1

The Direct Breaking of Internal Waves at Steep 1

Topography 2

Jody M. Klymak, Sonya Legg, Matthew H. Alford, Maarten Buijsman, 3

Robert Pinkel and Jonathan D. Nash 4 5

Abstract 6

7

Internal waves are often observed to break close to the bathymetry at which they are 8

generated, or from which they scatter. This breaking is often very spectacular, with 9

turbulent structures observed hundreds of meters above the seafloor, and driving 10

turbulence dissipations and mixing up to 10,000 times open-ocean levels. An overview is 11

given of efforts to observe and understand this turbulence, and to parameterize it near 12

steep “supercritical” topography (i.e. topography that is steeper than internal wave energy 13

characteristics). Using numerical models, we demonstrate that an important turbulence-14

producing phenomenon is arrested lee waves. Analogous to a hydraulic jump in water 15

flowing over an obstacle in a stream, these are formed each tidal cycle and break. Similar 16

lee waves are also seen in the atmosphere and shallow fjords, but in those cases their 17

wavelengths are of similar scale to the topography, whereas here they are quite small 18

compared to the water depth and topography. The simulations indicate that these lee 19

waves propagate against the generating flow (usually the tide) and are arrested because 20

they have the same phase speed as the oncoming flow. This allows us to estimate their 21

size a-priori, and, using a linear model of internal tide generation, estimate how much 22

energy they trap and turn into turbulence. This approach has proven to yield an accurate 23

parameterization of the mixing in numerical models, and these models are being used to 24

guide a new generation of observations. 25

2

Introduction 1

Internal waves are a mechanism by which energy is transmitted to the interior of the 2

ocean. In order to affect the mean flow and the mixing of nutrients in the ocean, these 3

waves must, by some mechanism, break and become turbulent. The resulting mixing 4

creates lateral density gradients that drive ocean currents, both local, and on the scale of 5

the global thermohaline circulation. It has been estimated that the deep ocean requires 2 6

TW of energy to maintain the overturning circulation observed, and that this energy can 7

be supplied approximately equally by internal waves forced by the wind and tides (Munk 8

& Wunsch, 1998). Tracking the energy pathways of internal waves generated by winds 9

is difficult because they are spatially variable and sporadic (Alford, 2003; D'Asaro, 10

1995), and accordingly recent attention has focused on understanding the energy balance 11

of tides. 12

generation generation

Surface Tide Surface Tide

low-mode propagation

reflection

wave-waveinteractions -> basin turbulence?

remoteturbulence

localturbulence

propagation

a) Time = 4 tidal periods

b) Time = 12 tidal periods

13 Figure 1: A numerical simulation of internal tide generation from a steep ridge (at x=0 km), and 14 impacting a remote continental margin (at x=1300 km). Velocity is colored. The upper panel is early 15 in the simulation to show the propagation of low-mode internal tides across the basin. The low-16 modes travel faster than the high-modes, which make up the “ray”-like character of the velocity 17 closer to the ridge. Eventually these rays would fill the whole basin if the resolution of the model 18 permitted. The full solution shows the interaction of the propagating and reflected waves, and 19 indicates the turbulence that would be observed at the remote continental slope. 20 21

Though simpler than wind-driven waves, the energy balance of internal tides is 22

still quite complex (Figure 1). Internal tides are generated when the surface tide pushes 23

density-stratified water over topography, creating internal pressure gradients that drive 24

3

the waves. In the bounded ocean, the waves propagate away from the topography in 1

vertical “modes.” The lowest mode has longest vertical wavelength, the strongest 2

velocities at the surface and seafloor, and a null in the middle of the water-column, and 3

propagates with a horizontal phase speed that is faster than higher modes. This can be 4

seen in Figure 1a, early in a numerical tidal simulation, where the low-modes have 5

traveled almost 500 km from the ridge at x=0, but the more complicated “high-mode” 6

structure is just starting to form near the ridge. The high modes may break into 7

turbulence, either locally where they are generated, at remote topography (Figure 1b, 8

Nash et al., 2004), or by wave-wave interactions in the interior (Henyey et al., 1986; 9

Polzin, 2009; MacKinnon & Winters, 2005). Low modes can reflect, interfere with one 10

another, and scatter into higher modes, though their fate is not yet well understood. 11

An important factor in the study of topographic interactions with internal waves is 12

the steepness of the topography (Garrett & Kunze, 2007) As seen emanating from the 13

topography in Figure 1, internal waves organize their energy along “beams” of energy 14

that propagate at characteristic angles that depend on the frequency of the wave and the 15

density stratification. If the topography is gentler than these characteristic slopes, it is 16

said to be “subcritical;” if steeper, it is said to be “supercritical”. Of course real 17

topography has regions of both, but the extremes are useful because mathematical 18

predictions of internal tide generation can be formulated for either assumption (for 19

subcritical topography see (Balmforth et al., 2002; Bell, 1975); for supercritical 20

topography see (Llewellyn Smith & Young, 2003; St. Laurent et al., 2003). 21

At regions of subcritical topography, such as the Brazil Basin, the generation of 22

internal tides is relatively linear. The nonlinear interaction of the waves cascades energy 23

to higher wavenumbers until they break, leading to enhanced dissipation above the sea-24

floor (Polzin et al., 1997; St. Laurent et al., 2001). This mechanism has been numerically 25

simulated recently (Nikurashin & Legg, 2011) and has been parameterized by Polzin, 26

(2009) based on the topography, the surface tide forcing, and the density stratification. 27

Using these approaches, a modest fraction of the locally generated energy dissipates near 28

such small amplitude but rough topography (about 30%, St.\ Laurent & Nash, 2003), and 29

the rest is believed to radiate away as low-modes. These approaches agree with available 30

4

observational evidence, though more observations and numerical tests of this process are 1

warranted. 2

This paper reviews our efforts to understand the other extreme of “supercritical” 3

topography, when the topography is relatively steep compared to the internal tide rays. In 4

this case, energy is believed to “break” directly near the topography due to non-linearities 5

in the generation process itself. This short-circuits the energy cascade we associate with 6

waves emanating from subcritical topography, and leads to spectacular localized 7

turbulence. This paper reviews observations of these local breaking processes, and 8

documents initial efforts to model and develop simple parameterizations of them. The 9

observed processes are very non-linear, and thus very sensitive to small changes in local 10

forcing, so a large number of caveats come into play before we fully understand 11

dissipation at supercritical sites. 12

Time on October 06, 2002

DEP

TH [m

]

04:00 05:00 06:00 07:00 08:00 09:00 10:00 11:00

400

450

500

550

600

650

700

750

¡ [m2 s−3]

−10

−8

−6

13 Figure 2: Observations of breaking waves made at Hawaii, on the ridge crest between Oahu and 14 Kauai (modified from Klymak et al 2008). Contours are density surfaces versus time during an 15 onslope phase of the tide. Colors are turbulence dissipation rate, a measure of turbulent strength, as 16 inferred from the size of the breaking overturns in the waves, which almost reach 250 m tall near 17 06:00. 18 19

Observations 20

Breaking of internal waves at abrupt topography has now been observed at a number of 21

locations, but the mechanism leading to this breaking has remained elusive. During the 22

Hawaiian Ocean Mixing Experiment, at the Hawaii Ridge, it was noted that there was 23

strong turbulence near the seafloor extending hundreds of meters into the water column 24

5

(Figure 2; Aucan et al., 2006; Klymak et al., 2008; Levine & Boyd, 2006). These 1

motions drove breaking internal waves that were up to 200 m tall, and dissipation rates up 2

to four orders of magnitude greater than the open-ocean values at those depths. 3

Considering an event in detail (Figure 2), the turbulence tends to be at the sharp 4

leading face of rising density surfaces (isopycnals). This leading edge breaks, in this 5

example in two patches (centered at 0500 and 0630). During the relaxation of the tide, 6

there is a rebound at mid depth (0700), and sharp oscillations deeper (though these only 7

have modest turbulence). In all, the strong turbulence lasts over three hours; the 8

remaining nine hours of the tidal cycle are relatively quiescent. 9

The main findings from the observations cited above at Hawaii were: 10

1. Breaking waves were phase-locked to the surface tide forcing. 11

2. Turbulence dissipation rates (a measure of the strength of the turbulence) scaled 12

cubically with the spring-neap modulation of the surface-tide forcing. 13

3. The turbulence that had these characteristics was confined to within a few 14

hundred meters of the seafloor; turbulence further aloft only varied weakly with 15

the tidal forcing. 16

Similar observations have been made at the Oregon slope (Figure 3, Nash et al., 17

2007) and the South China Sea Continental Slope (Klymak et al., 2011). The dissipations 18

at these locales were less that at Hawaii, but again depend on the tidal forcing. The 19

phenomenology along these slopes is complicated by both rougher topography, and the 20

presence of shoaling internal tides that originated from elsewhere. At the Oregon slope, 21

the strongest turbulence is during the transition from onslope to offslope flow, in contrast 22

to the ridge crest at Hawaii where the strongest turbulence is during the onslope flow. 23

The difference in the timing of the turbulence likely depends on the local curvature of the 24

topography and the mooring location relative to local generation sites, though a full 25

explanation of the phenomenology on complex topography has not yet been 26

accomplished. 27

6

1 Figure 3: Breaking waves on the Oregon continental slope. Driven by a combination of barotropic 2 and baroclinic tides that flow over small-scale bumps (of order 250-1000 m horizontal scale) that 3 overlay the otherwise steep continental slope, these waves occur on the leading edge of the down-4 slope phase of the tide. (modified from Nash et. al. 2007). Upper panel is turbulence dissipation rate 5 with isopycnals overlain; middle panel is isopycnal strain, measuring how stretched or compressed 6 parts of the water column are by the internal tide; bottom pane is the upslope semi-diurnal tidal 7 velocities. The velocity shows downward phase propagation with time, indicating upwards 8 propagation of energy from the sea floor. 9 10

A recent set of observations, were undertaken at Luzon Strait between Taiwan 11

and Luzon Island (Figure 4). Luzon Strait is unique in that it consists of two ridges, a 12

short one to the west, and a tall one to the east. It also is somewhat unique because the 13

daily (diurnal) tide is almost as strong as the twice-a-day (semi-diurnal) tide. Preliminary 14

results demonstrate a strong concentration of turbulence at the crest of these two ridges. 15

Interestingly, despite the strength of the diurnal tide, the turbulence at some locales is 16

most strongly phased with the semi-diurnal forcing, both in terms of the location of the 17

turbulence in the internal wavefront, and with the spring-neap cycle (Alford et al., 2011). 18

Preliminary calculations indicate that this is because the two ridges are a distance apart 19

7

that promotes resonance of the semi-diurnal internal tide between the ridges, and 1

dissonance for the diurnal frequency (Buijsman et al., 2012; Echeverri et al., 2011; Jan et 2

al., 2007). This leads to strong semi-diurnal signals like those shown in Figure 4. 3

4 Figure 4: Observations of breaking waves taken in Luzon Strait, between Taiwan and the 5 Philippines. These observations are taken from a mooring with many small temperature sensors 6 placed at the crest of a short ridge that is approximately half a mode-1 internal tide wavelength from 7 the main ridge. The response at this ridge is very dramatic, enhanced by a near-resonant response 8 between the two ridges. The upper plot is the barotropic forcing; the middle is the temperature time 9 series at the mooring; the bottom is the turbulence dissipation rate inferred from the size of breaking 10 overturns in the flow. Note the almost diagonal interleaving of warm and cold water starting at 11 about 0800. 12 13

Two-dimensional Modeling 14

Observations of dissipation near supercritical topography are quite difficult to link 15

to the generation mechanism. The turbulence varies by orders of magnitude over the tidal 16

cycle, occupations must last at least a tidal cycle in order to get an accurate mean. 17

8

However, observations from only one location are hard to relate to a turbulence 1

generation mechanism. To help with this interpretation problem, we have turned to 2

numerical modeling at relatively high resolutions compared to most tidal models, but still 3

coarse resolution compared to the turbulence (Klymak & Legg, 2010; Legg & Klymak, 4

2008). To date, we have simplified the problem by considering two-dimensional flows, 5

but recent work is extending these efforts to three dimensions. By choosing an 6

appropriate tidal forcing, stratification, and driving it over a two-dimensional topography, 7

be it realistic or idealized, a picture emerges of the phenomenology of these breaking 8

waves. 9

10 Figure 5: Simulated turbulence dissipation at a two-dimensional Hawaiian Ridge. Here the vertical 11 resolution is 15 m, and the horizontal 150 m, and the surface tide forcing 0.06 m/s in the deep water, 12 and 0.26 m/s over the ridge. The barotropic tide at ridge crest indicated with arrows and in text 13 under the time relative to peak left-going flow. Large breaking waves, over 200 m tall, set up just 14 after peak off-ridge flow, and then propagate across the ridge as the tide reverses direction. Note 15 that the peak turbulence is after peak flow, as it takes a finite amount of time for the lee wave to 16 grow. 17 18

9

Phenomenology 1

An example simulation (Figure 5) meant to represent flow over the Hawaiian 2

Ridge at Kauai Channel, shows the lee waves we believe are the primary turbulence-3

producing phenomena active at super critical topography. During strong off-ridge flow 4

(Figure 5b) a non-linear lee wave grows, and is arrested at the sill, reaching heights 5

exceeding 200 m, and widths over a few kilometers (Figure 5c). As the tide relaxes, the 6

wave is released and propagates on-ridge (Figure 5d). A similar structure forms on the 7

other flank during the strong flow in the other direction. The shock-like character of this 8

wave can be seen on its off-ridge edge, where the flow sharply rebounds analogously to 9

the flow over an obstacle in a stream. 10

The simulation indicates why the site of observation can change what phase of the 11

tide the turbulence can occur on. If observations are made downslope of the topographic 12

break (say x=3 km, Figure 5), then the turbulence will arrive with offslope flow. 13

Conversely, if the observations are upslope (say x=1 km, Figure 5), then the turbulence 14

will arrive during onslope flow as the trapped wave propagates on-ridge. At a simple 15

topography like the Hawaiian Ridge, this process is quite clear. At more complex 16

bathymetry like the Oregon Slope (Figure 3), the phasing is probably very dependent on 17

the exact placement of the observations relative to smaller scale changes in the 18

bathymetry. 19

A fine-resolution simulation details what a typical lee wave looks like (Figure 6). 20

The simulations are very reminiscent of observations in coastal fjords (Farmer & Armi, 21

1999; Klymak & Gregg, 2004) and simulations made in the atmosphere (Scinocca & 22

Peltier, 1989), except that this motion only takes up 300 m of a 2000 m deep water 23

column, and thus is a lot more “high-mode” than motions in those cases. A strong 24

downslope flow accelerates moving down the slope, but density layers peel off 25

sequentially with depth, driving structures that look like breaking hydraulic jumps. The 26

remainder of the jet separates part way down the slope, at the depth where it hits ambient 27

water, and rebounds with strong vertical oscillations, again reminiscent of a hydraulic 28

jump. 29

10

1 Figure 6: Detailed simulation of the breaking lee wave during peak off-ridge flow (this time to the 2 right). Here the vertical and horizontal resolution of the numerical model is 3 m, and the simulation 3 is non-hydrostatic. Both panels have a one-to-one aspect ratio, and the crest of a smooth Gaussian 4 topography is at 1000 m depth in a 2000-m deep water column. a) is temperature (proxy for density), 5 where large 200-m breaking waves can be seen both in the downslope flow zone, and where the 6 strong off-ridge flow separates from the topography. Note that the color has been shaded by the 7 vertical derivative of the temperature in order to accentuate the turbulent structure of the flow. b) 8 The cross-ridge velocity (red is off-ridge); the teal contours are temperature for comparison with (a). 9 Again, the velocity has been shaded with the first difference in the vertical to accentuate turbulent 10 structure. 11

Non-linear waves arrested at topography have a comprehensive literature for 12

situations where the flow is strong and the obstacle relatively short (both compared to the 13

strength of the stratification). Such flows tend to make waves that are “low mode” or 14

large compared to the water depth (Baines, 1995). For deep-ocean ridges, the flow is 15

weak compared to the height of the topography, and the vertical wavelength of the 16

breaking waves quite short compared to the water depth. It can be shown, however, that 17

the physics is the same (Klymak et al., 2010): if there is an off-ridge flow, the vertical 18

mode with a horizontal phase speed close to that of the flow speed at the ridge crest is 19

arrested. The size of this wave was predicted from internal wave theory, and compared 20

to the size of waves generated in numerical simulations, with very good agreement. The 21

time it takes for these waves to form depends on the topographic slope, with steeper 22

slopes launching waves faster than gentle slopes. For an oscillating flow such as the 23

tides, the finding is that the topography needs to be twice as steep as the internal wave 24

11

slopes for oscillating lee waves to be effectively trapped and dissipative (Klymak et al., 1

2010). 2

Recipe for turbulence at supercritical topography 3

Here we describe how the “arrested lee wave” concept allows for a prediction of 4

tidal dissipation and mixing near surpercritical topography based on the forcing, a 5

simplified bathymetry, and the ocean’s stratification. First, a linear theory can be used to 6

predict the internal response of tidal flow over sharp topography (Llewellyn Smith & 7

Young, 2003; St. Laurent et al., 2003). The resulting flow field has very sharp “beams” 8

of energy emanating from the ridge, and well-defined large-scale phase changes in the 9

velocity on either side of these beams. This theory predicts an internal wave energy flux 10

away from the topography, in each of the vertical modes (dashed line Figure 7d). 11

Non-linear numerical simulations of the same flow demonstrate similar features, 12

particularly on the large scales (Figure 7b and c). However, the “beams” in the 13

simulations are more diffuse, and some energy leaks out in higher-frequency (steeper) 14

beams. As the forcing is increased (compare Figure 7b to c) the beams become even 15

more diffuse. Diffuse beams indicate a loss of high-mode energy from the system, and 16

this is quantified by noting that the energy flux at high modes does not agree with the 17

linear theory (solid lines, Figure 7d and e). The stronger the forcing, the lower the mode 18

where the disagreement starts, indicating a larger fraction of energy is dissipated. 19

To arrive at a dissipation rate, the recipe uses the observations above to determine 20

the modes that dissipate. The strongest flows at the top of the topography are compared to 21

the phase speed of the modes. Fast modes are assumed to escape and form the radiated 22

signal; slow modes are assumed to be trapped and form the dissipation. The stronger the 23

forcing, the more modes are trapped (Klymak et al., 2010). When compared to numerical 24

simulations, this parameterization is quite successful over a range of tidal forcing, 25

bathymetry shape, stratifications, and latitudes (the Coriolis parameter strongly affects 26

the energy put into the waves), so long as the topography is sufficiently supercritical 27

(Figure 7f). 28

12

−20 −10 0 10 20

b) Uo = 4 cm/s

−20 −10 0 10 20

X [km]

c) Uo = 20 cm/s

−20 −10 0 10 20

0

500

1000

1500

2000

a) Linear Theory

DE

PT

H [m

]

U/U

o−0.1 0 0.1

10 20 30

10−6

10−4

10−2

Mode number

Fn/F

BT

d) Normalized Energy Flux

4 cm/s

20 cm/s

Linear

10 20 30

Fn/F

theory

e) Flux compared to Theory

Mode number

10−2

100

102

10−2

100

102

Dnumerical

[kW/m]

Dpara

mete

rized [

kW

/m]

f) Parameterization

h= 1700 m

1500 m

1000 m

500 m

u0= 0.08

0.16 m/s

0.24 m/s

1 Figure 7: a) snapshot of baroclinic (depth-mean removed) velocity from a linear generation from a 2 500-m tall knife-edge ridge in 2000 m of constant-stratification water, normalized by the barotropic 3 forcing velocity. b) Snapshot of the same configuration when the tidal forcing is 0.04 m/s, and c) 4 when it is 0.2 m/s. Note that the stronger forcing has less well-defined “beams” of energy radiating 5 from the topography than the weaker, more linear, forcing. d) Comparison of the energy flux 6 distributed by vertical mode number for a)-c) normalized by the energy flux of the barotropic wave. 7 The 0.2 m/s case rolls off at lower mode numbers than the 0.04 m/s, as also shown in e). f) The 8 parameterization of Klymak et al (2010) predicts this fraction of energy as a function of forcing, 9 topographic height, and stratification, usually within a factor of 1.5. 10

An important finding of this analysis is that the local dissipation, while 11

spectacular and locally important, is still a modest fraction of the energy removed from 12

the surface tide. This can readily be discerned from Figure 1 or Figure 7, in which the 13

full-water column motions that escape the ridges have significant energy. In fact, it 14

requires very strong and non-linear forcing for the local dissipation to even approach 15

10% of the energy budget at these sills. Thus, supercritical ridges are believed to be quite 16

efficient radiators of energy, at least in the idealized forms considered so far in the 17

13

modeling. A rough energy budget based on observations collected near Hawaii 1

corroborates that idea (Klymak et al., 2006). 2

More Complicated Systems 3

Since supercritical topography seems to be efficient at generating internal tides 4

without much local loss, the question stands as to the fate of that energy. Low-modes 5

will impact remote topography where they will scatter, reflect, or dissipate. If that 6

topography is supercritical, such as at other underwater ridges or continental shelves, the 7

same physics discussed above appears to apply: cross-topography flow generates 8

turbulent lee waves that are phase locked with the remote forcing, and a very similar 9

parameterization is very effective at predicting the dissipation in these lee waves 10

(Klymak et al, 2012 in review). Predicting the turbulence at a remote ridge can become 11

quite complicated if there is local generation, and an incoming wavefield (Kelly & Nash, 12

2010). The phase between the local and remote forcing can change the dissipation 13

predicted by an order of magnitude, as lee waves are either suppressed or enhanced by 14

the resonance. 15

Nowhere is this effect more clear than in a two-ridge system like Luzon Strait. In 16

two-dimensions, a two-ridge version of the linear model can be constructed that shows 17

strong interactions between the two ridges. The two ridges must be treated as a system, 18

and the relative heights of the ridges compared to the water depth, and the distance 19

between them compared to the gravest internal mode wavelength, greatly affect the 20

response. The potential for resonance means that a larger fraction of the internal tide 21

might go into local turbulence than for isolated ridges. Recent estimates of the 22

dissipation for Luzon Strait range from 40-20% (Alford et al., 2011; Buijsman et al., 23

2012). 24

Discussion 25

So far we have seen that the internal tide at abrupt topography often breaks, short 26

circuiting the process of cascading internal wave energy that is often envisioned to fuel 27

near-bottom turbulence over regions of small-scale topographic roughness (Polzin, 2009; 28

St. Laurent et al., 2002). This local breaking is vigorous, producing turbulent events that 29

reach hundreds of meters tall, and orders of magnitude greater than the open-ocean 30

turbulence. The breaking is also phase locked to the tide, and strongly dependent on the 31

14

strength of the forcing. Numerical modeling indicates that a source of this turbulence at 1

many locations are lee waves arrested at the topographic breaks during off-ridge tidal 2

flow, either driven by the local tide, or by remote tides. We have a parameterization for 3

this process that works under a large range of forcing and topographies. 4

This progress in understanding the lee wave process does not preclude the 5

importance of other processes that perhaps do not manifest themselves in idealized two-6

dimensional models. As indicated by (MacKinnon & Winters, 2005) and (Simmons, 7

2008), there is the potential for wave-wave interactions, particularly equatorward of the 8

critical latitude where the Coriolis frequency is half the tidal frequency. There is also 9

likely wave-wave interactions local to the Hawaiian Ridge, as indicated in observations 10

(Carter & Gregg, 2006), and as we have seen in our numerical models (Klymak et al., 11

2010). However, it remains to be seen how important this energy pathway is near 12

topography, and there are indications that it is modest in observations of the radiated tide 13

(Alford et al., 2007). 14

There are a number of outstanding questions about the lee wave process, and 15

dissipation near supercritical topography. The first, is how good are estimates of the 16

turbulent dissipation in these processes? There have been very few direct estimates of 17

turbulence in these breaking waves; instead the size of the wave is compared to the 18

stratification it encompasses, and a dissipation rate inferred following (Thorpe, 1977). 19

Detailed comparison between microconductivity and the size of the breaking waves at 20

Hawaii had encouraging results (Klymak et al., 2008), as did comparisons between shear 21

probes and breaking waves in Knight Inlet (Klymak & Gregg, 2004), but direct 22

observations of turbulence dissipation at the microscale in these large mid-ocean lee 23

waves are still needed. Similarly, questions arise as to the efficiency of the mixing in 24

these breaking waves; i.e. what fraction of energy goes towards mixing density rather 25

than turbulent friction processes. Some of these questions will be answered by 26

forthcoming observations, particularly those in Luzon Strait. Others will require more 27

detailed modeling, likely at the scale of large-eddy simulations, though these will be 28

particularly challenging given the broad scale of the internal tides (100s of kms), and the 29

small eddy sizes (a few meters). 30

15

Much work to date, as described here, has been on two-dimensional idealizations 1

of the flow. This is almost certainly an over simplification, and three-dimensional 2

models of these problems are being run (Buijsman et al, personal communication). 3

Funneling of flow by constrictions can greatly enhance the lee waves, so deciding on an 4

appropriate forcing and topography when applying the simple parameterizations 5

discussed above is not trivial. Similarly, we have considered flow that is largely 6

supercritical, except at the ridge crest. However, a lot of topography is more complicated 7

than this, with significant roughness that can interact with the tides, or substantial near-8

critical regions that can very efficiently convert tidal flows into turbulence (Eriksen, 9

1982; McPhee & Kunze, 2002). 10

Despite these substantial caveats, we are attempting to create a global map of 11

dissipation due to local dissipation at supercritcal topography (indicated in Figure 8). 12

There are estimates of tidal currents, stratification, and topography, so this problem 13

should be tractable. To make progress, there are a number of challenging assumptions 14

that need to be made when applying the parameterization suggested here globally: what 15

scale should the topography be smoothed over to decide on its height? What tidal 16

velocity should be used? How to account for the funneling of flows through 17

constrictions? How to account for narrow topography that may not see substantial 18

generation (Johnston & Merrifield, 2003)? Finally, if incoming internal tides from afar 19

are important for local turbulence, either enhancing or suppressing it, then accurate maps 20

of global internal tidal energy, perhaps at low modes, will be needed before a global 21

estimate can be made (Kelly & Nash, 2010). This will prove challenging as tides are 22

advected by large-scale currents and changes in stratification (Rainville & Pinkel, 2006), 23

so it is possible global coarse internal tide models will need to be used to quantify this 24

remote forcing (Simmons et al., 2004). 25

16

1 Figure 8: Regions of the global ocean with supercritical topography with respect to the semi-diurnal 2 tide (colored red). This image is smoothed to half a degree for presentation, and some isolated 3 supercritical regions exist elsewhere, but the major sources are indicated, i.e. the Hawaiian Ridge, 4 Luzon Strait, Aleutian Ridges, the SW Pacific, and the continental slopes all show up prominently. 5

6

Acknowledgments. 7

The observations presented here were part of the Hawaiian Ocean Mixing Experiment 8

(NSF) the Oregon Continental Slope Experiment (NSF), and the Internal Waves in Straits 9

Experiment (ONR). The modeling and parameterization were supported by the Office of 10

Naval Research grants N00014-08-1-0376, N00014-08-1-1039. Oregon Slope 11

observations were supported by NSF grants OCE-0350543 and OCE-0350647. Luzon 12

Strait observations were supported by the Office of Naval Research under Grants 13

N00014-09-1-021, N00014-09-1-0273, N00014-09-1-0281, N00014-10-1-0701, and 14

N00014-09-1- 0274. We thank Eric Kunze, Chris Garrett, Jennifer MacKinnon, and Sam 15

Kelly for the many helpful discussions. 16

17

Jody M. Klymak (jklymak@uvic.ca) is Assistant Professor, School of Earth and 18

Ocean Sciences, University of Victoria, Canada. Sonya Legg, is Research 19

Oceanographer, Atmospheric and Ocean Sciences Program at Princeton 20

17

University, Princeton, NJ, USA. Matthew H. Alford is Senior Oceanographer and 1

Associate Professor, Applied Physics Laboratory, University of Washington, 2

Seattle, WA, USA. Robert Pinkel is Professor and Associate Director, Marine 3

Physics Laboratory, Scripps Institution of Oceanography, La Jolla, CA, USA. 4

Jonathan D. Nash is Associate Professor, College of Earth, Ocean and 5

Atmospheric Sciences, Oregon State University, Corvallis, OR, USA. Maarten 6

Buijsman is Postdoctoral Research Scientist, Atmospheric and Ocean Sciences 7

Program at Princeton University, Princeton, NJ, USA. 8

9

Bibliography 10

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