1. Introduction

The shelf break is a transition zone between the shallow shelf and the deep ocean, and their respective property fields. Because coastal flows are predominantly along-isobath, the cross-shelf transports of sediments, nutrients, and physical tracers are difficult to measure. As a consequence, the processes responsible for these transports are difficult to deduce from direct observations.

Compounding this problem is the inherent difficulty in modeling the flow over and around steep topography in the presence of background stratification, rotation, and turbulent mixing (itself poorly understood). In situ measurements having sufficient spatial and temporal resolution to serve as datasets for the development and testing of numerical models are few. The goals of the present research include a better understanding of the effect of seafloor topography—in particular, submarine canyons—on coastal flows and a demonstration of the utility of laboratory datasets for the testing and improvement of coastal ocean models.

Submarine canyons are known to be critical factors in determining the nature of coastal current systems. A background on our present understanding of the effects of canyons from an observational standpoint can be obtained by referring to the papers of Hickey (1995, 1997), and the papers of Klinck (1996) and She and Klink (2000) are good introductions to the theoretical and numerical aspects of the problem.

Hickey's (1997) study of Astoria Canyon reported on datasets obtained from an 18-element moored array and concurrent CTD surveys. She was able to characterize spatial patterns of the three-dimensional velocity and temperature fields and to deduce vorticity patterns resulting from upwelling- and downwelling-favorable winds. Nevertheless, the Astoria dataset by itself is insufficient to define completely many important elements of the three-dimensional circulation and dynamics.

She and Klinck (2000) apply a hydrostatic primitive equation model to a smoothed version of Astoria Canyon. For conditions of steady upwelling- and downwelling-favorable winds, their results are consistent with the general nature of Hickey's (1997) observations. Nevertheless, quantitative replication of Hickey's Astoria dataset using coastal ocean models has thus far proven difficult.

Allen et al. (2003) recently considered an upwelling event in both a laboratory model and a numerical model [the S-Coordinate Rutgers University Model (SCRUM)]. They find poor agreement between the laboratory and numerical models for the homogeneous case owing to nonhydrostatic effects, and, although the agreement is better for the stratified case, the differences are attributed to truncation errors in the terrain-following numerical model.

In a recent paper by the three of us, comparisons of the flow fields obtained by a series of laboratory experiments and a numerical model were made (Pérenne et al. 2001a, henceforth referred to as PHB). The motivations for this study were (i) the recognition that current numerical models when applied to the same forcing and boundary conditions can provide qualitatively different results when strict one-to-one comparisons are made (Haidvogel and Beckmann 1998), (ii) the difficulty of field programs providing adequate data in space and time to test the models, (iii) the understanding that most of the physical aspects of the coastal ocean can be modeled to some degree in the laboratory, and (iv) the realization that modern data acquisition techniques such as particle tracking (PTV) and particle image velocimetry can provide datasets having characteristics similar to those generated by numerical models.

It should be made clear at the outset that there is no substitute for comparing models with oceanic data. The view here is that laboratory experiments can isolate individual physical processes and can be a cost-effective way of aiding in the early development of numerical models. Laboratory datasets, therefore, complement datasets obtained under realistic oceanic conditions, and both are valuable for model testing and validation.

Several other studies by the authors offer additional background and perspective on the problem studied here. Boyer et al. (2000, henceforth BZP) report on the theoretical aspects of the physical system being considered, as well as the results from some of the initial experiments. Pérenne et al. (2001b) consider impulsively started, upwelling- and downwelling-favorable flows past a submarine canyon; this study depicts clearly some of the fundamental differences between upwelling and downwelling flows (e.g., an upwelling flow spins down far more slowly than its downwelling counterpart). A companion article by Haidvogel (2004, hereinafter HAI) examines the issue of cross-shelf transport in this physical system.

The PHB study considered only a single set of system parameters. The experiments with this set are referred to as the “central case” and are designated as experiment 01 below. Four additional laboratory experiments are reported in this communication (expts. 02–05). Last, a number of other parameter sets are considered in the numerical simulations, thus allowing for a wider coverage of parameter space.

The plan for the paper is as follows. We give in section 2 a brief description of the physical system and the laboratory and numerical techniques employed. We discuss the reasons for the quantitative differences between the laboratory and numerical model results for the central case, as presented in PHB, in section 3. The sensitivity to changes in the system parameters, and a discussion of the physical processes responsible for the changes, are presented in section 4. Conclusions are given in section 5.

2. The laboratory and numerical model

The present study has two principal purposes: (i) to explain the reasons for the quantitative differences between the laboratory and numerical experiments in PHB and (ii) to determine variations of the characteristic flow features of the system owing to changes in the governing parameters. This work is a natural follow-on to PHB, and the reader is thus referred to that paper for details concerning the background for the study and the laboratory and numerical models employed; only a brief synopsis of that study is given here.

The physical system considered is shown schematically in Fig. 1. An annular coast (vertical), shelf (horizontal), and continental-slope model, incised by a single submarine canyon, is placed in the central portion of a circular test cell; the deep ocean is between the continental slope and the outside wall of the test cell. The test cell is filled with a linearly stratified fluid, and the initial background rotation of the tank is set at f/2, where f is the Coriolis parameter.

After the fluid has reached a state of solid-body rotation, experiments are initiated by modulating sinusoidally the test cell at an amplitude ΔΩ and frequency ω₀ about the background rotation rate f/2. The amplitude ΔΩ is chosen so as to obtain the desired amplitude u₀ of the oscillatory velocity at the shelf break; the designation of ω₀ specifies the temporal Rossby number of the background flow. Tables 1 and 2 are expanded versions of those given in PHB and reflect the experiments reported herein. Note that oceanic prototype values are given to indicate that the region of parameter space for the ocean is similar to that being investigated in the laboratory.

The laboratory observations in PHB, and those to be discussed herein, employed PTV to obtain the time-dependent horizontal velocity field [u′(t), υ′(t)] at selected vertical levels. PHB, to obtain an estimate of the errors involved in the laboratory experiments, conducted a set of four experiments at as close as possible to the same set of system parameters; the data were processed using Digimage proprietary software (Dalziel 1992). These data were phase-averaged over 10 cycles, the start of which occurred 10 cycles after initiation of the oscillatory modulation of the tank. These phase-averaged data were then time-averaged to obtain the measured time-mean velocity field (u_m, υ_m) for each experiment. The data from the four experiments are then ensemble-averaged to yield the mean measured residual velocity field (u_me, υ_me). A spatially averaged uncertainty for the experiments was then determined by calculating a normalized difference between the kinetic energy per unit mass of the mean and that of each realization; that is,

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where N is the number of points in the display window and e = (u² + υ²)/2. The experimental error for the laboratory experiments was then taken to be the ensemble average of these four experiments and was found to be approximately 5%.

The numerical model used is the Spectral Element Ocean Model (SEOM; Haidvogel and Beckmann 1999; Iskandarani et al. 2003). SEOM solves the hydrostatic primitive equations using a higher-order finite-element approximation. The use of the finite-element decomposition offers several practical advantages. For example, a spatial finite-element grid can be constructed that exactly follows both the sloping bottom and the circular sidewalls of the test cell. This, together with the seventh-order spatial approximation being applied, guarantees an accurate solution to the physical problem. For these experiments, SEOM is configured to exactly match the geometric and physical forcing conditions prevailing in the laboratory. A more detailed description of the SEOM implementation is given in PHB.

It is important here to note how the viscosity and bottom boundary condition were handled in the numerical simulations. In the vertical direction, a combination of laminar viscosity (0.01 cm² s⁻¹) and a bottom stress parameterization to represent the laminar bottom boundary layer are used. The parameterization employed assumes that the shear stress τ₀ along the boundary is a linear function of the free-stream velocity outside the boundary layer u_h and is given by

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where ρ₀ is the density and ν is the kinematic viscosity.

Relation (2.2) is an exact solution for the streamwise shear stress exerted on the fluid by the floor for the laminar flow of a homogeneous incompressible fluid along a smooth horizontal surface in a rotating frame of reference. In the numerical experiments presented in PHB, the horizontal viscosity is enhanced by a factor of 100 above its true (molecular) value. The numerical model is not able to obtain solutions unless enhanced viscosities are used, owing to the finite numerical resolution employed. (The finest horizontal and vertical grid spacing used in the SEOM simulations are on the order of 1–2 mm.)

3. On the differences between the PHB numerical and laboratory model results

Although comparisons of the qualitative features of the time-mean or residual flows in the laboratory and numerical model in the PHB study were highly satisfactory, the quantitative differences were too large to be ignored. For example, Table 6 in PHB indicates differences ranging from 33% to 79% in the normalized kinetic energy difference parameter given by (2.1).

In recognizing that the shear-stress condition (2.2) used on the model floor in the numerical simulations is a parameterized boundary condition that depends only on the Coriolis parameter, the flow speed outside the boundary layer, the mean fluid density, and the viscosity (e.g., it does not depend on the background stratification and the slope of the topography), this aspect of the model was considered a strong candidate for contributing to the quantitative model–model differences.

To explore this matter further, SEOM was modified to provide for the application of a true no-slip condition along the solid bottom boundary of the canyon, in place of the parameterized shear-stress condition (2.2). As for the stress condition, the use of the no-slip condition also required enhanced viscosities for numerical stability, and again the vertical and horizontal viscosities employed were the same as and 100 times that of water, respectively. Figures 2a–c are the residual vorticity and divergence fields at the shelfbreak level for the PHB laboratory experiment (Fig. 2a), a numerical run using the parameterized shear-stress boundary condition (Fig. 2b), and a numerical run for the no-slip condition (Fig. 2c), respectively.

In comparing the vorticity and divergence plots for the laboratory experiment and the stress-law and no-slip numerical calculations, one notes the following: (i) the no-slip calculation better captures the symmetry across the canyon axis of the laboratory experiments, (ii) the no-slip run more accurately predicts the maximum of the vorticity and the horizontal divergence, and (iii) the no-slip model better represents the anticyclonic pattern in the upper-left side of the canyon.

The principal difficulty with the no-slip run is the thick boundary layer of anticyclonic vorticity along the canyon walls. This layer owes to the enhanced horizontal viscosity. In nonrotating boundary layer theory (e.g., aerodynamics) the boundary thickness scales as δ_H/L ∼ Re^−1/2, where Re = Lu₀/ν is the Reynolds number, δ_H is the dimensional boundary layer thickness, and L is a streamwise length scale. Taking L = W = 20 cm (W is canyon width), u₀ = 1.0 cm s⁻¹, and the enhanced viscosity ν = 1.0 cm² s⁻¹, one obtains δ_H/W ∼ 0.22, which is indeed of the same order of magnitude as the noted boundary layer thickness in the numerical model.

Owing to conservation-of-mass constraints, these thick boundary layers in the no-slip case force the maxima of the vorticity and divergence farther away from the boundaries than would occur for solutions using the viscosity of water (if they could be obtained). Nevertheless, the numerical simulations away from the boundaries for the no-slip model agree considerably better with the laboratory experiments than do the parameterized shear-stress runs. However, the use of the no-slip condition requires far more computer time than does the use of the parameterized shear-stress law.¹ The need for an improved parameterization for the laminar Ekman layer for unsteady stratified flows along sloping surfaces is thus clearly evident.

To examine the sensitivity of SEOM solutions to changes in the horizontal and vertical viscosities, a number of additional experiments were run using the shear-stress condition and different combinations of these viscosities. At the smallest spatial resolution employed in PHB, it was not possible to use SEOM to reach a stable result for the case in which both the horizontal (ν_H) and vertical (ν_V) viscosities had the value of the kinematic viscosity of water; we designate this viscosity combination as (1, 1) and use this notation in the following discussion.

Figure 2b (left panel) shows the velocity and vorticity fields at the shelfbreak level for the central-case experiment using the (100, 1) combination; Figs. 3a,b depict the velocity and vorticity fields for the viscosity coefficient combinations (50, 50), and (10, 10), respectively. [An attempt was also made to obtain numerical solutions for the viscosity combination (1, 100). This attempt failed. An interpretation of these results is that the sidewall boundary layers were unresolved on our finite numerical grid for lateral viscosity values as low as that of water.]

The strongest maximum vorticity and largest characteristic time-mean flow occur for the lowest-viscosity experiment [(10, 10); Fig. 3b]; this is in general agreement with the scaling analysis given below. Furthermore, it is clear that Fig. 3a depicts the weakest maximum vorticity and time-mean kinetic flow. In keeping with the scaling arguments, the (50, 50) viscosity combination must therefore represent the “largest” viscosity pair. Indeed, this viscosity pair has the largest value projected normal to the bottom boundary over the shelf and upper slope. These regions are shown to dominate in the vorticity and energy cycles [see below and in the companion paper (HAI)].

The intent, of course, in introducing eddy viscosities is to stabilize the numerical calculation, with the view that the flow patterns obtained are relatively insensitive to the exact choice of the coefficients. It is concluded that the use of enhanced eddy viscosities, as evidenced in the present experiments, must be done with caution because such viscosities can affect the basic flow patterns.

In summary, owing to the use of enhanced viscosities and a simplified shear-stress boundary condition along the model floor, the model–model comparisons for the time-mean flow in the present study, while demonstrating good qualitative agreement, showed significant quantitative differences. It is emphasized that these comparisons were for the first-order residual velocity field (the oscillatory forcing is considered to be the zeroth order) and thus are a severe test for the model-to-model comparison.

4. A sensitivity study to parameter changes

We now discuss how the residual flow fields found for the central case of PHB are altered as selected parameters are varied.

a. Scaling arguments

BZP considered certain aspects of the theory underlying the physical system studied here. It is useful to address the underlying assumptions made by BZP because they are important in developing the scaling arguments for the characteristic horizontal component of the residual velocity that follows.

A cylindrical coordinate system (r, θ, z) is employed, with z = 0 being at the tank center at what would be the level of the free surface were it not for the presence of the coastal region.² Further we introduce the perturbation density,

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where ρ(r, θ, z, t) is the density field,

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is the background density distribution before the turntable modulation is initiated, ρ₀ is now the density along the free surface, the dependence of ρ_b on the radial coordinate is ignored, Δρ is the density difference from the free surface to the tank floor in the deep water, and h_D is maximum depth. The perturbation pressure is similarly defined as

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where p(r, θ, z, t) is the pressure field,

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p_b is the background pressure field, p_atm is the atmospheric pressure, and g is the acceleration due to gravity.

The dimensionless variables are defined as

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where asterisks designate dimensionless quantities, δr is the characteristic scale of variations in the dependent parameters in the r direction, (υ_r, υ_θ, w) are the Eulerian velocity components, and the remaining terms are defined in Table 1. The dimensionless parameters that govern the system are then the

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The resulting governing equations are given in BZP and are not reproduced here.

To address the question of scaling and in keeping with the parameter values given in Table 2, we assume a priori that

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On the basis of (4.7), as in BZP, it is reasonable to assume that the dependent parameters, including the three velocity components and the perturbation density and pressure, can be expressed as power series in the Rossby number. To the lowest order (i.e., Ro⁰), BZP show that the laboratory flow exhibits a balance among the unsteady acceleration, the Coriolis “force,” the pressure gradient, and the periodic forcing that owes to the time-dependent rotation rate of the turntable (i.e., the turntable's angular acceleration). This last term can be interpreted as a time-varying body force. The lowest-order flow is periodic, with the time mean of the velocity components and the perturbation density and pressure being zero. The time-mean flow thus appears as a first-order term (i.e., ∼Ro¹).

It is helpful to define a control volume for the canyon and its surroundings as follows, where one recalls that a control volume is an imaginary surface in space through which a fluid can flow freely. The control volume used here is given schematically in Fig. 4 and is composed of two parts. The upper part is the volume between the shelf and the surface and whose boundaries follow the shelf break as defined by the intersection of the shelf with the canyon; the boundary at the mouth of the canyon is given by the straight line connecting the two extremes of the canyon mouth (see Fig. 4b). The lower part is defined by the canyon walls on the shoreward side and the surface formed by horizontal straight lines connecting the extremes of the canyon along its mouth (see Fig. 4c). The other symbols on the figure will be discussed below.

The preliminary experiments of BZP include time series measurements of the horizontal velocity components at various elevations along vertical traverses at selected horizontal locations within and in the vicinity of the canyon. The BZP data show that the velocity components υ_r and υ_θ (i) reach an approximate periodic state almost immediately (i.e., roughly within the first oscillation); (ii) scale as the forcing amplitude u₀, with, as might be expected, υ_θ > υ_r; (iii) exhibit harmonics, which are most readily apparent in the υ_r component; and (iv) lead to a mean or rectified horizontal motion field whose scale is much smaller than u₀. The vertical velocity field is not measured directly, but it clearly is weak.

Owing to the nature of the topography and the stratification, consider first the expected magnitude of the vertical velocity component within and in the vicinity of the canyon. A study of the canyon topography leads one to conclude that the maximum vertical velocities would most likely occur in the region neighboring the line of intersection of the shelf and canyon. Along this line, fluid parcels advect over the shelf break with the expectation that vortex lines passing into the canyon will be stretched, leading to the generation of cyclonic vorticity. Fluid parcels advecting along the continental slope and then encountering the canyon are able to move along the canyon isobaths and are not strongly forced to move vertically against the buoyancy forces.

The kinetic energy per unit mass of the fluid passing over the rim has a maximum value of u²₀/2. If we assume that the maximum depth Δz to which fluid parcels with this kinetic energy can penetrate is the distance over which all of the kinetic energy is transformed into potential energy, (NΔz)², one can readily show that

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where the numerical factor has been absorbed. Note also that, although fluid parcels might penetrate only a small distance Δz*, the actual disturbance might be felt at a considerably larger distance, but that distance should also scale as (4.8).

Now let the characteristic time-mean or residual horizontal velocity be U₁; that is, (U₁ ∼ υ_r1, υ_θ1) is of order unity because of the assumed expansion in Ro. We then consider the system behavior after the flow has reached an approximate periodic state. First, consider fluid parcels advecting into the canyon beginning with the zero velocity time in the cycle and continuing for one-half of a flow cycle. During this time interval, fluid columns, before passing over the shelf break, have weak relative vorticity but gain cyclonic vorticity by vortex stretching as they advect across the shelf break. (During the next half cycle, cyclonic vorticity is formed by the same process on the opposite side of the canyon.) This vorticity is produced along a line of length ∼ L characterized by the length of the canyon. Owing to stratification effects, it is assumed that the stretching will be limited to the vertical scale given by (4.8). This length is smaller than the drop in elevation of the topography after the column has advected only a short distance into the canyon. The stretching can thus be assumed to be almost instantaneous as the fluid column passes across the break.

If one then assumes that the potential vorticity for the short columns passing over the shelf is conserved, we can write

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where α is a scaling factor and ζ is the dimensional vorticity. After simplification, (4.9) can be written as

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The scaling for the total cyclonic vorticity Z generated within the control volume of Fig. 4 for the left-to-right and right-to-left phases of one oscillation cycle, and for the full length of the canyon, can then be written as

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Using (4.10) and integrating (4.11), one obtains

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At the time a periodic state has been obtained there must be a balance between the vorticity produced during one oscillation cycle and that dissipated. The former is given by (4.12), and the latter can be estimated by the dissipation in the boundary layers along the surface where the rectified flow directly encounters the lower bounding surface; see material on Ekman boundary layers on flat and sloping surfaces in Pedlosky (1979). We take the total area over which this so-called Ekman suction is in effect to be scaled by the product of the length and width of the canyon. Owing to stratification effects, vertically oriented vortex lines passing over the shelf break do not penetrate to the lower surface over the full canyon area, and thus the effective area over which significant dissipation occurs is smaller than LW. Nevertheless, the effective area for dissipation should scale as LW, and thus this value is used in the following.

The total dissipation of vorticity D for a single cycle for the time-mean flow thus scales as

D^1/2ζ_BWL,(4.13)

where ζ_B ∼ U₁/W is the basin-scale vorticity. Equating (4.12) and (4.13) and simplifying, one obtains

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where λ is defined as the nondimensional scaling parameter. In the present experiments, the parameters h_S/ h_D and Ek are not varied, and thus these aspects of the scaling cannot be tested. Last, it is instructive to solve (4.14) for the dimensional value of the characteristic rectified flow; one obtains

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Relation (4.15) demonstrates the influence of the dimensional parameters on the scale of the time-mean flow. The laboratory and numerical model results and their comparisons are discussed next.

b. The laboratory experiments and the numerical simulations

Laboratory experiments were conducted for five different sets of dimensionless parameters. These are listed in Table 3 and are labeled experiments 01–05. Experiment 01 is the same as that reported in PHB and is henceforth designated the “central experiment.” Three of the laboratory experiments (expts 02, 03, and 04) have the same geometrical parameters as the central experiment, but each differs from the central case in one of the dynamical parameters. The fifth (expt 05), while having the same dynamical parameters as the central experiment, has a canyon of the same depth but only one-half as wide; this experiment will be referred to as the “narrow canyon” experiment. In applying the Digimage software (Dalziel 1992) to the present configuration, it was found convenient to introduce a local rectangular, Cartesian coordinate system.

Although there are aspects of the numerical model that make it impossible to obtain precise simulations of the laboratory experiments, the numerical model is nevertheless compared with the laboratory results with the aim of determining whether the flow trends resulting from changes in the dynamical parameters are in consonance with the changes observed in the laboratory. In addition to simulating four of the five experimental cases (i.e., all but the narrow canyon), a number of other numerical experiments were conducted to expand the coverage of the parameter space investigated.

1) Case studies

To illustrate the temporal nature of the oscillatory flow, consider experiment 02, the case with the smallest Burger number (i.e., the weakest stratification). Figures 5a–d are instantaneous flow fields obtained by phase averaging the PTV data over 10 oscillation cycles. The data are shown for the shelfbreak level in the particular phases of flow as indicated in the caption. The most striking feature of these illustrations is the asymmetries in the flow fields, which are 180° out of phase. For example, Figs. 5a and 5c are for the cases in which the background speed is zero and the impending motion is in the opposite sense. Note the asymmetric nature of the fields and the significant residual motions even though the forcing velocity is zero. The differences in Figs. 5b and 5d illustrate nicely some of the most important physical processes at work in the present system. Note, for example, in Fig. 5b that the flow at the shelfbreak level passes over the break with little change in direction. This behavior owes to the fact that the anticyclonic motion associated with the flow following depth contours is countered by the cyclonic vorticity produced by the vortex stretching of fluid parcels passing over the shelf break. In contrast, Fig. 5d shows a sharp cyclonic turning of the fluid parcels passing over the shelf break. This behavior is because the relative vorticity introduced by the fluid parcels following the depth contours of the canyon and that generated by vortex stretching of fluid passing over the shelf break are both cyclonic.

Figures 6a–c are plots of the laboratory and numerical-model time-mean velocity fields for experiment 01 (the central case of PHB) for the observation levels z* = {−0.1, −0.2, −0.4}, respectively. In these and the like figures to follow, the velocity vectors and vorticity contours, to the extent possible, have the same scale. Those cases with different scales are identified.

The large-scale time-mean flow field for the laboratory experiment for the central case is given by the left-side illustrations of Figs. 6a–c. In Fig. 6a (left) one notes a large anticyclonic vortex structure having a canyon-scale horizontal dimension centered near the left side of the canyon near the mouth. This feature does not appear in the numerical simulations of Fig. 6a (right). In fact, it is concluded below that this feature owes to the presence of a surface shear stress exerted by the quiescent laboratory air on the fluid–air interface. Additional evidence on this point is given below. Note also that the numerical simulations in Fig. 6a (right) show that the flow at this level “feels” the presence of the canyon as evidenced by the two fields of cyclonic vorticity.

The time-mean flow at the shelf break (Fig. 6b) is strong relative to the layers above and below (Figs. 6a, c). The observation that the strongest time-mean flows are cyclonic and occur near the shelf break is a ubiquitous finding of the present study. The dominance of cyclonic flow and the narrow vertical scale of the time-mean flow qualitatively are consistent with the scaling arguments advanced.

Figures 7a–c are the velocity fields obtained from the laboratory experiments and the numerical model for experiment 02 in which the dimensionless parameters have all been held fixed with the exception of the Burger number, which has been decreased by a factor of 4 from the central case; i.e., the only change from Fig. 6 is that the buoyancy has been significantly reduced. One obvious effect with the decrease in buoyancy is the increased vertical extent of the time-mean flow and the fact that its vorticity and its characteristic speed are increased substantially. This result is in qualitative agreement with the scaling argument given by (4.14), which suggests that the characteristic mean flow velocity should scale as Bu^−1/2. The qualitative agreement between the laboratory and the numerical runs is good at all levels in that all of the major flow features are captured by the numerical simulation.

Figures 8a and 8b are the time-mean velocity and vorticity fields at the shelfbreak level for the laboratory experiments and the numerical model for experiments 03 and 04, respectively. These experiments have the same parameters as the central case with the exception that experiment 03 has a smaller temporal Rossby number, Ro_t = 0.25, and experiment 04 has a superinertial Ro_t = 1.25. Comparing first the central experiment with the run with the lowest temporal Rossby number (i.e., Figs. 6b and 8a, respectively) one notes that the qualitative agreement is satisfactory but that the intensity of the residual flow is greater for the experiment with smaller Ro_t, in keeping with the scaling argument (4.14). There is also qualitative agreement between the laboratory and numerical runs of Fig. 8a (left and right, respectively).

Figure 8b depicts the velocity and vorticity fields at the fields shelfbreak level for the superinertial case. Comparing the vorticity fields in Figs. 6b and 8b, one notes distinctly different structure for the superinertial case; that is, the superinertial case has a multicell structure that differs markedly from the subinertial cases, which are dominated by a single cyclonic cell. The multicell structure owes to the very short tidal excursions in these experiments. That is, from Table 3, one notes that the normalized tidal excursion for the superinertial case is X* = 0.16, while those for the subinertial cases of Figs. 6b and 8a are 0.38 and 0.80, respectively. Thus on each one-half cycle only small patches of cyclonic vorticity are formed on opposite sides of the upper levels of the canyon, and these retain separate identities rather than merging into a single canyon-scale cyclonic structure as is true for the subinertial runs.

A reviewer has noted, quite rightly, that the differences between the laboratory and numerical results appear to be accentuated in the superinertial experiment (Fig. 8b, left and right). Although the available data do not allow us to explain conclusively the enhanced dissimilarity, it is likely that the explanation resides once again in the heightened viscosities used in the numerical simulation. At superinertial frequencies, one expects preferential generation of smaller-scale motions (e.g., trapped, rather than propagating, modes), as is indeed suggested by the multicellular response found in the laboratory. These smaller-scale motions are in turn preferentially acted upon by the heightened viscosity used in the numerical model, further reducing the agreement.

No laboratory experiments were performed for varying Rossby numbers. SEOM, however, explored this aspect, and the results for the three separate Rossby numbers are given in Fig. 6b (right) and Figs. 9a and 9b. Note that the velocity scales differ from other illustrations discussed above. These experiments show that the scale of the time-mean flow increases with the increase in the Rossby number, in agreement with the scaling argument.

The final experiment (expt 05) was conducted using a narrower canyon; in fact, the streamwise scale was made one-half of the size used for the other experiments. The various dimensional parameters were adjusted so that all of the dynamical dimensionless parameters were held fixed. Figures 10a–c are plots of the velocity and vorticity fields obtained from the laboratory experiments for the z* = −0.2 (shelf break), −0.4, and −0.6 levels. We note first that this experiment does not satisfy the scaling arguments advanced in obtaining (4.14) because the geometrical parameters have been changed, and the experiment thus does not satisfy the requirement of geometrical similarity.

The vorticity fields show a strong region of cyclonic vorticity at the shelfbreak level (Fig. 10a), which follows the slope of the canyon head and can still be clearly identified at the z* = −0.4 level deeper in the canyon. Regions of anticyclonic and cyclonic vorticity are similarly found near the canyon mouth along the left and right sides of the canyon, respectively.

The narrow-canyon case simultaneously demonstrates the two vorticity-generating mechanisms discussed in this communication. At the shelfbreak level, the oscillating forcing flow continuously injects cyclonic vorticity into the region, by the mechanism of vortex stretching which at large times is balanced by dissipation. This cyclonic vortex core is seen to be strong at the shelfbreak level and, although weaker, also can be identified at the z* = −0.4 level.

One also notes at the z* = −0.4 level that cyclonic (anticyclonic) regions of roughly the same strength are found on the right (left) of the canyon facing the deep water. These vortices are also in evidence at the z* = −0.6 level on Fig. 10c. At this level deep in the canyon, the vortex stretching mechanism is no longer effective because the canyon walls prevent the advection of fluid parcels from directly upstream. The vortices are present because, on each stroke of the forcing current, cyclonic (anticyclonic) vorticity is advected into the right (left) sides of the canyon mouth; this vorticity is not advected out of the canyon on the return stroke, but rather gives a mean cyclonic (anticyclonic) region that remains near the mouth of the canyon; this vorticity advection into the canyon at equilibrium is balanced by dissipation.

A general finding in this study in both the laboratory and the numerics is that no strong upstream influence of the canyon is noted at any of the levels investigated. The downstream flow, however, is strongly affected by the presence of the canyon as evidenced by the along-slope mean flow that extends several canyon widths downstream of the canyon.

2) Sensitivity to changes in parameters

It is of interest to introduce a number of metrics that address the question of the sensitivity of the overall flow field to changes in selected parameters. It is, of course, beneficial to have these metrics relate to flow aspects of interest in ocean applications. To this end, we focus on the large-scale mass (volume) transport of the residual flow in the vicinity of the canyon. It is recognized that conservation of mass requires zero net flow through the control volume of Fig. 4; nevertheless the total mass balance for the control volume involves inflows and outflows over its entire surface.

The following integrals define the normalized volume transport per unit depth into the control volume through the canyon mouth, over the shelf break on the left side of the canyon facing toward the deep water, and over the shelf break on the right side of the canyon, respectively (see Fig. 4 in which all quantities are positive for flow into the canyon and negative for flow out); that is,

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where y_m and y_h are the y coordinates of the mouth and the head of the canyon at the calculation level, respectively.

Although the vertical velocity cannot be measured directly in the laboratory, it can readily be obtained from the numerical model results. Therefore Qz*, the vertical volume flux through the respective levels of the control volume, was calculated at each level for all of the numerical experiments; in particular

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where A_CV is the cross-sectional area of the control volume at the elevation in question.

Although it is not possible to obtain Qz* directly in the laboratory, it is possible to determine the integral of the horizontal divergence D* over a specified horizontal surface; that is,

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It then follows that

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Because Qz* is zero along the free surface, it is possible, in principle, to obtain the vertical volume flux through each level of the control volume of Fig. 4 by integrating D* with respect to z from the free surface. In the present experiments, however, the vertical spacing is too large, and, as described above, the vertical velocity gradients are too steep to make accurate calculations.

Last, the normalized kinetic energy per unit mass,

u²υ²u²₀(4.22)

gives some sense of the strength of the time-mean current; this quantity will be used as a measure of U₁ to test the scaling prediction of relation (4.14).

The measurements (laboratory) and calculations (numerical model) of the normalized horizontal volume fluxes (Q1*, Q2*, Q3*), the integral of the horizontal divergence (D*, laboratory only), the vertical volume flux (Qz*, numerical model only), and the maximum and average kinetic energy per unit mass (KE^*_max, KE^*_ave) as a function of the vertical coordinate z* are given in Tables 4 and 5, respectively.

These numerical values are given so that investigators who may wish to develop models to test against the laboratory data have the requisite data available. The tables by themselves, however, are difficult to interpret. To depict graphically the nature of changes in the characteristic flow fields to changes in the system parameters, the data are plotted in three sets. These include all of the laboratory experiments and numerical simulations for variations in the Burger (Figs. 11a–c), temporal Rossby (Figs. 12a–c), and Rossby (Figs. 13a–c) numbers, respectively. In all of the plots, the data have been normalized by the value of the physical quantity under consideration calculated at the shelfbreak level of the central case, experiment 01. The choice of normalizing with a numerical model–generated quantity rather than a laboratory one owes to the fact that the mean upwelling or downwelling could only be obtained for the numerical experiments. The curve fitting was accomplished in Figs. 11, 12, and 13 by employing polynomials of the same order as the number of data points for the fit. The plots all use the same symbols for each of the four experiments under consideration; closed symbols are the laboratory results, and open ones are the numerical simulations. The remaining symbols are used for each of the numerical runs for which a laboratory experiment has not been conducted.

Figure 11a depicts the normalized mean kinetic energy in horizontal planes within the control volume against the dimensionless vertical coordinate for the laboratory experiments and numerical simulations conducted at various Burger numbers. Comparing first the laboratory and the numerical simulation results for the same Bu, one notes substantial variations at virtually every z*. On the positive side, however, the qualitative nature of the plots is similar, and the trends of KE^*_ave in the present instance with decreasing Bu are in accord. In addition, these plots support, at least qualitatively, the scaling argument (4.14) that the intensity of the residual current scales as ∼Bu^−1/2. The data here also support the notion that the vertical length scale of the residual motion is limited as discussed in the derivation of (4.14).

Figure 11b is a plot of the normalized mean vertical transport Qz* through horizontal sections within the control volume as a function of z*. While Qz* was not calculated at z* = 0, on physical grounds it must vanish there, and thus this point was included on this and each of the other Qz* graphs to follow. Because only numerical data are available for Qz*, no comparisons could be made with the laboratory studies. The numerical data, however, show that the magnitude of the upwelling increases with increasing Rossby number. Weak downwelling is found deep in the canyon.

Figure 11c depicts the normalized volume transport per unit depth through the mouth of the canyon as of function of −z*. For the smallest Burger number, the flow, as obtained from the numerical simulation, is into the canyon at all depths, whereas the laboratory measurements indicate a flow out of the canyon in the vicinity of z* = −0.4. The agreement between the laboratory and the numerical simulations is reasonable for the central experiment 01 for the elevations below the shelf break, but a large discrepancy is found in the region above the shelf, with the laboratory flow being substantially larger than its numerical counterpart. This discrepancy is due to the wind shear on the surface of the fluid. Because the table rotates cyclonically, wind shear is in the anticyclonic or clockwise direction. Such an azimuthal shear stress drives fluid toward the center of the tank (i.e., through the mouth of the canyon). Then, by conservation of mass, the fluid is forced vertically downward, leading to an anticyclonic flow tendency in the interior of the canyon. It is noted that the metric Q1* drops from its maximum near the shelf break in all of the numerical runs considered. It is thus concluded that the discrepancies between the laboratory experiments and the numerical simulations at the z* = −0.1 level owe to the wind shear on the fluid surface, a factor not included in the numerical model. No laboratory data are available for experiment 02 at z* = −0.1. The data above the canyon middepth show a tendency to flow into the canyon for all parameter values. The laboratory flow for experiment 02 is outward for level z = −0.4.

Figure 12a shows that, although the numerical runs and the laboratory experiments do not quantitatively coincide, the data show clearly that the mean kinetic energy of the residual flow increases with decreasing temporal Rossby number, other parameters being fixed. This is in consonance with the scaling arguments above. These data also support the notion that the time-mean flow has a restricted vertical length scale in the vicinity of the shelf break.

The numerical runs in Fig. 12b show that the upwelling flow in the upper layers of the fluid increases with decreasing temporal Rossby number, again in accord with the scaling analysis. Note that the downwelling occurs in the lower portions of the canyon and is weak and relatively insensitive to the temporal Rossby number.

The dependence of the volume flow rate into the canyon through its mouth as a function of Ro_t is depicted in Fig. 12c. As noted, there is a tendency at the shelf break and above for the upwelling flow to be stronger, with smaller temporal Rossby numbers. Note that the numerical and laboratory solutions diverge considerably above the shelf. This divergence between the laboratory and numerical results owes to the surface shear stress as imposed by the wind.

Last, Fig. 13 illustrates the not-too-surprising result that all three residual fluxes increase with increasing Ro. In particular, upwelling strength (Fig. 13b) gains dramatically with an increase in Ro. Note also that, for the lowest Rossby number, downwelling occurs in the lower parts of the canyon. Further discussion of upwelling strength and cross-shelf exchange is given in HAI.

3) Parameterization of the strength of the residual flow

To test the scaling argument leading to relation (4.14), it is hypothesized that an objective measure of a characteristic U₁ is the square root of the average kinetic energy per unit mass of the horizontal time-mean flow within the canyon at the shelf break level. Figure 14 is a plot of the measured U₁/u₀ against the scaling parameter, λ ∼ [Ro(h_S/h_D)⁻¹Ro⁻¹_tBu^−1/2Ek^−1/2] for the laboratory experiments (except for expt 05) and the numerical simulations (except for expt 08). The least squares fit to a straight line of the remaining data is plotted on Fig. 14; that is,

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where the root-mean-square (rms) deviation from (3.24) is ±0.029; an error bar depicting a plus or minus one rms deviation is included on Fig. 14. Note that the choice of the characteristic velocity of the residual flow being defined in a single plane for an inherently 3D problem would tend to lead to increased scatter of data in a plot such as Fig. 14 because 3D physics is not readily captured in a 2D metric. There are insufficient data in the laboratory to employ a 3D analysis of the characteristic velocity of the secondary flow. Nevertheless, the good agreement with the scaling argument advanced is encouraging.

5. Discussion and concluding remarks

This study has addressed the question of the degree to which laboratory models might be used in the development of geophysical models and, in particular, those involving coastal currents and their interaction with bottom topography such as the shelf, continental slope, and isolated submarine canyons. The study is a precursor to the study of the effects of turbulence on these flow–topography interactions. The results have been both encouraging and at the same time somewhat worrisome. They are encouraging in the sense that SEOM was able to obtain good qualitative representations of the laboratory flows and that the combined laboratory–numerical approach provided a desired level of checks and balances on the other. For example, the laboratory results encouraged the numerical side of the effort to seek a better understanding of why some of the numerical results did not better simulate aspects of the laboratory measurements. The numerical results brought into question the nature of the laboratory flows in the surface layer where wind shear stress effects and surface cooling seemed to be distorting the near-surface flow.

Given the “friendly environment of the laboratory,” the fact that the flows considered were laminar, and that the initial state and the fluid forcing and boundary conditions in the laboratory were well defined, it is nevertheless a bit disconcerting that the agreement between the laboratory and numerical results was not any better. It is plausible to assume that additional numerical simulations having sufficient numerical resolution, low enough (i.e., molecular) viscosity, and some provision for the “wind” forcing effects encountered in the laboratory, would more closely replicate the laboratory results. Nonetheless, one must wonder what our experience portends for numerical simulation of the true coastal ocean wherein such factors as (i) the nature, distribution, and role of turbulence; (ii) the parameterization of smaller-scale processes; and (iii) the lack of extensive in situ data in space and time are complicating issues.

One of the principal findings of the present study is that, for the range of parameters investigated, upwelling in the upper regions of the canyon (i.e., z* ≳ −0.3) to the free surface is a ubiquitous feature of periodic along-shelf forcing in the vicinity of an isolated canyon. The numerical and laboratory models also demonstrate that weak downwelling (relative to the magnitude of the characteristic vertical mass flux of the upwelling) is found deeper in the canyon. These upper-level upwelling flows are associated with a ubiquitous flow into the canyon through its mouth; that is, the canyon acts as a “suction pump” for the upper-level fluid. The models also show that the flow above the shelf is generally away from the canyon along its flanks, with the magnitude being typically larger in the direction to the left of the canyon facing the deep water (see Tables 4 and 5). This study, therefore, reinforces previous numerical and observational studies that portray the upper regions of canyons as locations of systematic upwelling.

The models also support the scaling that the normalized magnitude of the time-mean flow of the time-mean current generated is given by U₁/u₀ ∼ [Ro(h_S/ h_D)⁻¹Ro⁻¹_tBu^−1/2Ek^−1/2]. This finding is also considered to have potentially important oceanic implications because it forms the basis for estimating the magnitude of residual currents that might be anticipated in terms of the governing parameters.

The laboratory–numerical findings also suggest that, for coastal regions having canyons (or other such topographic features), sharp shelf breaks are regions in which the vertical gradient in flow properties such as the rectified current is predicted to be large over a narrow (as compared with the fluid depth) vertical distance that scales as RoBu^−1/2h_D. Last, the models also suggest that superinertial flows generate residual motions that scale as predicted for subinertial ones. The superinertial forcing, however, does not produce mean flows either far upstream or far downstream of the canyon.

One of the referees raised the question as to the relationship of the residual cyclonic and anticyclonic eddies found at and below the shelf-break level in the narrow-canyon experiment depicted in Figs. 10b and 10c to those residual eddies found to the left and right, respectively, of a headland (facing offshore) studied in the field (Geyer and Signell 1990) and model studies (Signell and Geyer 1991). Noting that the left side of a canyon near its mouth is similar in overall geometry to the left-to-right flow past a headland and that the right side of the canyon is similar to the right-to-left flow past a headland, there clearly is a similarity between the current laboratory numerical studies and these earlier headland investigations. Signell and Geyer (1991) argue that the effects of the earth's rotation are not important in the headland studies because the relative vorticity of the headland flows is significantly larger than the Coriolis parameter. In the present experiments (Figs. 10a– c), the Coriolis parameter is important at the shelfbreak levels and above but is not important deep in the canyon where vortex stretching must be minimal because the vortex lines do not advect significantly over the depth contours of the bottom topography. The deep-canyon flows are thus qualitatively akin to the referenced headland flows because rotational effects are unimportant. Support for the Signell and Geyer (1990) assumption of the neglect of Coriolis effects is that the residual cyclonic and anticyclonic eddies on the left and right of the headland, respectively, in their Fig. 9 are symmetric. A closer look at their Fig. 9 suggests that the cyclone may be a bit stronger, thus indicating some effect of vortex stretching in enhancing the cyclone and weakening the anticyclone. This must be considered a conjecture in that other factors (e.g., bottom topography) might also lead to similar differential eddy strength.

Careful comparisons between the laboratory and numerical models have shown the importance of having a good parameterization of the boundary layer along the coastal floor. Although this importance was shown for the laminar flow considered herein, one finds it difficult to believe that the presence of boundary and interior turbulence will make the problem easier. The model-to-model comparisons show that the use of enhanced viscosities in the numerical model may cause difficulties in certain regions of the flow field, especially along coastlines where large horizontal shears may be expected.

In summary, one concludes that laboratory experiments can be a source of data for testing aspects of numerical ocean models. Model-to-model concurrence on phenomenological results, such as, for example, the suggestion that upwelling is to be expected in canyon flows as determined herein, gives far more credibility to the result than if it was predicted by either of the models alone.