Virtual Seafloor Reduces Internal Wave Generation by Tidal Flow

Likun Zhang* and Harry L. Swinney†

Department of Physics and Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712, USA(Received 19 August 2013; published 11 March 2014)

Our numerical simulations of tidal flow of a stratified fluid over periodic knife-edge ridges and randomtopography reveal that the time-averaged tidal energy converted into internal gravity wave radiation arisesonly from the section of a ridge above a virtual seafloor. The average radiated power is approximated by thepower predicted by linear theory if the height of the ridge is measured relative to the virtual floor. Theconcept of a virtual floor can extend the applicability of linear theory to global predictions of the conversionof tidal energy into internal wave energy in the oceans.

DOI: 10.1103/PhysRevLett.112.104502 PACS numbers: 47.35.Bb, 47.55.Hd, 92.10.hj, 92.10.Hm

The oceans are stratified as a consequence of decreasingtemperature and increasing salinity with depth. Buoyancyprovides a restoring force for density perturbations that areproduced, for example, by tidal flow over bottom topog-raphy [1–3]. Tidal flow over topography produces internalgravity waves, called internal tides [4,5], which play a rolein ocean mixing and circulation [6–8].An understanding of the energy budget of the oceans

requires a determination of the efficiency of conversion oftidal energy into internal gravity wave energy [9–14]. Theconversion rate for topography of small slope and smallheight H in a deep ocean (weak topography) is given bylinear theory, which predicts the main topographic depend-ence as H2 [15]. This H2 dependence has also been foundto describe steep isolated ridges in the deep ocean, exceptfor a prefactor that is about twice as large as for smallslopes [16–20]. In contrast, recent studies of periodic steeptopography suggest that wave interference between neigh-boring ridges can suppress the H2 dependence [21–23].Complex topographies can also suppress tidal conversion[24] and generate wave attractors [25]. Thus, considera-tions of weak or isolated topography are not sufficient forpredicting tidal conversion in the deep ocean [26].Here we examine in numerical simulations how wave

interference suppresses tidal conversion for periodic knife-edge and random topography, and we explore an extensionof linear theory to such topographies. The simulations areconducted for tidal flow of a uniformly stratified fluid withconstant buoyancy frequency, N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−ðg=ρ0Þðdρ=dzÞp

[with ρðzÞ the vertical density distribution, g, the gravita-tional acceleration, ρ0, a reference density]. The internalwaves radiated by the topography have a beam slope(relative to the horizontal direction) [1–3],

SIW ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1 − ðf=ωÞ2�=½ðN=ωÞ2 − 1�

q; (1)

where ω is the tidal frequency and f is the Coriolisparameter. We vary SIW for fixed topography and examinethe wave drag and the time-averaged radiated energy flux.

We find that for both periodic and random topographies, thetime-averaged internal wave power is generated only abovean elevated virtual seafloor between neighboring ridges,and this reduces the tidal conversion.Methods.—We conduct two-dimensional direct numerical

simulations of the Navier-Stokes equations in the Boussinesqapproximation using the CDP-2.4 code (subgrid modeling isdisabled) [27], modified to include buoyancy effects [28] andvalidated in studies of internal waves [28–31]. Tidal flow isproduced by adding to the momentum equation a horizontalbody force,FðtÞ ¼ ρ0U0ω cosðωtÞ, where ρ0 ¼ 103 kg=m3,U0 ¼ 0.14 cm=s, and ω ¼ 1.4 × 10−4 rad=s (the M2 tidalfrequency).Theresultantbarotropic tidecloselyapproximatesa horizontal oscillating flow UðtÞ ¼ U0 sinðωtÞ with tidalexcursion A ¼ U0=ω ¼ 10 m, sufficiently small to avoidoverturning and turbulence. We neglect rotation, i.e., f ¼ 0.A periodic array of knife edges is modeled by narrow top

hats of height H ¼ 100A and width W ¼ 1.6A, separatedby L ¼ 200A; some computations are also made withðW;H; LÞ=A ¼ ð1.6; 50; 150Þ and (1.6, 50, 200). Also,random topography is generated from an ensemble ofGaussian random processes with a spectrum, SðnÞ ¼4H2

0ð42 þ n2Þ−5=2 (with mode number n ≤ 32) [16], assuggested by statistical modeling of the seafloor [15,32].We choose L ¼ 1000A and heightH0 ¼ 33A, which yieldsa topographic peak-to-peak height H ¼ 100A.The beam slope SIW is changed over a wide range by

varying stratification N, which leads to a correspondingvariation of the Froude number, Fr¼U0=NH∼10−2–10−1.The fluid is isothermal with a large viscosity, ν ¼ 10−2 m2=s,to save computational expense, but the viscosity has anegligible effect on the tidal conversion in laminar flow[31]. The Reynolds number is Re ¼ HU0=ν ∼ 100. The saltdiffusivity, κ ¼ 2 × 10−5 m2=s, has a negligible effect for theduration of our simulations because of the large Schmidtnumber, ν=κ ¼ 500.Our computations are for a domain of width −L=2 <

x < L=2 and height 0 < z < H þ 1450A with periodicboundary conditions on the sides. The topography is

PRL 112, 104502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

14 MARCH 2014

0031-9007=14=112(10)=104502(5) 104502-1 © 2014 American Physical Society

located at the bottom with a no-slip bottom boundarycondition; the knife edge is centered at x ¼ 0. To absorbupward-propagating internal waves, a sponge layer abovez ¼ H þ 450A is made by applying a Rayleigh dampingforce (proportional to the velocity deviation from the tidalvelocity) that smoothly increases upward to the topboundary. The knife-edge (random topography) domainhas a structured grid of 1 × 106 (3 × 106) control volumes.The horizontal resolution for the knife-edge domain variessmoothly from Δx ¼ 0.08A at the center to Δx ¼ 0.8A atthe sides; for random topography the resolution is uniform,Δx ¼ 0.625A. The vertical resolution is Δz ¼ 0.2A at thebottom (z ≤ H þ 50A) and gradually stretches to Δz ¼ 2Aat the top boundary. To achieve a steady state, eachsimulation is run for 20 tidal periods with 2000 time stepsper period. Numerical convergence is checked by doubling(halving) both spatial and temporal resolution, which leadsto computed wave power changes smaller than 1% (3%).Virtual seafloor.—The radiated power per ridge com-

puted for a periodic array of knife edges is shown in Fig. 1as a function of the ratio of the valley slope, Svalley≡2H=L,to the wave beam slope. The power is computed from thetime-averaged work rate [33], P ¼ hDðtÞUðtÞi, whereDðtÞ ¼ R

H0 Δp0ðz; tÞdz is the wave drag per ridge,

Δp0ðz; tÞ is the drop of the wave pressure p0 across theknife edge from x ¼ −W=2 to x ¼ W=2, and the bracketsdenote the average over a tidal cycle. For internal wavebeams (Svalley=SIW ≪ 1), the power per ridge P approachesthat radiated by a single isolated ridge in the deep ocean,Pisolated ¼ 1

4πρ0U2

0H2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN2 − ω2

p[17]. For shallow internal

waves, constructive and destructive interference leads tooscillations in the power ratio with maxima and minima atSvalley=SIW ¼ 2n − 1 and 2n, respectively (where n is an

integer) [22]. The successive maxima, rather than beinginfinite as in [22], are finite and decay with increasing sloperatio (see also [23]).We find the time-averaged power in the successive

maxima is generated only above an elevated virtual seafloor(cf. dot-dashed lines in Fig. 1). To identify the virtual floorassociated with the conversion of tidal energy into waveenergy, we compute the power input by the tides over thepart of a ridge below a height z from the wave drag,PinðzÞ≡ hDðz; tÞUðtÞi, with Dðz; tÞ≡ R

z0 Δp0ðζ; tÞdζ. For

Svalley=SIW ¼ 1, the power PinðzÞ increases from zero at theocean bottom (z ¼ 0) to the total power at the top of theridge (Fig. 2, top panel in the center). For the two lowerpanels, the power drops to zero at the virtual floor z ¼ z0,like that at z ¼ 0, while above the virtual floor, the powerincreases with the height until reaching the total powerat z ¼ H.The converted tidal energy is radiated by internal waves

with a time-averaged energy flux Φ ¼ hp0u0i (Fig. 2,left-hand panels), where the wave velocity u0ðx; z; tÞ is thedifference between the fluid velocity uðx; z; tÞ and thebarotropic tidal velocity ubaroðx; z; tÞ [33]. The radiatedwave power crossing a horizontal plane, PoutðzÞ¼R L=2−L=2Φzðx;zÞdx, as shown by the red curves (center panels),is nearly equal to the input power PinðzÞ for z ≤ H; the smalldifference arises from local dissipation. The upward radiationstarts from the virtual floor, as no time-averaged energycrosses this floor.The tidal conversion depends on the phase ϕðzÞ of the

pressure drop Δp0 relative to the tidal flow UðtÞ. The phasefactor cosϕðzÞ is shown in the right-hand panels of Fig. 2.For the first maximum (top), cosϕðzÞ > 0 along the wholeridge surface; thus, the entire ridge surface acts as a sourcewith the time-averaged energy flux radiating away from theridge (left panels). For higher maxima (two lower panels),cosϕðzÞ is positive above the virtual floor, while below thevirtual floor cosϕðzÞ is alternately positive or negativealong the surface; correspondingly, the time-averagedenergy flux effectively radiates away from or towardsthe ridge (left panels).The successivepanels from top tobottom inFig.2wouldbe

equivalent if the part below the virtual floors was deleted andthe height of the ridge above the virtual floor, ðH − z0Þ ¼H=ð2n − 1Þ, was scaled to the full heightH. Thus, the powerin successive maxima is P=Pisolated∝ ½ðH−z0Þ=H�2¼1=ð2n−1Þ2, as confirmed by the normalization ð2n − 1Þ2 ofthe flux and power ratio in Fig. 2 (except for small differencesdue to dissipation and boundary layers).Avirtual floor, where the tidal conversion below that floor

is zero, exists for general values of Svalley=SIW, not only forthe power maxima. This virtual floor is close to the levelwhere downward beams generated by neighboring ridgesfirst intersect (for Svalley=SIW ≥ 1; see sketch in Fig. 3),

z0=H ¼ 1 − ðSvalley=SIWÞ−1; (2)

0 1 2 3 4 5 60

1

2

FIG. 1 (color online). The radiated internal wave power perridge for a periodic array of knife edges exhibits peaks (bluediamonds) due to constructive interference, but the successivepeaks are suppressed as a consequence of an elevated virtualseafloor (horizontal dot-dashed lines); the radiated power orig-inates only above this floor. The peaks are located at slope ratiosSvalley=SIW ¼ 1, 3, 5, where SIW is the slope of the wave beams(red lines). The peaks have relative values 1∶1=32∶1=52 when theradiated power is normalized by the power radiated by an isolatedknife edge. The dashed lines in the inset illustrate the valley slopeSvalley, which is unity for the data plotted.

PRL 112, 104502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

14 MARCH 2014

104502-2

while forSvalley=SIW ≤ 1, z0=H ¼ 0. As a result, the powerPoscillates about the power P0

isolated radiated by a singleisolated ridge with height (H − z0); see Fig. 3. The threesets of data in Fig. 3, corresponding to different Re and Fr,agree except at the maxima there is a few percent differenceassociated with nonlinearity. Further, the average ofP=P0

isolated between successive maxima is about 1=2 [34],in accord with the result from linear theory for a seafloorshifted from the real one at z ¼ 0 to the virtual one at z ¼ z0.Virtual floor model and random topography.—The

concept of a virtual floor can extend the applicability oflinear theory for weak topography to arbitrary topography.Internal wave beams can be constructed to obtain a virtual

floor between each pair of adjacent topographic ridges.Then linear theory can be applied to the topography abovethe virtual floor to predict the conversion rate. The lineartheory for weak topography gives the total wave powerconverted by tidal flow of an uniformly stratified fluid inthe deep ocean [15],

Plinear ¼ ðρ0U20ωÞ½ðN=ωÞ2 − 1�1=2½1 − ðf=ωÞ2�1=2gtopo; (3)

where the topographic dependence gtopo ¼ ðL=2πÞ ×R∞0 kj ~hðkÞj2dk is computed from the Fourier transform~hðkÞ≡ R∞

−∞ expð−ikxÞzðxÞdx of the topography zðxÞ withextent L in the x direction and unit length in the y direction.We illustrate the virtual floor model using sinusoidal

topography, which serves as a Fourier basis for an arbitraryseafloor. For a one-dimensional sinusoidal basis zðxÞ ¼12H cosðkxþ φÞ, the power radiated for each topographic

period (L≡ 2π=k) with small kH is given by linear theoryEq. (3), Plinear ∝ H2. Our virtual seafloor model simplyreplaces the topographic height H with the height (H − z0)measured relative to the virtual floor z0 [cf. Fig. 4(a) inset].In other words, we multiply the energy flux from lineartheory by a weighting function, ð1 − z0=HÞ2. The virtualfloor height is z0=H ¼ 0 for Svalley=SIW ≤ 1, where thevalley slope, Svalley ¼ ð0.3623ÞkH, is computed from thegeometrical definition, Svalley ≡max ½ðzðxÞ − zðxmÞÞ=ðx − xmÞ� with zðxmÞ ¼ min ½zðxÞ�. For Svalley=SIW > 1,the virtual floor height,

z0=H ¼ 1 − ½ϵ −ffiffiffiffiffiffiffiffiffiffiffiffiϵ2 − 1

pþ π − sin−1ðϵ−1Þ�=2ϵ; (4)

[where ϵ ¼ ð1.38ÞSvalley=SIW ¼ 12kH=SIW] follows from a

determination of the point where the wave beam is radiated.To test the proposed use of the virtual floor concept in

linear theory, we compare in Fig. 4(a) our result forsinusoidal topography, P ¼ Plinearð1 − z0=HÞ2, with anexact Green’s function analysis (magenta curve) [23]. Thetwo results agree on the transition at Svalley=SIW ¼ 1, and

0 1 2 3 4 5 60

1

2

FIG. 3 (color online). The sketch on the left illustrates theaverage virtual floor height z0 for an arbitrary beam slope. Whenthe virtual floor is included in the normalization, P0

isolated≡Pisolated½ðH − z0Þ=H�2, the radiated wave power per knife-edgeridge oscillates about unity, and the average value between twomaxima is about 1=2 (horizontal dashed line), in good accordwith linear theory prediction for a ridge of height (H − z0). Thedata points show results for three topographies: Svalley ¼ 1 (blackcircle), 2=3 (red square), and 1=2 (green triangle).

FIG. 2 (color online). The virtual floor at z0 (horizontal dot-dashed lines) for the constructive maxima at Svalley=SIW ¼ 2n − 1(top panels: n ¼ 1; center: n ¼ 2; bottom: n ¼ 3) is illustrated bythe time-averaged energy flux in the left panels (arrows: direction ofthe flux; colors: vertical component of the flux, Φz, whereΦisolated ≡ Pisolated=L), and by the power in themiddle panels (blackcurves: power generated by the tides below height z; red curves:power radiated by internal waves crossing height z). The normali-zation ð2n − 1Þ2 is the squared ratio of ridge height above thevirtualfloor to the full ridge height. The ratio of the ridge height above thevirtual floor to the full ridge height, ðH − z0Þ=H ¼ 1=ð2n − 1Þ,results in a suppression given in the abscissa normalization,ð2n − 1Þ2. The tidal conversion is proportional to cosϕðzÞ (right-hand panels), whereϕðzÞ is the phase of the pressure drop across theridge relative to the tidal flow.

PRL 112, 104502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

14 MARCH 2014

104502-3

both approach the linear theory prediction for Svalley=SIW ≪1 (small kH). For Svalley=SIW > 1 (large kH) the predictionfrom the virtual floor approach agrees (within 5%) with theGreen’s function result for the average power betweensuccessive maxima. The virtual floor prediction yields

P ¼ ðαρ0U20ωÞ½ðN=ωÞ2 − 1�−1=2½1 − ðf=ωÞ2�3=2k−2; (5)

which recovers the independence of H in [22,23]. In Eq. (5)we have used ð1−z0=HÞ2¼αðkH=SIWÞ−2, where α ∈½3; π3=8� follows from Eq. (4) [for periodic knife edges, α ¼π3=8 follows from Eq. (2)]. The suppression in radiatedpower for Svalley=SIW ≫ 1 is far more significant than theincrease around Stopo=SIW¼1 (corresponding to Svalley=SIW ¼ 0.7246 in Fig. 4(a) with the topographic slope Stopo ≡max½dzðxÞ=dx� [16]).The time-averaged radiated internal wave power is

suppressed as a consequence of virtual floors between

adjacent topographic peaks, as Fig. 4(b) illustrates forinternal waves generated by random topography that coversa wide range of slope ratio (Svalley=SIW ∈ ½0; 1.94�,Stopo=SIW ∈ ½0; 3.43�, and SIW ¼ 1). For this small hori-zontal scale and steep topography, the total radiated power,Poutðz ¼ HÞ in Fig. 4(c), is 55% of the power predicted bylinear theory, Plinear in Eq. (3). An average over a largenumber of realizations of random topography wouldprovide an accurate determination of power suppressionby the virtual floor.Discussion.—Our examination of wave drag and energy

flux for periodic knife-edge topography reveals that thetidal conversion into internal wave radiation arises onlyfrom topography that lies above an elevated virtual floor.The average radiated power is found to be given by lineartheory for weak topography with the actual floor replacedby a virtual floor. The virtual floor picture is validated for asinusoidal basis by comparison with a Green’s functionanalysis. The virtual floor is further shown to be applicableto random topography.This study provides insight into the wave interference

that suppresses the conversion of tidal energy for smallhorizontal scales and steep topography. The elevated virtualfloor height z0 for topographic valley slope greater than theinternal wave slope leads to four consequences for theradiated wave power: (i) The power is approximatelyindependent of H because the height (H − z0) measuredrelative to the virtual floor is almost independent of H.(ii) To leading order the power has a k−2 dependence thatrepresents the reduction of the effective ridge heightðH − z0Þ2 when the ridge separation (2π=k) narrows.(iii) The power decreases with increasing stratification Nbecause the virtual floor is elevated when the beam slopebecomes shallower [cf. Eq. (1)]. In contrast, linear theorypredicts the power increases with increasing N [Eq. (3)].(iv) Rotation suppresses the power relative to linear theorybecause increasing the Coriolis parameter f reduces thebeam slope [Eq. (1)] and, hence, elevates the virtual floor.The virtual floor concept can extend the applicability

of linear theory to global random topography. High-resolution data for ocean topography [35] and stratification[36] could be used to map the seafloor to a virtual floor, andthen linear theory could be applied to the virtual floor. Thevirtual floor for small scales, say less than 1 km whereseafloor topographic data are scarce [32], could be includedin the linear theory analysis by multiplying the seafloorspectrum by ½1 − z0ðkÞ=HðkÞ�2 [cf. Eq. (4)]. The virtualfloor approach provides a nonlocal correction to thelinear theory and thus should yield better estimates ofthe global energy flux distribution than the local correctionin [26].In the future the virtual floor approach should be

extended to more realistic oceanic conditions by includingthree dimensionality [28,37–40], finite depth [17,21], andnonuniform stratification [41,42]. The virtual floor concept

10−1

100

101

10−4

10−3

10−2

10−1

100

(a)

(b)

−0.5 0.5 0

1

(c)

0 1

−2 0 2

FIG. 4 (color online). Application of the virtual floor concept tosinusoidal and random topography. (a) For sinusoidal topographyofheight H and virtual floor height z0 (cf. inset), the internal wavepower predicted using the virtual floor concept (black curve), P ¼Plinearð1 − z0=HÞ2, agrees well with the power predicted by aGreen’s function approach (magenta curve, cf. Fig. 2 in [23]), ifthe comparison for Svalley=SIW > 1 is made by averaging betweenadjacent maxima in the Green’s function result. The inset illustratesthevalley slopeSvalley (slopeof dashed line through the lowest point)and the internal wave beams (red lines) radiated from the points (reddots) where the beam slopes equal the mountain slopes. (b)-(c) Forrandom topography the virtual floor is illustrated by (b) the verticalenergy flux Φz, where no time-averaged wave power crosses thevirtual floor between adjacent ridges (horizontal dot-dashed lines),andby (c),where the total power radiated upwardPoutðzÞ starts fromvirtual floor (dot-dashed line) and increases up to the total power atz ¼ H (for larger z the power decreases due to dissipation). Both thepower and the flux are normalized by predictions of linear theory(where Φlinear ≡ Plinear=L).

PRL 112, 104502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

14 MARCH 2014

104502-4

could also be useful in the parametrization of internal wavebreaking and dissipation in systems with multiple steepridges [43–46], abyssal hills [26], rough topography [47],and internal tide scattering [48].

We thank Amadeus Dettner, Matthew Paoletti, andBruce Rodenborn for discussions, and Neil Balmforthfor providing the Green’s function data in Fig. 4(a). Theresearch was supported by the Office of Naval ResearchMURI Grant No. N000141110701. L. Z. acknowledges thesupport of the 2013-14 F. V. Hunt Postdoctoral ResearchFellowship in Acoustics. The computations were done atthe Texas Advanced Computing Center.

*lzhang@chaos.utexas.edu†swinney@chaos.utexas.edu

[1] P. G. Baines, Topographic Effects in Stratified Flows(Cambridge University Press, Cambridge, England, 1995).

[2] V. Vlasenko, N. Stashchuk, and K. Hutter, Baroclinic Tides:Theoretical Modeling and Observational Evidence(Cambridge University Press, Cambridge, England, 2005).

[3] B. R. Sutherland, Internal Gravity Waves (CambridgeUniversity Press, Cambridge, England, 2010).

[4] H. P. Zhang, B. King, and H. L. Swinney, Phys. Rev. Lett.100, 244504 (2008).

[5] B. Gayen and S. Sarkar, Phys. Rev. Lett. 104, 218502(2010).

[6] W. Munk and C. Wunsch, Deep-Sea Res., Part I 45, 1977(1998).

[7] G. D. Egbert and R. D. Ray, Nature (London) 405, 775(2000).

[8] C. Wunsch and R. Ferrari, Annu. Rev. Fluid Mech. 36, 281(2004).

[9] S. R. Jayne and L. C. St. Laurent, Geophys. Res. Lett. 28,811 (2001).

[10] L. C. St. Laurent and C. J. R. Garrett, J. Phys. Oceanogr. 32,2882 (2002).

[11] J. Nycander, J. Geophys. Res. 110, C10028 (2005).[12] E. D. Zaron and G. D. Egbert, J. Phys. Oceanogr. 36, 1019

(2006).[13] N. V. Zilberman, J. M. Becker, M. A. Merrifield, and G. S.

Carter, J. Phys. Oceanogr. 39, 2635 (2009).[14] J. A. M. Green and J. Nycander, J. Phys. Oceanogr. 43, 104

(2013).[15] T. H. Bell, J. Fluid Mech. 67, 705 (1975); J. Geophys. Res.

80, 320 (1975).[16] N. J. Balmforth, G. R. Ierley, and W. R. Young, J. Phys.

Oceanogr. 32, 2900 (2002).[17] S. G. Llewellyn Smith andW. R. Young, J. Fluid Mech. 495,

175 (2003).[18] L. St Laurent, S. Stringer, C. Garrett, and D. Perrault-Joncas,

Deep Sea Res. 50, 987 (2003).[19] F. Pétrélis, S. G. Llewellyn Smith, andW. R. Young, J. Phys.

Oceanogr. 36, 1053 (2006).[20] C. Garrett and E. Kunze, Annu. Rev. Fluid Mech. 39, 57

(2007).[21] S. Khatiwala, Deep Sea Res. 50, 3 (2003).

[22] J. Nycander, J. Fluid Mech. 567, 415 (2006).[23] N. J. Balmforth and T. Peacock, J. Phys. Oceanogr. 39, 1965

(2009).[24] L. R.M. Maas, J. Fluid Mech. 684, 5 (2011); S. G. Llewellyn

Smith, J. Fluid Mech. 684, 1 (2011).[25] P. Echeverri and T. Peacock, J. Fluid Mech. 659, 247 (2010);

P. Echeverri, T. Yokossi, N. J. Balmforth, and T. Peacock,J. Fluid Mech. 669, 354 (2011).

[26] A. Melet, M. Nikurashin, C. Muller, S. Falahat, J. Nycander,P. G. Timko, B. K. Arbic, and J. A. Goff, J. Geophys. Res.:Oceans 118, 6303 (2013).

[27] F. Ham and G. Iaccarino, Annual Research Briefs (Centerfor Turbulence Research, Stanford, CA, 2004), p. 3; F. Ham,K. Mattsson, and G. Iaccarino, Annual Research Briefs(Center for Turbulence Research, Stanford, CA, 2006),p. 243.

[28] B. King, H. P. Zhang, and H. L. Swinney, Phys. Fluids 21,116601 (2009).

[29] B. King, H. P. Zhang, and H. L. Swinney, Geophys. Res.Lett. 37, L14606 (2010).

[30] M. S. Paoletti and H. L. Swinney, J. Fluid Mech. 706, 571(2012).

[31] A. Dettner, H. L. Swinney, and M. S. Paoletti, Phys. Fluids25, 116601 (2013).

[32] J. A. Goff and T. H. Jordan, J. Geophys. Res. 93, 13 598(1988).

[33] S. M. Kelly, J. D. Nash, and E. Kunze, J. Geophys. Res.:Oceans 115, C06014 (2010).

[34] Recall that the linear theory for weak topography predictsa power about half of that radiated by an isolated knifeedge.

[35] K. M. Marks and W. H. F. Smith, Marine Geophys. Res. 27,19 (2006), see also www.gebco.org.

[36] B. King, M. Stone, H. P. Zhang, T. Gerkema, M. Marder,R. B. Scott, and H. L. Swinney, J. Geophys. Res. 117,C04008 (2012).

[37] P. E. Holloway and M. A. Merrifield, J. Geophys. Res.:Oceans 104, 25 937 (1999).

[38] J. R. Munroe and K. G. Lamb, J. Geophys. Res. 110, C02001 (2005).

[39] P. G. Baines, Deep Sea Res. Part I 54, 1486 (2007).[40] B. Voisin, E. V. Ermanyuk, and J.-B. Flór, J. Fluid Mech.

666, 308 (2011).[41] S. G. Llewellyn Smith and W. R. Young, J. Phys. Oceanogr.

32, 1554 (2002).[42] M. Zarroug, J. Nycander, and K. Döös, Tellus 62A, 71

(2010).[43] J. Xing and A. M. Davies, J. Geophys. Res. 116, C04022

(2011).[44] M. H. Alford and Coauthors, J. Phys. Oceanogr. 41, 2211

(2011).[45] M. C. Buijsman, S. Legg, and J. Klymak, J. Phys. Oceanogr.

42, 1337 (2012).[46] J. M. Klymak, M. Buijsman, S. Legg, and R. Pinkel, J. Phys.

Oceanogr. 43, 1380 (2013).[47] M. Nikurashin and S. Legg, J. Phys. Oceanogr. 41, 378

(2011).[48] O. Bühler and M. Holmes-Cerfon, J. Fluid Mech. 678, 271

(2011).

PRL 112, 104502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

14 MARCH 2014

104502-5