The Direct Breaking of Internal Waves at Steep Topography

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INTroDucTIoN
Internal waves transmit energy to the ocean's interior.In order to affect the mean flow and the mixing of nutrients in the ocean, these waves must, by some mechanism, break and become turbulent.The resulting mixing creates lateral density gradients that drive ocean currents, both local and on the scale of the global thermohaline circulation.It has been estimated that the deep ocean requires 2 TW of energy to maintain the observed overturning circulation, and that this energy can be supplied approximately equally by internal waves forced by wind and by tides (Munk and Wunsch, 1998).Tracking the energy pathways of internal waves generated by winds is difficult because they are spatially variable and sporadic (D' Asaro, 1995;Alford, 2003), but recent progress has been made on understanding the energy balance of tides.
Though simpler than wind-driven waves, the energy balance of internal tides is still quite complex (Figure 1).
Internal tides are generated when the surface tide pushes density-stratified water over topography, creating internal pressure gradients that drive the waves.
In the bounded ocean, the waves propagate away from the topography in vertical "modes." The lowest internal wave mode has the longest vertical wavelength that can fit between the surface and the seafloor, and is analogous to the resonant note in a pipe organ or guitar string.In an internal wave, this means that the lowest mode has the strongest velocities at the surface and seafloor and null velocity in the middle of the water column, and it propagates with a horizontal phase speed that is faster than higher modes.Higher modes have more zero crossings and travel at slower speeds.This response can be seen in the top panel of Figure 1, early in a numerical tidal simulation, where the low modes have traveled almost 500 km from the ridge at x = 0, but the more complicated "high-mode" structure is just starting to form near the ridge.The high modes may break into turbulence locally where they are generated (Figure 1, lower panel), at remote topography (Nash et al., 2004), or by wave-wave interactions in the interior (Henyey et al., 1986;St. Laurent et al., 2002;Polzin, 2009;MacKinnon and Winters, 2005).Low modes can reflect, interfere with one another, and scatter into higher modes, though which processes dominate at any given location is not understood.
Steepness of the topography is an important factor in the study of topographic interactions with internal waves (Garrett and Kunze, 2007).As can be seen emanating from the topography in Figure 1, internal waves organize along "beams" of energy, which propagate at characteristic angles that depend on the frequency of the waves and the density stratification.If the topography is gentler than these characteristic slopes, it is said to be "subcritical;" if steeper, it is said to be "supercritical." Of course, real topography has regions of both, but the extremes are useful because mathematical predictions of internal tide generation can be formulated for either assumption (for subcritical topography, see Bell, 1975, andBalmforth et al., 2002; for supercritical topography, see Llewellyn Smith andYoung, 2003, andSt. Laurent et al., 2003).
At regions of subcritical topography, such as the Brazil Basin, internal tide generation is relatively linear.Nonlinear interactions between the waves drive energy to higher wavenumbers until they break, leading to enhanced turbulence far above the seafloor (Polzin et al., 1997;St. Laurent et al., 2001).This mechanism has been numerically simulated by Nikurashin and Legg (2011) and has been parameterized by Polzin (2009)  Mixing Experiment, at the Hawaiian Ridge, it was noted that there was strong turbulence near the seafloor and extending hundreds of meters into the water column (Figure 2; Aucan et al., 2006;Levine and Boyd, 2006;Klymak et al., 2008).These motions drove breaking internal waves that were up to 200 m tall, and mixing rates inferred from the size of the overturns were up to four orders of magnitude greater than open-ocean values at those depths.
Considering an event in detail (Figure 2), the turbulent overturns tend to be at the sharp leading face of rising density surfaces (isopycnals).This leading edge breaks, in this example, in two patches centered at 05:00 and 06:30.
During the relaxation of the tide, there is a rebound at mid-depth (07:00) and sharp oscillations deeper (though these produce only modest turbulence).In all, the strong turbulence lasts over three hours; the remaining nine hours of the tidal cycle are relatively quiescent.
The main findings from the observations at Hawaii were: 1. Breaking waves were phase-locked to the surface tidal forcing.
2. Turbulence dissipation rates (a measure of the strength of the turbulence) scaled cubically with the springneap modulation of the surfacetide velocities.
3. The turbulence that had these characteristics was confined to within a few hundred meters of the seafloor; turbulence further aloft varied only weakly with the tidal forcing.
Similar observations have been made at the Oregon slope (Figure 3; Nash et al., 2007;Martini et al., 2011) and the South China Sea continental slope (Klymak et al., 2011).The turbulence at these locales was less than at Hawaii, but again depended on the tidal Jody M. Klymak (jklymak@uvic.ca)Situated between Taiwan and luzon Island of the Philippines, luzon Strait generates some of the largest-amplitude internal tides in the world (alford et al., 2011).These internal tides result from the combination of strong barotropic tides (> 1 m s -1 ) and ridge geometry that intersects the ocean's main thermocline at a depth where it most efficiently generates the energy-containing low modes.luzon Strait is somewhat unique in that it consists of two ridges: a short one to the west, and a tall one to the east.In addition, the daily (diurnal) tide is stronger than the twice-a-day (semidiurnal) tide.While, in principle, a strong response at both diurnal and semidiurnal frequencies should ensue, the spacing of the two ridges promotes resonance of the semidiurnal internal tide between the ridges, and dissonance for the diurnal frequency (Jan et al., 2007;echeverri et al., 2011;Buijsman et al., in press).as a result, the semidiurnal internal tide dominates the energetics between the ridges, producing waves that exceed 600 m in vertical displacement near steep topography.
figure B1 shows the internal wave response during a 25-hour period in June 2011 (recent work of author Nash and colleagues).at this phase of the spring/neap cycle, the surface tide has strong diurnal inequality, with two westward velocity pulses every day, but just one to the east.The response of internal temperature surfaces reflects this imbalance.although the wave (or low-frequency) nature of the flow is predominantly semidiurnal, turbulent overturning develops primarily after the strong diurnal velocity pulse to the east, producing 600 m tall waves and unstable overturns that exceed 500 m in height.In addition to these turbulent billows that pass our mooring as the barotropic flow reverses direction (e.g., 08:00-10:00 on June 16), also evident are short-period undulations higher in the water column that only occur once per day and are likely associated with instability of the strongest downslope flows between 04:00 and 08:00.This example highlights how semidiurnal and diurnal internal tides add together to produce a very strong wave and turbulent response that scales nonlinearly with the forcing.for these waves to form depends on the topographic slope, with steeper slopes launching waves faster than gentler slopes.For an oscillating flow such as the tide, the finding is that the topography needs to be twice as steep as the internal wave slopes for oscillating lee waves to be effectively trapped and dissipative (Klymak et al., 2010b).
recipe for Turbulence at Supercritical Topography Here, we describe how the "arrested lee wave" concept allows for an analytical prediction of tidal turbulence and mixing near surpercritical topography based on the forcing, a simplified bathymetry, and the ocean's stratification.First, a linear theory can be used to predict the internal response of tidal flow over sharp topography (Llewellyn Smith and Young, 2003;St. Laurent et al., 2003).
The predicted flow field has very sharp "beams" emanating from the ridge and well-defined large-scale phase changes in the velocity on either side of these beams (Figure 6a).This theory predicts an internal wave energy flux away from the topography in each of the vertical modes (dashed line in Figure 6d).To arrive at a dissipation rate, our recipe uses the observations above to determine the modes that break (Klymak et al., 2010b).The strongest flows at the top of the topography are compared to the phase speed of the modes.Fast modes are assumed to escape and form the radiated signal; slow modes are assumed to be trapped and form the dissipation.The stronger the forcing, the more modes are trapped (Klymak et al., 2010b).All the energy contained in the trapped modes is then assumed to transfer to turbulence locally.When compared to numerical simulations, this parameterization for the local turbulence dissipation rate is quite successful over a range of tidal forcing, bathymetry shape, stratifications, and latitudes (the Coriolis parameter strongly affects the energy put into the waves), so long as the topography is sufficiently supercritical (Figure 6f).
An important finding of this analysis is that the local turbulence, while 10 -2 10 -1 10 0 figure 6.(a) Snapshot of baroclinic (depth-mean removed) velocity from linear generation over a 500 m tall knife-edge ridge in 2,000 m of constantstratification water, normalized by the barotropic forcing velocity.(b) Snapshot of the same configuration when the tidal forcing is 0.04 m s -1 , and (c) when it is 0.2 m s -1 .Note that the stronger forcing has less well-defined "beams" of energy radiating from the topography than the weaker, more linear forcing.(d) comparison of the energy flux distributed by vertical mode number for (a)-(c) normalized by the energy flux of the barotropic wave.The 0.2 m s -1 case rolls off at lower mode numbers than 0.04 m s -1 , as also shown in (e).(f) The parameterization of Klymak et al. (2010b) predicts this fraction of energy as a function of forcing, topographic height, and stratification, usually within a factor of 1.5.spectacular and locally important, is still a modest fraction of the energy removed from the surface tide.This can readily be discerned from Figure 1 or Figure 6, in which the full water-column motions that escape the ridges have significant energy.In fact, it requires very strong and nonlinear forcing for the local dissipation to even approach 10% of the internal tide energy budget at these sills.
Thus, supercritical ridges are believed to be quite efficient radiators of energy, at least in the idealized forms considered so far in the modeling.A rough energy budget based on observations collected near Hawaii corroborates that idea (Klymak et al., 2006).

More complicated Systems
Because supercritical topography seems to be efficient at generating internal tides without much local loss, the question stands as to the fate of that energy.Low modes will impact remote topography where they will scatter, reflect, or dissipate.If that topography is supercritical, such as at other underwater ridges or continental slopes, the same physics discussed above appears to apply: crosstopography flow generates turbulent lee waves that are phase-locked with the remote forcing, and a very similar parameterization is very effective at predicting the turbulence in these lee waves (recent work of author Klymak and colleagues).Predicting the turbulence at a remote ridge can become quite complicated if there is also local generation (Kelly and Nash, 2010).The phase between the local and remote forcing can change the turbulence predicted by an order of magnitude, and breaking lee waves are either suppressed or enhanced by the resonance.
Nowhere is this effect more clear than in a two-ridge system like Luzon Strait.
In two-dimensions, a two-ridge version of the linear model can be constructed that shows strong interactions between the two ridges.The two ridges must be treated as a system, and the relative heights of the ridges and the distance between them greatly affect the response.
The potential for resonance means that a larger fraction of the internal tide might go into local turbulence than might be the case for isolated ridges.Recent estimates of the local dissipation for Luzon Strait range from 20 to 40% of the energy lost from the surface tide (Alford et al., 2011;Buijsman et al., in press).

DIScuSSIoN
So far, we have seen that the internal tide at abrupt topography often breaks, short circuiting the process of cascading high-mode internal wave energy.This local breaking is vigorous, producing turbulent events that reach hundreds of meters tall, and orders of magnitude greater than open-ocean turbulence.The breaking is also phase-locked to the tide and strongly dependent on the strength of the forcing.Numerical modeling indicates that sources of this turbulence at many locations are lee waves arrested at the topographic breaks during off-ridge tidal flow, either driven by the local tide, or by remote tides.We have a parameterization for this process that works under a large range of forcing and topographies (i.e., Klymak et al., 2010b).This progress in understanding the lee wave process does not preclude the importance of other processes that perhaps do not manifest themselves in idealized two-dimensional models.As indicated by MacKinnon and Winters (2005) and Simmons (2008), there is potential for wave-wave interactions, particularly equatorward of the critical latitude where the Coriolis frequency is half the tidal frequency.It is also likely that there are wave-wave interactions local to the Hawaiian Ridge, as indicated in observations (Carter and Gregg, 2006) and as we have seen in our numerical models (Klymak et al., 2010b).However, it remains to be seen how important this energy pathway is near topography, and there are indications that it is modest in observations of the radiated tide (Alford et al., 2007).
There are a number of outstanding questions about the lee wave process and turbulence near supercritical topography.
The first is, how good are estimates of the turbulent dissipation in these processes?
There have been very few direct estimates of turbulence in these breaking waves; instead, the size of the wave is compared to the stratification it encompasses, and a dissipation rate inferred following Thorpe (1977).Detailed comparison between microconductivity and the size of the breaking waves at Hawaii had encouraging results (Klymak et al., 2008), as did comparisons between shear probes and breaking waves in Knight Inlet (Klymak and Gregg, 2004) However, a lot of topography is more complicated than this, with significant roughness that can interact with the tides, or substantial near-critical regions that also convert tidal energy efficiently into turbulence (Eriksen, 1982;McPhee and Kunze, 2002).Despite these substantial caveats, we are attempting to create a global map of dissipation due to local forcing at supercritical topography.There are estimates of tidal currents, stratification, and topography, so this problem should be tractable.
To make progress, a number of challenging assumptions need to be made when applying the parameterization suggested here globally.What scale should the topography be smoothed over to decide on its height?What tidal velocity should be used?How to account for the funneling of flows through constrictions?How to account for narrow topography that may not result in substantial generation (Johnston and Merrifield, 2003)?Finally, if incoming internal tides from afar are important for local turbulence, either enhancing or suppressing it, then accurate maps of global internal tidal energy, perhaps at low modes, will be needed before a global estimate can be made (Kelly and Nash, 2010).This task will prove challenging as tides are advected by large-scale currents and changes in stratification (Rainville and Pinkel, 2006), making their predictability questionable (Nash et al., 2012, in this issue), so it is possible that global coarse internal tide models will need to be used to quantify this remote forcing (Simmons et al., 2004, andArbic et al., 2012, in this issue).
figure 1. a numerical simulation of internal tide generation from a steep ridge (at x = 0 km) and a remote continental margin (at x = 1,300 km).Weaker generation from the continental slope reflects the weaker cross-slope tides there.Velocity is colored.The upper panel is early in the simulation to show the propagation of low-mode internal tides across the basin.The low modes travel faster than the high modes, which make up the "ray"-like character of the velocity closer to the ridge.eventually, these rays would fill the whole basin if the resolution of the model permitted.The full solution shows the interaction of the propagating and reflected waves, and indicates the turbulence that would be observed at the remote continental slope.

PhenomenologyFigure 4
figure 2. observations of breaking waves made on the ridge crest between oahu and Kauai, hawaii (modified fromKlymak et al., 2008), with the local surface tide plotted in the upper panel.The mooring was in 1,050 m of water; the bottom 250 m of the water column was not measured.contours are density surfaces versus time during an onslope phase of the tide.colors indicate turbulence dissipation rate, a measure of turbulent strength, inferred from the size of the breaking overturns in the waves, which reach almost 250 m tall near 06:00.
figure B1. observations of breaking waves 2 km downslope from the 1,500 m deep heng-chun ridge in luzon Strait (at the 2,100 m isobath; 20°35.39'N,121°2.03'e),which is approximately half a mode-1 diurnal internal tide wavelength from the main east ridge.Measurements were taken from a mooring with many small temperature sensors spanning the lower 1,500 m of the water column.The response at this ridge is very dramatic, enhanced by a near-resonant response between the two ridges.The top panel (a)shows barotropic velocity at a nearby ridge.Panel (b) shows a time series of temperature, from which the turbulence dissipation rate inferred from the size of breaking overturns in the flow (panel c); contours indicate constant temperature surfaces.Note the almost diagonal interleaving of warm and cold water starting at about 08:00.for context, panel (d) shows a two-dimensional numerical simulation byBuijsman et al. (in press), with colors and contours respectively representing velocity and temperature; the mooring location is in a region of intense wave breaking, as indicated by the dashed line.Note that the slope is extremely steep at this location, dropping approximately 300 m per kilometer!

figure 4 .
figure 4. Simulated turbulence dissipation at a two-dimensional hawaiian ridge.The resolution is 15 m in the vertical and 150 m in the horizontal.The surface tide forcing is 0.06 m s -1 in the deep water and 0.26 m s -1 over the ridge.The barotropic tide at the ridge crest is indicated with arrows and in text under the time relative to peak left-going flow.large breaking waves, over 200 m tall, set up just after peak off-ridge flow, and then propagate across the ridge as the tide reverses direction.Note that the peak turbulence is after peak flow, as it takes a finite amount of time for the lee wave to grow.
figure 5. Detailed simulation of a breaking lee wave during peak off-ridge flow (this time to the right).here, the vertical and horizontal resolution of the numerical model is 3 m, and the simulation is nonhydrostatic.Both panels have a one-to-one aspect ratio, and the crest of a smooth gaussian topography is at 1,000 m depth in a 2,000 m deep water column.(a) Temperature (proxy for density), where large 200 m breaking waves can be seen both in the downslope flow zone, and where the strong off-ridge flow separates from the topography.Note that the color has been shaded by the vertical derivative of the temperature in order to accentuate the turbulent structure of the flow.(b) The cross-ridge velocity (red is off-ridge); the teal contours are temperature for comparison with (a).again, the velocity has been shaded with the first difference in the vertical to accentuate turbulent structure.
on the large scales (Figure6b,c).However, the "beams" in the simulations are more diffuse, and some energy leaks out in higherfrequency (steeper) beams.As the forcing is increased (compare Figure6cto b), the beams become even more diffuse.Diffuse beams indicate a loss of high-mode energy from the system, and this loss is quantified by noting that the energy flux at high modes does not agree with the linear theory (solid lines, Figure6d and e).The stronger the forcing, the lower the mode where the disagreement starts, indicating a larger fraction of energy is dissipated.
, but direct observations of turbulence dissipation at the microscale in these large mid-ocean lee waves are still needed.Similarly, questions arise as to the efficiency of the mixing in these breaking waves (i.e., what fraction of energy goes toward mixing density rather than turbulent friction processes?).Finally, the arguments in the papers summarized depend on numerical modeling that also assumes the turbulence is a function of the size of the breaking waves rather than more direct estimates of dissipation, which require much more computing power.Some of these questions will be answered by forthcoming observations, particularly those planned in Luzon Strait.Others will require more detailed modeling, likely at the scale of large-eddy simulations, though these will be particularly challenging, given the broad scale of the internal tides (hundreds of kilometers) and the small eddy sizes (a few meters).Much work to date, as described here, has been on two-dimensional idealizations of the flow.They are almost certainly over-simplifications, and threedimensional models of these problems are being run (recent work of author Buijsman and colleagues).Funneling of flow by constrictions can greatly enhance the lee waves, so deciding on appropriate forcing and topography when applying the simple parameterizations discussed above is not trivial.Similarly, we have considered flow over topography that is largely supercritical with respect to the tide, except at the ridge crest.