JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Topographic coupling of tides on a continental slope:

Internal-tide generation and scattering

S. M. Kelly1, J. D. Nash1, K. I. Martini2, M. H. Alford2, and E. Kunze3

Abstract. Observations of the semidiurnal internal tide on the Oregon slopeindicate 500 W/m-coastline of (i) surface-tide energy loss and (ii) turbulent kineticenergy dissipation. Comparison with numerical simulations suggest the observedinternal tide is a superposition of locally-generated and shoaling internal tides.Here, we derive a new expression for linear tide-topography coupling that includesboth internal-tide generation and scattering. Quantifying these terms, we find thatsurface-tide energy loss and turbulent kinetic energy dissipation are consistent withtwo observed forms of tide-topography coupling. These results suggest shoalinginternal tides are unlikely to survive reflection from continental slopes and tide-topography coupling may contribute significantly to deep-ocean tidal-energydissipation.

1. Introduction

Munk [1966] suggested that tidally-driven mixing inthe deep-ocean may be important for maintaining abyssalstratification. An ensuing challenge has been to identifythe deep-ocean distribution of tidal dissipation. WhileEgbert and Ray [2001] mapped surface-tide energy sinks,most surface-tide energy is lost to low-mode internal-tides[Garrett and Kunze, 2007], which are observed to propagateO(1000 km) [Ray and Mitchum, 1996] before dissipating.

Tides have historically been described as a linearsuperposition of uncoupled normal modes [Wunsch, 1975].However, because the astronomical tide-generating forceonly acts on the surface tide (mode 0) [Hendershott ,1981], some form of modal coupling is required to powerthe observed internal tides (modes n ≥ 1). Zeilon

[1912] demonstrated topographic internal-tide generation inlaboratory experiments, but it was not until Rattray [1960]that the first mathematical description of tide-topographycoupling was published.

More recently, observations [e.g., Rudnick et al., 2003]and global tide models [e.g., Jayne and St. Laurent ,2001] have demonstrated that surface-tide energy lossesin the deep ocean are predominately due to internal-tidegeneration, which can be approximated by linear dynamicswhere surface-tide excursions are small [Garrett and Kunze,2007]. Mathematically, linear internal-tide generation canbe written as a generic expression, C (Section 5.1), whichhas been verified in numerical simulations [e.g., Niwa and

Hibiya, 2001; Kurapov et al., 2003; Carter et al., 2008] andobservations [Kelly and Nash, 2010].

Tide-topography coupling may also explain energylosses from the low-mode internal tides. Theory andobservations suggest that low-mode internal tides aredissipated during reflection and scattering at continentalmargins [Eriksen, 1998; Nash et al., 2004, 2007; Alford

1College of Oceanic and Atmospheric Sciences, OregonState University, Covallis, Oregon, USA.

2Applied Physics Lab, University of Washington, Seattle,Washington, USA.

3School of Earth and Ocean Sciences, University ofVictoria, Victoria, British Columbia, Canada.

Copyright 2010 by the American Geophysical Union.0148-0227/10/$9.00

and Zhao, 2007], but not during propagation in the openocean [St. Laurent and Garrett , 2002; Alford et al., 2007].In addition, numerical simulations have demonstratedthat tide-topography coupling efficiently scatters low-modeinternal tides to higher modes [Johnston and Merrifield ,2003; Johnston et al., 2003], which may subsequentlydissipate locally [St. Laurent and Garrett , 2002; Muller

and Buhler , 2009]. Unfortunately, it has been difficultto quantify the importance of tide-topography couplingwithout a general mathematical expression for internal-tidescattering. In Section 5.1, we derive such an expression,Cm,n, which like C is valid where ever tidal excursions aresmall.

If tide-topography coupling is the primary mechanism forlow-mode internal-tide energy losses, continental margins,which catch much of the radiating internal tide, could bethe predominant sink of low-mode tidal-energy in the deep-ocean. In the present study, we examine observations andnumerical simulations of tidal dynamics on the Oregoncontinental slope at 43.2◦ N (which is steep, rough, turbulent[Moum et al., 2002] and characteristic of western NorthAmerica, Fig. 1). Here, Egbert and Ray [2001] infer500 W/m-coastline of surface-tide energy loss, which weassociated with internal-tide generation (Fig. 2), and Nash

et al. [2007] observe 500 W/m-coastline of tidally-modulatednear bottom turbulent kinetic-energy dissipation, which weassociate with tidal-energy dissipation. In addition, Martini

et al. [2011a] presents evidence that shoaling internal tidestransform into near-bottom turbulent bores at stationsMP3 and MP4 (Fig. 1, b), driving turbulent kinetic-energydissipation.

Despite estimates of net internal-tide generation anddissipation, characterizing internal-tide dynamics on theOregon slope has been difficult because of the simultaneouspresence of locally generated and shoaling internal tides.Examining 40-day mooring records, Martini et al. [2011b]identify intermittent periods of net onshore and offshoreinternal-tide energy fluxes, complex interference patterns,and evidence of both locally-generated and shoaling internaltides.

Here, we complement Martini et al. [2011b], bycomparing observations over a brief 2-day period with 3-D numerical simulations of internal-tide generation andshoaling. The goals of this investigation are to (i) identifythe presence of locally-generated and shoaling internal tides,(ii) quantify internal-tide generation and compare it withobserved surface-tide energy loss, and (iii) quantify the

1

X - 2 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

conversion of tidal energy into high modes and compare itwith observed turbulence.

The remainder of the paper is organized as follows:In Section 2, we present an observational snapshot ofthe semidiurnal internal tide’s structure across the Oregonslope. In Section 3, we present two three-dimensionalnumerical simulations of tidal dynamics on the Oregon slope.One simulation is designed to isolate local internal-tidegeneration, and the other mode-1 internal-tide shoaling. InSection 4, we compare observed and simulated velocities andpressures, looking for evidence of both locally-generated andshoaling internal tides. In Section 5, we derive and evaluateenergy balances that include inter-modal energy conversion.We quantify internal-tide generation and scattering to highmodes, which we compare with surface-tide energy loss andturbulent dissipation, respectively. In Section 6, we estimatethe amplitudes of onshore and offshore propagating internaltides. In Section 7, we review tidal energetics on the Oregonslope and consider the relevance of this study to tides as awhole.

2. Observations

During a spring tide in September 2005, the Oregoncontinental slope, which runs north-south, was sampled inten locations at 43.212◦ N (Fig. 1, a and b). Samplingstations were spaced approximately 3 km apart, betweenthe shelf-break and abyssal plain, to produce a cross-slopepicture of tidal dynamics. Three sampling schemes wereemployed to obtain almost full-water-column measurements[see also Nash et al., 2007; Martini et al., 2011b, a]:

(i) At three locations, McLane Moored Profilers (MPs)continuously traversed the water column, obtaining verticalprofiles of velocity, temperature, and salinity every 3 h orless over a 40-day deployment. (Two additional moorings,MP1 and MP6, are omitted from this analysis because theirwater-column coverage is inadequate for the vertical-modedecomposition described in Section 5.2.)

(ii) At six locations, eXpendable Current Profilers(XCPs) collected depth-profiles of velocity and temperature,during a 24-hour survey, while the ship steamed from stationto station. XCP-measured velocities are relative to a depth-independent constant and are made absolute using GPS-referenced 75-kHz shipboard ADCP velocities between 20and 300 m. Because XCPs do not measure conductivity,salinity was inferred from the unambiguous temperature-salinity relationships at nearby MPs and CTDs.

(iii) At one location, a CTD/Lowered-ADCP package(LADCP) was yo-yoed vertically over the side of the shipto obtain an 18-h timeseries of velocity, temperature, andsalinity. (Several additional LADCP timeseries are omittedfrom this analysis because they were obtained several daysearlier or later, during different conditions of surface tideand shoaling internal-tide forcing [Martini et al., 2011b].)

In order to create an across-slope snap-shot of the semi-diurnal internal tide, MPs records were only analyzed duringa 3-d window coincident with the XCP survey and LADCPstation (Fig. 1c). While the 3-day MP records may includeadditional temporal variability, the strength of the surfacetide is approximately constant over this spring-tide period,and the lengthened records reduce errors in semi-diurnal fits.Although LADCP measurements were taken several hoursafter the XCP survey, there inclusion provides valuableinformation about tidal dynamics on the lower-slope.

3. Numerical simulations

Over the 40-day mooring records, Martini et al. [2011b]identified periods dominated by locally-generated and

shoaling internal tides. However, during the cross-slopesurvey they concluded that both tides are present and noteasily deconvolved to reveal the independent dynamics ofeach tide. To aid in our interpretation of these observations,we conduct two three-dimensional diagnostic numericalsimulations with realistic 250-m topography [courtesy J.Chaytor and C. Goldfinger, Romsos et al., 2007] and 20-m stratification. The generation simulation is forced atthe boundaries by realistic surface-tide velocities [TPXO7.2Egbert , 1997] to reproduce the locally-generated internaltide. The shoaling simulation is forced at the westernboundary by an onshore (eastward) propagating internal-tide in lieu of a surface tide to approximate a shoalinginternal tide.

Because the vertical structure, direction, amplitude,and phase of the true shoaling internal tide are notknown a priori, we force the shoaling simulation in thesimplest manner that is still relevant for interpretingobservations. The prescribed vertical structure is mode-1because it dominates observed energy flux (Section 5.4) andis associated with long-distance propagation [e.g., Ray and

Mitchum, 1996]. J. Klymak (personal communication) hasfound that prescribing an additional mode-2 shoaling wavecan drastically alter the dynamics of mode-1 scattering.Here, we avoid this layer of complexity by prescribinga single mode-1 wave. The prescribed direction iseastward. Although Martini et al. [2011b] report similaritiesbetween observed energy fluxes and those of a northward-propagating obliquely-shoaling internal tide, an eastwardpropagating tide is less ambiguous to prescribe numericallybecause topography to the west is approximately flat. Inaddition, we expect a northward-propagating tide to refractin this direction as it shoals (like a surface wave on abeach). Finally, the prescribed phase and amplitude of theshoaling internal tide are chosen so that simulated velocity,pressure, and onshore energy flux are qualitatively similarto observations (i.e., onshore energy flux is approximately1000 W/m-coastline, Sections 4 and 6). Given theapproximations required to force the shoaling simulation,we expect it to be illustrative rather than predictive.

Numerically, both simulations are conducted with theMIT general circulation model [MITgcm, Marshall et al.,1997]. The model is configured with a linear free surface,free-slip boundary conditions, f -plane planetary rotation,two-thousand time-steps per tidal period and constant eddyviscosities of 1×10−1 and 1×10−2 m2/s in the horizontal andvertical, respectively. The eddy viscosities are large becausewe have simplified the model’s energy balance by avoidinga complicated turbulence closure. The full model domaincontains 400 × 240 × 155 points in the x, y, and z directionsand ranges between 42.8-43.7◦ N and 124.2-127◦ W (Fig. 1,a). Horizontal resolution begins to telescope smoothly from250 to 1000 m 25 km north and south of 43.2◦ N and overthe abyssal plain. A flow-relaxation scheme is applied atthe lateral boundaries to prevent the reflection of internalwaves. We sample velocity, temperature, and salinity duringthe tenth tidal-cycle when the energy balance has reachedan approximate steady state.

4. Velocity and pressure

Because Oregon slope internal tides are dominantly largescale, velocity and pressure can be approximated as a linearsuperposition of locally-generated and shoaling wavefields.The observed structures of velocity and pressure containqualitative similarities to both numerical simulations.

4.1. Theory

Large-scale tidal motions are dominated by linear,hydrostatic, mechanics. Combining the vertical-momentum, buoyancy, and continuity equations with

KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING X - 3

these approximations, reduces the equations of motion to:

�−iω −f

f −iω

�u = −∇p (1)

∇ · u =�pzt

N2

�

z

, (2)

where u is horizontal velocity, p reduced-pressureperturbation, ω the tidal frequency, f the Coriolisfrequency, N the buoyancy frequency, ∇ the horizontalgradient operator, and the domain extends betweenz = [0, H(x)].

4.2. Methods

Temperature and salinity at each observational stationare used to calculate depth-profiles of instantaneous (ρ)and tidally-averaged (ρ0) density. Following Desaubies and

Gregg [1981], ρ0 is calculated as the average depth of eachisopycnal rather than the average isopycnal at each depth.Internal-tide pressure is obtained from the hydrostaticbalance by integrating buoyancy, b = −g(ρ − ρ0)/ρ0, andremoving the depth-average:

p� =

�z

0

b dz −��

z

0

b dz

�, (3)

where �·� denotes a depth average [Kunze et al., 2002].Surface-tide pressure, which requires an absolutemeasurement of sea-surface displacement, is unknownin the observations and therefore not analyzed. Kelly et al.

[2010] identified a depth-dependent component of surface-tide pressure, which arises from isopycnal heaving bymovement of the free surface, and can lead to O(50 W/m-coastline) errors in internal-tide energy flux. We correctinternal-tide pressures using surface-tide displacementsfrom TPXO7.2 [Egbert , 1997] and Kelly et al. [2010]’s (27)and (28). Surface and internal-tide velocities are definedu0 = �u� and u� = u− u0, respectively.

Semidiurnal fits to all available measurements of velocityand pressure are estimated over 50-m vertical intervalsby least-squares regression. Results are scaled by

√2

to produce concise notation for tidally-averaged kineticenergy (u2

/2) and energy flux (Re[u�∗p�]). Phases in the

observations and generation simulation are referenced fromthe time of maximum northward surface-tide velocity. Phasein the shoaling simulation is defined to maximize qualitativesimilarities with the observations. To limit the introductionof error, phases are not reported where velocity is less than0.01 m/s and pressure is less then 5 Pa.

4.3. Discussion

Surface-tide velocities are 0.05 m/s along the slope andO(0.01 m/s) across the slope (Fig. 1b). Because topographybelow the shelfbreak is largely super critical (i.e., s/α ≥ 1,where s is the topographic gradient, α =

�ω2 − f2/N the

slope of a tidal characteristic, Fig. 1b), locally-generatedinternal tides in this region can propagate up or down.Where the bottom is super-critical, shoaling internal tidescan be reflected to the deep ocean. Where the bottomis near-critical, shoaling internal tides can be criticallyreflected and turbulently dissipated [e.g., stations MP4 andX4.3 Nash et al., 2007; Martini et al., 2011b, a]. Althoughtopography does not exhibit a large-scale slope, Strongalong-slope surface-tide velocities are expected to producecomplicated three-dimensional internal tides over 1-3 kmscale bumps.

Observed internal-tide velocities are elevated 500 m abovethe bottom near MP3 and MP5; and in the upper 1000 macross the slope (Fig. 3a). Generation-simulation velocitiesare smaller and elevated 500 m above the bottom at MP5

and near the bottom at L2.5, MP3, and X3.3 (Fig. 3b).Shoaling-simulation velocities are more comparable to theobservations and are large within 500 m of the bottomand in the upper 1000 m (Fig. 3c). However, the shoalingsimulation does not predict the large velocities observed500 m above the bottom at MP3 and MP5. The phaseof observed internal-tide velocity bears little resemblance tothat predicted by the generation simulation, but comparesslightly better with the shoaling simulation (Fig. 3d-f). Boththe observations and shoaling simulation indicate a 180◦

phase shift between the upper and lower halves of the watercolumn.

Observed and simulated internal-tide pressures (Fig. 4)have less vertical structure than internal-tide velocities(Fig. 3). Observed pressure is predominantly low mode,but not mode-1; i.e., the upper-water pressure maximumoccurs at 500 m depth rather than the surface (Fig.3a). Neither simulation predicts the observed amplitudeof pressure (Fig. 3b-c). However, the simulations dopredict the observed phase of bottom pressure (Fig. 3a-c), which determines internal-tide generation [Kelly and

Nash, 2010]. At the base of the slope, the observedphase of bottom pressure is consistent with the generationsimulation, suggesting observed internal-tide generation willhave the same sign as the generation simulation. Near theshelfbreak, the phase of bottom pressure is predicted by theshoaling simulation, which is opposite that of the generationsimulation. Therefore, near the shelfbreak, we expectobserved internal-tide generation to have the opposite signof that predicted by the generation simulation.

By examining current ellipses, Martini et al. [2011b]concluded that the internal tide during this periodwas multidirectional. Our present comparison of theobservations with two diagnostic simulations also suggests asuperposition of locally-generated and shoaling wavefields.In particular, (i) the generation simulation, which is basedupon realistic local forcing, does not predicted observedvelocity and pressure, and (ii) the shoaling simulation, whichroughly represents remote forcing, is consistent with thephase of observed pressure.

5. Energetics

Modal energy equations that include inter-modal energyconversion are used to quantify internal-tide generation andscattering to high modes on the Oregon continental slope.

5.1. Theory

Previous studies have derived surface and internal-tideenergy equations by separating terms involving depth-averaged and residual quantities [e.g., Kurapov et al., 2003;Carter et al., 2008]. These studies essentially extenda partial “flat-bottom” normal-mode decomposition overarbitrary topography. For instance, the surface tide, whichis defined by depth-averages, is the 0th mode, and theinternal tide, which is defined as residuals, is the sum ofmodes n ≥ 1 (Appendix A). Here we complete this analogyby deriving energy equations for individual normal modes[see also Llewellyn Smith and Young , 2002; Simmons et al.,2004; Griffiths and Grmishaw , 2007].

Tidally-averaged modal-energy equations are obtained bymultiplying (1) by u∗

n, multiplying the complex conjugate of(2) by pn, adding both expressions, and depth integrating:

H

2

�u2n +

�1− f

2

ω2

�p2n

c2n

�

t

+∇ · (Hu∗npn) =

∞�

m=0

Cn , (4)

where cn is the group velocity, and hats indicate modalamplitudes (e.g., pn = pnφn, where φn is the vertical

X - 4 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

structure function). From left to right, the terms in (4)represent time change in energy, energy-flux divergence, andinter-modal energy conversion:

Cm,n = H�u∗m ·∇pn − u∗

n ·∇pn

�, (5)

which quantifies the rate of work done by mode m on moden.

Internal-tide generation [e.g., Niwa and Hibiya, 2001;Kurapov et al., 2003] is the rate of work done by the surfacetide on the internal tide, and can be obtained from (5) bysetting m = 1 and summing n from 1 to ∞:

C = ∇H · u∗0p

�|z=H . (6)

In the deep ocean, where bottom drag is weak, internal-tidegeneration is a good proxy for surface-tide energy loss [Jayneand St. Laurent , 2001].

Because high mode tides are expected to dissipate quasi-locally [St. Laurent and Garrett , 2002; Muller and Buhler ,2009], the rate of work done on them may be a reasonableproxy for turbulent kinetic energy dissipation. For thepurpose of this study, we define scattering to high modesas:

C� =2�

m=0

∞�

n=3

Cm,n , (7)

where n ≥ 3 are considered high modes.An important property of (4), is that all terms are

nonlinear, meaning the energetics of the generation andshoaling simulations cannot be linearly superimposed. Thephasing of two wavefields determines thier covariance overa tidal cycle, which impacts energy, energy flux, andinter-modal energy conversion. Kelly and Nash [2010]recently highlighted the importance of phase in internal-tidegeneration. But, the effects of phasing are also importantfor energy conversion to higher modes.

5.2. Methods

We calculate normal modes and group velocities ateach horizontal location by numerically solving the relevanteigenvalue problem (A1) with a second-order finite differencematrix and horizontally-uniform N

2(z) profile [e.g., Smyth

et al., 2010]. To minimize the occurrence of spurious fits, thefirst twenty-five modal amplitudes of velocity and pressureare fit one-by-one using least-squares regression. Each fittedmode is removed before fitting the next. Only stationscontaining observations over 75% of the water column aredecomposed into modes.

To remove dependence on ∇pn, which is not measured,inter-modal conversion is rewritten equivalently as thematrix:

Cm,n = H �u∗mφm · pn∇φn − u∗

nφn · pm∇φm� , (8)

which only depends on spatial gradients of normal modes(which can be computed from topography and backgroundstratification). In the analysis of observations andsimulations, we calculate horizontal gradients of the normalmodes using 2-km central differences in both horizontaldirections. Additionally, surface-tide pressure, which isnot generally measured, does not appear in (8) because itsvertical-structure is constant everywhere.

5.3. Energy

Observed internal-tide energy (Fig. 5a and d) has anaverage across-slope magnitude of 1500 J/m2, an order ofmagnitude smaller than observed at Kaena Ridge, Hawaii[Nash et al., 2006], but similar to that observed 500 km fromthe Hawaiian Ridge [Zhao et al., 2010]. Internal-tide energy

is greater than either numerical simulation (and more thandouble that of the generation simulation), further suggestinga superposition of locally-generated and shoaling internaltides.

High-mode tides are likely to dissipate locally becausethey propagate slowly and have large velocity shear [St.Laurent and Garrett , 2002]. The observed internal tidecontains more high-mode energy (i.e., modes 3-10) thanthe shoaling simulation, which in turn, contains more high-mode energy the the generation simulation. Therefore,we expect that the observed internal tide may dissipatedmore energy than either simulation. Additionally, weexpect that shoaling internal tides will dissipate moreenergy than locally-generated internal tides. High modeenergy in the shoaling simulation is evidence of topographicscattering. For this simulation, the across-slope peakin high-mode energy occurs at MP4, where the slope ispredominantly near-critical. A similar peak is not found inthe generation simulation, suggesting near-critical reflectionis less important for dissipating locally-generated internaltides.

5.4. Energy flux

Observed net internal-tide energy fluxes areapproximately 500 W/m-coastline onshore (Fig. 6aand d), contrasting the generation simulation (500 W/m-coastline offshore, Fig. 6b and e), and complementing theshoaling simulation (500 W/m-coastline onshore, Fig. 6cand f). Because energy propagates at the group speed,which decreases with mode number, internal-tide energyflux is dominated by low modes.

While observed onshore energy flux indicates the presenceof a shoaling internal tide, it does not preclude thesimultaneous presence of a reflected or locally-generatedoffshore propagating internal tide. For instance, thegeneration simulation suggests the presence of a 500 W/m-coastline offshore-propagating internal-tide, but this featurecannot be verified simply from observed net energy-fluxsince net energy-flux is subject to interference and cannotgenerally be obtained by simply adding energy fluxes fromindividual wavefields [e.g., Nash et al., 2004; Martini et al.,2007; Rainville et al., 2010].

The observed decrease in energy flux with distanceonshore indicates the Oregon slope is a sink of internal-tide energy. Unfortunately, the convergence of net energyflux alone does not determine whether internal-tide energyis scattered into along-slope propagating components,returned to the surface tide [Kelly et al., 2010], orturbulently dissipated [Kunze et al., 2002]. However, MP4,a location of large energy-flux convergence (Fig. 5d), isalso associated with high-mode energy in the shoalingsimulation (Fig. 5f), near-critical topography (Fig. 1b),and observed near-bottom turbulent bores [Martini et al.,2011a]. Therefore, the onshore convergence of internal-tideenergy flux is likely explained by turbulent dissipation.

5.5. Internal-tide generation

Surface-tide ellipses on the Oregon continental slope areoriented along large-scale isobaths (Fig. 1b); therefore, thegeneration simulation predicts most internal-tide generationover 1-3 km scale along-slope bumps (Fig. 7a). Much ofthe small-scale generation produces along-slope energy-fluxdivergences and convergences, which average-out over largerregions. Therefore, internal-tides that are generated overlarge-scale across-slope topography and propagate offshoremay be more important to net energy balances. Offshoreenergy-flux in the generation simulation is about 500 W/m-coastline (Fig. 6a), consistent with the average surface-tideenergy loss estimated by Egbert and Ray [2001] over a

KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING X - 5

larger 80-km region. The shoaling simulation displays littleinternal-tide generation (Fig. 7c) and loses O(100) W/m-coastline to the surface tide (Fig. 7e).

The observed across-slope structure of internal-tidegeneration indicates a maximum at X3.3, which is notpresent in the shoaling simulation (Fig. 7e). Near theshelfbreak, the shoaling tide affects observed generation byaltering the phase of bottom pressure [Kelly and Nash,2010]. At X5.7 observed generation is negative becausebottom pressure is opposite that predicted by the generationsimulation. The observed across-slope integral of internal-tide generation at 43.2◦ N is 450 W/m-coastline, similar tothat inferred from satellite observations [Egbert and Ray ,2001].

5.6. Scattering to high modes

Although C has been previously decomposed by mode[e.g., Simmons et al., 2004; Zilberman et al., 2009],and Johnston and Merrifield [2003] inferred inter-modalconversion from energy-flux divergences, the analysispresented here may be the first to directly quantify energyconversion within the internal tide. Here we interpretscattering to high modes, C�, as an imperfect proxy forturbulent kinetic energy dissipation. A benefit of C� is thatit may be computed from sparser, large-scale, observationsand pertains directly to cascade of tidal energy. A drawbackof C� is that it may be positive or negative, unlike trueenergy dissipation, which is always positive.

In the generation simulation (Fig. 7b), C� has nearlyequal regions of positive and negative values, indicating thelocal surface tide produces little net transfer of energy tohigh modes. At 43.2◦ N, the across-slope integral is slightlynegative, indicating that high-mode energy is providingsome power to the low-mode tide (Fig. 7f).

In the shoaling simulation (Fig. 7d), C� is positive overthe lower portion of the slope, indicating the shoaling mode-1 tide is scattering to higher modes as it propagates onshore.At 43.2◦ N, C� is positive everywhere on the lower slope(except X3.3) and has an across-slope integral of 450 W/m-coastline. Large C� in the shoaling simulation suggests thatshoaling tides on the Oregon slope may dissipate energymore efficiently than locally generated tides.

Observed C� (Fig. 7f) is maximized across the lowerslope and has an across-slope integral of 470 W/m-coastline,consistent with the 500 W/m-coastline of observed turbulentkinetic energy dissipation [Nash et al., 2007]. Martini et al.

[2011a] also associated the lower-slope (MP3 and MP4) withstrong internal-tide shoaling and the generation of turbulentnear-bottom bores.

6. Onshore and offshore energy fluxes

In the study of multidirectional internal tides, there isno general (i.e., three-dimensional) method for resolvingopposing and partially-offsetting energy fluxes. Here,we estimate onshore and offshore-propagating internaltides for each vertical mode by assuming along-slopeuniformity. Some errors associated with this approximationare addressed in Appendix C.

6.1. Theory

In Appendix B, we show that tide-topography couplingcan be confined to infinitely-thin topographic steps.Between topographic steps the bottom is flat and un andpn can be written:

un = Aneiknx +Bne

−iknx (9)

pn = cnAneiknx − cnBne

−iknx, (10)

where An and Bn are complex amplitudes of harmonicfits that represent onshore and offshore-propagating waves.pn has been solved from un by modally-decomposing (1)and using the relation cn = ω/kn(1 − f

2/ω

2). Transversevelocity, vn is non-zero because of planetary rotation, butalso varies sinusoidally in x.

Replacing un and pn with wave amplitudes, the two-dimensional tidally-averaged energy equation (over thelocally-flat bottom) becomes:

H�A

2n +B

2n

�t+

�HcnA

2n −HcnB

2n

�x= 0 . (11)

Internal-tide energy flux may be arbitrarily small when An

and Bn have similar magnitudes (i.e., form a standing wave).Therefore, weak internal-tide energy flux is not uniquelyassociated with weak internal-tide energy. Although energyflux in the direction transverse to wave propagation is non-zero, it does not have a physical interpretation because it isnon-divergent and does not appear in the energy budget.

6.2. Methods

Tidally-averaged magnitudes (but not phases) of onshoreand offshore-propagating internal tides are obtained for eachmode by inverting (9) and (10):

A2n =

14|un + pn/cn|2 (12)

B2n =

14|un − pn/cn|2 . (13)

6.3. Discussion

Separating observations into onshore and offshore-propagating internal tides indicates that approximately1000 W/m-coastline is shoaling and 500 W/m-coastline isradiating offshore (Fig. 8a, d, and g). The across-slopestructure of the simulations is evident in the observations.The generation simulation predicts the observed peaks inoffshore energy flux near regions of strong internal-tidegeneration (e.g., near X3.3 and X5.3, Fig. 8b, e, and h).Observations and the shoaling simulation also suggest aqualitatively similar decrease in onshore energy flux withdistance onshore (Fig. 8c, f, and i). A relative lack of offshoreenergy flux in this simulation indicates that the shoalinginternal tide is scattered/dissipated rather than reflected.

The shoaling internal tide appears to deposit 1000 W/m-coastline on the Oregon slope, roughly twice the observedturbulent kinetic energy dissipation [Nash et al., 2007]. Wepropose, 3-D effects, i.e., along-slope energy flux divergence,may explain the excess 500 W/m-coastline of across-slopeenergy flux convergence at 43.2◦ N. Unfortunately, thesedynamics are difficult to assess without (i) along-slopeobservations and (ii) knowledge of the true shoaling tide.However, both numerical simulations display along-slopeenergy-flux divergences that vary in sign and have amplitudeO(500 W/m-coastline).

7. Conclusions

The Oregon continental slope is home to locally-generatedand shoaling internal tides that interact vigorously withtopography. We have illuminated these processes using anew theory of linear tide-topography coupling (Section 5).Direct observations indicate the surface tide scatters 450W/m-coastline to the internal tide, which approximatelyequals satellite-derived surface-tide energy loss [Egbert andRay , 2001]. In addition, surface and low-mode internaltides scatter 470 W/m-coastline to high-mode internal tides,

X - 6 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

which approximately equals observed turbulent kinetic-energy dissipation [Nash et al., 2007]. Lastly, we haveestimated onshore and offshore energy fluxes, but are unableto close the energy budge without considering 3-D dynamics.

Although the results presented here are specific to theOregon slope, the importance of linear tide-topographycoupling likely pertains to continental margins as a whole.These features provide a large surface area to catch andscatter shoaling internal tides. Like previous investigations,we have found evidence that shoaling internal tides are notelastically reflected [Alford and Zhao, 2007], but insteadscattered to higher modes and dissipated quasi-locally [Nash

et al., 2004].

Appendix A: Normal modes

Without loss of generality, the depth-structure of pressureand velocity can be isolated in a set of functions, φn(z),reducing (1) and (2) to an eigenvalue problem at each

horizontal location:�φnz

N2

�

z

+

�1− f

2

ω2

�φn

c2n= 0 , (A1)

where phin and c2n are eigenfunctions and eigenvalues,

respectively. The eigenfunctions are orthonormal:

�φmφn� = δm,n (A2)

and complete, so that projections of pressure and velocityonto this basis are uniformly convergent (i.e., varianceconserving) even when φn do not obey local boundaryconditions:

(u, p) =∞�

n=0

(un, pn) , (A3)

where un = u(x)φn(z) and pn = p(x)φn(z). To connectour analysis to an extensive body of existing literature, φn

are chosen to obey a rigid-lid (φnz = 0 at z = 0) and flatbottom (φnz = 0 at z = H) [e.g., Gill , 1982].

Although a normal-mode decomposition does not alwaysconverge point-wise at the bottom boundary, it alwaysconserves energy, regardless of the topographic slope. Usingnormal modes to decompose flow over sloping topographyproduces effects that are analogous to Fourier transforminga finite-length timeseries with an endpoint mismatch. TheFourier Transform conserves energy, but the mismatchcauses leakage and redistributes energy into a characteristicf−2 shape. In regions of sloping topography, we expect

internal-tide energy to exhibit an analogous n−2 shape.

Appendix B: A Topographic step

Here, we show that decomposing tidal flow with normalmodes is equivalent to approximating topography aspiecewise-constant. Consider a tidally-averaged internal tidein steady-state: at any topographic step, inter-modal energyconversion must equal energy-flux divergence:

Cn =�H

+u∗+n p

+n −H

−u∗−n p

−n

�/δx , (B1)

where δx is an arbitrarily small distance and superscriptsdenote variables on the left (−) and right (+) sides of thestep. Transverse velocity may be nonzero due to planetaryrotation but produces no energy-flux divergence.

Jump conditions for u and p are obtained by conservingmass and momentum:

H−u− = H

+u+ (B2)

p− = p

+, (B3)

respectively. (Note that tidally-averaged pressure gradientsare zero because tidally-averaged accelerations are zero.)Using the orthogonality condition (A2) these conditions aresubstituted in (B1) to produce:

Cn =�H

−u∗−

p+n −H

−u∗−n p

+�/δx . (B4)

Adding and subtracting H−u∗−n p

−n /2 (and invoking the

orthogonality condition) results in:

Cn = H−�u∗− p

+n − p

−n

δx− u

∗−n

p+ − p

−

δx

�, (B5)

which reduces to the general expression for inter-modalenergy conversion (5) as δx → 0. Physically, this derivationconfirms that flat-bottom dynamics such as modal wave-speeds are descriptive everywhere and conceptually alltide-topography coupling is confined to infinitely-thintopographic steps.

Appendix C: Sensitivity of onshore and

offshore energy flux estimates

We conduct two sensitivity experiments to examineerrors associated with assuming that internal tides arepropagating in the across-slope plane. The first experimentconsiders only a single-mode, 500 W/m-coastline, offshore-propagating internal tide (Fig. 9a). The direction of thetide’s propagation is varied between θ ± 90◦ from due west.As the angle deviates from zero, actual offshore energy fluxdecreases proportional to cos θ. Estimated onshore andoffshore energy fluxes are computed from (12) and (13).We find the two-dimensional approximation predicts actualoffshore energy flux within 20% provided |θ| < 60◦.

The second experiment considers a single-mode, 1000W/m-coastline, onshore-propagating tide is superimposedon a single-mode, 500 W/m-coastline, offshore-propagatingtide (Fig. 9b). The direction of the offshore-propagatingtide is fixed (as due west), but the direction of theonshore-propagating tide is varied between θ ± 90◦ fromdue east, which affects estimates of offshore energy flux.Additionally, the phase difference between the onshore andoffshore-propagating tides also affects the accuracy of thetwo-dimensional approximation. The worst possible phaserestricts accurate energy-flux estimates to situations where|θ| < 15◦. However, the median phase allows energy-fluxestimates for |θ| < 50◦. These sensitivity experimentssuggest that estimated onshore and offshore energy fluxesare generally within 20% of the true values, even wheninternal tides are propagating at at |θ| ≈ 50◦ to the across-slope plane.

Acknowledgments. We thank Jody Klymak for sharing hisinsights on the separation of multidirectional waves. This workwas supported by the Office of Naval Research and NationalScience Foundation.

References

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Alford, M. H., J. A. MacKinnon, Z. Zhao, R. Pinkel, J. Klymak,and T. Peacock (2007), Internal waves across the pacific,Geophys. Res. Lett., 34, L24,601, doi:10.1029/2007GL031,566.

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Carter, G. S., M. A. Merrifield, J. M. Becker, K. Katsumata,M. C. Gregg, D. S. Luther, M. D. Levine, T. J. Boyd, andY. L. Firing (2008), Energetics of M2 barotropic-to-baroclinictidal conversion at the Hawaiian Islands, J. Phys. Oceanogr.,38, 2205–2223.

Desaubies, Y., and M. C. Gregg (1981), Reversible andirreversible finestructure, J. Phys. Oceanogr., 11, 541–556.

Egbert, G. D. (1997), Tidal data inversion: Interpolation andinference, Prog. in Oceanogr., 40, 81–108.

Egbert, G. D., and R. D. Ray (2001), Estimates of M2 tidalenergy dissipation from TOPEX/POSEIDON altimeter data,J. Geophys. Res., 106, 22,475–22,502.

Eriksen, C. C. (1998), Internal wave reflection and mixing atFieberling Guyot, J. Geophys. Res., 103, (C2) 2977–2994.

Garrett, C., and E. Kunze (2007), Internal tide generation in thedeep ocean, Annu. Rev. Fluid Mech., 39, 57–87.

Gill, A. E. (1982), Atmosphere-ocean dynamics, Academic Press.Griffiths, S. D., and R. H. J. Grmishaw (2007), Internal tide

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Hendershott, M. C. (1981), Long waves and ocean tides, inEvolution of physical oceanography: Scientific Surveys inhonor of Henry Stommel, edited by B. A. Warren andC. Wunsch, pp. 292–341, The MIT Press.

Jayne, S. R., and L. C. St. Laurent (2001), Parameterizing tidaldissipation over rough topography, Geophys. Res. Lett., 28,811–814.

Johnston, T. M. S., and M. A. Merrifield (2003), Internal tidescattering at seamounts, ridges, and islands, J. Geophys. Res.,108 (C6), 3180, doi:10.1029/2002JC001,528.

Johnston, T. M. S., M. A. Merrifield, and P. E. Holloway (2003),Internal tide scattering at the Line Islands Ridge, J. Geophys.Res., 108 (C11), 3365, doi:10.1029/2003JC001,844.

Kelly, S. M., and J. D. Nash (2010), Internal-tide generation anddestruction by shoaling internal tides, Geophys. Res. Lett.,accepted.

Kelly, S. M., J. D. Nash, and E. Kunze (2010), Internal-tide energy over topography, J. Geophys. Res., p.doi:10.1029/2009JC005618.

Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg(2002), Internal waves in Monterey Submarine Canyon, J.Phys. Oceanogr., 32, 1890–1913.

Kurapov, A., G. Egbert, J. S. Allen, R. N. Miller, S. Y. Erofeeva,and P. M. Kosro (2003), The M2 internal tide off Oregon:Inferences from data assimilation, J. Phys. Oceanogr., 33,1733–1757.

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Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey(1997), A finite-volume, incompressible Navier-Stokes modelfor studies of the ocean on parallel computers, J. Geophys.Res., 102, 5753–5766.

Martini, K., M. H. Alford, J. D. Nash, E. Kunze, and M. A.Merrifield (2007), Diagnosing a partly standing internal wavein Mamala Bay, Oahu, Geophys. Res. Lett., 34, L17,604,doi:10.1029/2007GL029,749.

Martini, K. I., M. H. Alford, E. Kunze, J. D. Nash, and S. M.Kelly (2011a), Observations of remotely-generated internaltides breaking on the oregon continental slope, in progress.

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Internal tide reflection and turbulent mixing on the continentalslope, J. Phys. Oceanogr., 34, 1117–1134.

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Nash, J. D., M. H. Alford, E. Kunze, K. I. Martini, andS. M. Kelly (2007), Hotspots of deep ocean mixing on theOregon continental slope, Geophys. Res. Lett., 34, L01,605,doi:10.1029/2006GL028,170.

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S. M. Kelly,104 COAS Admin. Bldg., Oregon State University,Corvallis, OR 97331, USA. (skelly@coas.oregonstate.edu)

X - 8 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

a)

b)

−125 −124

42

42.5

43

43.5

44

44.5

MP3 MP4 MP5

Latitude

Long

itude

10 km

−125.3 −125.2 −125.1 −125 −124.9

43.1

43.2

0 1000 2000 3000

Depth [m]

0 0.5 1 1.5 2

s/α

260 261 262

−0.05

0

0.05

year day

u 0 [m

/s]

XCPs LADCPc)

Figure 1: Oregon-slope topography and tides. (a) Large-scale topography (contours every 500 m). The model domain isoutlined in black and the location of observational study in red. (b), Local east-west topographic gradient normalized bythe slope of a semidiurnal tidal ray (contour are every 250 m). Dots indicate location of observations and ellipses representtidal excursions amplified by 10. (c), Timeseries of across-slope (solid) along-slope (dashed) surface-tide velocities duringXCP and LADCP measurements (dark gray) and sub-sampled MP measurments (light gray).

KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING X - 9

Figure 2: Schematic of tidal energetics on the Oregon slope. Surface tides and shoaling internal tides supply energy to thelocal internal tide, which partially radiates and partially dissipates. Our goals are to quantify the amplitudes of shoalingand radiating internal tides and determine whether surface tides or shoaling internal tides fuel local turbulence.

Figure 3: Amplitudes (a-c) and phases (d-f) of internal-tide velocity. From left to right columns contain observations, thegeneration simulation, and the shoaling simulation. Black lines indicate example tidal characteristics.

X - 10 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

Figure 4: Amplitudes (a-c) and phases (d-f) of internal-tide pressure. From left to right columns contain observations, thegeneration simulation, and the shoaling simulation. Black lines indicate example tidal characteristics.

0

500

1000

1500

2000

2500

3000

L2.5A MP3

X3.3X3.7MP4X4.3X4.7MP5

X5.3X5.7

10 km

Energy

5 J/m3

Dep

th [

m]

a) b) c)

−125.4−125.3−125.2−125.1 −125 −124.90

500

1000

1500

2000

2500

3000

Longitude

∫ E d

z [J

/m2 ]

d) Depth−integrated energy e) f)

Observations Generation Simulation Shoaling Simulation

Figure 5: Depth-structure (a-c) and depth-integrals (d-f) of internal-tide energy. From left to right columns containobservations, the generation simulation, and the shoaling simulation. Gray integrals indicate contributions by modes 1and 2.

KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING X - 11

0

500

1000

1500

2000

2500

3000

L2.5A MP3

X3.3X3.7MP4X4.3X4.7MP5

X5.3X5.7

10 km

Energy flux

2 W/m2

Dep

th [

m]

a) b) c)

−125.4−125.3−125.2−125.1 −125 −124.9−1000

−500

0

500

1000

Longitude

∫ F d

z [W

/m]

d) Depth−integrated energy flux e) f)

Observations Generation Simulation Shoaling Simulation

Figure 6: Depth-structure (a-c) and depth-integrals (d-f) of internal-tide energy flux. From left to right columns containobservations, the generation simulation, and the shoaling simulation. Gray integrals indicate contributions by modes 1and 2.

X - 12 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

Latit

ude

10 km 1000 W/m

a) C Generation sim.

43.1

43.15

43.2

43.25

43.3

Latit

ude

b) Cε Generation sim.

43.1

43.15

43.2

43.25

43.3

Latit

ude

c) C Shoaling sim.

43.1

43.15

43.2

43.25

43.3

Latit

ude

d) Cε Shoaling sim.

43.1

43.15

43.2

43.25

43.3

−125.3 −125.2 −125.1 −125 −124.9

−0.2

−0.1

0

0.1

0.2

Longitude

C [

W/m

2 ]

e) C Observations

L2.5AMP3

X3.3X3.7MP4X4.3X4.7MP5

X5.3X5.7

ObservationsGeneration Sim.Shoaling Sim.

450 W/m−coastline570 W/m−coastline110 W/m−coastline

−125.3 −125.2 −125.1 −125 −124.9−0.1

−0.05

0

0.05

0.1

Longitude

Cε [

W/m

2 ]

f) Cε Observations

L2.5AMP3

X3.3X3.7MP4X4.3X4.7MP5

X5.3X5.7

ObservationsGeneration Sim.Shoaling Sim.

470 W/m−coastline−130 W/m−coastline

460 W/m−coastline

−0.25 0 0.25

Internal−tide generation, C [W/m2]

−0.1 0 0.1

Conversion to high modes, Cε [W/m2]

Figure 7: Spatial maps of inter-modal energy conversion. (a) and (c), internal-tide generation in the generation and shoalingsimulations, respectively. Arrows indicate internal-tide energy fluxes. (b) and (d) scattering to high modes (n ≥ 3) ingeneration and shoaling simulations, respectively. Arrows indicate high-mode energy fluxes. (e) and (f), observed andsimulated internal-tide generation and scattering to high modes, respectively, at 43.2◦ N.

KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING X - 13

a) Onshore energyL2

.5A MP3X3.3X3.7MP4

X4.3X4.7MP5X5.3X5.7

0

1000

∫ E d

z [J

/m2 ]

d) Offshore energy

−125.4−125.3−125.2−125.1 −125 −124.9−1000

−500

0

500

1000

Longitude

∫ F d

z [W

/m]

g)

Onshore and offshore energy fluxes

b)

e)

h)

c)

f)

i)

Observations Generation Simulation Shoaling Simulation

Figure 8: Onshore- and offshore-propagating internal tides. From left to right, columns contain observations, the generationsimulation, and the shoaling simulation. Depth-integrated energy associated with onshore (a-c) and offshore-propagating(d-f) internal tides. Depth-integrated energy flux (g-i) associated with onshore- and offshore-propagating internal tides.In all graphs, gray indicates contributions by modes 1 and 2.

X - 14 KELLY AND NASH: TIDE-TOPOGRAPHY COUPLING

a)

−1500 −1000 −500 0 500 1000 1500−90

−45

0

45

90

∫ F dz [W/m−coastline]

φ [d

eg]

b)

Figure 9: Sensitivity of the two-dimensional approximation. In both (a) and (b) thick black lines indicate true onshoreand offshore energy fluxes. Green and red regions indicate estimated onshore and offshore energy fluxes. Dark green andred indicate regions where estimated energy flux is with 20% of true energy flux. In (a) the propagation angle of a 500W/m-coastline tide is varied between θ ± 90◦ from due west. In (b) a 500 W/m-coastline tide propagates due west whilethe propagation angle of a 1000 W/m-coastline tide is varied between θ± 90◦ from due east. The phase difference betweenthe onshore and offshore-propagating tides affects the estimated fluxes, the median estimate is denoted by a dashed lineand best and worst-case scenarios are denoted by thin black lines.