1. Introduction
The flow of barotropic tides over ocean topography may lead to the generation of internal tides and mixing, a topic of much recent interest following observations that about 30% of tidal dissipation occurs in the open ocean (Egbert and Ray 2000). It is believed that some of this energy loss results in mixing that could drive the thermohaline circulation (Wunsch and Ferrari 2004). Several recent field programs have examined the role of tide–topography interactions in leading to this tidally generated mixing at oceanic topography—which may take the form of multiple irregular ridges, such as the fracture zones around the Mid-Atlantic Ridge (Polzin et al. 1997), or the form of isolated steep ridges, such as the Hawaiian Ridge (Rudnick et al. 2003; Klymak et al. 2006)—and on continental slopes (Nash et al. 2007). These observations have shown that tidal energy is both converted into internal waves, which radiate away from the topography, and used for local mixing at the topography. To develop new energetically consistent parameterizations of tidal mixing for global climate models, the physical processes governing the transfer of energy from barotropic tide to baroclinic motion to mixing need to be fully understood. This article describes one physical process by which tidal flow over isolated topography leads to local mixing, namely internal hydraulic jumps, and explores the parameter regime in which this behavior is possible.
While theoretical studies have proved useful in predicting the energy conversion rate, they are unable to examine regimes with finite-amplitude barotropic forcing or the partitioning of energy between local mixing and radiated internal tides. In Legg (2004) and Legg and Huijts (2006), the parameter space of large U0/(ωL) combined with small U0/(Nh0) has been shown numerically to lead to the possibility of nonlinear hydraulic effects, which may be one source of local mixing. Another source of local mixing is the rapid dissipation of internal tides of small vertical scale, which are generated when γ > 1, with a greater proportion of energy in smaller wavelengths if h0/H is small.
In this article, we further examine the possibility for mixing associated with internal hydraulic jumps in a somewhat different regime, where U0/(ωL) is small but where strongly nonlinear transient lee waves with overturning similar to internal hydraulic jumps may form if the Froude number associated with the vertical tidal excursion distance is small.
The parameter space we are concerned with in this paper is motivated by observations made close to topography on Kaena Ridge during the Hawaiian Ocean Mixing Experiment (Levine and Boyd 2006; Aucan et al. 2006; Klymak et al. 2008). The top left of Figs. 1, 2 show two tidal cycles of observations made from R/P FLIP near the top of the shelf break. While the total water depth in this location is about 1000 m, R/P FLIP observations only extended down to 800 m. Interesting features of the observations include a sudden increase in density around the time the current, dominated by the M2 tide, changes sign, accompanied by almost vertical isopycnals with overturning and high values of dissipation. The enhanced dissipation and steepened isopycnals extend well above the topography and are clearly linked to the tidal cycle. Because the observations are made only in a single location, it is difficult to identify the physical mechanism by which the tide leads to this overturning and dissipation from these observations alone. We are therefore motivated to carry out a numerical and theoretical study, whose purpose is to identify these mechanisms, linking the processes taking place at this location with those at other locations around this topography and identifying the necessary criteria for this phenomenon to exist, so that the understanding gained from the Hawaiian Ridge can be generalized to other areas of the global ocean.
The Hawaiian Ridge falls into the region of parameter space characterized by γ > 1, U0/(ωL) ≪ 1, and U0/(Nh0) ≪ 1, that is, region 5 in the parameter space diagram of Garrett and Kunze (2007). The specific values of these parameters when the barotropic flow amplitude is around 5 cm s−1 are γ ≈ 4, U0/(ωL) ≈ 0.01 (corresponding to a horizontal tidal excursion of around 350 m) and U0/(Nh0) ≈ 0.006. Here, h0/H is large (i.e., h0/H ≈ 0.8).
2. Model setup
In our model study we employ the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997), integrating the nonhydrostatic Boussinesq equations in a 2D configuration similar to that described in Legg and Huijts (2006) and Khatiwala (2003). A nonhydrostatic model is preferable because previous studies (e.g., Legg and Adcroft 2003) have shown that highly nonlinear internal bores associated with rapid increases in density, which may be responsible for the density perturbations seen in the observations, are poorly represented by hydrostatic models. Simulations in only two dimensions are a reasonable first step for this topography because at the Hawaiian Ridge the flow is constrained to go over the topography through a channel between islands and cannot go around the topography instead. The relative inexpensiveness of 2D simulations also allows us to perform a comprehensive parameter space investigation, so that we can identify the controlling physical processes.
The barotropic tidal forcing U = U0sin(ωt) is applied through a body-forcing term at the M2 tidal frequency (ω = 1.41 × 10−4s−1), with a deep-water velocity amplitude U0 of 2, 5, or 10 cm s−1. Observational values vary from 2 to 5 cm s−1 over the course of the spring-neap cycle. Three other calculations have a forcing frequency of 2×, 3×, and 0.5× the M2 frequency (in the latter case, Coriolis is reduced so that we remain in the regime where propagating waves can exist at the forcing frequency).
The resolution is variable in both horizontal and vertical directions, as shown in Fig. 4, with a minimum grid size over the top of the slope of Δx = 115 and Δz = 8.8 m. The maximum Δx near the boundaries of the domain is 3 km, while the maximum Δz at depth is 49.3 m, with a total number of grid points in the horizontal and vertical directions of nx × nz = 1000 × 300. Laplacian friction is employed in the momentum equations, with horizontal and vertical components of viscosity set to νh = 10−1 and νυ = 10−2 m2 s−1, these being the values necessary to prevent grid-scale noise in the velocity fields. These values are certainly much larger than realistic ocean values, being chosen for numerical rather than physical reasons, and the simulations will therefore be at a lower Reynolds number than the ocean, implying more viscous damping of the flow. Comparison with observations will allow us to evaluate this aspect of the model. The Superbee flux-limiting advection scheme for tracers introduces numerical diffusivity where needed to eliminate grid-scale noise, so the explicit tracer diffusivities are set to zero. Again, the numerical diffusion is introduced for numerical reasons rather than as an attempt to model the physical mixing, and hence we do not expect the model mixing efficiency to match that which would be observed. For this reason we do not use the model to calculate quantities such as diapycnal mixing in this study. Previous studies (Legg and Adcroft 2003) have shown that the model mixing efficiency is considerably reduced compared to values measured for similar scenarios in the laboratory. These limitations of the model should not, however, affect its ability to model nonlinear waves, and we therefore focus on this aspect of the simulations.
No-flux boundary conditions are applied to tracers at the free-surface and bottom topography, while no-slip boundary conditions are applied to momentum at the topography. The side boundaries employ an Orlanski radiative boundary condition for the baroclinic flow and specified boundary values for the barotropic flow (Khatiwala 2003). All simulations are started from rest and continued for 5.6 M2 tidal periods. The Coriolis parameter has a constant value of f = 8.0 × 10−5 s−1 (except for the ω/2 case where f = 2.0 × 10−5 s−1). This is larger than the actual value of Coriolis in the Kaena Ridge region (5.4 × 10−5 s−1) because parametric subharmonic instability (PSI, which may occur for 2f < ω; Gerkema et al. 2006) is not the focus of this particular study, and so f is set high enough to prevent it from occurring in the M2-forcing frequency calculations (however, the two calculations at higher forcing frequency are susceptible to PSI). Other aspects of the real ocean that we are ignoring for this study are horizontal variations in stratification and multiple tidal frequencies. Obviously a full understanding of the tide–topography interaction would require examination of these complications; however, our purpose here is to understand a single process in isolation.
3. Results
a. Realistic simulation
We first focus on the comparison between the observations and the results from the simulation with realistic topography and stratification, shown in the top center of Figs. 1, 2. Like the observations, the model results show sudden increases in density at the time the flow changes direction from off- to onslope, coinciding with steep isopycnals and enhanced dissipation. This suggests that the model is to first order capturing the phenomena responsible for this overturning and dissipation. A principal difference between the model and observations is the reduced magnitude of the isopycnal displacements and dissipation in the model, probably a result of excessive numerical damping in the model. Despite these quantitative differences, the model results can be regarded as qualitatively capturing the features and can therefore provide guidance on their origin.
The source of these features can be identified by examining the behavior simulated by the model farther down the slope. Figure 5 shows a sequence of snapshots of the cross-slope flow and density, beginning near maximum offslope flow and ending near maximum onslope flow. At maximum offslope flow, there is a strong downslope flow, with isopycnals plunging sharply downward adjacent to the slope, followed by a sudden rebound. This feature is strongly reminiscent of an internal hydraulic jump or a strongly nonlinear quasi-stationary lee wave (Farmer and Smith 1980). When the flow begins to relax, the jumplike feature moves toward the slope, and in successive snapshots, it propagates up the slope as an internal bore associated with strong convergent flow and overturning isopycnals. This internal bore seems the likely cause of the overturning and dissipation seen in the observations from FLIP above the shelf break.
To further examine the behavior associated with the passage of the internal bore on the slope, we show time–depth profiles of cross-slope velocity, dissipation, stratification, and shear at a location down on the slope where the water depth is −1150 m (Fig. 6). This is located within the region of plunging isopycnals during the downslope flow. At this deeper location, there is a noticeable bottom enhancement in the offslope flow. High dissipation is found near the boundary, which near the time of the flow reversal extends about 250 m above the topography. This vertical extent is considerably greater than the height of the maximum shear in the downslope flow (about 50 m above the bottom). The high dissipation extending above the bottom coincides with greatly weakened stratification, including overturned isopycnals, centered on about −1000-m depth.
To summarize, the results for this particular topography show enhanced downwelling on the left flank of the topography, associated with a hydraulic jumplike feature. When the flow relaxes, the flow reversal is associated with low stratification (including overturning characterized by negative stratification) and high dissipation. The flow reversal and associated overturning and mixing propagate up the slope as an internal bore, weakening as they go.
b. Sensitivity of nonlinear features to physical parameters
These hydraulic jumplike features and the overturning and dissipation associated with them depend on the external parameters of topographic slope, stratification, flow amplitude, and forcing frequency. In Figs. 1, 2 the velocity, density, and dissipation are shown for model simulations with idealized topography and varying slopes and tidal forcing amplitude. The SteepU5 simulation has linear stratification and an idealized topography, with values chosen so that the stratification is equal to that at the shelf break in the realistic simulation, and the maximum topographic steepness is also equal to that in the realistic simulation. The results are very similar to those in the realistic simulation, suggesting that the idealized simulation reproduces the main physical parameters responsible for this behavior. The snapshots of cross-slope velocity and density shown in Figs. 7a,b also confirm the similar behavior in the realistic and SteepU5 simulations, with overturning internal bores visible in both.
With the idealized topography, we can examine the sensitivity of the behavior to the topographic steepness, by varying the parameter L in Eq. (3). For the medium slope, similar internal bore features are seen (Figs. 7c, 1, bottom left); however for the gentle slope, there is no evidence of internal bores or enhanced dissipation associated with the change in flow direction (Figs. 7d, 1, bottom center). Slope steepness is therefore evidently a necessary ingredient for the formation of the internal bores.
Tidal flow amplitude also influences the response. For weak-flow SteepU2, although overturns and enhanced dissipation are still seen (Figs. 7e, 1, bottom right), they are of smaller amplitude. For stronger-flow SteepU10, the vertical extent of the overturning is greatly enhanced (Fig. 7f).
In the interest of brevity, we do not show snapshots or time series for the simulations with different stratification and forcing frequency, because these all look similar to other simulations. Simulations with varied stratification all show internal hydraulic jumps and overturning but on length scales that increase as stratification is decreased. Simulations with increased forcing frequency show less marked overturns, so that with a frequency of 3ω, the structures have many similarities to those seen with the gentle slope.
At a single location on the slope, the passage of the internal bores is seen by a rapid decrease in buoyancy, followed by a more gradual increase. A quantification of this asymmetry is the skewness of the time derivative of buoyancy, shown for all simulations in Fig. 8. A negative skewness in ∂b/∂t corresponds to rapid decreases in buoyancy, with more gradual increases. At h = −1150 m there is no consistent tendency for the skewness, while at h = −1028 m, all except two cases show a negative skewness over the bottom 200 m of the profile, near the topography. This is similar to that seen in Legg and Adcroft (2003) when upslope propagating bores are generated by reflection of low-mode internal tides from near-critical slopes. This supports the proposal suggested by the snapshot sequence that the internal hydraulic jumps/lee waves generated during the offshore flow give rise to upslope propagating bores when they encounter the slope following the flow relaxation. The simulations that do not show any negative skewness are those where bores are not seen, that is, when the slope is very gentle (GentleU5) or the forcing frequency is very large (3ω).
A measure of the vertical extent of the overturning associated with the passage of the internal bores is the region affected by a low gradient Richardson number (Fig. 9). The highest location where Ri = 1 is shown as a function of time for a location of topographic depth h = −1150 m, for all simulations. Steeper slopes and higher-amplitude forcing are associated with greater vertical extent of the low Richardson number region. For increasing ω the height of low Ri increases with time, and for 3ω it is displaced above the bottom, associated with middepth shear, possibly due to the development of PSI, not the jumplike regions seen in other simulations. Reducing (increasing) stratification leads to a greater (lesser) vertical extent of overturning, as the displacement scale increases (decreases).
Dissipation, calculated from νi(∂uj/∂xi)2, shown in Fig. 10, is concentrated along the internal tide characteristics emanating from the critical points at the top of the two flanks of the topography and following reflection, from the surface. In most of the calculations, an additional region of enhanced dissipation is seen at the top of the left flank, where the slope is close to critical. Exceptions to this occur when the slope is closely aligned with the wave slope (Fig. 10d) or the forcing amplitude is small (Fig. 10e).
In summary, weaker amplitude forcing (SteepU2) leads to a smaller region of low Richardson number, as well as a smaller peak in dissipation, with dissipation at the top of the topography not significantly larger than on the internal tide beams. Stronger forcing (SteepU10) enhances the features seen in SteepU5 and extends the overturning over a greater vertical extent. The slope steepness, s, has a strong influence. In the gentle slope case, the flow is much more linear with only a small low–Richardson number region, lower dissipation, and no negative skewness in the buoyancy time derivative. Both significant steepness downstream of the topographic peak and large amplitude forcing therefore appear to be necessary to produce significant local overturning. Stratification and forcing frequency also influence the development of internal jumps, with decreasing stratification leading to larger vertical length scales and increasing ω reducing the appearance of the jumps.
4. Interpretation in terms of internal hydraulics
Having shown qualitatively that both large steepness and large amplitude flow enhance the strength of overturning internal bores, we will now quantify these relationships by developing theoretical predictions for (i) the regime in which tidally generated internal hydraulic jumps may occur and (ii) the magnitude of the vertical displacements associated with the tidal flow.
In single-layer hydraulic control theory, a flow is supercritical if the flow speed is greater than the single-layer long-wave phase speed: U >
Continuously stratified flows over topography are considerably more complicated, because the internal wave horizontal phase velocity may take on an infinite number of values. The maximum phase velocity corresponds to the lowest vertical mode; if the flow speed exceeds this maximum phase velocity, all modes will be unable to propagate against the flow. The Hawaiian Ridge topography does not fall in this regime for realistic barotropic velocities. Instead it may fall into a regime in which larger wavelengths can propagate while smaller vertical wavelengths are arrested. An alternative viewpoint in which to consider this regime is that of the lee waves that occur in steady flow over topography. It is instructive to consider the conditions under which lee waves become highly nonlinear and overturn: when the Froude number Fr = U/(Nh0) (where h0 is the amplitude of the topography) is equal to or less than unity, a regime of highly nonlinear, breaking lee waves occurs, in which the overturning bears some resemblance to an internal jump. Durran (1986) has shown that the flow structure in the breaking region also resembles a supercritical flow. Our scenario differs strongly from these atmospheric studies in that our forcing velocity is time dependent, and our topography extends close to the surface of a finite-depth ocean rather than through a small extent of a very deep atmosphere. Nonetheless, it is useful to consider our jumplike features as perhaps transient analogs of the highly nonlinear overturning lee waves seen when Fr < 1.
In Fig. 11 we show the maximum vertical displacement Δh of isopycnals originating above the shelf, nondimensionalized by U0/N, plotted against Fr−1Zω. The Δh has been diagnosed from all the different simulations in which U0, ω, dh/dx, and N are varied. The schematic shown in Fig. 12 shows an example of the measured Δh. In the region of (11) where Fr−1Zω < 3, there is a linear relationship, as predicted in Eq. (11). For low Fr−1Zω, therefore, this scaling holds, and the correlation coefficient between Δh and Zω is R = 0.97 for these low Fr−1Zω points. However, for that part of Fig. 11 where Fr−1Zω > 3, a different scaling applies: in this regime ΔhN/U0 is approximately constant and has a value of about 10. For these high Fr−1Zω points, the correlation coefficient between Δh and U0/N is R = 0.94.
There are therefore two distinct regimes, with Fr−1Zω being the regime-controlling parameter, as suggested by the theoretical scaling analysis. For low Fr−1Zω (i.e., shallow slopes, weak stratification, and high-frequency forcing), the tide reverses before the stratification suppresses vertical motions, and Δh is just set by the vertical tidal excursion. The waves in this regime are linear with little overturning. For high Fr−1Zω (i.e., steep slopes, strong stratification, and low-frequency forcing), the tide lasts long enough for stratification to work against the downslope motions, and Δh scales like U0/N. The U0/N scaling gives a maximum value of Δh permitted on energetic grounds; when the Zω scaling for Δh exceeds this maximum value, then Δh is capped at the energetically permitted value. Not coincidentally, in the high-Fr−1Zω regime when the maximum displacement is proportional to U0/N, the Froude number local to the displacement region is of order unity, the scenario under which highly nonlinear jumplike lee waves occur. In our particular scenario, this regime transition occurs at Fr−1Zω ≈ 3, and it is for Fr−1Zω > 3 that we observe nonlinear jumplike behavior in our simulations. We expect that the U0/N scaling for Δh will only hold so long as U0/N < h0, the topographic height, and U0/N < H − h0, the fluid depth above the topography. If either of these limits is exceeded, then the finite height of the topography or the finite depth of the fluid will influence the motion.
Given this scaling analysis and regime diagram, we predict the principal requirement for internal jump behavior driven by tidal flow is large relative steepness, that is, large dh/dx(N/ω). However, given that the jump magnitude, Δh, scales like U/N, large U/N is a second requirement for jumps of significant amplitude. In nondimensional terms, we would not expect the mechanism of internal hydraulic jumps to be important if U/N is smaller than the frictional bottom boundary layer, so we can express this as a type of Reynolds number constraint: ReN = U2/(Nυυ) ≫ 1 for internal jumps to dominate over frictional processes. (In our simulations ReN varies from 17 in U2 to 420 in U10. In U2 the jumps are therefore much more strongly affected by friction.) Interestingly, stratification can play a dual role: increasing stratification increases Fr−1Zω, making jumps more likely to occur, but it reduces U/N, reducing the vertical scale of the isopycnal displacements associated with the jumps.
Finally, we can reexamine our previous qualitative results in terms of the regimes delineated by Fr−1Zω. Low Fr−1Zω simulations (e.g., gentle slope or 3ω) did not show any evidence of borelike behavior, in the snapshots, time–depth profiles, or skewness of the temperature time derivative. Borelike behavior was seen only for high Fr−1Zω.
One question might be raised: How important is a rapid transition from flat shelf to steep slope? Note that our idealized topography, by including a smooth slope transition on the left and a discontinuous slope transition on the right, helps to answer this: the discontinuous slope transition on the right does not lead to any internal hydraulic jumps, whereas the smooth transition (to a steeper slope) on the left does. Hence, it is the steepness of the slope, not the sharpness of the transition, that determines whether an internal hydraulic jump appears. However, the mechanism we have discussed does presume that the flow encounters the steep slope within the tidal cycle, and so the slope transition must be rapid enough that a steep slope, such that Fr−1Zω is large, is reached within a vertical tidal excursion.
It should be noted that the steepness γ approaches Fr−1Zω in the limit f < ω < N, and so the criterion that Fr−1Zω > 3 for borelike behavior could also be expressed as γ > 3. In practice, for most of the parameters we chose, where ω is not very much greater than f, γ is slightly less than Fr−1Zω.
5. Discussion and conclusions
In this paper, motivated by observations from the Kaena Ridge in Hawaii, we have carried out a series of simulations of tidal flow over an isolated ridge, exploring the factors that contribute to overturning behavior on small scales near the shelf break. We have identified two regimes of behavior near the shelf break, with the regime transition dependent on Fr−1Zω = (dh/dx)N/ω. For large Fr−1Zω, highly nonlinear jumplike lee waves are found at maximum offslope flow, with a vertical displacement scale proportional to U/N, such that the jump Froude number is of order unity. When the flow relaxes, the internal jump features propagate toward the slope, leading to well-defined bores propagating up the slope, characterized by negative skewness in the buoyancy time derivative and associated with overturning and enhanced dissipation. By contrast, for small Fr−1Zω, the flow response consists of linear waves, with no jumps or overturning, and the vertical displacement scale is proportional to the vertical tidal excursion, (U/ω)dh/dx. The FLIP observations from the Hawaiian Ridge fall into the large Fr−1Zω regime of internal jumps and overturning.
Having determined the parameter regime in which the features observed over the Hawaiian Ridge may exist, we now examine whether such features might be expected to have a widespread global distribution. Figure 13a shows the vertical excursion inverse Froude number, Fr−1Zω = dh/dxN/ω, calculated on a 1/2° global scale for the M2 tidal component. The N is calculated at the level of the bottom topography from the 1/2° gridded World Ocean Circulation Experiment (WOCE) dataset of Gouretski and Koltermann (2004). This dataset was chosen because particular attention was paid to ensuring stable stratification at depth in the interpolation process. Many of the major ocean ridge systems are associated with sufficiently steep topography that Fr−1Zω > 3, and the vertical tidal excursion distance is therefore sufficiently large that internal hydraulic jumps could result. The other prerequisite for such features is large amplitude flow. Figure 13b shows the amplitude of the M2 barotropic flow component projected onto the direction of the topographic gradient. This calculation was made using the global inverse tide model TPXO6.2 (Egbert and Erofeeva 2002). Regions of both large-amplitude flow and Fr−1Zω > 3 include the Hawaiian Ridge, the Mascarene Ridge, the Kerguelen Plateau, and the entrance to the East China Sea. This examination of the global scale is, however, limited by its low resolution but nonetheless provides motivation for examination of tidally driven internal hydraulic jumps in many more locations. One such location is the Oregon continental slope, where nonlinear effects on the seafloor have recently been reported by Nash et al. (2007). Similar tidally driven internal hydraulic jumps have also been observed in fjords (Klymak and Gregg 2004; Inall et al. 2005), although the dynamics in these regions is modified by the shallow depth of the sill.
In this study we have focused on the qualitative features of the internal hydraulic jump and quantified the parameter space within which this regime occurs. By doing so, we have identified one mechanism by which mixing can be generated local to the topography by the tides. It should be stressed that there are many other mechanisms for generating local tidal mixing, including shear instability in narrow internal tide beams, and wave–wave interactions, especially parametric subharmonic instability. Ultimately, we would like to be able to parameterize the mixing generated by tides in an energetically consistent fashion. Simulations such as these cannot provide a complete quantitative answer; a particular caveat is that the ratio of total dissipation to rate of energy conversion in these simulations has been found to be highly sensitive to the viscosity coefficients and resolution of the model. In addition, 2D numerical simulations are known to omit important turbulence-generating instabilities. For this reason, our focus has been on understanding the mechanism of generating the internal hydraulic jumps, not on quantifying the mixing due to such features. To fully understand the effect of the internal hydraulic jumps on the energetics of tidal dissipation will therefore require progressively higher-resolution 3D simulations until an asymptotic regime is approached. The simulations in the present study nonetheless provide guidance as to a particular parameter regime where local mixing is important.
Acknowledgments
We thank Rob Pinkel, Steve Garner, and Bob Hallberg for their constructive feedback, and two anonymous reviewers whose comments led to a substantial improvement in the manuscript. JK was supported by NSF Grant OCE-98-19529 and ONR Grant N00014-05-1-0546. SL was supported by Office of Naval Research Grant N00014-03-1-0336 and by Award NA17RJ2612 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the U.S. Department of Commerce.
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Time–depth sections showing the cross-slope component of velocity, with density contours overlain at a location near the top of the shelf break. (top left) Values observed from R/P FLIP, while other panels show model-simulated values, with details of the different simulations given in Table 1 and in section 2.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Time–depth sections showing the kinetic energy dissipation, with density contours overlain, at a location near the top of the shelf break. (top left) Values observed from R/P FLIP, while other panels show model-simulated values, with details of the different simulations given in Table 1 and in section 2.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
(a) The stratification N 2 and (b) the topography used in the different simulations. In (a), the solid line shows the real N 2 profile, while the dashed line shows the constant N 2 profile used in most of the idealized calculations, which matches the real N 2 value at the top of the topography. Four other idealized calculations have N 2 = 2×, 4×, 8×, and 1/4 of this value. In (b), the thick solid line shows the real topography, and the other lines show different idealized topographies described in the text. Note that all idealized calculations have the same topography for x > 0, with much gentler slope than the real topography.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
The resolution used in all the simulations: (a) Δx and (b) Δz. High resolution is concentrated within the region −50 km < x < 50 km, −2000 m < z < 0.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Snapshots of the cross-shelf velocity (color, m s−1) and density for the realistic simulation, near the shelf break, over the course of approximately half a tidal cycle. Maximum offslope tide is at 46.4 h, while maximum onslope tide is at 52.6 h.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Time–depth plots from the realistic simulation at a location of depth h = −1150 m, to the left of the ridge, with density contours overlain.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Snapshots of the cross-slope velocity field (color) and buoyancy (black contours) at a time 1.12 tidal periods after the beginning of the simulation for (a) realistic topography and stratification, U0 = 5 cm s−1 , (b) steep slope, U0 = 5 cm s−1, (c) medium slope, U0 = 5 cm s−1, (d) gentle slope, U0 = 5 cm s−1 (e) U2, U0 = 2 cm s−1, and (f) U10, U0 = 10 cm s−1.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
The skewness of the temperature time derivative, shown as a function of depth for two locations where the topographic depth is (a) −1150 and (b) −1028 m, shown for all idealized topography runs. Dashed lines indicate simulations with FrZω > 0.3 (see section 4).
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Comparison of the height at which gradient Ri = 1 for all runs, at a location where the topographic depth is −1150 m. (a) Dependence on variations in topography and U0, (b) dependence on variations in N 2, and (c) dependence on variations in ω.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Time-averaged dissipation ε: (a) realistic topography and stratification, U0 = 5 cm s−1; (b) steep slope, U0 = 5 cm s−1; (c) medium slope, U0 = 5 cm s−1; (d) gentle slope, U = 5 cm s−1; (e) SteepU2, U0 = 2 cm s−1; (f) SteepU10, U = 10 cm s−1. The M2 internal tide characteristic emanating from the shelf break for uniform stratification is shown by a black line in (b). The color scale shows units of log10 (ε), where ε has units of m2 s−3.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
The maximum downward displacement of isopycnals originating over the shelf (Δh), nondimensionalized by U0/N, shown plotted against Fr−1Zω. The two lines show least squares best fits calculated separately for the points with Fr−1Zω < 3 (solid) and Fr−1Zω > 3 (dashed). For Fr−1Zω < 3, there is a linear relationship: ΔhN/U0 ≈ 3.7 Fr−1Zω, while for Fr−1Zω > 3, ΔhN/U0 is independent of Fr−1Zω, and approximately constant, with a value of ΔhN/U0 ≈ 10.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
A schematic, based on the steep slope simulation, showing the maximum downward vertical displacement of the isopyncals Δh and the slope of the lee wave, defined by the horizontal velocity maximum. Horizontal velocity is shown in color, buoyancy in black contours.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
(a) The M2 tidal excursion inverse Froude number Fr−1zω = dh/dxN/ω on 1/2° spatial scale, where N is the stratification just above the topography, obtained using Koltermann and Gouretsky 1/2° gridded WOCE hydrography. (The color scale has been truncated to highlight regions where Fr−1 > 3.) (b) The M2 tidal velocity amplitude (cm s−1), projected onto the topographic gradient. (The color scale has been truncated to highlight midocean values.) In both figures the Hawaiian Ridge is marked by a white oval.
Citation: Journal of Physical Oceanography 38, 9; 10.1175/2008JPO3777.1
Values of topographic and flow parameters for the numerical simulations.