pressure difference and the total back pressure due to the tilted isopycnals gives: Z 2 1 þ Z b Z ¼ fulf N 2 where is a proportionality constant. [36] The amount of upwelling in the canyon, Z, is dependent on the over lying pressure gradient due to flow cross the canyon and pressure effect of tilted isopycnals along the canyon. In the absence of a shelf slope, the process just above the canyon is the same; there are no new dimensional scales to influence Z b differently than Z and so they must scale proportionally. This gives Z bf ¼ Z where Z bf is Z b in the Flat Slope case and is a proportionality constant. The water upwelled up, on to the shelf is advected downstream by the shelf flow U. [37] However, if the shelf is sloped, this dense water will naturally tend to move upstream against the flow at a speed c ¼ g 0 s=f, where is a proportionality constant, and g 0 is ð6þ ð7þ the reduced gravity [Nof, 1982]. The top of upwelled water at the head of the canyon is Z b above the head and this isopycnal is relatively flat. However, at the bottom at the head, the water originally was Z deeper than it is now. Thus, its density perturbation is d =dzz. So here g 0 ¼ N 2 Z=2 because g 0 should be an average over the depth of the upwelled water [Whitehead et al., 1990]. The speed then becomes c ¼ N 2 Zs 2f [38] Thus, with a shelf slope, the upwelled water will be slowed to a speed U c compared to a speed U in the absence of a shelf slope. As the water is moving away more slowly, it will tend to accumulate in the region of the canyon head, thus increasing the thickness of the upwelled water in the slope case so that Z b ðu cþ ¼ Z bf U. This can be rewritten as: ð8þ Z b ¼ Z 1 c ð9þ U [39] The denominator does not actually go to zero because c is proportional to Z which is dependent on U. Equation (9) implies that a shelf slope causes the deflection above the canyon Z b to be a larger proportion of Z. [40] Substituting (9) into (7) gives Z 2 1 þ 1 2 SZ ¼ Zo 2 ð10þ p where Z o ¼ ffiffiffiffiffiffiffiffiffiffiffiffi fulf=n and S¼N 2 s=fu. p [41] If S¼0 then Z ¼ a 1 Z o where a 1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = ð1 þ Þ. As the slope increases the upwelled water advects more slowly away from the canyon and its thickness increases, increasing the back pressure due to the upwelled water on the shelf near the head. The overlying along canyon pressure gradient, due to the shelf flow crossing the canyon, provides both the pressure due to upwelling in the canyon and this pressure due to the upwelled water on the shelf. If the latter is increased, the former must decrease. Thus, the presence of a shelf slope that causes accumulation of the upwelled water near the head of the canyon reduces Z, the depth of upwelling. Specifically as S increases, Z/Z o decreases (10). [42] Expanding (10) gives a cubic equation for Z. 2 SZ3 þ ð1 þ ÞZ 2 þ 2 SZ2 o Z Z2 o ¼ 0 ð11þ [43] The ratio of term 1 to term 3 in (11) is Z 3 Z 2 o Z ¼ Z2 Z 2 o ð12þ which is small for large S. So two approximations can be made: (1) for large S that the first term of (11) is small and (2) for small S that Z a 1 Z o. Both approximations along with the quadratic formula and Taylor expansions lead to the scale for the depth of upwelling: 5821

9 Z ¼ a 1 Z o a 1SZ o ð13þ where is 1 for the first approximation and = ð1 þ Þ for the second. Neglecting this small variation and normalizing this equation with the depth scale D h ¼ fl=n and combining 1=4a 1 as a 2 the depth of upwelling scale can be written as: Z ¼ a 1 ðfr L Þ 1=2 sn 1 þ a 2 D h f HOWATT AND ALLEN: SHELF SLOPE IMPACT ON UPWELLING CANYONS! F 1=2 þ a 3 R L ð14þ where a 1, a 2, and a 3 are coefficients to be fit using the Levenberg-Marquardt algorithm for curve fitting with the numerical model results Upwelling Flux [44] The upwelling flux onto the shelf,, through the canyon is given as the width of the upwelling stream times its depth, which scales with Z, times its velocity denoted U [AH]. [AH] took the width to be the width at the mouth of the canyon, W sb, but as U is a mean velocity in the canyon, caused by the along-canyon pressure gradient, a midcanyon width W is more appropriate; we find scaling with W better fits the numerical results. Thus, the formula for the flux can be expressed as ¼ WZU. As (14) is a better estimate of the upwelling stream depth Z compared to that estimated by [AH] who neglected variations in the shelf slope, (14) should be used find the upwelling flux. In addition, the velocity of the upwelling stream is affected by changes in Z. [45] The flow speed of the upwelling stream varies with depth as the pressure gradient does. The average flow speed will be in balance with the average pressure gradient which will occur at about Z/2 below the canyon head. The barotropic pressure gradient force is o fuf which is the sum of the baroclinic pressure gradient forces due to upwelled water above the canyon, Z b as well as tilted isopycnals within the canyon, Z. The pressure gradient at the head of the canyon, H h, is found by subtracting the contribution from Z b from the barotropic pressure gradient force. Thus, the pressure gradient force at H h is equal to the pressure gradient force due to Z. [46] The pressure gradient is zero at depth Z and at the head depth, H h, the pressure gradient is o N 2 Z 2 [AH]. The upwelling pressure gradient force is then equal to one half the pressure gradient force at H h. [49] Normalizing this equation with WUD h and combining 1=4a 1 as b 2 the upwelling flux scale can be written as: ¼ b 1 F 3=2 R 1=2 L WUD h 1 þ b 2 sn f! F 1=2 3 þ b 3 ð18þ where b 1, b 2, and b 3 are coefficients fit using the Levenberg-Marquardt algorithm for curve fitting with the numerical model results Deep Water Stretching [50] Vorticity deep in the canyon is determined by the amount of stretching which is S ¼ Z=D W where D W ¼ fw sb =2N is a depth scale deep in the canyon [AH]. However, comparing with numerical results suggests that a width scale of W works better than W sb in the scaling. The deep water stretching can then be written as: S ¼ f ¼ d 1ðF R L Þ 1=2 sn 1 þ d 2 f R L! F 1=2 R L L W þ d 3 ð19þ where d 1, d 2, and d 3 are coefficients fit using the Levenberg-Marquardt algorithm for curve fitting with the model results. 4. Results 4.1. Depth of Upwelling [51] There are two possibilities for Rossby numbers to describe the ability of the incoming shelf-flow to cross the canyon: Ro, based on the radius of curvature, R, of the upstream shelf-break isobath [AH] and R W, based on the width of the canyon, R W [Klinck, 1989] (Table 3). Figures 8 and 9 show the scaling for the depth of upwelling using Ro and R W, respectively. The norm of residuals is smaller (0.27) using R W than using Ro (0.43), thus the model suggests that R W describes the turning of the flow over the canyon better than Ro. We will return to this issue in the discussion, but all further results will be based on a scaling using R W. Then scaling for the depth of upwelling can be written as: Z ¼ 1:8ðF W R L Þ 1=2 ð1 0:42S E Þþ0:05 ð20þ D h o fu 1 o N 2 Z 2 2 L [47] Rearranging, U can be scaled as: U / N 2 Z 2 2fL ð15þ ð16þ [48] So now the upwelling flux is proportional to Z 3 where Z is defined in (13) / Wa3 1 N 2 2fL ðf ULf Þ 3= a 1SZ o ð17þ N 3 where F W is a function similar to F but uses Rossby number R W and where S E ¼ sn=f ðf W =R L Þ 1=2 is the nondimensional slope effect. Table 4 contains the 95% confidence levels for the fitted coefficients a 1, a 2, and a 3. [52] The scaling was applied to results from real canyons: Astoria, Barkley, Carson, Quinault, and Tidra Canyons [AH]. The scaling for these canyons were found to match closely to the model canyons (Figures 8 and 9). However, the scale using the Rossby number Ro appears to be a better scale than that based on R W as the points lie closer to the one-to-one line Upwelling Flux [53] The flux is determined by differences in the volume upwelled onto the slope between 25 and 30 s. Figure

10 Table 3. Nondimensional Numbers and the Vertical Scale Depth for the Different Cases, With Corresponding Error a Case R L ¼ U=fL Ro ¼ U=f R R W ¼ U=f R W F W D h (cm) S E Fr ¼ U c =NH s Basic Basic High U Basic Low U Basic Medium High U Basic Medium Low U Basic Low N Basic High N Basic Low f Basic High f Small Big Big Width Small Width Big L Small L Big W sb Small W sb Big W Small W Sharp Entry Smooth Entry Double Slope Double Slope High U Double Slope Low U Double Slope Low N Double Slope High N Double Slope Low f Double Slope High f Half Slope Half Slope High U Half Slope Low U Half Slope Extra Low U Half Slope Low N Half Slope High N Half Slope Low f Half Slope High f Flat Slope Flat Slope High U Flat Slope Low U Flat Slope Extra Low U Flat Slope Low N Flat Slope High N Flat Slope Low f Flat Slope High f Big Slope Big Slope High U Big Slope Low U Big Slope Low N Big Slope High N Big Slope Low f Big Slope High f Small Slope Small Slope High U Small Slope Low U Small Slope Low N Small Slope High N Small Slope Low f Small Slope High f a Blank entries have the same values as the base case. HOWATT AND ALLEN: SHELF SLOPE IMPACT ON UPWELLING CANYONS shows the volume of water upwelled by three different processes: the time-dependent peak is symmetrical over the canyon and in many cases this peak is decreasing over the last 5 s; a peak due to upwelling near the mouth is found slightly downstream; and an advection peak is found further downstream. 5823

11 Table 4. 95% Confidence Intervals for Coefficients in the Scaling of the Depth of Upwelling, Deep Water Vorticity, and Flux Coefficient Name Lower Bound Value Upper Bound Depth of upwelling using Ro a a a Depth of upwelling a using R W a a Upwelling flux b b b Deep water vorticity d d d Figure 9. As Figure 8 but with the scaling based on R W. The norm of residuals is [54] The extra source of upwelling in the downstream corner of the canyon at the shelf-break was unexpected (Figure 6, top view). Full characterization of this upwelling is beyond the scope of this paper; however, we note that it occurs in cases where the Froude number, Fr ¼ U c =NH s > 0:5. U c is velocity at the downstream corner of the canyon. Looking at salinity cross sections at the shelfbreak and at half length of the canyon of the Flat Slope High U case (Figure 10) it can be seen that the internal waves have a greater amplitude at the shelf-break compared to further up the canyon; at the shelf-break a portion of water upwelled stays localized near the canyon downstream rim due to the localized uplift by internal waves, whereas at half length the internal wave effect is small and almost all the upwelled water has been transported further downstream by advection. For Fr < 0:3 this extra upwelling does not occur. For 0:3 < Fr < 0:4 most cases do not show it, but Flat Slope cases did. [55] Most of the upwelling seen in Figure 6 is found downstream of the canyon and due to advection. The scaling is based on the advection phase of an upwelling event Figure 8. Observed versus scaled plot for the depth of upwelling with the scaling based on Ro. The Slopes referenced in the legend are cases where only the shelf slope was varied (Table 1). The black points to the bottom left have steeper slopes and those to the top right have flatter slopes. The geometry cases involve cases where the canyon shape was varied with Small representing small length, width, etc., and sharp entry and Big representing big length, width, etc., and smooth entry. The U, N, and f cases refer to cases where these parameters were varied (Table 1) along with the shelf slope. Similar to the Slopes cases, those to the lower left of any color grouping have steeper slopes than those the upper right. Some real canyons from [AH] and the one-to-one line are also shown. The norm of residuals is Figure 10. Salinity cross sections for the Flat Slope High U case taken at 25 s at (top) the shelf-break and (bottom) half length of the canyon. 5824

12 and so the flux was calculated based solely on changes of the advection peak. The scaling for the upwelling flux can be written as: WUD h ¼ 0:91F 3=2 HOWATT AND ALLEN: SHELF SLOPE IMPACT ON UPWELLING CANYONS W R1=2 L ð1 1:21S E Þ 3 þ 0:07 ð21þ [56] Comparisons of the calculated flux values from the model and the scaled values have a norm of residuals of 0.37 (Figure 11). The Double Slope High N case is an outlier and has a negative scaled flux value. The Flat Slope cases tend to lie above the one-to-one line and the Double Slope cases tend to lie below. The coefficients in front of the slope a 2, b 2, and d 2 are expected to be the same for the depth of upwelling, vorticity, and flux; however, b 2, is larger than the depth of upwelling coefficient, a 2 (Table 4) Deep Water Stretching [57] The maximum cyclonic vorticity due to variations in the radial velocity generated by the canyon (3) (Figure 7, bottom) is deeper in the canyon than the maximum total vorticity (Figure 7, top) and can be seen on the upstream side at approximately 3 cm depth in the Flat Slope High U case. [58] Comparing the calculated values from the model output using (3) with the scaled values (Figure 12) leads to: f ¼ 2:0 ð F W R L Þ 1=2 ð1 1:04S E Þ L þ 0:48 ð22þ W [59] Here the fit has a norm of residuals of 3.0 (Figure 12). The Double Slope High N is an outlier from the other points. In general the high U cases tend to lie above the one-to-one line and the low U cases tend to lie below the one-to-one line. As already mentioned the coefficients in front of the slope for the depth of upwelling, vorticity, and flux are expected to be the same. The coefficient d 2 is the same as the flux coefficient, b 2, within the 95% confidence intervals; however, it is larger than that for the depth of upwelling, a 2 (Table 4). Figure 11. Observed versus scaled plot for the normalized upwelling flux. For explanation of the legend see the caption of Figure 8. The norm of residuals is Figure 12. Observed versus scaled plot for the stretching within the canyon. For explanation of the legend see the caption of Figure 8. The norm of residuals is 3.0. [60] The scaling was applied to results from real canyons: Astoria, Barkley, and Quinault Canyons [AH]. The scaling for these canyons was found to match closely to the model canyons (Figure 12) as the points lie close to the one-to-one line. 5. Discussion [61] A closer examination of the scaling done by [AH] shows that the shelf slope plays an important role in the depth of upwelling, upwelling flux, and deep water vorticity. Including the effect of the additional pressure due to water already on the shelf and of the reduced velocity of this water on a sloped shelf, these upwelling quantities can be scaled more accurately. The few data for the depth of upwelling and the deep vorticity of real canyons follow this scaling Dynamics [62] An advantage of having algebraic formulations (20), (21), and (22) is that one can interpret the results dynamically. In each of the three relations, there are three parts: (1) the dynamical scale which is used to nondimensionalize the quantity, (2) the basic dynamical response in the absence of the slope, and (3) the effect of the slope. We will treat each part separately. [63] 1. The dynamical scales are based on the incoming flow speed U as a velocity scale, the Coriolis parameter f as a vorticity scale, the width of the canyon W as the horizontal scale and the stratified scale depth D h ¼ fl=n as the vertical depth scale. [64] 2. As the flow on the shelf approaches the canyon, if it has a low Rossby number it is constrained to follow the topography around the canyon whereas if it has a large enough Rossby number it will flow directly across. If the flow follows the topography, there is no pressure gradient in the canyon and thus no upwelling. If the flow goes directly across the canyon, the pressure gradient along the canyon is the same as it would be away from the canyon. Thus, as the Rossby number increases from 0 to infinity, the fraction of the pressure gradient realized over the canyon goes from 0 to 1. This is expressed as the function F. 5825