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Internal‐tide energy over topography

S. M. Kelly,1 J. D. Nash,1 and E. Kunze2

Received 6 July 2009; revised 24 November 2009; accepted 3 February 2010; published 19 June 2010.

[1] The method used to separate surface and internal tides ultimately defines propertiessuch as internal‐tide generation and the depth structure of internal‐tide energy flux. Here,we provide a detailed analysis of several surface‐/internal‐tide decompositions overarbitrary topography. In all decompositions, surface‐tide velocity is expressed as the depthaverage of total velocity. Analysis indicates that surface‐tide pressure is best expressed asthe depth average of total pressure plus a new depth‐dependent profile of pressure, whichis due to isopycnal heaving by movement of the free surface. Internal‐tide velocity andpressure are defined as total variables minus the surface‐tide components. Correspondingsurface‐ and internal‐tide energy equations are derived that contain energy conversionsolely through topographic internal‐tide generation. The depth structure of internal‐tideenergy flux produced by the new decomposition is unambiguous and differs from that ofpast decompositions. Numerical simulations over steep topography reveal that thedecomposition is self‐consistent and physically relevant. Analysis of observations overKaena Ridge, Hawaii; and the Oregon continental slope indicate O (50 W m−1) error indepth‐integrated energy fluxes when internal‐tide pressure is computed as the residual ofpressure from its depth average. While these errors are small at major internal‐tidegeneration sites, they may be significant where surface tides are larger and depth‐integrated fluxes are weaker (e.g., over continental shelves).

Citation: Kelly, S. M., J. D. Nash, and E. Kunze (2010), Internal‐tide energy over topography, J. Geophys. Res., 115, C06014,doi:10.1029/2009JC005618.

1. Introduction

[2] Astronomical forcing of the ocean produces surfacetides which propagate as shallow‐water edge waves. Whensurface tides encounter topography, they deform and perturbstratified fluid, generating internal tides. To obtain ameaningful expression of surface‐to‐internal‐tide energyconversion, the surface and internal tides must be correctlyidentified. Deriving a relevant and practical method of iso-lating internal tides is an important step to understandingtheir physics.[3] Over a flat bottom, a simple method for identifying

internal tides is normal‐mode decomposition [e.g., Wunsch,1975]. This solution identifies the depth structure and phasespeed of internal tides, but does not provide an expressionfor topographic internal‐tide generation, because it does notapply over a sloping bottom.[4] Baines [1982] suggested that internal tides could be

identified over arbitrary topography by removing motionreproduced in an identical but unstratified fluid. Niwa andHibiya [2001], Khatiwala [2003], and Legg and Huijts[2006] used this method to identify internal tides in

numerical experiments. Lu et al. [2001] and Di Lorenzo etal. [2006] criticized this method for ignoring the effect ofinternal‐wave drag on the surface tide. Additionally, inter-preting oceanographic observations is not possible with thisdecomposition.[5] As a practical solution, Kunze et al. [2002] identified

internal tides as the residuals of pressure and horizontalvelocity after subtracting depth averages. By default, thisdecomposition identifies surface tides as depth averages ofpressure perturbation and velocity. Conveniently, surfacetides can vary by an unknown constant without altering thecorresponding internal tides, allowing the decomposition tobe applied to ocean observations and numerical simulations[e.g., Kunze et al., 2002; Alford, 2003; Althaus et al., 2003;Simmons et al., 2004; Nash et al., 2004, 2005; Alford et al.,2006; Nash et al., 2006; Carter et al., 2006, 2008; Dudaand Rainville, 2008]. Gerkema and van Haren [2007] crit-icized this decomposition for being inconsistent over slopingtopography. However, we dispute this assertion in section 3.2.More importantly, Kurapov et al. [2003] found that thisdecomposition results in spurious surface‐to‐internal‐tideenergy conversion over both flat and sloping topography.Here, we eliminate spurious conversion by removing pressuredue to isopycnal heaving by movement of the free surface. Luet al. [2001] made an analogous surface‐/internal‐tidedecomposition but also removed pressure due to isopycnalheaving by the surface tide flowing over topography. Thedecomposition of Lu et al. [2001] is examined here and found

1College of Oceanic and Atmospheric Sciences, Oregon StateUniversity, Corvallis, Oregon, USA.

2School of Earth and Ocean Sciences, University of Victoria, Victoria,British Columbia, Canada.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2009JC005618

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to provide unphysical distributions of surface‐to‐internal‐tideconversion.[6] It is the purpose of this note to identify a complete,

consistent, and practical method for identifying the internaltide. In section 2, we begin deriving a surface‐/internal‐tidedecomposition which depends on an arbitrary profile ofpressure. In section 3, we examine five different methods ofdecomposing pressure. The first is that implied by Baines[1982], the second by Gerkema and van Haren [2007],the third by Kunze et al. [2002], and the fifth by Lu et al.[2001]. The fourth decomposition (section 3.4) is similarto that of Kunze et al. [2002] but includes a correction toaccount for pressure due to isopycnal heaving by movementof the free surface. During this investigation, an emphasis isplaced on the implications of each physical assumption.Careful examination of the momentum budget refutesobjections raised by Gerkema and van Haren [2007],indicating that their derivation predetermines an alternativesurface‐/internal‐tide decomposition which is unphysical andincompatible with that of Kunze et al. [2002]. In section 4,numerical simulations and observations are used to examinethe depth structure of internal‐tide energy flux and confirmthe accuracy of the tidally averaged internal‐tide energyequations. The decomposition proposed here eliminatesspurious energy conversion and provides the most physicallyrelevant description of internal‐tide energetics. In section 5,conclusions are presented.

2. Beginning a Decomposition

[7] Although surface‐/internal‐tide decompositions areroutinely employed in oceanographic literature, they arerarely discussed and/or motivated in detail. Here, we developseveral decompositions noting the physical implications ofeach assumption.

2.1. Requisites of a Decomposition

[8] Before attempting to decompose flow, it is useful todefine some requisites for success. (1) The first requisite iscompleteness: the sum of the surface and internal tideshould equal the total flow. (2) The second requisite isconsistency: surface‐ and internal‐tide variables shouldreduce to normal‐mode solutions over a flat bottom. Thisrequisite precludes dynamically balanced modes (i.e., modesthat do not contain topographic internal‐tide generation)because energy fluxes over a flat bottom routinely identifysteep topography as a source of internal‐tide energy viathedivergence theorem. (3)The third requisite ispracticality:the decomposition should be physically interpretable andapplicable for interpreting observations and numericalsimulations.

2.2. Total Equations of Motion

[9] We start with linear, Boussinesq, f plane, hydrostatic,momentum, continuity, and buoyancy equations:

ut � fv ¼ �px ð1Þvt þ fu ¼ 0 ð2Þ

0 ¼ �pz þ b ð3Þ

ux þ wz ¼ 0 ð4Þ

bt þ wN2 ¼ 0; ð5Þ

where partial derivatives are denoted with subscripts.Buoyancy perturbation is b = −g(rtot − �tot)/r0, where �ð Þdenotes a time average, rtot total density and r0 a constantreference density. ptot is total pressure and p = (ptot − ptot)/r0pressure perturbation scaled by reference density (i.e.,reduced pressure). N2(z) is the time‐averaged buoyancygradient (assumed to be horizontally uniform). For sim-plicity, uniformity is assumed in the y direction, (·)y = 0.[10] The surface is located at z = h and the bottom z = −h(x).

Surface displacement is given by the hydrostatic relationgh = p∣z=0. The upper boundary condition is a linear free‐surface w∣z=0 = ht, which can be applied at z = 0 providedh � h. The lower‐boundary condition is an impermeablebottom w∣z=−h = −hxu∣z=−h (note that −hx indicates a positivebottom slope).

2.3. Depth Averaging Over Topography

[11] Decomposing surface and internal tides requires depthaveraging variables and horizontal gradients of variables. Tosimplify notation, we introduce h·i as the depth‐averageoperator:

�h i ¼ 1

h

Z 0

�hð�Þdz0: ð6Þ

For consistency, all depth‐averaged quantities will be de-noted with uppercase letters and residuals by primes.[12] Over sloping topography, taking the depth average of

a horizontal gradient requires the use of Leibniz’s theorem[i.e., Kundu and Cohen, 2002] and the product rule. Theresulting identity is

ð�Þx� � ¼ �h ix þ

hxh

�h i � ð�Þjz¼�h

� �: ð7Þ

The first term on the right‐hand side is the horizontalderivative of the depth‐averaged quantity. If h does notdepend on x, this is the only term on the right‐hand side.The first part of the second term on the right‐hand side isdue to applying the product rule after integration, since thenormalization factor 1/h depends on x. The last part of thesecond term on the right‐hand side is due to the x depen-dence in the bottom limit of integration.

2.4. Decomposing Horizontal Velocity

[13] We use depth averaging to decompose horizontalsurface‐ and internal‐tide velocity as:

ðU ;V Þ ¼ uh i; vh ið Þ ð8Þ

ðu0; v0Þ ¼ u� U ; v� Vð Þ; ð9Þ

respectively. Although the decomposition defined by (8)and (9) has not been formally justified over topography

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(to our knowledge), it is widely accepted/applied by theoceanographic community.[14] Approximating surface‐tide velocity as a depth

average assumes that the surface tide produces all volumeconvergence and surface displacement. This property isillustrated by depth‐integrating (4) using (7) to obtain

ðHUÞx þ �t ¼ 0: ð10Þ

This approximation is consistent with both normal modesover a flat bottom and the shallow‐water equations com-monly employed in surface‐tide models containing complextopography [e.g., Egbert and Ray, 2000]. A convenience ofthis decomposition is that surface‐tide momentum balancescan be obtained by depth averaging (1) and (2).[15] For the internal tide, (10) is a rigid‐lid boundary

condition. Although smaller than surface tides, it is wellknown that internal tides produce surface displacements [e.g.,Ray and Mitchum, 1996]. Provided these displacements aresmall, they can be estimated from internal tide pressure atz = 0 using the hydrostatic relation [e.g., Gill, 1982].Requiring internal‐tide velocity to have zero depth averageover topography is consistent with the modes derived byWunsch [1969] for internal waves in awedge‐shaped domain.

2.5. Decomposing Pressure

[16] The decomposition of pressure into surface andinternal tide components has been debated for over a decade[Holloway, 1996; Lu et al., 2001; Kunze et al., 2002;Gerkema and van Haren, 2007] because it defines bothinternal‐tide energy flux and surface‐to‐internal‐tide energyconversion. Here, we examine how an arbitrary decompo-sition influences the surface‐ and internal‐tide momentumbalances. Without loss of generality, we define surface‐ andinternal‐tide pressure as:

psðx; zÞ ¼ P þ p; ð11Þ

piðx; zÞ ¼ p0 � p; ð12Þ

respectively, where p(x, z) is an arbitrary pressure field, P(x) =hpi and p′(x, z) = p − P. The benefit of writing pressure usingP, p′, and p is that terms involving depth integrals are inter-preted easily and role of p is isolated (which contains anydepth dependence of the surface tide). Pressure terms anddescriptions are listed in Table 1.

2.6. Decomposing the Equations of Motion

[17] Surface‐ and internal‐tide momentum equationsilluminate the implications of choosing a pressure decom-position. The surface‐tide x momentum equation is formedby first depth averaging (1) using (7), then both adding andsubtracting px. The surface‐tide z momentum and buoyancyequations are produced by kinematically decomposing (3)and (5):

Ut � f V ¼ �psx � D� pxð Þ ð13Þ

0 ¼ �pz þ b ð14Þ

bt þ N 2w ¼ 0; ð15Þ

whereD is the internal‐wave drag due to sloping topography:

DðxÞ ¼ �hxh

p0 jz¼�h: ð16Þ

Internal‐tide equations are obtained by differencing (13)–(15) with (1), (3), and (5), respectively:

u0t � fv

0 ¼ �pix þ D� pxð Þ ð17Þ

0 ¼ �piz þ bi ð18Þ

bit þ N 2wi ¼ 0 ð19Þ

where bi = b − b and wi = w − w.[18] The surface‐ and internal‐tide x momentum equations

are dynamically coupled through D − px, which representsinternal‐wave drag and an arbitrary pressure gradient.Although it is expected that this term will transfer momentumfrom the surface to internal tide, it is also possible for ashoaling internal tide to induce a surface tide. For instance,the internal‐wave modes in a wedge‐shaped domain derivedby Wunsch [1969] have U = 0 but P ≠ 0 and D ≠ 0.

3. Choosing a Decomposition

[19] It is possible to characterize several surface‐/internal‐tide decompositions by their designation of p. Here, wereview four previously used decompositions and suggest afifth (section 3.4).

3.1. Using an Unstratified Fluid

[20] Baines [1982] suggested that the surface tide could bedetermined from an unstratified flow. Such a decompositionrequires p = pB where:

pBðxÞ ¼ P0 � P ð20Þ

and P0(x) is the pressure obtained from an identically forcedbut unstratified fluid. Although this decomposition has beenemployed in several studies [e.g., Niwa and Hibiya, 2001;Khatiwala, 2003; Legg and Huijts, 2006], it is physicallyproblematic because it implies the surface tide is indepen-dent of the internal tide and internal‐wave drag, which aredetermined by stratification. In practice, this decomposition

Table 1. Summary of Pressure Terms

Term Description

p(x, z) totalP(x) depth averagep′(x, z) = p − P residualps(x, z) = P + p surface tidepi(x, z) = p′ − p internal tidep(x, z) arbitrary profileP0(x) identical but unstratified fluidpB(x) unstratified minus depth averagepG(x) balances wave dragph(x, z) due to free‐surface isopycnal heavingph(x, z) due to topographic isopycnal heaving

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requires that the surface tide not exchange energy with theinternal tide. Although this might be a reasonable approxi-mation for the surface tide, even a small error in surface‐tidepressureO(104 Pa) could result in a massive error in internal‐tide pressureO(102 Pa). In situations where the surface tide isboundary forced and/or freely propagating, it is subject tointernal‐wave drag and cannot be obtained accurately from anunstratified fluid.

3.2. Eliminating Wave Drag

[21] Gerkema and van Haren [2007] assumed a priori that(13) and (17) are not coupled through internal‐wave dragand implicitly required px = D. This decomposition results inp = pG, where

pGðxÞ ¼Z x

x0

hxhp0jz¼�hdx: ð21Þ

Gerkema and van Haren [2007] fail to recognize that theirsurface‐ and internal‐tide momentum equations predeter-mine internal‐tide pressure (and its constant of integration),leading them to incorrectly conclude that p cannot generallybe zero over sloping topography. Equations (13) and (17)include coupling, and contradict their findings by indicat-ing that p can equal zero.[22] pG is problematic because it depends on a horizontal

integral that is path dependent. For instance, starting in a flatregion and evaluating (21) along a path where p′∣z=−h > 0 upa slope to another flat region results in a net decrease in pG.Such a decrease is unphysical because normal modes requirepG = 0 in both flat regions. Because of this inconsistency(which is further demonstrated in section 4 using numericalsimulations), this decomposition does not meet the requisitesof section 2.1.

3.3. Depth Averaging

[23] Kunze et al. [2002] defined internal‐tide pressure astotal pressure minus its depth average, effectively setting p =0. Such a decomposition has been used to interpret the depthstructure of internal‐tide energy flux [e.g., Nash et al., 2005]and surface‐to‐internal‐tide conversion in numerical simu-lations [e.g., Kurapov et al., 2003]. In order to examine thephysical implications of this decomposition, it is useful toderive separate surface‐ and internal‐tide energy balances.Depth‐integrated energy equations are produced by dottingsurface‐ and internal‐tide momentum equations with totalvelocity and integrating [e.g., Simmons et al., 2004; Carteret al., 2008]. The resulting equations are similar to thoseobtained by Kurapov et al. [2003] (with the omission ofnonlinear terms and terms involving the depth structure ofw, which is zero here):

h

2U2 þ V 2 þ g�2

h

� �t

¼ � hUPh ix�CT � CS ð22Þ

h

2u02þ v02þ b2

N2

� t

¼ � hu0p0h ix þCT þ CS : ð23Þ

The left‐hand sides contain time change in kinetic energy(u2 + v2)/2, and two forms of potential energy: b2/(2N2) andgh2/2. The right‐hand sides contain energy‐flux divergence

and energy conversion. The equations are attractive becausevariables are simple to calculate, and potential energy andenergy flux are naturally decomposed.[24] The first energy conversion term,

CT ðxÞ ¼ �hxUp0jz¼�h ð24Þ

represents topographic internal‐tide generation and can betraced back to internal‐wave drag in the momentum equa-tions. CT is a generic expression for work done on topog-raphy by tidal flow and has been used in numerous studiesof internal tides [Niwa and Hibiya, 2001; Llewellyn Smithand Young, 2002; Khatiwala, 2003; Gerkema et al., 2004;Kurapov et al., 2003; Legg and Huijts, 2006; Carter et al.,2008]. It depends on the topographic gradient −hx, surfacetide velocity U, and internal‐tide pressure at the bottomboundary p′∣z=−h. Although p′∣z=−h can be extracted fromnumerical simulations and observations, linear theory hasalso been used to obtain analytical estimates of CT [e.g.,Bell, 1975; Khatiwala, 2003; Llewellyn Smith and Young,2002]. These linear analytical estimates can be calculatedover arbitrary topography following a procedure outlined byNycander [2005], which Zilberman et al. [2009] foundcompared favorably with conversion estimates obtainedusing p′∣z=−h from high‐resolution numerical simulations.[25] The second energy conversion term,

CSðxÞ ¼ ��tp0jz¼0 ð25Þ

has been identified in several studies [Craig, 1987; Kurapovet al., 2003; Carter et al., 2008] and arises because thesurface and internal tides do not individually satisfy thelinear free‐surface boundary condition. Specifically, sur-face‐ and internal‐tide motions are dynamically coupledthrough (10), where (hU)x is a surface‐tide term and ht =wi∣z=0 an internal‐tide term. Kurapov et al. [2003] identifiedCS as an error in the surface‐/internal‐tide decomposition,pointing out that it is nonzero even over a flat bottom andcontradicts the physics of normal modes. Craig [1987] alsoidentified CS but simply considered it an error in the inter-nal‐tide energy budget. The surface‐/internal‐tide decom-position suggested by Kunze et al. [2002] thus provides aphysically interpretable conversion term CT but also con-tains an error represented by CS.

3.4. Including Surface Motion

[26] Here, we amend the surface‐/internal‐tide decompo-sition presented in section 3.3 to eliminate spurious energyconversion through the free surface, CS. Our choice of p ismotivated from the energy equations for normal modes overa flat bottom, where CS is canceled by a cross‐terminvolving internal‐tide velocity and surface‐tide pressure(Appendix A). Over arbitrary topography, this cross‐termcan be reproduced by requiring p = ph and defining:

w�ðx; zÞ ¼ �tzþ h

h

� �ð26Þ

b�ðx; zÞ ¼ �N 2�zþ h

h

� �ð27Þ

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p�ðx; zÞ ¼Z z

0b�dz

0 þ c: ð28Þ

The constant c(x) is determined by requiring hphi = 0,assuring that, like the normal‐mode solution, internal‐tidepressure has zero depth average. Sample time series of p′and ph (Figures 1b and 1c) from a numerical simulation (seesection 4.3 for details) illustrates that ph is surface intensi-fied and has constant phase with depth. The magnitude of ph

is largest at high and low tides. Subtracting ph from p′ alsochanges the depth profiles of instantaneous and tidallyaveraged energy flux (Figures 1f and 1g), the magnitudes ofwhich depend on how u′ is phased with ph.[27] In order to remove CS from the energy equations, it is

also required that (u′ph)x ≈ u′xph. This relation arises from

scaling the normal‐mode energy equations and indicates thatthe surface tide has much longer wavelength than theinternal tide. This approximation must be applied with

caution over steep topography. However, the applicability ofthe linear free surface is also at stake if the horizontalwavelength of the surface tide becomes too short. Setting pi =p′ − ph, ps = P + ph, and bi = b − bh results in the energyequations:

h

2U2 þ V 2 þ b�2

N 2þ g�s2

h

� t

¼ � hUpsh ix �CT ð29Þ

h

2u02þ v02þ bi2

N 2

� t

¼ � hu0pi� �

xþCT ; ð30Þ

where ghs = ps∣z=0 and CT is given by (24). Equation (30) wasalso simplified using hhwhbii = htp

i∣z=0, from integration byparts. This surface/internal‐tide decomposition is idealbecause it is consistent with normal modes and all energy

Figure 1. (a) Time series of h from Oregon topography simulation; vertical line indicates high tide.(b) Time series of p′. (c) Time series of ph. (d) u′ at high tide. (e) p′ − ph (black), p′ (red), and ph (blue)at low tide. (f) Energy flux at high tide. (g) Energy flux averaged over a tidal period.

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conversion occurs through topographic internal‐tide genera-tion, CT.

3.5. Including Topographic Heaving

[28] Lu et al. [2001] effectively set p = ph + ph, whichincludes both pressure due to isopycnal heaving by move-ment of the free surface ph, and pressure due to isopycnalheaving by the surface tide at sloping topography ph. Theeffect of topographic heaving is described by:

whðx; zÞ ¼ hxUz

hð31Þ

bhðx; zÞ ¼ N 2

i!hxU

z

hð32Þ

phðx; zÞ ¼Z z

0bhdz0 þ c; ð33Þ

where c(x) is determined by the requirement hphi = 0.Although Lu et al. [2001] chose their decomposition pri-marily because it removed spurious energy conversion overa flat bottom (i.e., CS), they asserted that topographicheaving is part of the “external mode of response” andshould be included in the surface tide.[29] The energy budgets are slightly more complicated

than those in section 3.4, because pxh is large over steep

topography and it cannot be assumed that (u′ph)x ≈ u′xph.

The resulting surface‐ and internal‐tide energy equations areidentical to (29) and (30) on the left‐hand sides, but, containdifferent expressions for energy flux and energy conversionon the right‐hand side. The right‐hand sides are now written:

¼ � h up� u0pi �� �

x�CT � CF ð34Þ

¼ � hu0pi� �

xþCT þ CF ð35Þ

where pi = p′ − (ph + ph), CT is given by (24), and CF is a newsurface‐to‐internal‐tide energy conversion term:

CFðxÞ ¼ � hu0ph� �

x: ð36Þ

A complication in (34) is that surface‐tide energy flux isdefined as the total minus the internal‐tide component. Such adivision is more ad hoc than physically intuitive. CF is sus-picious because the divergence theorem requires that it inte-grate to zero over isolated topographic features. Therefore,CF

changes the distribution of topographic internal‐tide genera-tion but has no net contribution. In the next section, we findthat including ph in surface‐tide pressure results in unphysicaldistributions of energy flux and internal‐tide generation forthree numerical simulations.

4. Assessing the Decompositions

[30] In the following, we assess the merits of eachdecomposition presented in section 3 by evaluating themusing numerical simulations and observations of tidal flowover topography.

4.1. Tidally Averaged Energy Equations

[31] Most time dependence in the internal‐tide energyequations is periodic. When forcing is constant, tidallyaveraging (30), (23), and (35) produces:

hu0ðp0 � p�Þh ix ¼ CT ð37Þ

hu0p0h ix ¼ CT þ CS ð38Þ

hu0ðp0 � p� � phÞh ix ¼ CT þ CF ; ð39Þ

respectively, where ð�Þ represents a tidal average. Theseequations can be assessed using observed full‐depth timeseries or numerical simulations.

4.2. Calculating p′, pB, pG, ph, and ph

[32] Although total pressure is calculated explicitly inmodels, it cannot be obtained from observations because

pðx; zÞ ¼Z z

0b dz0 þ c ð40Þ

has an undetermined constant of integration c(x) which de-pends on surface displacement. However, determining p′from full‐depth profiles of b is unambiguous even withoutmeasuring the absolute surface displacement. Starting withthe measured quantity

pmðx; zÞ ¼Z z

0b dz

0; ð41Þ

the practical expressions for pressure are

p;P; p0ð Þ ¼ pm þ c; pmh i þ c; pm � pmh ið Þ: ð42Þ

so that p′ does not depend on the unknown constant ofintegration.[33] But, the internal tide may also depend on a depth‐

dependent profile of pressure p(x, z), which can be moredifficult to compute. For instance, pB is calculated for ournumerical simulations by conducting an identical butunstratified simulation. In the analysis of observations, pB

would be impossible to determine accurately. Additionally,calculating pG is ambiguous in both observations andnumerical simulations because the starting point and path ofthe integral are undetermined. In our analysis, we examineintegrals originating from the left‐ and right‐hand sides ofthe domain, denoted pGl and pGr, respectively. Calculatingph requires an estimate of h. Models calculate this termexplicitly. In observations, it could be extracted from eitherthe measured bottom pressure or estimated from a surface‐tide model such as TPXO [Egbert and Ray, 2000]. Finally,although, ph appears straightforward to compute in bothnumerical simulations and observations, the appropriatehorizontal scale for evaluating hx is unclear. Our pragmaticapproach for numerical simulations is to evaluate ph at thefinest resolution available.

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4.3. Numerical Simulations

[34] Simulations were performed using the MIT generalcirculation model (MITgcm), a nonlinear, nonhydrostatic,z coordinate, implicit free‐surface, finite‐volume model[Marshall et al., 1997]. For simplicity, the domains are uni-form in one horizontal direction (i.e., 2‐D) but contain f planerotation. Horizontal resolution is 250 m near sloping topog-raphy and smoothly transitions to 1000 m near the boundaries.Realistic stratification from the Oregon slope is used with auniform vertical resolution of 20 m. Eddy viscosities of 5 ×10−1 m2 s−1 in the horizontal and 5 × 10−2 m2 s−1 in the verticalwere employed for stability.[35] The first two simulations contain idealized subcritical

and supercritical continental slopes. The slopes are pre-scribed so that s/a = 0.75 and 1.25, respectively, where s isthe topographic slope and a =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 � f 2ð Þ= N2 � !2ð Þp

theslope of an M2 internal‐tide characteristic. The third simu-lation contains realistic topography from the site of anobservational study of internal tides on the Oregon slope[Nash et al., 2007; K. Martini et al., manuscript in prepa-ration, 2010]. To minimize the effects of numerical lateralboundaries, the full model domains extend 400 km from theregion of topographic interest.[36] Although realistic values are used in the simulations,

two‐dimensional domains make realistic forcing impossible.Simplified forcing is accomplished by prescribing a 0.05,0.05, and 0.025 m s−1 semidiurnal depth‐constant velocityalong the open boundary for the subcritical, supercritical,and Oregon topographies, respectively. The amplitudes offree‐surface oscillations are 1.8, 1.8, and a realistic 0.9 m forthe subcritical, supercritical, and Oregon topographies,respectively. Two thousand time steps are used per tidalperiod. A flow‐relaxation scheme applied at the boundariesprevents reflection of internal waves. All modeled quantitiesare averaged over the tenth tidal cycle when the energybalance has reached a nearly steady state.

4.4. Depth Structure of Energy Flux

[37] The depth structure of internal‐tide energy flux, u0pi,is useful for interpreting numerical simulations and observa-tions [e.g., Kunze et al., 2002; Althaus et al., 2003; Nash etal., 2004, 2005, 2006; Carter et al., 2006; Gerkema and vanHaren, 2007; Duda and Rainville, 2008]. The profilesdefined by sections 3.4 and 3.3, u0 p0 � p�ð Þ and u0p0 , arephysically intuitive (Figures 2a–2f) because they indicatethat all internal‐tide energy is propagating away from steeptopography, the expected region of internal‐tide generation.As suggested by theory [Craig, 1987], these profiles alsorepresent beam‐like energy flux to deep water. The beamsreflect off the bottom near the base of the slope and off thesurface near −75 km.[38] The other four representations of internal‐tide energy

flux produce distributions that are unphysical and confusing.For instance, the energy flux where topographic heaving hasbeen removed from the internal tide, u0 p0 � p� � phð Þ,contains a discontinuity in energy flux along a vertical lineextending upward from the base of the slope (Figures 2g–2i). There are also patches of onshore energy flux observedin the upper water column over the slope (0 to 25–50 km)that do not originate from a known source of the internaltide. Similarly unexplained energy fluxes are obtained when

an unstratified simulation is used to determine the surfacetide, u0 p0 � pBð Þ (Figures 2j–2l) [Baines, 1982]. In particu-lar, this surface‐/internal‐tide decomposition indicatesinternal‐tide energy propagating from regions where thebottom is flat and so there should be no internal‐tide gen-eration. The energy fluxes obtained by decomposing pres-sure to cancel out internal‐wave drag, u0 p0 � pGlð Þ andu0 p0 � pGrð Þ (Figures 2m–2r) have similar unphysicalenergy flux directed toward the expected generation region.Additionally, the renditions of this decomposition thatintegrate (21) from the left‐hand and right‐hand sides of thedomain are drastically different, indicating sensitivity to theintegral starting point. In both cases, near the start of theintegral, the energy flux is similar to that of u0p0 but, as pG isintegrated over steep topography, the flux becomes un-physical, again indicating that internal‐tide energy is origi-nating over a flat bottom.[39] Because pi = p′ − ph is identified in section 3.4 as the

most physically relevant pressure decomposition, deviationsfrom the resulting energy flux are quantified for each sim-ulation. Percentages were calculated of root‐mean‐squaredifference in energy flux, normalized by the root‐mean‐square of u0 p0 � p�ð Þ averaged over the domain shown.These deviations, reported in Figure 2, indicate that u0p0 isthe most similar to u0 p0 � p�ð Þ, especially over steeptopography. We also note that these two fluxes converge assurface displacement decreases and ph → 0. Other dis-tributions of internal‐tide energy flux have larger deviationsfrom u0 p0 � p�ð Þ, which range from 42% to 197%, and donot necessarily converge as h → 0.

4.5. Conversion and Energy‐Flux Divergence

[40] Internal‐tide conversion and energy‐flux divergenceare important quantities for understanding internal‐tideenergetics. Although many studies equate these quantities,they may be substantially different due to energy dissipation[Carter et al., 2008]. The expressions for energy‐fluxdivergence and energy conversion (37)–(39) are comparedin Figure 3 for each topography.[41] Although the spatial structure of surface‐to‐internal‐

tide conversion is different for each decomposition, it isalways balanced by energy‐flux divergence (to first order),indicating that the balances are self‐consistent. Errors in theenergy balance are attributed to time variability in internal‐tide energy, energy advection, and dissipation. Distribu-tions of energy conversion when pi = p′ − ph and pi = p′(Figures 3d–3i) are similar over topography, indicatinginternal‐tide generation all along the slope. Conversely, whenpi = p′ − (ph + ph) (Figures 3j–3l), conversion is concentratedat the base of the slope as found by Lu et al. [2001]. Becauseinternal‐tide generation is expected in regions of large hxN

2

[Baines, 1982], which is constant in the subcritical andsupercritical simulations, decompositions which result inconversion all along the slope are more physically intuitivethan the decompositions where conversion is isolated at thebase of the slope.[42] Over a flat bottom, normal modes require the surface

and internal tides be dynamically uncoupled (Appendix A).Craig [1987], Lu et al. [2001], and Kurapov et al. [2003]all identified energy conversion through the free surface intheir energy budgets and attributed it to an error in the

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surface‐/internal‐tide decomposition. This error causesspurious oscillations in energy flux that coincide withsurface and bottom reflections of the internal tide.Removal of these oscillations may be important forquantifying nonlinear effects associated with internal‐tidereflection [Althaus et al., 2003; Gerkema et al., 2005].Here, the amplitude of spurious energy flux oscillations is10–60% that of the total flux due to topographic internal‐tide generation (Figures 3a–3c). An important consequence

of defining internal‐tide pressure as pi = p′ − ph is theelimination of spurious energy conversion and the isolationof topographic internal‐tide generation.

4.6. Role of ph

[43] In the subcritical topography simulation (Figures 3a–3c), removing ph from p′ corrects internal‐tide pressure atthe peak (x = −74 km) and trough (x = −126 km) of anenergy‐flux oscillation. In both of these locations (Figure 4),

Figure 2. Depth structure of tidally averaged internal‐tide energy flux for subcritical, supercritical, andOregon slope topographies. (a–c) Spatial structure of u0 p0 � p�ð Þ. (d–r) Spatial structure of u0p0 ,u0 p0 � p� � phð Þ, u0 p0 � pBð Þ, u0 p0 � pGlð Þ, and u0 p0 � pGrð Þ. pGl and pGr are integrated from the left‐and right‐hand sides of the domain, respectively. Percentages indicate average root‐mean‐square differ-ence between the plotted energy flux and u0 ðp0 � p�Þ, normalized by the root‐mean‐square of u0 ðp0 � p�Þ.

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ph is surface‐intensified. For example, at low tide, ph

dominates pi = p′ − ph at x = −74 km (because p′ � ph), andit represents half of pi at x = −126 km. Correcting internal‐tide pressure increases time‐averaged offshore energy fluxat x = −74 km and halves offshore energy flux at x =−126 km (Figure 4a). This result indicates that computationof energy flux without accounting for ph (i.e., followingKunze et al. [2002]) produces unphysical energy‐fluxdivergences and convergences that have the potential forbeing misinterpreted as nonlinear processes and/or turbulentdissipation.

4.7. Consequences for the Analysis of Observations

[44] Here, we examine the effect of ph in two observa-tional studies of internal tides. Energy fluxes were correctedby using pi = p′ − ph rather than pi = p′ at four stations atKaena Ridge, Hawaii [Nash et al., 2006], and five stations atthe Oregon continental slope [Nash et al., 2007; K. Martiniet al., manuscript in preparation, 2010] (Table 2). Surfacedisplacements estimated from TPXO are six times smaller at

Kaena Ridge (0.14 m) than the Oregon slope (0.92 m) butthe tidally averaged depth‐integrated internal‐tide energyflux at Kaena Ridge O (10 kW m−1) is much larger than thaton the Oregon slope O (500Wm−1). In both locations, energyflux corrections (defined as −hhu0p�i) are 10–50 W m−1. AtKaena Ridge, large depth variability in p′ marginalizes therole of ph in both the depth profile of internal‐tide pressureand energy flux (Figures 5a–5d). On the Oregon slope, ph hasa noticeable impact on the depth profile of internal‐tidepressure and energy flux, particularly at the surface(Figures 5e–5h).

5. Conclusions

[45] Five surface‐/internal‐tide decompositions have beenexpressed using different divisions of pressure. The mostsatisfactory decomposition results from defining internal‐tide pressure as pi = p′ − ph, where ph is due to isopycnalheaving by movement of the free surface. However, otherdecompositions also provide insight into the physics of

Figure 3. (a–c) Tidally averaged depth‐integrated energy fluxes. (d–l) Energy‐flux divergence (gray)and conversion (black), see equations (37)–(39), respectively. All trends are smoothed by a doublerunning average with span of 10 km. (m–o) Subcritical, supercritical, and Oregon slope topographies.

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topographic internal‐tide generation. Inconsistencies arisingwhen the surface tide is obtained from an unstratified flow[Baines, 1982] reveal the important effects of internal‐wavedrag on the surface tide. Small perturbations from the sur-face tide create significant changes in the depth structure ofinternal‐tide energy flux, making it impossible to interpretphysically. Similarly, when pressure is decomposed toeliminate internal‐wave drag [Gerkema and van Haren,2007], the resulting depth structure of internal‐tide energyfluxes are both sensitive to an undetermined limit of inte-gration, and indicate energy propagating from regions wherethere is no internal‐tide generation.[46] Using a simple depth average to decompose pressure

results in spurious surface‐to‐internal‐tide energy conver-sion over a flat bottom. Closely examining the energetics ofnormal modes (Appendix A) motivates the inclusion of apressure profile ph in the surface tide which accounts formovement of the free surface, even over a sloping bottom.The careful elimination of energy conversion related to thefree surface confirms the interpretation that CS is an error inthe surface‐/internal‐tide decomposition and does not havephysical relevance.[47] Including pressure due to topographic heaving ph in

the surface tide results in spatial distributions of internal‐tideenergy flux and topographic internal‐tide generation that areinconsistent with theory. This finding indicates that alltopographic heaving is a component of the internal tide, notthe surface tide.[48] Derivation and assessment of these surface‐/inter-

nal‐tide decompositions also leads to several practicalconclusions:[49] 1. The most satisfactory surface‐/internal‐tide

decomposition is described in section 3.4 and definesinternal tide pressure as pi = p′ − ph (33). The inclusion of ph iscrucial for eliminating spurious surface‐to‐internal‐tide energyconversion, CS. In other words, convergences in hhu0p0 icomputed following Kunze et al. [2002] are not entirely

associated with turbulent dissipation, but also have a contri-bution from ph through hhu0p�i.[50] 2. In the tidally averaged internal‐tidal energy

equation (37), depth‐integrated internal‐tide energy‐fluxdivergence is balanced by topographic internal‐tide gener-ation CT = −hxUp′∣z=−h, which arises from internal‐wavedrag.[51] 3. The depth structure of internal‐tide energy flux, u′

(p′ − ph), is unambiguous and can be computed as long as anestimate of surface displacement and full‐depth time seriesof velocity and buoyancy are available. When surface dis-placements are small ph ≈ 0 and internal‐tide energy flux isequivalent to u′p′ as proposed by Kunze et al. [2002].[52] 4. Observations over Kaena Ridge, Hawaii, and the

Oregon continental slope indicate ph alters depth‐integratedinternal‐tide energy flux by 10–50 W m−1. Changes in thedepth structure of energy flux due to ph are unimportant overKaena Ridge, but nonnegligible near the surface over theOregon slope. Over continental shelves, where internal‐tideenergy flux and topographic generation are reduced, ph canhave increased importance [Kurapov et al., 2003].[53] Separating the surface and internal tides correctly

over steep topography is challenging. Although thedecomposition suggested here will likely be improved byfuture investigations, it currently provides a complete,consistent, and practical method for examining internal‐tideenergetics.

Appendix A: Modes Over a Flat Bottom

[54] Normal modes over a flat bottom provide a physicaldescription of surface and internal tides as dynamicallyuncoupled waves [e.g., Wunsch, 1975; Gill, 1982; Pedlosky,2003]. Here we review the formulation of the problem,approximate the solutions, and scale the modal‐energyequations.

Figure 4. Effect of ph at two locations in the subcritical simulation. (a) At low tide, −ph dominates p′ −ph. (b) Time‐averaged energy flux is dominated by −hu0p�i. (c) At low tide, ph represents half of the mag-nitude of pi. (d) Subtracting hu0p�i reduces the time‐averaged energy flux by half.

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[55] All variables are represented as a series of modesusing separation of variables:

u; v; pð Þ ¼X1n¼0

un; vn; pnð ÞFn;z ðA1Þ

ðw; bÞ ¼X1n¼0

wn; bn �

Fn ðA2Þ

where hats denote the modal amplitude, which contains thex dependence, and Fn,z(z) is the z derivative of the n

th vertical‐structure function. Each mode satisfies the following differ-ential equations in frequency space (i.e., (·)t = −iw(·)):

Fn;zz þ 1

c2nN2Fn ¼ 0 ðA3Þ

wn þ 1

c2ni!pn ¼ 0 ðA4Þ

where cn are modal phase speeds and 1/cn eigenvalues. Theboundary conditions are a linear free surface: Fn,z = g/cn

2F atz = 0 and flat bottom:Fn = 0 at z = −h. This derivation assumesconstant stratification, but can be extended to situations ofslowly varying N2(z) by applying WKBJ normalization[Munk, 1981]. One set of solutions to (A3) and (A4) is:

Fn ¼ cnNsin

N

cnzþ hð Þ

� �ðA5Þ

where cn must be determined by the surface boundarycondition:

tanNh

cn

� �¼ N 2h

g

� �cnNh

; ðA6Þ

and the nondimensional parameter � = N2h/g is normallymuch less than 1% in the deep ocean [e.g., Pedlosky,2003]. Although cn may be solved numerically in (A6),they may also be approximated analytically when cn/(Nh)is very small or large. When cn/(Nh) is O(1), the right‐

hand side of (A6) is nearly zero and cn = Nh/(np) where n ≥ 1.These modes represent the internal tide and are consistentwith a rigid‐lid boundary condition. However, if n = 0, thencn/(Nh) � 1, and we should employ the small‐angleapproximation in (A6) to obtain c0 =

ffiffiffiffiffigh

p. The 0th mode

describes the surface tide and is not consistent with a rigid‐lid boundary condition because it does not have infinitephase speed.[56] Using the approximated modal phase speeds and the

nondimensional parameter, �, the surface and internal tidesare expanded in terms of pn. Because the vertical wave-length of the surface tide is much greater than h, the cosinesand sines of this mode are expanded using a Taylor series toO(�). The surface‐tide solution is approximated:

p0 ¼ p0 1� �1

2

zþ h

h

� �2" #

ðA7Þ

b0 ¼ ��p0h

zþ h

h

� �ðA8Þ

u0 ¼ p0c0

1� !2

f 2

� ��1

1� �1

2

zþ h

h

� �2" #

ðA9Þ

v0 ¼ � ip0f

c0!1� !2

f 2

� ��1

1� �1

2

zþ h

h

� �2" #

ðA10Þ

w0 ¼ � ip0!h

c20

zþ h

h

� �ðA11Þ

and the internal‐tide solution:

pn ¼ pn cosn�

hzþ hð Þ

h iðA12Þ

bn ¼ � pnn�

hsin

n�

hzþ hð Þ

h iðA13Þ

un ¼ 1ffiffi�

p pnn�

c01� !2

f 2

� ��1

cosn�

hzþ hð Þ

h iðA14Þ

vn ¼ � 1ffiffi�

p ipnfn�

c0!1� !2

f 2

� ��1

cosn�

hzþ hð Þ

h iðA15Þ

wn ¼ � 1

�

ipn!hn�

c20sin

n�

hzþ hð Þ

h i; ðA16Þ

where (A12)–(A16) can be summed over all n to account forthe complete internal tide.[57] Next, modal‐energy equations are formed by depth

integrating the dot product of the nth mode and totalvelocity. Horizontal derivatives can be written (·)x = iw/cn

Table 2. Summary of Reanalyzed Observations of Energy Flux atSelected Sites From Kaena Ridge and the Oregon Slopea

Site Depth (m) h (m) Energy Flux (W m−1) Correction (W m−1)

KR 2 2,048 0.14 −10,294 −7KR 3 1,300 0.14 −4,086 +10KR 4 969 0.14 −63 +43KR 5 970 0.14 −3,058 +50OR 2.5 2,204 0.92 332 −35OR 3.3 1,714 0.91 535 −57OR 4.3 1,241 0.92 123 −31OR 5.3 1,025 0.92 54 −13OR 5.7 752 0.92 −75 −33

aKR, Kaena Ridge [Nash et al., 2006]; OR, Oregon Slope [Nash et al.,

2007]. Energy flux was computed as hu0 p0 � p�ð Þi. The correction represents−hu0p�i.

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(which is proportional 1/ffiffi�

pfor n > 0). Using (A7)–(A16)

the first‐order surface and internal tide energy balances are:

h

2u20 þ v20 þ

g�20h

� t

¼ � hu0p0h ix � h unp0h ixþ h wnb0h i þ Oð ffiffi

�p Þ ðA17Þ

h

2u2n þ v2n þ

b2nN2

� t

¼ � hunpnh ix � �tpnjz¼0

þ h wnb0h i þ Oð ffiffi�

p Þ; ðA18Þ

respectively, where the relation hwnb0i = hw0bni has beenused.[58] These equations have several interesting properties.

First, the surface tide contains all the free‐surface energy buthas no internal energy (i.e., hb02/N2i / �). Furthermore, free‐

surface energy is only determined by the surface‐tide pres-sure at z = 0 (i.e., h0 not simply h), consistent with theinternal tide obeying a rigid lid. Secondly, it is nonintuitivethat curvature is important in the depth structure of p0 butnot u0. For instance, hunp0ix / 1 while hu0pnix /

ffiffi�

p. These

scalings are a result of p0 being a factor of c0 larger than u0,while pn is only a factor of c0

ffiffi�

p/(np) larger than un. Lastly,

it is interesting that two modal cross‐terms are present ineach equation. These represent energy conversion; however,the modes are dynamically independent because the termscancel each other. Summing over all n, and using thenotation of section 3.4, these relations can be written:

�tpijz¼0 ¼ h u

0p�

D Ex¼ h wib�

� �; ðA19Þ

where CS = −htp′∣z=0 has been previously interpreted asspurious surface‐to‐internal‐tide energy conversion.

Figure 5. Effect of ph in observations. Kaena Ridge, deep location: (a) p′ − ph (black) is indiscerniblefrom p′ (red) at low tide. (b) Time‐averaged energy fluxes are indiscernible. Kaena Ridge, shallow loca-tion: (c) pressures and (d) energy fluxes are indiscernible. Oregon slope, deep location: (e) ph impacts p′ −ph near the surface at low tide. (f) Offshore energy flux is enhanced by subtracting hu0p�i in the upper100 m. Oregon slope, shallow location: (g) ph is largest near the surface at low tide. (h) Subtractinghu0p�i nearly doubles the offshore energy flux in the upper 100 m.

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[59] Acknowledgments. This work was supported by NSF grantOCE‐0350543. The authors thank Alexandre Kurapov and Emily Shroyerfor their helpful suggestions. We also thank two anonymous reviewers,whose thoughtful comments greatly improved this manuscript.

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S. M. Kelly and J. D. Nash, College of Oceanic and Atmospheric Sciences,Oregon State University, 104 COAS Admin. Bldg., Corvallis, OR 97331,USA. (skelly@coas.oregonstate.edu)E. Kunze, School of Earth and Ocean Sciences, University of Victoria,

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