1. Introduction
Exchange of water and solutes between the coastal and open ocean is key to understanding global biogeochemical budgets and their response to climate change and human activities (Jordi et al. 2008). Moreover, the specific spatial distribution of tracers like dissolved oxygen can impact benthic and demersal communities (Keller et al. 2010). In general, exchange between the deep ocean and the continental shelf is limited, as homogeneous, geostrophic flow is restricted to follow isobaths along the continental shelf (Taylor–Proudman theorem), such that deep-ocean exchange occurs only when ageostrophic dynamics occur (Allen and Durrieu de Madron 2009). Submarine canyons can induce ageostrophic motions because the canyon is a region of higher Rossby number relative to the slope, meaning that near the canyon, advection of momentum is an important driver of the flow. On a regional scale, submarine canyons are known to modify or enhance shelf–slope mass exchange and regional currents (Hickey 1995).
Both the distribution and on-shelf inventory of nutrients and oxygen can have relevant biological consequences for the shelf system. A recent numerical study of the coast of Washington State estimated that seasonal nitrate input from the slope to the shelf associated with three nearby canyons was between 30% and 60% of that contributed by local wind-driven upwelling (Connolly and Hickey 2014). They also found that changes in near-shelf-bottom oxygen concentrations in the presence of the canyons matched levels of hypoxia in the region. These changes were large enough to have an ecological impact if compared to levels of severe hypoxia associated with mortality in marine organisms. Moreover, it has been reported that on the West Coast of the United States, small changes in dissolved oxygen concentrations in already-hypoxic waters can cause large changes in the total and species-specific catch of demersal fish (Keller et al. 2017).
In addition to enhancing upwelling, submarine canyons can enhance mixing within their walls by focusing internal waves and tides (Gordon and Marshall 1976). Although turbulence has been measured in only a few canyons, average diapycnal diffusivity values in the surveyed ones are very high compared to levels outside [e.g., Monterey Canyon 
There has been extensive research on the upwelling circulation within submarine canyons (e.g., Allen and Hickey 2010, hereafter AH2010; Howatt and Allen 2013, hereafter HA2013; Freeland and Denman 1982; Klinck 1996; Kämpf 2007). However, the slope–shelf flux of biologically relevant tracers such as oxygen or nitrate is less understood. The objective of this work is to study the impact of an idealized submarine canyon on the cross-shelf exchange and on-shelf distribution of a passive tracer, taking into account the effect of locally enhanced mixing. We quantify cross-shelf fluxes of a passive tracer and develop scaling estimates for the tracer flux upwelled onto the shelf. To quantify the impact of locally enhanced mixing, we have designed numerical experiments that represent mixing in the form of enhanced vertical diffusivity and viscosity using different geographical distributions.
In the following sections, we explain the numerical configuration and experiments (section 2); we describe the flow dynamics of the base case (section 3a) and the effect of locally enhanced diffusivity on the dynamics of the flow and tracer transport from the canyon to the shelf; and we look at the tracer evolution within the canyon (section 3b), cross-shelf transports (section 3c), and upwelling through the canyon (section 3d). In section 4, we scale the advection–diffusion equation and provide justification for choosing the parameter space we explored in the numerical experiments. Furthermore, we develop a scaling estimate for the tracer flux onto the shelf as the product of a characteristic concentration and the canyon-upwelled water flux derived in previous scaling estimates (AH2010; HA2013) with a modification to account for the effect of enhanced mixing. Finally, in section 5, we provide a summary and discussion of our results.
2. Methodology
We use the Massachusetts Institute of Technology general circulation model (MITgcm) (Marshall et al. 1997) to simulate a system consisting of a sloping continental shelf cut by a submarine canyon (Fig. 1), with incoming flow from the west (upwelling favorable), parallel to the shelf. The range of stratifications, incoming shelf currents, and Coriolis parameters selected for all runs represents realistic oceanic conditions over continental shelves around the world, and in this sense, they constitute typical dynamical settings for a submarine canyon. We explore a wider range of parameter space for vertical mixing, as we go from low values of vertical diffusivity to the extreme values observed both in magnitude and vertical distribution.
(a) Cross-shore section through the canyon axis; the dashed line marks the shelf bottom, which can be identified with the canyon rim. (b) Top view of the canyon. The shaded area corresponds to the LID section, across which vertical transport was calculated. The solid black line is the shelf break at 149.5 m. Canyon dimensions: 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
The simulation starts from rest. A shelf current is spun up by applying a body force on every cell of the domain directed westward, along shelf with a similar effect as changing the rotation rate of a rotating table (Spurgin and Allen 2014). The body forcing ramps up linearly during the first day, stays constant for another day, and ramps down to a minimum, after which it remains constant and just enough to avoid the spindown of the shelf current. This forcing generates a deeper shelf current, less focused on the surface, than the coastal jet generated by wind-forced models (Fig. S1 of the online supplemental material). The model was run for 9 days.
The domain is 280 km along shelf and 90 km across shelf divided in 616 × 360 cells horizontally. The cell width increases smoothly along shelf and cross shelf, from 115 m over the canyon to 437 m at the north boundary, and to 630 m at a distance of 60 km upstream and downstream of the canyon, and then is uniform to the downstream boundary. Vertically, the domain is divided in 90 z levels spanning 1200 m, with grid sizes varying smoothly from 5 (surface to below shelf) to 20 m at depth. The time step used was 40 s, with no distinction between baroclinic and barotropic time steps. The experiments ran in hydrostatic mode. Some runs were also repeated in nonhydrostatic mode, with no significant differences in the results.
The canyon was constructed from a hyperbolic tangent function. Geometric parameters of the canyon (Fig. 1b) are similar to those of Barkley or Quinault Canyons, with geometric and dynamical nondimensional groups representative of numerous short canyons, as will be discussed in section 4e [AH2010; Allen (2000) for short canyon discussion]. The domain has open boundaries at the coast (north) and deep ocean (south). Open boundaries use Orlanski radiation conditions and no sponge. Bottom boundary conditions are free slip with a quadratic drag with coefficient 0.002, while vertical walls on the bathymetry steps have a free-slip condition. East and west boundaries are periodic. The domain is long enough that water does not recirculate through the canyon during the simulation. However, barotropic Kelvin waves, first- and second-mode baroclinic Kelvin waves, and long-wavelength shelf waves do recirculate through the domain as in previous studies with similar configurations (e.g., She and Klinck 2000; Dinniman and Klinck 2002). Subinertial shelf waves of wavelength likely to be excited by the canyon (40 km; Zhang and Lentz 2017) are too slow to recirculate. The gravest mode has a wave speed of approximately 0.5 m s−1 against the flow [calculated using Brink (2006)].
The initial fields of temperature and salinity vary linearly in the vertical and are horizontally homogeneous (Fig. 1e). For all runs, temperature decreases and salinity increases with depth, but their maximum and minimum values are changed to generate the different stratifications used in the simulations. A passive tracer was introduced from the beginning of the simulations with a linear vertical profile that increases with depth, intended to mimic a nutrient such as nitrate (Fig. 1f). The maximum and minimum values of the profile come from data collected during the Pathways Cruise in summer 2013 in Barkley Canyon (Klymak et al. 2013).
We use the GMREDI package included in MITgcm for diffusing tracers. Since the mesoscale eddy field is resolved, we have no need to characterize the transport due to these structures. However, it is desirable to numerically handle the effects of tilting isopycnals that are intrinsic to canyon upwelling dynamics (Allen et al. 2001). Mixing and stirring processes are better described within the canyon as being along isopycnal and cross isopycnal, rather than horizontal and vertical. Inside the canyon, vertical mixing is set to be larger than outside (see below), so diapycnal tracer transport will be enhanced. Considering this, we use the scheme for isopycnal diffusion (Redi 1982) but did not use the skew flux parameterization (Gent and McWilliams 1990). In sum, the vertical effective diffusivity on the tracer is determined by the prescribed vertical eddy diffusivity 
Patterns of enhanced diapycnal mixing within submarine canyons vary spatially and temporally. For example, Ascension Canyon has sides and axis slopes supercritical to M2 internal tides, with maximum dissipation zones near the bottom, just below the rim, and larger average dissipation rates during spring tides (Gregg et al. 2011). On the other hand, Gaoping Canyon is subject to strong barotropic and baroclinic (first mode) tides, and at critical frequencies, there is a turbulent overturning due to shear instability and breaking of internal tides and waves; diapycnal diffusivity varies seasonally due to changes in stratification (Lee et al. 2009).
Diapycnal diffusivity profiles along Ascension Canyon’s axis show a sharp gradient near rim depth, close to the head, and the mean profile shows a clear step in diffusivity at rim depth (Fig. S2, bottom row). We also see sharp but less intense variations of diapycnal diffusivity 
Upwelled water on the shelf has been estimated previously by finding water originally below shelfbreak depth based on its salinity (HA2013). We take the same approach but use the tracer concentration at shelfbreak depth as the criterion to find water on shelf that was originally below shelfbreak depth. Enhanced diffusion may cause our algorithm to underestimate the amount of upwelled water on shelf. To minimize this error, we added a second tracer with the same linear profile as the original but with smaller explicit diffusivity. This allows us to find upwelled water on the shelf without the effects of enhanced diffusivity on concentration, only keeping the dynamical effects of enhanced 
We explore the effects of vertical eddy diffusivity 
All runs in the dynamical experiment have a corresponding no-canyon run and constant vertical diffusivity as in the base case in Table 2. For all runs, parameters 
All runs in the mixing experiment have the same dynamical parameters as the base case in Table 1. All runs reported have a corresponding no-canyon run. Only values changed from the base case (boldface entries in first row) are shown. Values of 
3. Results
a. Description of the flow
The body forcing generates an upwelling-favorable shelf current that slightly accelerates after the initial push (Fig. 2e). These conditions tilt the sea surface height down toward the coast. During spinup (days 0–3), the upwelling response is intense on the shelf and through the canyon. This time-dependent response is linear and thus directly proportional to the forcing (Allen 1996) and will not be discussed further (time-dependent phase). We focus on the next stage, after day 4, when the current has been established, baroclinic adjustment has occurred, and advection dominates the dynamics in the canyon (advective stage).
Main characteristics of the flow during the advective phase. Average day 3–5 contours of tracer concentration (color) and σθ (solid black lines; kg m−3) at an along-shelf section close to the (b) canyon mouth and (a) along the canyon axis. (c) Along-shelf and (d) cross-shelf components of velocity. (e) Evolution of along-shelf component of incoming velocity U, calculated as the mean in the gray area delimited in (c). The dashed line marks the beginning of the advective phase. (f) Speed contours and velocity field at 127.5-m depth with shelf break in white. (h) Tracer concentration on the shelf bottom averaged over days 3–5. (g) Evolution of average bottom concentration on the downstream shelf, bounded by the yellow rectangle in (h).
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
The highest along-shelf velocities can be found on the slope at about 200 m (Fig. 2c), but the scale velocity for canyon upwelling is on the upstream-canyon shelf, between shelf break and canyon head, close to shelf bottom but above the bottom boundary current (gray box; Fig. 2c). This constitutes the incoming velocity scale U (section 4). In that area, U stays between 0.35 and 0.37 m s−1 during the advective phase for the base case (Fig. 2e).
Circulation over the canyon is cyclonic. An eddy forms near the canyon rim (Fig. 2f), and incoming flow deviates toward the head on the downstream side of the canyon and offshore on the downstream shelf. Within the canyon, below shelfbreak depth, circulation is also cyclonic. Water comes into the canyon on the downstream side (positive υ) and out on the upstream side (Fig. 2d). This circulation pattern is consistent with previous numerical investigations (Spurgin and Allen 2014; HA2013; Dawe and Allen 2010), observations (Allen et al. 2001; Hickey 1997), and laboratory experiments (Mirshak and Allen 2005).
Upwelling within the canyon is forced by an unbalanced horizontal pressure gradient between canyon head and canyon mouth (Freeland and Denman 1982). In response, a balancing, baroclinic pressure gradient is generated by rising isopycnals toward the canyon head. The effect on the density field drives a similar response on the tracer concentration field (Fig. 2a). Near the canyon rim, pinching of isopycnals occurs on the upstream side (Fig. 2b). This region is associated with stronger cyclonic vorticity generated by incoming shelf water falling into the canyon, stretching the water column (not shown). This well-known feature has been observed in Astoria Canyon (Hickey 1997) and numerically simulated (e.g., HA2013; Dawe and Allen 2010).
Most water upwells onto the shelf over the downstream side of the rim, near the canyon head. This upwelled water has higher tracer concentration than the water originally on shelf since the initial tracer profile increases with depth (Fig. 1f). As a result, a “pool” of water with higher tracer concentration than background values forms near shelf bottom (Fig. 2h). This pool grows rapidly during the time-dependent phase and more slowly during the advective phase (Animation S1). A similar feature was seen in a numerical study of canyon upwelling on the shelf of Washington (Connolly and Hickey 2014). The average concentration near shelf bottom increases quickly during days 0–3 (by 1.5 μmol L−1) and more slowly during the next 6 days for the base case (Fig. 2g).
We isolate the canyon effect on the on-shelf tracer distribution by subtracting the corresponding no-canyon run. We look at the near-bottom tracer concentration anomaly (BC anomaly), defined as the concentration difference near the shelf bottom between the canyon and no-canyon cases, normalized by the initial concentration near the bottom and expressed as a percentage. Contours of BC anomaly for the base case show a region of positive anomaly or higher tracer concentration relative to the no-canyon case downstream of the canyon (Animation S2). The vertical extent of the pool can be between 10 and 30 m above the shelf bottom. The formation dynamics, extension, and persistence of the pool will be characterized in future papers.
b. Vertical gradient of density and tracer
During an upwelling event, isopycnals and isoconcentration lines near the canyon rim are squeezed as they tilt up from mouth to head (Fig. 3a). Close to the canyon head, on the downstream side where most upwelling occurs, stratification 
Concentration contours averaged over days 4 and 5 are plotted along canyon axis for (a) the base case and locally enhanced diffusivity cases with (b) Kcan = 10−3 m2 s−1, (c) Kcan = 10−2 m2 s−1, and (d) Kcan = 10−2 m2 s−1, 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
The tilting of isopycnals and thus, the increase in stratification near the head, is a baroclinic response to the unbalanced pressure gradient on the shelf. AH2010 showed that the pressure gradient along the canyon is 
When diffusivity is locally enhanced, there is an additional effect on stratification. Within the canyon, enhanced diffusivity Kcan is acting on the density gradient, which was sharpened by the canyon-induced tilting of isopycnals, more rapidly than it is being diffused above the rim. So stratification near the rim but within the canyon is lower than it would be if the diffusivity profile was uniform, and stratification above the rim is higher than for the case with uniform diffusivity. The effect increases with Kcan (blue and solid green lines in Figs. 3a–c), and it is maximum when the 
Isoconcentration lines mimic isopycnals (Figs. 3a–d, f). Vertical tracer gradients sharpen at rim depth as upwelling evolves, similar to stratification (Fig. 3f). Compared to the base case, lower 
Tracer concentration is relatively higher above rim depth with higher Kcan and lower below rim depth (all green and blue lines vs black line above and below rim depth). This increased concentration is related to the gradient spike above rim depth (Fig. 3g; green and blue lines).
c. Cross-shelf transport of water and tracer
To determine the pathways of water and tracers onto the shelf, we calculate their cross-shelf (CS) and vertical transports. We define CS transport of water as the volume of water per unit time that flows across the vertical planes (CS1–CS6) that extend from the shelf break in the no-canyon case to the surface (Figs. 1a,c), while vertical transports flow across the horizontal plane (LID) delimited by the shelfbreak depth in the canyon case and the canyon walls (Figs. 1a,b).
We define the net or total water and tracer transport onto the shelf (TWT and TTT, respectively) as the mean during the advective phase of the sum of the water and tracer transports through cross sections CS1–CS6 and LID, and the vertical water transport (VWT) and tracer transport (VTT) onto the shelf as the mean transport through LID during the advective phase (days 4–9).
Tracer transport is divided into advective and diffusive contributions. The advective part is defined as 
The canyon effect in cross-shelf fluxes is the anomaly between canyon and no-canyon cases. Negative transports generally mean that either water or tracer is leaving the shelf; it is only near the shelf bottom, where shelf upwelling is onshore, that negative transports mean that transport for the no-canyon case is larger than in the canyon case.
Patterns of cross-shelf transport anomaly of tracer and water are similar. Both tracer (Fig. 4c) and water (not shown) anomaly fluxes are onto the shelf through CS3, close to the downstream side of the canyon mouth and through LID (vertical flux; Fig. 4e). Tracer and water transport anomalies flux off the shelf, downstream of the canyon, close to canyon mouth (CS4), and both transport anomalies are mainly offshore through CS1, CS2, CS5, and CS6. These agree with shelfbreak upwelling suppression in the presence of a canyon. Deeper-than-shelfbreak-depth water comes into the canyon through the downstream side and leaves through the upstream side, consistent with cyclonic circulation (Fig. 4d).
Base case (a) tracer and (b) water cross-shelf and vertical transport anomalies during the simulation through cross sections defined in Fig. 1. (c) The horizontal, cross-shelf tracer transport anomaly through sections CS2–CS4 and averaged over days 4–9. (d) A full-depth version including cross-shelf transport anomaly through the canyon. (e) VTT averaged over days 4–9.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
Positive transports through LID and CS3 indicate that tracer and water upwell onto the shelf throughout the simulation. The vertical upwelling response is maximum at the same time that the body forcing is maximum and then decreases to a steady value of 20% of the maximum. Cross-shelf transport through CS3 reaches its maximum at day 4 and then decreases to a steady value of 40% of that maximum (Figs. 4a,b). In contrast, transport anomaly through CS4, CS1, CS2, CS5, and CS6 is offshore throughout the 9 days. Offshore transport at CS4 is the main balance to the onshore transports, especially during the time-dependent phase. Its response has similar timing as that of the vertical transport, and it also decreases to a quasi-steady value after reaching its maximum on day 2.5. This offshore transport is consistent with the offshore steering of the flow described in section 3a.
Overall, the TTT anomaly is onto the shelf (Fig. 4a), and the TWT anomaly is zero (Fig. 4b). During the advective phase, there is a constant supply of tracer onto the shelf induced by the canyon (TTT). This result could change if the initial tracer profile was not linear or would reverse if it decreased with depth.
Changing dynamical parameters 
Mean VTT, VATT, and TTT anomalies through cross sections CS1–CS5 and LID, as well as VWT and TWT anomalies throughout the advective phase with corresponding standard deviations calculated as 12-h variations for selected runs. Results for all runs are available in Table S1.
Upwelling through the canyon is well characterized by the vertical transport through LID. VTT is dominated by advection over diffusion (VTT and the advective component are equal to two significant figures for runs in dynamical experiments; not all shown). Nonetheless, the vertical advective tracer transport (VATT) component is modified by enhanced vertical diffusivity through modifications to the density field. Larger diffusivities weaken the density gradients near the rim, which allows more water to upwell onto the shelf. During the advective phase, VATT tends to increase when diffusivity is enhanced and can be as much as 25% larger than in the base case (Table 3). VTT can be higher by 25%–37% for the largest two Kcan used (Table 3) and can almost double for smoother 
d. Upwelling flux and upwelled tracer mass
Upwelled water on the shelf has been estimated previously by finding water originally below shelfbreak depth based on its salinity (HA2013). We take the same approach but use the tracer concentration at shelfbreak depth as the criterion to find water on shelf that was originally below shelfbreak depth. For this, we use the low-diffusivity tracer described in section 2.
Water upwells onto the shelf on the downstream side of the canyon rim. The upwelled-water volume anomaly Vanom [(2)] along shelf (integrated in the cross-shore direction) at day 3.5 is concentrated on the shelf, on the downstream side of the canyon rim (Fig. 5a). Water continues upwelling through the canyon, and at the same time, the bulge of upwelled water is advected downstream. On the upstream shelf, shelfbreak upwelling is suppressed as water is redirected to upwell through the canyon, as indicated by negative values of upwelled water volume anomaly (Fig. 5a). The upwelled tracer mass anomaly Manom as in (3) follows a similar pattern along shelf (Fig. 5b).
(a),(b) Vertically integrated upwelled water volume and tracer at day 3.5 along shelf. Dashed lines show the position of the canyon. (c)–(e) Volume of water on the shelf with concentration values initially below shelfbreak depth. (f)–(h) Upwelled tracer mass on shelf. Canyon cases are shown in (c) and (f), no-canyon cases in (b) and (g), and the difference between these in (e) and (h). The boundaries of the shelf box are the wall that goes from shelf break to surface in the no-canyon case, along-shelf wall at northern boundary, and cross-shelf walls at east and west boundaries.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
The upwelled water volume Vcan through the canyon is larger than that upwelled on a straight shelf and increases throughout the simulation. In the canyon case, water upwelling is dominantly canyon induced; at day 9, it accounts for between 24% and 89% of Vcan throughout the runs and between 25% and 90% of upwelled tracer mass Mcan [(4)], except for the lowest U case with enhanced background diffusivity, where canyon-induced upwelling accounts for 0.8%.
(a) Comparison between the mean flux of water and the mean flux of tracer upwelled through the canyon during the advective phase of upwelling for all runs. Error bars correspond to standard deviations. (b) Upwelled tracer flux increases (darker, larger markers) with increasing Rossby number 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
Mean water and tracer upwelling fluxes [
Locally enhanced diffusivity moderately increases the tracer upwelling flux and the water upwelling flux, which increase by 27% and 19%, respectively, compared to the base case for the highest Kcan. Moreover, high Kcan combined with a smooth 
Given that the tracer we added had an initial linear nitrate profile, the difference in the total on-shelf nitrate inventory 
4. Scaling analysis
There are two main processes acting to transport tracer onto the shelf: mixing and advection. The mixing contribution, represented by locally enhanced diffusivity within the canyon, has been described in the results and is scaled in this section, while the advective part is driven by the upwelling dynamics described and scaled by AH2010 and HA2013. Additionally, we found that enhanced mixing within the canyon can have an effect on advection through modifications to the density field near the canyon head, and we include a correction for it.
There were two criteria that guided our choice of the dynamical parameter space: (i) to have realistic values of U, 
a. Advection–diffusion equation in natural coordinates
Let (
The coordinate system (τ, η, b) corresponds to the trihedron that moves along the upwelling current (blue line), and (s, z, n) corresponds to its horizontal projection (natural coordinate system). So s is the projection of 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
Let us consider the deepest streamline that upwells within a submarine canyon and assume that the isopycnal plane is associated with the 
b. Relevant parameter space
The relevant dynamical variables in (12) are the horizontal and vertical velocities u and w, scaled by the horizontal upwelling velocity 
In total, there are 10 parameters (
Nondimensional groups constructed for the tracer scaling. To calculate these scales, we took geometric parameters reported by Allen et al. (2001) for Barkley Canyon (L = 6400 m, 
We estimate the scales 
Horizontal advection will dominate over isopycnal diffusivity (
Nondimensional numbers 
c. Stratification and tracer gradient evolution
In our system, the evolution of isopycnals during canyon-induced upwelling is very similar to that of tracer isoconcentration lines, as shown in section 3b; thus, vertical tracer gradient and stratification evolve similarly.
(a) Isopycnals (gray lines) tilt toward the canyon head during a canyon-induced upwelling event. This tilt is proportional to the upwelling depth Z, defined as the displacement of the deepest isopycnal to upwell onto the shelf (heavy, green line). Locally enhanced vertical diffusivity Kcan compared to the background value Kbg further squeezes isopycnals above rim depth (canyon rim represented by the dashed line) and in turn further stretches isopycnals below rim depth. The squeezing effect is proportional to the characteristic length Zdif in (19). (b) Zoom in of the red square keeping only two isopycnals: the heavy, dark green line is the deepest isopycnal that upwells onto the shelf, and the gray one is a reference isopycnal. The light green line represents the deepest isopycnal that upwells when diffusivity is homogeneous everywhere (base case). The extra displacement of this isopycnal when diffusivity is locally enhanced is the scale Zdif.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
Scaling estimates of (a) maximum stratification Nmax above the canyon, (b) minimum stratification below rim depth Nmin, (c) tracer concentration just above rim depth 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
d. Average tracer concentration
e. Upwelling and tracer fluxes
Scaling estimates of upwelling flux of (a) water and (b) tracer through a submarine canyon. Dashed lines correspond to ±1 mean squared error. The run legend is as in Fig. 9.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
We included five runs with 
5. Discussion and conclusions
Advection-induced upwelling of water through a canyon is the dominant driver of on-shelf transport of tracer mass from the open ocean; however, the tracer concentration profile and enhanced vertical diffusivity within the canyon contribute considerably to the amount and spatial distribution of the tracer on shelf. The main characteristics of canyon-induced tracer upwelling are the following (Fig. 11):
- The upwelling flux carries tracer onto the shelf near the head and the downstream side of the canyon rim to be further spread on the shelf; with decreasing
- Locally enhanced mixing weakens the stratification below rim depth. A smaller stratification increases the vertical advective transport of water and thus of tracers. The mechanism is that isopycnals close to the head are squeezed due to upwelling, which generates a local increase in stratification proportional to the isopycnal tilting generated by upwelling. However, enhanced diffusivity acts against temperature and salinity gradients, thus reducing this density gradient and locally reducing stratification below the rim. The combined effect of lower N and higher diffusivity below the rim via a smoother
- Enhanced mixing within the canyon increases the tracer concentration near rim depth. Just above rim depth, where the value of Kcan changes, the tracer gradient increases. This means that concentration isolines are elevated higher, compared to the situation with uniform diffusivity, and in turn, isolines of higher concentrations will be reaching rim depth. This water with higher tracer concentration will upwell. Together, this mechanism and the two above increase the tracer flux onto the shelf. For instance, taken together, both contributions can increase tracer upwelling flux by 27% when Kcan is locally enhanced by three orders of magnitude.
- The upwelled water spreads out on the shelf, downstream of the rim, and generates a region of relatively larger tracer concentration near the bottom.
Schematics of tracer transport through a submarine canyon. 1) The upwelling current (blue arrow) brings tracer-rich water onto the shelf, generating an area of relatively higher tracer concentration than the upstream shelf (point 4). Enhanced vertical diffusivity within the canyon (2 and 3) increases the tracer concentration near rim depth and weakens the stratification. These two effects enhance canyon-induced tracer flux onto the shelf.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
For comparison, Messié et al. (2009) estimated that the wind-driven nitrate supply for the northern Washington Shelf is 6.4 mmol s−1 m−1. This corresponds to 153 mol s−1 across a shelf section of length 
a. Implications on internal waves
The second mechanism is the transition between a partly standing wave during pre-upwelling conditions to propagating during upwelling conditions. This effect has been observed (Zhao et al. 2012) and modeled (Hall et al. 2014) for the M2, mode-1 internal tide in Monterey Canyon. During pre-upwelling conditions, the pycnocline was located below rim depth, which increased the supercritical reflections (down canyon) of the up-canyon-propagating internal tide. During upwelling conditions, the pycnocline rose above rim depth, decreasing the stratification, and with it the supercriticality of the canyon walls. This decreased stratification decreased the reflection of the up-canyon-propagating tide. The comparatively large reflection during pre-upwelling conditions allowed for a horizontally, partly standing wave setup, while upwelling conditions caused a progressive up-canyon wave to dominate.
In our model, maximum stratification within the canyon and near the rim is a consequence of shelfbreak- and canyon-induced upwelling where isopycnals tilt toward the canyon head, squeezing closer together around rim depth, not too far above the canyon walls. This enhanced stratification could push the reflecting characteristics of the canyon walls or bottom toward the supercritical regime, as results from Zhao et al. (2012) and Hall et al. (2014) suggest. Moreover, our results show that having elevated diffusivity within the canyon will erode the increased, canyon-induced stratification below rim depth and enhance it above rim depth. If we assume that the stratification that matters for criticality occurs around rim depth, then the competition between squeezing and stratification erosion will determine the change in criticality. Close above the rim, we see stratification 
Upwelling in short canyons is stronger on the downstream half of the canyon, and thus, the eroding effect of enhanced diffusivity over increased stratification will also be stronger there due to the large upwelling-generated gradients. So, the change in criticality will be impacted by this asymmetry, too. A larger shift toward supercriticality is to be expected on the downstream side of the canyon, close to the head, and strongly modulated by the difference in diffusivity below and above rim depth. This shift will also influence the location of internal wave breaking and, as a consequence, where vertical diffusivity is enhanced.
b. Extension to other canyons
The diffusivity-driven weakening of vertical gradients is a function of time. There is a natural time scale in which diffusivity acts on vertical gradients, given a characteristic length scale (e.g., the upwelling depth). The larger the diffusivity, the smaller the time scale given the same length scale. We find that diffusivities of around O(10−3) m2 s−1 or above are sufficiently high to noticeably weaken stratification and tracer gradient in the first 4 days. This means that when the flow enters the advective phase, the effects of high Kcan are already noticeable. Enhanced diffusivity continues to act on the gradients during the advective phase, but the effect weakens as it is proportional to the gradient itself. In canyons such as Monterey, where diffusivities are on the order of 10−2 m2 s−1, the weakened gradients would be considerable after only 11 h, assuming a depth of upwelling of about 20 m.
Our results and overall scaling scheme are valid only for short canyons, which are canyons for which the canyon head occurs well before the coast (Allen 2000). This criterion removes some of the most iconic canyons, like Monterey and Nazaré Canyons. For canyons not in the AH2010 scaling, we expect that, provided there is squeezing of isopycnals and a difference in diffusivity above and below the rim, the same effect of nonuniform diffusivity would occur: the differentiated diffusivity will act to further enhance the stratification above the rim and further decrease it below the rim. The tracer part of the scaling would be similar, but an appropriate depth of upwelling Z and fitting parameters would need to be found. For less idealized bathymetries, the overall upwelling pattern is expected to be very similar, provided that the incoming flow is along the shelf, perpendicular to the canyon axis, and relatively uniform along the length of the canyon. Scaling of the upwelling flux and depth of upwelling is robust enough that it has been successfully applied to real, short canyons like Astoria, Barkley, and Quinault Canyons (AH2010) and in one of the limbs of Whittard Canyon [depth of upwelling in Porter et al. (2016)].
Runs with longer canyons (2 times and 1.5 times longer than our original canyon) show that the general circulation pattern and evolution of the upwelling event is similar, as seen in HA2013. Isopycnals and isoconcentration lines tilt toward the canyon head similarly for both canyons, so that squeezing of isopycnals happens close to the head in both cases. The stratification evolution near canyon head, on the downstream side of the canyon, is also similar for longer canyons. Moreover, having locally enhanced diffusivity within the canyons has the same effect on isopycnal squeezing near the canyon head. Locally enhanced diffusivity increases the near-rim-depth concentration in all three cases, compared to the case with uniform diffusivity, and the concentration is well predicted by (25) with root-mean-square error 0.04, compared to 0.03 for the single canyon. These runs show that the effect of diffusivity can be applied to other canyons whenever there is isopycnal squeezing and different diffusivities above and below the rim.
The tracer mass flux scaling estimated in this work is restricted to flows that follow the same conditions as AH2010 and HA2013 because it depends on their upwelling flux estimation, and as such, it can only perform as well as their estimate. The main contribution of our scaling scheme is the estimation of tracer concentration and stratification within the canyon. Our scaling preformed reasonably well when we used it on runs with 
c. Significance to upwelling nutrients
Connolly and Hickey (2014) identified a similar feature to the pool. They estimated that canyon-exported nitrate onto the shelf after 2 months during an upwelling season can be about 1–2 × 107 kg 
Future work will consider scaling for realistic profiles of nutrients and oxygen as well as characterizing the pool of upwelled water and tracers that forms on the downstream shelf. Some of the key features to consider are the slope and curvature of the profile, as suggested by the scaling of the advection–diffusion equation, and the location of the nutricline and oxygen-minimum zone.
The authors thank J. Klymak and S. Waterman for sharing insightful comments about the project, A. Waterhouse and G. Carter for providing measurements of diffusivity profiles, and D. Sheinbaum for fruitful discussions. Computing power was provided by WestGrid and Compute Canada. This work was funded by NSERC Discovery Grant RGPIN-2016-03865 to SEA and UBC through a 4-yr fellowship to KRM. The model configuration and postprocessing scripts can be consulted from our repository at https://bitbucket.org/canyonsubc/tracer_upwelling_paper.
APPENDIX A
Advection–Diffusion Equation in Natural Coordinates
APPENDIX B
1D Model of Diffusion
We use a 1D model of diffusion through two layers of water with different diffusivities to illustrate the effect of a sharp diffusivity profile and progressively smoother versions of that step described by the smooth Heaviside function in (1). Increasing ϵ increases the depth where the concentration is changing due to a mismatch in the flux (Figs. B1a–c), and at the interface (
(a)–(c) Tracer concentration difference from the initial profile for runs from the 1D diffusion model varying (a) Kcan, (b) 
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0174.1
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