a. Physical system and external parameters of the problem

The numerical model was configured to match, as closely as possible, all aspects of the laboratory experiments: the physical system [Fig. 1; see also Boyer et al. (1999) for a detailed description] is the same for both models and consists of a circular continental shelf/slope geometry, interrupted by a single canyon; the equations were solved at the laboratory scale and the laboratory forcing was implemented in the numerical model (section 4). In order to focus on the concept of laboratory–numerical model comparisons, only one set of parameters is considered in this article. The consequences of varying these parameters will be the subject of a subsequent paper.

2) Background parameters

The mean rotation rate of the turntable was not varied during the experiments; the model is rotating counterclockwise (seen from above), leading to a positive (Northern Hemisphere) value of the Coriolis parameter:f = 0.50 s⁻¹. The typical Brunt–Väisälä frequency of the experiments is N = 2.5 s⁻¹. The resulting Burger number,

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has a value close to 10 (Table 2).

The importance of viscous effects can be characterized by an Ekman number defined as

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which is based on the depth above the continental shelf and either the horizontal or vertical viscosity. The viscosity that applies in the physical model is of course the same in all directions (i.e., the laminar viscosity of water; cf. Table 1). In contrast, the finite spatial resolution of the numerical model requires enhancement of the viscosity above molecular levels. In principle, the numerical model seeks to represent all important scales present in the laboratory flow, with the possible exception of the bottom frictional boundary layer. This goal would seem to be achievable here because the flow appeared to be quite laminar in the laboratory experiments, which did not exhibit significant turbulence at scales so small that they would obviously be missed by the numerical model. In the vertical direction, we used a combination of (i) laminar viscosity (ν = 0.01 cm² s⁻¹) and (ii) a bottom stress parameterization to represent the (laminar) bottom boundary layer that occurs in the laboratory flow. The parameterization used is a linear function of the horizontal velocity:

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This is inspired by a scaling of the stress that would occur in a sheared flow where the typical velocity is u_h and the typical length scale is D_E = [(2ν)/f]^1/2, the height of the Ekman boundary layer. (Here, D_E = 2 mm in the physical flow.) Due to computational stability constraints, however, it was not possible to set the horizontal numerical viscosity to the laminar viscosity of water; it was enhanced by a factor of 100 (Table 1).

3) Forcing parameters

The laboratory forcing is realized by modulating the rotation rate of the turntable. This is equivalent to a body force, described in Boyer et al. (1999), which was introduced into the numerical model. This body force has no vertical or azimuthal dependence, but is proportional to the distance r from the center of the turntable (cf. the appendix). It is shown that, in the absence of the canyon and neglecting viscosity effects, this forcing will lead to a purely azimuthal oscillatory flow, when observed in a reference frame that is fixed to the topography. Two parameters then define the forcing used in the models: (i) the period T of the modulation, and (ii) the amplitude U of the velocity, which is forced at the shelf-break radius, far from the canyon.

The experiments presented here were performed using T = 24 s and U = 1 cm s⁻¹. One can then define a temporal Rossby number,

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which is the ratio of the forcing frequency to the Coriolis parameter, and a Rossby number,

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The values of Ro_t and Ro are specified in Table 2. With these values, the typical particle excursion is E = UT/(2π) = 3.8 cm. A Reynolds number based on the forcing velocity U and the excursion E is Re = UE/ν = 380. This is too small to ensure a transition to turbulence over a flat plate, in accordance with the apparently laminar behavior of the fluid in the laboratory experiments.

b. Oceanic similitude

The determination of the conditions required for our laboratory and numerical experiments to simulate the oceanic situation is quite complex. The question of similitude is important, however, and is now discussed because (i) it helps to place the results in an observational and theoretical perspective, and (ii) the geophysical relevance of the models is an important motivation of this study (laboratory–numerical model comparisons are more common for industrial applications). In order to discuss the oceanic significance of the experiments, we will first define a simplified prototype.

2) Similitude of the model with the simplified prototype

The fractional depth (h_S/h_D) is well reproduced by the models (Table 2), although there is of course some degree of arbitrariness in the specification of h_D for the oceanic prototype. In contrast, the models have a much greater vertical aspect ratio (h_D/W) than the prototype, because the amplification of the vertical scale is necessary for several reasons when performing laboratory experiments. Provided that the flow is hydrostatic, however, the ratio (h_D/W) disappears from the nondimensional equations of motion and thus is no longer a similarity requirement. A set of conditions for the flow to be hydrostatic was derived by Boyer et al. (1999) in the same experimental context as for the present study and is

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From Table 2, these nondimensional quantities equal 4.1 × 10⁻⁴ and 2.1 × 10⁻³, respectively, which, at least from the point of view of the external flow parameters, indicates that the laboratory experiment is approximately hydrostatic. Hence, oceanic similitude can be realized approximately in spite of the fact that the vertical aspect ratio of the model is elevated.

The horizontal aspect ratio (W/L) is significantly greater in our models than in the prototype (Table 2), meaning that the model canyon is wider than the “typical” oceanic canyon defined in section 2b(1). This might have some important dynamical consequences. A canyon with a smaller relative width will be considered in the future.

The Burger number of the prototype canyon is close to 20 (Table 2); this is mainly a consequence of the fact that the topographic scale W is much smaller than the internal radius of deformation. This important specification of the canyon flow problem seems to be quite realistic. [See, for instance, the values mentioned in Hickey (1997).] The model is in reasonable Burger similitude with the ocean since its Burger number is also much greater than 1.

The horizontal Ekman numbers for the oceanic prototype and the laboratory models are both very small although the laboratory flow is more viscous (Table 2). The vertical Ekman numbers have the same order of magnitude (both quite small), if one assumes a reasonable amount of tidally induced turbulence in the prototype, as explained in section 2b(1). The smallness of these Ekman numbers allows one to argue that the flow is not driven by frictional/diffusion effects “at first order”; however, the residual currents (a “higher order” flow feature), might depend on these parameters, even if small. In the numerical model, the horizontal Ekman number is larger than those of both the laboratory model and the oceanic prototype. In the vertical direction, the Ekman number is the same as in the laboratory but recall that the numerical model also includes a bottom stress parameterization. This parameterization, however, should introduce dissipation effects of the same order of magnitude as those corresponding to a real Ekman–Stokes layer, as argued in section 2a(2).

The temporal Rossby number is subinertial (Table 2), in accordance with the timescales induced by, say, wind fluctuations. In the future, we plan to consider higher and lower forcing frequencies. The Rossby number is somewhat low; nonetheless, we will see that the flow is substantially nonlinear.

The similitude of our models with the simplified prototype implies the following rules for conversion of laboratory–numerical dimensional results into oceanic quantities: a model horizontal (vertical) distance of x centimeters corresponds to a prototype distance of (0.35)x kilometers [(4.8)x meters], a duration of 1 s in the model corresponds to 1.4 h in the prototype, a model horizontal or vertical velocity of x centimeters per second corresponds to a prototype velocity of (7)x centimeters per second, and finally a vorticity or divergence of x per second in the model corresponds to a value of (2 × 10⁻⁴)x s⁻¹ in the prototype.

a. Time series of density fluctuations

In order to monitor the density field (the establishment of which, using saltwater, will be described in section 3c), an array of density probes was set up in the canyon, and time series of density fluctuations were recorded at various (fixed) locations. These probes (Head 1983) actually measure the conductivity of the water, which is related to its salinity and thus to its density. They are manufactured by Precision Measurement Engineering. The output from the probes is amplified using electronics built at Arizona State University and digitized on a PC computer featuring a digital acquisition system board. This output was directly calibrated against density using solutions of various salinities, the densities of which were known by hydrometer measurement; the densities used in these experiments typically ranges from ρ ∼ 1 g cm⁻³ (nearly freshwater) to ρ ∼ 1.1 g cm⁻³ (corresponding to a salinity of S ∼ 130‰). Over such a range, the calibration curves (voltage output as a function of density) depart slightly from a straight line, but this departure was taken into account.

The locations of the probes are indicated in Fig. 2, which also defines the coordinate system used for the presentation of the velocity field data. The Cartesian coordinate system used here is such that the y axis is along the canyon axis, with y = 0 standing where the shelf break would be located if there were no canyon. The x axis is tangent to the circle of radius r = 55 cm (the radius of the shelf break) and x = 0 is located on the canyon axis. The y coordinate increases in the offshore direction and the x coordinate increases in the direction that will be later referred to as the downstream direction, the direction that leaves the shallow water on its right.

The (horizontal) array of four conductivity probes was placed at various depths inside the canyon. The rack used for this purpose has a spacing of Δx = Δy = 2.5 cm. Probes 2 and 4 are on the canyon axis while probes 1–3 are on the x axis (Fig. 2a). Vertical exploration of the density field was achieved by repeating the same experiment several times and varying the depth of the probes. Three depths will be considered in this article: z = z₁ = −h_S/2 is the upper level, located at middepth on the model continental shelf; z = z₂ = −h_S is the level of the shelf break; finally, z = z₃ = h_S − (h_D − h_S)/4 is the lower level, located below the canyon rim but still relatively high in the water column (Fig. 2b).

The time at which the forcing is initiated was recorded along with the signals from the four probes, thanks to a timing signal derived from the PC controlling the rotation rate of the turntable. The phase of the density fluctuations, relative to that of the forcing, is thus known accurately. The length of the records was typically 900 s in the experiments reported here; the sampling rate was 10 Hz but, in order to remove acquisition noise, the signal was then filtered using a time-averaging window of width equal to 1 s.

The conductivity probes were also used to measure the density profiles just before the experiments (i.e., while the rotation rate of the table is still constant). This background state will be discussed in section 3c, where we will see that for various reasons the density stratification is not linear from top to bottom. However, one can use the background density gradient measured prior to the experiment at the vertical location where the time series will be recorded to convert the density fluctuations into a measure of the vertical displacement of iso- pycnals:

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where ρ(z) is the background density profile. This quantity is useful in physical interpretations and/or attempts of comparisons with oceanic observations.

b. Particle-tracking velocimetry

A horizontal light sheet was projected in the fluid at the various levels defined in the previous section, using a standard light source. This light sheet is a few millimeters thick. The fluid was seeded with neutrally buoyant particles and their motion was filmed from above and recorded on an S-VHS video tape recorder. The video tapes were then processed using the Digimage software (Dalziel 1992), which allows for (i) identification of the particles present in the field of view; (ii) tracking¹ of these particle using a sophisticated, cost-minimizing algorithm; and (iii) interpolation/averaging of the randomly spaced Lagrangian measurements onto a regular (Eulerian) grid.

Among all the parameters one has to specify in order to operate Digimage, it is important to mention a few matters regarding step iii. The grid spacing chosen for the interpolation was Δx = Δy = 1 cm. Digimage was also configured so that the (randomly spaced) Lagrangian measurements contributing to the estimation of the Eulerian velocity at a given grid point are those that are located within a circle of radius 2 cm centered on that grid point. (A weight function linearly decaying from 1 to 0 over 2 cm is used, so that the more distant measurements have a lesser contribution.) Thus although many scales might be present in the physical flow, the particle-tracking velocimetry (PTV), as implemented here, samples only the length scales larger than ∼1 cm.

A calibration of the Digimage settings was performed using a plate on which some of the particles used in our experiments were glued, the concentration of these particles being close to the one we were able to achieve in the actual experiments. The plate was then rotated with a known vorticity (again, typical of the vorticities observed in the actual experiments) and the velocity and vorticity were recomputed using the PTV. In this situation, the error was found to be ∼5% on velocity and ∼15% for vorticity.

The tracking was performed on polystyrene particles of density 1.04 ± 0.005 g cm⁻³. The fact that all these particles do not have exactly the same density was especially useful. Since they float at various levels in the continuous density stratification, there were always particles in the light sheet in spite of the vertical excursions experienced by the fluid during the experiments. To explore the fluid at the various levels, the mean density of the fluid was adjusted so that the particles would float around the level of interest (the vertical density gradient being kept constant). The disadvantage of these particles, however, is their rather large size (diameter ∼ 1 mm), which limited to approximately 1000 the number of particles present in the field of view (30 × 25 cm²);this limitation is due to the fact that the coalescing tendency of the particles is too strong² to allow for higher concentrations. Finally, we note that the measurements at the z = z₁ level proved to be difficult because a significant fraction of the particles would then float to the surface.

The Digimage tracking operates better if some stationary “reference lights” are present in the field of view of the video camera. These reference lights were also used to provide the time reference of the video records: the light emitting diodes used for this purpose were actually lit according to a signal derived from the PC controlling the rotation rate of the table and a Digimage utility was then be used to begin the tracking of the particles at very accurate positions on the video tape. Figure 3 defines the various kinds of temporal averages performed during the processing of the velocity data. “Instantaneous” velocity fields are first computed using a time-averaging window of 1 s. (This is done inside the Digimage software.) Then, various phase-averaged velocity fields are constructed using some of the instantaneous velocity fields of cycles 11–20 after the start of the forcing (Fig. 3a). Four phases will be considered here (Fig. 3b): when the offshore flow is maximum in the downstream direction (“phase 1”), and then each quarter of a forcing cycle thereafter. The experimental residual velocity field is defined here as the time mean of all the instantaneous velocity fields measured between the beginning of the 11th cycle and the end of the 20th cycle.

c. Establishment of the density stratification in solid-body rotation

The now standard “two bucket” method (Oster and Vamamoto 1963) was used to establish a linear density stratification using varying concentrations of salt. The table was then accelerated very slowly until the background rotation rate was reached (f = 0.5 s⁻¹). Once established, the rotating stratified fluid was used to perform a number of experiments. As mentioned earlier, vertical density profiles were taken before each experiment. Figure 4 shows that the various experiments induced no significant mixing, thereby allowing several experiments with the same stratification. However, one can see in Fig. 4 that the density profiles are not quite linear from top to bottom. This is due to a number of effects, including the following: (i) the two-bucket method does not give a linear profile when the wet area is a function of the vertical coordinate; (ii) the spinup of the table induces some mixing (filling the tank while it is already rotating would be an improvement); (iii) the free surface of the test tank, if considered as an insulated surface, would lead to a condition of no vertical density gradient; and (iv) the slow but persistent evaporation along the free surface leads to convection having the same effect as number iii above. In Fig. 4, it appears that although effect (i) would lead to a stronger stratification in the upper layers, the other effects weaken substantially the stratification stability close to the surface. Thus, although the overall stability is close to that expected (N = 2.5 s⁻¹), a polynomial formulation of the mean background density profile (Fig. 4) was computed for use in the numerical model. In light of Fig. 4, however, one must note that there is some variability in the actual density profile measured in the test tank prior to a given experiment; it is acknowledged that this variability contributes to the experimental uncertainty quantified in the repeatability study of section 5c (where it is shown that its level remains acceptable). Furthermore, numerical simulations performed using either the polynomial stratification or a linear one, did not exhibit fundamentally different residual flow fields at the shelf-break level (section 6c).

Another issue concerning the background state of the fluid is the quality of the solid-body rotation that is observed before starting a given experiment. To prevent the immobile air of the laboratory from exerting a stress on the free surface of the rotating fluid, a lid was installed a few centimeters above it. With this setup, it appeared that the most deleterious effect was due to the imperfect alignment of the axis of rotation with the gravity vector. The bottom of our rotating turntable was modified in order to ensure a alignment of less than 10⁻⁴ rad. Some (weak) periodic motions were nevertheless observed in the fluid even in the absence of forcing; their period is equal to the rotation period of the table and they are quite well detected by the conductivity probes, which indicated that the vertical amplitude of the background isopycnal oscillations is ∼0.1–0.3 mm. This did not lead to any significant topographic rectification, however, as demonstrated by velocity measurements performed in the absence of forcing and exhibiting background residual currents of amplitude ∼0.2–0.3 mm s⁻¹ only.

a. Density time series

Figure 6 depicts the vertical dependence of the amplitude and phase of the density oscillations during forcing cycles 11 and 12, at horizontal location 2 (see Fig. 2). Relation (2) has been used to transform the density data into measures of the time-dependent displacement of isopycnals. The main frequency appears to be that of the forcing (period equal to 24 s). The phase at the shelf-break level (z = z₂ = −2.5 cm) is such that the maximum upward elevation is reached slightly before the forcing velocity vanishes (increasing in the downstream direction); recall that the oscillations cycles have a period of 24 s and begin by an increase in the downstream direction (Fig. 3). The isopycnal vertical displacement thus leads the alongshore forcing by roughly 90°; therefore, the vertical velocity is in phase with the alongshore current. (The isopycnals are moving upward when the offshore current is oriented so as to leave the coast on its left, which is indeed the “upwelling favorable” direction.) The oscillation at the upper level leads the oscillation at the shelf-break level, and lags the oscillation at the lower level. The amplitude is maximum at the shelf-break level, where it reaches ∼1 mm;according to the scaling rules mentioned in section 2b(2), it amounts to ∼5 m only in oceanic dimensions. The physical model actually exhibited very small vertical isopycnal displacements at all the locations systematically studied in our experiments. These locations are relatively far from the topography (see Fig. 2) and one additional experiment was performed with a probe located at the number-free open-circle symbol in Fig. 2. The amplitude of the oscillations recorded there (along the canyon axis, close to the shelf break) exceeded 6 mm (amounting to 60 m for a crest-to-crest oceanic value). In this experiment the significant vertical excursions thus appear to be confined to the immediate vicinity of the topography.

Such observations, relevant to the temporal evolution at the forcing frequency, proved to be experimentally repeatable (although no quantitative error analysis of the conductivity measurements was conducted). This is not the case of the low-frequency evolution, however. For example, one can see in Fig. 6 that the mean displacement during cycles 11 and 12 is negative at all depths; the details of the low-frequency evolution, of which the mean displacement at cycles 11 and 12 is an indication, were not found to be very reliable (i.e., experimentally repeatable). The mean isopycnal displacement, however, was consistently found to be small (of the order of 1 mm), at all the locations investigated, including the canyon head, and over the full length of the experiments (20 forcing cycles).

The conclusions to be drawn from the density time series data measured in the laboratory are the following. First, no dramatic residual displacements of the isopycnals are observed. Second, the vertical excursion is weak when translated into oceanic values, except very close to the canyon head.

b. Velocity fields

The velocity fields at levels z₁, z₂, and z₃ in Fig. 2 were investigated in the region extending between (i) x = −10 cm and x = 15 cm in the alongshelf direction, and (ii) y = −15 cm and y = 8 cm in the cross-shelf direction. As noted previously, the alongshore direction of increasing x will be referred to as downstream.

c. Laboratory repeatability

The experimental uncertainty is due to the imperfect control of the experimental conditions and to the noise introduced by the measurement technique. The repeatability of the velocity measurements was carefully examined by conducting four independent experiments. This analysis was restricted to the shelf-break level owing to the amount of work involved. A “mean laboratory realization” was defined by taking, at each point, the average of the measurements obtained in the four (independent) realizations and the uncertainty relative to this ensemble average was deduced from the departures observed in each particular realization.

This uncertainty will be discussed using spatially averaged (“statistical”) quantities in the display window defined at the beginning of section 5b. The same quantities will also be used when comparing quantitatively the numerical and laboratory results (section 6). It is thus important to know the experimental “error bars” that apply to them.

The statistics considered here are spatial averages, at a given level, of (i) the u and (ii) the υ components of the (Eulerian) residual velocity in the reference area, and (iii) the spatial average of the residual “kinetic energy” [e_c = (u² + υ²)/2]; these spatially averaged quantities will be noted as u, υ, and e_c, respectively. The values thus obtained (at the shelf-break level) for the mean laboratory realization are listed in Table 3, where the spatially averaged residual currents appear to be oriented in the onshore (negative υ) and downstream (positive u) direction. The statistics obtained for the phase-averaged flow are also mentioned in Table 3, where it appears, for instance, that the amplitude of the spatially averaged alongshore velocity, u, is larger at phase 1 than at phase 3, suggesting a greater topographic drag when the flow is in the upstream direction. Such a conclusion, however, remains tentative as long as the uncertainty pertaining to the values mentioned in Table 3 has not been specified. This experimental uncertainty was quantified by defining the following nondimensional ratios:

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where N is the number of points in the display window, (u_n, υ_n) is the residual velocity at each point, and the overbar designates a spatial average. The superscript“ref” corresponds to the data of reference, in this case the mean laboratory realization.

Thus for each realization, the departure from the mean realization was measured by three error coefficients: Δu, Δυ, and Δe_c. The rms values of these coefficients (defined on the ensemble of the four realizations) are listed in Table 4. Concerning the phase-averaged currents, the relative uncertainty becomes smaller as the quantity investigated gets larger: Δu is only 1% at phases 1 and 3 but rises to 20% at phases 2 and 4, when u is reduced ∼1 mm s⁻¹. We would expect the same error to apply on the cross-shore velocity υ, which is always ∼1 mm s⁻¹, but it actually appears to be more repeatable at most phases. The relative error on the kinetic energy statistic is small at all phases (Table 4). Concerning the residual currents, there is a substantial relative uncertainty for u and υ although the various experimental realizations of the residual flow field compare well, visually, to each other (not shown). This relatively high uncertainty on u and υ is certainly due to the fact that these quantities are dimensionally very small (<0.4 mm s⁻¹ see Table 3) and are thus easily contaminated by background currents, even if the latter are small (section 3c). The spatially averaged, residual kinetic energy, however, is a more reproducible quantity: its experimental variance is 5% only.

a. Isopycnal displacements

The isopycnal elevation time series from the numerical model, like the laboratory ones, exhibited only small vertical excursions and residual elevations at the locations considered. This is shown, for example, in Fig. 12, which depicts the isopycnal elevations at horizontal location 2, obtained from the numerical model. These numerical results can be compared directly with the laboratory measurements of Fig. 6. The phase relationships between the density fluctuations at the various levels are approximately the same in both models, but the numerical curves lead their laboratory counterparts at all levels. While the amplitudes have a similar magnitude in the laboratory and numerical models, there are differences in the details of the observations: for instance, the maximum amplitude is observed at the shelf-break level in the laboratory but the amplitude increases with depth in the numerics.

b. Velocity fields

c. Quantitative comparison to the laboratory results

The differences between the numerical and laboratory solutions of the canyon flow problem can be further quantified by applying, to the numerical results, the procedure that was used in section 5c to study the experimental uncertainty. This will provide a global estimation of the departure, from the average laboratory realization (i.e., Table 3), of the numerical results, and allow us to see whether or not this departure lies within the error bar of the laboratory measurements (i.e., Table 4). The numerical values of u, υ, and e_c are thus given in Table 5 for the phase-averaged and residual currents, at the shelf-break level. The difference between numerical and laboratory results is also documented by the values of Δu, Δυ and Δe_c that one gets when applying the formula of Eq. (3) to the numerical solution (the data of reference then being the mean laboratory realization).

Concerning the spatially averaged alongshore component of the phase-averaged velocity fields, Table 5 shows quantitative agreement between both models when the currents are strongest (Δu ∼ 5% at phases 1 and 3) but a relatively strong disagreement at the other phases. At phases 2 and 4, however, u is small (<1 mm s⁻¹), and the very large percentage error at phase 2 is due to the fact that the sign of u at this particular phase is actually different in the laboratory and numerical models; nevertheless, the two fields compare favorably from a visual point of view (Figs. 7 and 13) and the coefficient Δe_c remains reasonable (35%). Table 5 also shows that the amplitude of the spatially and phase averaged cross-shore current is always smaller in the numerical model (negative Δυ at all phases) although the alongshore forcing is well represented (actually, the numerical u exceeds slightly the laboratory value at phases 1 and 3; see Table 5): the currents have a tendency to be less deflected by the topography in the numerical model than in the laboratory experiments. Finally, one can note that the statistic e_c, which was found to be the less dependent on the laboratory uncertainty (section 5c), is also the most robust when comparing quantitatively the numerical and laboratory data in these experiments. As far as phase-averaged flows are concerned, the relative difference of e_c (between both models) ranges from ∼4% at phases 1 and 3 to ∼40% at phases 2 and 4, the latter value being much larger than the corresponding laboratory uncertainty (Table 4). So there is a significant discrepancy between the laboratory and numerical results at phases where the forcing currents are weak (although both models are in qualitative agreement at all phases): the models are not in perfect agreement concerning the details of the temporal evolution of the flow, as already noticed in section 6a when discussing the density time series.

The statistics for the residual current, also given in Table 5, show that the numerical model generates a greater alongshelf residual current: u exceeds the average laboratory value by 34%. (This component leaves the shallow waters on its right in both models.) This difference, however, might not be meaningful owing to the laboratory uncertainty of the u measurement (Table 4). Concerning the y component of the residual flow, the spatial average is directed toward the coast in both models, and the same kind of agreement is found, apart from the fact that now the numerical model tends to slightly underestimate its amplitude. The kinetic energy statistic of the residual flow field, however, is significantly greater in the numerical results (Δe_c = 36%, whereas the laboratory error bar on e_c is only ∼5%); this owes to the “additional” numerical residual circulation already shown in Fig. 15.

This quantitative analysis was restricted to the shelf-break level because the study of the experimental uncertainty was performed at this level only and because, as discussed earlier (section 6b), the laboratory and numerical residual velocity fields are qualitatively different at the other levels (the error statistics are most useful when a qualitative agreement has been observed in the first place).

Before closing this section, we mention here some results obtained in the course of the numerical implementation of the submarine canyon model. Several numerical configurations, numbered 1–5, were actually explored before the “final” one was chosen. Runs 1 and 2 used a linear density profile (N = 2.5 s⁻¹) whereas runs 3–5 included the polynomial profile mentioned in section 3c. For runs 1 and 3, a free-slip bottom boundary condition was applied and no vertical viscosity was introduced. Run 2 was an attempt to solve the frictional boundary layer (no-slip bottom boundary condition and laminar vertical viscosity). Finally, run 4 made use of the bottom stress law defined in section 2a(2) and run 5 used it together with a laminar vertical viscosity coefficient. The output from run 5 was also processed in order to assess the importance of the spatial filtering introduced by the PTV (as implemented in our experiments). For that purpose, the residual velocity field given by run 5 was filtered using the same averaging procedure utilized in the laboratory processing to generate a regular grid of Eulerian measurements with the randomly spaced Lagrangian estimations (as explained in section 3b). Note that all the plots presented in this article were extracted from the direct results of run 5 (the final configuration), apart from Fig. 15, which derives from the filtered output of the same run.

Table 6 shows that the best agreement, based on the statistics for the residual velocity field, is reached when the bottom stress parameterization is used together with a laminar value of the vertical viscosity (run 5, i.e., the final configuration). It is interesting to note that the residual flow pattern from run 2 was found to be very different from the other ones. A cyclonic eddy then appeared close to the downstream edge of the canyon (not shown). This is most probably due to the fact that the vertical grid spacing used in the numerical computations is not small enough to adequately resolve the frictional boundary layer. Apart from that, the residual currents at the shelf-break level do not appear to be very sensitive to the details of the background stratification (compare runs 1 and 3 in Table 6, for instance). Finally, it was observed that the filtering of the residual velocity field given by run 5 leads mainly to a lesser intensification of the residual current along the upstream edge of the canyon (not shown). Such a velocity field is then more comparable with the laboratory one, supporting the idea that the strong residual currents exhibited along the upstream edge by the numerical model are also present in the laboratory experiments, but that they are smoothed out by our acquisition procedure. However, this does not explain all the differences between the laboratory and numerical results, as shown by Table 6 where the nondimensionalised kinetic energy of the difference field still amounts to 33% when the numerical results are filtered (compared to 36% for the nonfiltered data).