JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. C10, PAGES 22,475-22,502, OCTOBER 15, 2001

Estimatesof tidal energy dissipationfrom TOPEX/Poseidon altimeter data

Gary D. Egbert Collegeof Oceanicand Atmospheric Sciences, Oregon State University, Corvallis, Oregon, USA

RichardD. Ray NASA GoddardSpace Flight Center,Greenbelt, Maryland, USA

Abstract. Most of the tidal energydissipation in the oceanoccurs in shallowseas, as haslong been recognized. However, recent work has suggested that a significantfraction of the dissipation,perhaps 1 TW or more,occurs in the deepocean. This paperbuilds furtherevidence for that conclusion.More than6 yearsof datafrom the TOPEX/Poseidon satellitealtimeter are usedto map the tidal dissipationrate throughoutthe world ocean. The dissipationrate is estimatedas a balancebetween the rate of workingby tidal forces andthe energyflux divergence,computed using currents derived by leastsquares fitting of the altimeterdata and the shallowwater equations. Such calculations require dynamical assumptions,in particularabout the nature of dissipation.To assesssensitivity of dissipation estimatesto inputassumptions, a large suite of tidal inversionsbased on a wide rangeof dragparameterizations and employing both real andsynthetic altimeter data are compared. Theseexperiments and Monte Carlo errorfields from a generalizedinverse model are used to establisherror uncertainties for the dissipationestimates. Owing to the tight constraints on tidal elevationfields provided by the altimeter,area integrals of the energybalance are remarkablyinsensitive to requireddynamical assumptions. Tidal energydissipation is estimatedfor all major shallowseas (excluding individual polar seas)and comparedwith previousmodel and data-basedestimates. Dissipation in the openocean is significantly enhancedaround major bathymetricfeatures, in a mannerconsistent with simpletheories for the generationof baroclinictides.

1. Introduction earlier work the presentpaper addressesthe questionsof The problemof how and where the tides dissipatetheir "how much" and "where" tidal energyis dissipated,but not energy was listed by Wunsch[1990, p. 69] as one of four "how." Yet the answerto "where" holdssome obvious sug- "significanthard problems[in physicaloceanography] to be gestionsas to "how." solvedin the nextcentury." The problemcertainly has a long The total amountof tidal energy being dissipatedin the and frustratinghistory. That it continuesto attractattention Earth-Moon-Sunsystem is now well determined.The meth- is a reflectionof bothits intrinsicfascination and its impor- ods of spacegeodesy--altimetry, satellite laser ranging, lu- tance. The geophysicalimplications are far-reaching,rang- nar laser ranging--have convergedto 3.7 TW, with 2.5 TW ing from the historyof the Moon [Hansen,1982] to the mix- (for more precisionand error bars, see below) for the prin- ing of the oceans[Munk, 1997]. cipal lunar tide M2 [Cartwright, 1993; Ray, 1994; Kagan In a recent short paper [Egbert and Ray, 2000] (here- and Sandermann,1996]. Certainly,the bulk of this energy inafterreferred to asE-R] we addressedthe problemin light dissipationoccurs in the oceans.And within the oceansthe of accurateglobal estimatesof tidal elevationsmade avail- principal sink is most likely bottom boundarylayer dissi- able by the TOPEX/Poseidon(T/P) satellitealtimeter. That pation in shallow seas,the traditionalexplanation since the paperconcludes that approximately1 terawatt(TW), or 25- work of Jeffreys[1920]. There is much evidenceto support 30% of the globaltotal, is dissipatedin the deepoceans. The this view [Munk, 1997], including long experiencewith hy- presentpaper reexamines the subjectwith considerablymore drodynamicmodels [Le Provost and Lyard, 1997] and our thoroughanalysis of both data and methods, and it builds own recentestimates from satellitealtimetry (E-R). Another further evidence for the basic conclusions in E-R. Like our possibleenergy sink, conversionof energyinto internaltides and other baroclinicwaves, has long attractedattention but has proven extremely difficult to quantify [Wunsch,1975]. Copyright2001 by the American Geophysical Union. Baines [1982; see also Huthnance, 1989] concluded that Paper number2000JC000699. generationof internal tides at the continental slopesis an 0148-0227/2000JC000699509.00 insignificantsink (approximately12 GW for M2 overthe en-

22,475 22,476 EGBERT AND RAY: TIDAL ENERGY DISSIPATION tireglobe). But scattering bydeep-ocean bottom topography some work and flux termsfrom first principles.Section 4 may be moreimportant than generation at continentalslopes providesa brief overview of the methodsused for estimat- [Sj6bergand $tigebrandt,1992; Morozov, 1995]. The possi- ing tidal currentsfrom the T/P elevations.In section5 we ble importanceof thismechanism to deep-oceanmixing has provideestimates of energyfluxes, work terms, and dissipa- recentlybeen discussed by Munkand Wunsch[ 1998]. tion derivedfrom a numberof T/P tidal solutions,including As is well known [e.g., Lambeck,1977; Platzman, 1984], a completetabulation of estimatesof energyfluxes into all global charts of tidal elevationssuffice to determinethe shallowseas, and dissipation in selected deep-ocean areas. globalrate of workingof tidal forceson the ocean(see also In this sectionwe also considerin detail the sensitivityof section2). To deducelocalized estimates of energyfluxes ourresults to theprior assumptions about dissipation and andenergy dissipation requires corresponding charts of tidal bathymetryrequired to estimatecurrents. Section6 com- currentvelocities. Direct measurementsof currentsare, of paresour estimatesof shallow-seadissipation to empirical course,inadequate to the task;they are too sparse,gener- andmodel estimates from a numberof previousauthors. The ally too noisy,and often contaminatedby barocliniceffects. geophysicalimplications of our resultsare brieflyexplored To makeprogress, one must invoke dynamics to infer cur- in section 7. rentsfrom elevations,and this unfortunatelymeans making Asin E-R,discussion islimited to the principal lunar con- assumptionsabout dissipation. The problem appears inher- stituentM2. The principalsolar constituent S2 is confounded ently circular. by insolationand atmospheric effects that considerably com- Assessingthe degreeto which dissipationestimates can plicatethe main issue, while the major diurnal tides are less be madeinsensitive to dynamicalassumptions is the heart well determined.As notedabove, M2 accountsfor approxi- of the problem. It shouldbe clear that simplyfitting a nu- matelytwo thirdsof the total tidal dissipation. mericaltide modelto satellitemeasurements and evaluating themodel's dissipative terms is toosimplistic an approach. 2. GlobalDissipation Rate for Earthand Theresulting dissipation would be overly sensitive to model Oceans assumptionsand parameterizations.For example,using the typicalbottom drag dissipation(parameterized as quadratic The primarypurpose of thispaper is to determineempiri- in velocity)would force all dissipationinto shallowseas for callythe distribution of tidal energydissipation in theworld anyplausible specification of frictioncoefficients. ocean.The presentsection is a prologue,acting to establish The basic approachtaken in E-R and here is to esti- thetotal disgipation rate within the entireplanet and the par- mate currentsby fitting the dynamicalequations and the tition of this total amongthe solidEarth, oceans,and atmo- T/P elevationsusing weighted least squares, calculat e en- sphere.We reviewwhat kinds of measurementsor theories ergy fluxes, and then form a balancebetween the rate of constrainthe partition and what the current uncertainties are, workingof tidal forcesand the flux divergence.In general, Thepartitioning is not as accurately known as the planetary the resultsOf this calculationwill not equalthe dissipation total. in the assumeddynamical model, becausethe tidal eleva- tionsare tightlyconstrained to satisfythe satelliteobserva- 2.1. Planetary DissipationRate tions and thereforecannot, in general,also exactlysatisfy The theoryof the planetarytidal dissipationwas laid out the model equations. The implied "dynamicalresiduals" in comprehensiveand elegant form by Platzman[1984]. We are in some sensean additionalforcing term whosework- takeit as axiomaticthat the meanplanetary dissipation rate ing correctsthe assumeddissipation to be consistentwith equalsthe mean rate of workingby tidalforces throughout thealtimetrically constrained elevations (see below) [Zahel, theplanet. Platzman showed that this rate of workingcan be 1995; Egbert, 1997]. We showhere that with sufficiently expressedas a simplesurface integral involving the primary tight elevationconstraints and rational weighting of the mo- astronomicalpotential cI) and the secondary (induced) poten- mentumand massconservation equations, dissipation maps tial cI)',the latterresulting from tidal displacementswithin computedwith thisapproach are robustto a wide rangeof thesolid and fluid components of theEarth. For anysemidi- dynamicalassumptions. The near-globaltidal observationsUrnal tide like M2 the integraltakes the followingform: byT/P thusprovide a powerfulframework for addressingthe dissipationproblem. These ideas are discussedat length in section5, where D,o•=Wt,• =(5/4rcGR) fs(• O•'/Ot) dS,(1) the main resultsare given. Sections2-4 establishnecessary preliminaries.Section 2 reviewsthe data and theories that where G is the gravitationalconstant and R is the mean constrainthe total energydissipation rate in the ocean;any radiusof the Earth which forms the surface$. The angle localizeddissipation estimates must integrate to thesewell- bracketsdenote averaging in time over a tidal cycle. As determinedglobal totals. Section3, followingprimarily Platzmanpoints out, the simplicityof (1) is somewhatde- Hendershott[•972], forms energy balance equations used ceptive,because disentangling and understanding the com- for localizeddissipation estimates, and it reviewshow vari- ponentsof • requiresa mixtureof theoryand accurate ousterms simplify when inserted into surface integrals. Be- globalmeasurements which, although much improved since causeof thewide variety of energybalance equations that Platzman'swork in 1984, are still in somemeasure inade- havebeen used in the past,we find it usefulto reexamine quate to the task. EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,477

We supposethat both potentials ß and•t referto M2 accountingfor differencesin certaingeodetic and astronom- only. ThencI> is a degree2, order2 sphericalharmonic ical constantsand in mathematicalformulations used by the [Cartwrightand Tayler,1971] differentgroups, the weightedmean estimates are foundto be cI>(0,99, t) = g•/(5/96rr) P22(cos0) cos(rot q-2rp), D•2= 3.2954- 0.016 cm •2 = 128'69ø4-0'280, where(0, qo)are spherical polar coordinates, g is theaccel- erationof gravity, P22 is an associated Legendre function, ro implying isthe M2 frequency (1.405 x 10-4 s-l), and• isthe M2 D•o• = 2.536 4- 0.016 TW (4) potentialamplitude in lengthunits (63.194 cm). Because4> for the planetaryM2 dissipationrate. takesthis form, time averagingin (1) andthe orthogonality If similar analysesare made for all other lunar tides, in- of sphericalharmonics imply that evaluation of Dtota•requires cludingthe lunarparts of K1 andK2, thenthe total may be onlythe degree 2, order2 componentof 4>' that is in quadra- comparedwith estimatesfrom lunar laser ranging,which tureto cI>[cf. Hendershott,1972; Lambeck, 1977; Platzman, accuratelymeasures the Moon's secularacceleration as in- 1984]. ducedby tidalfriction. This has been done [e.g., Cartwright, The mostdirect route to determiningcI> ' is by analyzing 1993; Ray, 1994], and the resultsare in reasonablygood the orbital perturbationsof artificial satellites,which result agreement.Note that the lunar rangingmeasurements can- from forcing by the entire planetarypotential. The order 2 not establishthe rate for M2 alone,although some progress terms of cI>•, which include the one of interesthere, are es- has beenmade in separatingdiurnal from semidiurnaltidal peciallywell determinedfor semidiurnaltides, because they contributions [Williams et al., 1992]. The lunar acceleration induce long-periodperturbations that are easily observable is also sensitive to a small, but somewhat uncertain, contri- with presenttracking systems [Lambeck, 1977]. bution from tidal friction in the Moon itself [Williams et al., AlthoughcI> • is a combined(solid + ocean+ air) effect, 2001]. nearlyall publishedorbit analysesparameterize cI> • as if it arisesfrom a strictly elasticbody tide and a small residual 2.2. Partition Among Solid and Fluid Tides oceantide [e.g., Christodoulidiset al., 1988]. This parame- terization,although formally incorrect,does reflect the dom- As has been known for decades, "most" of the 2.54 TW inant tidal forces on a satellite, and it providesa fully satis- of the M2 tide is dissipatedin the oceans[Munk and Mac- factoryestimate of cI>• needed for evaluating(1) numerically. Donald, 1960], but establishingan exactsolid/fluid partition The potentialdue to the oceantide is usefullyexpressed in (accurateto the 4-0.016 TW uncertaintyof the total) is not terms of a seriesof waves,each a sphericalharmonic com- yet possible. Let us considereach of the major nonoceanic ponentof someunique degree and order. As notedabove, components. we requirehere only one of thesewaves: the degree2, order The M2 atmospherictide dissipatesabout 10 GW = 0.01 2 prograde("prograde" meaning moving in the directionof TW, accordingto Platzman [1991]. This estimateis based theMoon). Thenfor ourpurposes cI> ' reduces to on observationaldata, namely, surfacebarometer measure- ments at 104 meteorologicalstations, which were gridded ß ' = k2• + (4rr/5)GRp(1+ k•) x and subjectedto sphericalharmonic analysisby Haurwitz 22(cosø) cos(ot+ 2v- (2) and Cowley [1969]. A similar estimate,relying exclusively on numericalsimulations, was madeby Kagan and Shkutova The first term on the right is the potentialinduced by de- [1985], who arrive at 17 GW. Evidently,the atmosphereac- formationsof the body tide, the secondby deformationsof countsfor lessthan 1% of the M2 planetarydissipation rate. the oceantide and its load. Here k2 is the body tide Love Dissipationby the Earth's body tide (i.e., the part of the number,k} isthe degree 2 loading number, and p isthe den- solidtide forcedonly by the astronomicalpotential) has been sityof seawater.Nominal, real values for k2,k 2 areassumed estimatedin variousways by a numberof authors.Platzman in most orbit analyses,with the effect of anelasticityof the [1984] obtained an estimate of 32 GW based on the com- solid Earth (as well as all atmospherictides) absorbedinto plex Love numberk2 as calculatedby Zschau [1978]. But theocean tide amplitude D•2 and phase lag •2 throughfit- Zschauhad employeda solidearth Q appropriateto seismic tingto thetracking data. Inserting (2) withreal k2, k• into frequencies.A complexk2 computedrecently by B. Buf- (1) andsetting gR 2 = GM whereM isthe mass of the Earth fett and P.M. Mathews (personalcommunication, 1999, but yields cited by McCarthy [1996]) assumesa solid Q proportional to cod where a = 0.15, which appearsto be supportedby Dto•a•= p(1 q- k})ooGM•(24:r/5)l/2D•2 singt•2, (3) nutation measurements; their k2 = 0.3010 - 0.0013 i, im- plying a body tide dissipationof 130 GW. In another ap- an expressionallowing evaluation of the planetarydissipa- proach, Zschau [1986] used observationsof the Chandler tionrate from satellite tracking estimates ofD•2, •2' Wobble Q to boundthe dissipationto the interval [70, 140] A setof six recent estimates ofD•2, •b•2 for M2, deduced GW with "most probable"value of 120 GW. In yet another by severaldifferent groups using data from a varietyof satel- approach,Ray et al. [1996] combinedaltimetry and track- lites, hasrecently been compiled by Ray et al. [2001]. After ingestimates ofD•2 sin •P•2 to arrive at a figure recently up- 22,478 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

Table 1. Global OceanTide EnergyIntegrals

Model D•2,cm •P•2,deg Wtou,,TW

TPXO.4a 3.223 4- 0.017 129.99 4- 0.30 2.435 4- 0.017 TPXO.4b 3.211 129.86 2.430 TPXO.4c 3.229 130.06 2.437 GOT99.0 3.200 129.57 2.432 CSR4.0 3.229 129.41 2.460

datedto 110q- 25 GW [Ray et al., 2001]. From this evidence (Eo theequilibrium ocean tide height (allowing for the it appearssafe to concludethat the M2 bodytide dissipation bodytide "reductionfactor"). rateis well lessthan 10% of theplanetary rate and probably •^,• an equilibrium-typetide as inducedby self- close to 5%. attractionand loading. Dissipationin the solid earthload tide is more difficult to We assumethat the barotropic ocean tide satisfies the tidal determinebecause it dependson all sphericalharmonic com- equationsof Laplace,modified to includeeffects of an elas- ponentsof theocean tide through complex loading numbers tic Earthand a self-gravitatingocean. Writing these equa- k} [Platzman,1984]. Platzman concludes that the dissipa- tions in termsof volumetransports U (= uH whereu is the tionis anorder of magnitudesmaller than the body tide, and depthaveraged velocity and H is waterdepth) offers certain we areaware of no evidenceto disputethis. Some empirical advantages.The equationsthen take the form boundson the imaginary components of k} wouldbe highly desirable. 0U + f x U = -gHV([ -[m - [s•) - f', (5) In summary,a processof eliminationsuggests that the •)t oceantide mustaccount for roughly95% of the M2 plan- etarydissipation rate of 2.54TW. Althoughit is perhapsnot • = -V. U (6) Ot ' as well determinedas one would like, the nonoceanicdis- sipationis clearly small. In a discussionof local oceanic wherethe gradientand divergenceoperators are assumed energybalances, which is our main topic, errorsare suf- two dimensional,and the Coriolisparameter f is assumed ficiently large that considerationof the small nonoceanic orientedto thelocal vertical, g is gravity,and •' is a generic componentbecomes an unnecessarycomplication. There- frictionalor dissipative stress, including possibly a Reynolds forethroughout the remainder of thispaper we ignoreany stressthrough some turbulent viscosity scheme. The equilib- nonoceaniccomponents of energyloss and take all Love riumtides •m and• aregiven by and loadingnumbers to be real. Underthat assumption (m = •'2•/g the total oceanicdissipation rate, equalingthe totalrate of workingon the oceanby bothgravitational and mechanical = •'nOtn•n (solidfide) forces, can be expressed by thesame formula (3) n givenabove for the planetaryrate [cf. Hendershott,1972; 3 Cartwrightand Ray, 1991] but with the parameters D•2and an= 2n+ 1Pe' •2 determinedbysatellite altimetry. Table 1lists the global M2 ratefor theprimary ocean tide models used extensively where•'2 = (1 q-k2 - h2)and •,• = (1 q-k} - h•) forLove below.The quoteduncertainty for theinverse model is based numbersh2, k2 and loading numbers h•, k•, andwhere p, Pe on the error covariancesdescribed in section4; this uncer- arethe meandensities of seawaterand Earth, respectively. tainty is slightlytoo small, sinceit doesnot accountfor the Asidefrom the (asyet unspecified)dissipation term, (5) is smallsystematic error from assuming k2and k• arestrictly linear.At leastone of themodels discussed below employs real. additionalterms in (5)-(6), includingadvection and turbu- lenthorizontal viscosity, but these features are peripheral to the main discussionof energeticsin the openocean where 3. Tidal Energy Balance:Theory • << H andso are excluded from the equations above. Thisdiscussion requires a numberof differenttidal height Equation(5) is sometimeswritten [Hendershott, 1972] variables,which are best summarized at thebeginning: 0U + f x U = -gHV(( + (s - r/g) - y, (7) ( the usualocean tide heightas measuredby a tide •t gauge.It hasa sphericalharmonic decomposition whereF is the completetidal potential,corresponding to givenby • = En •n. bothastronomical and induced forcing: (•, the bodytide height. (l the load tide height. F = (1+ k2)(I>+ E(1 + k;)gOtn•'n. (s the solidEarth tide height:(s = (t, + (t. n EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,479

That is, F comprisesfour parts: the astronomicalpotential Aftersubtracting therest state potential energy -«pgH 2, ß , a perturbationk2• inducedby the body tide, a pertur- themean potential energy density over a tidalcycle is bation• k•ngOtn•'ninduced by theload displacements and a perturbation• gOtn•'nfrom self-attraction.Equation (7) (PE)= «pg ((•2)+ (2•s)). (9) follows from (5) because Thisexpression agrees with Hendershott [1972]. g(•m+ •s^L)= (1+ k2)•+ gE(1 + ktn)Otn•n- Considernow the vertically integrated time-averaged en- ergyflux P at a pointwhere the tidal current velocity is u. h2cI>- g E h'nan•'n Againfollowing Taylor, the energyflux acrossa fixed(Eu- lerian)surface ,_q bounding a columnof wateris the sumof = F - g(b -- g(l - F - thework done on the water mass by thehydrostatic pressure p of the surroundingocean, and the energy (potential (PE) Notice that •' + •'s in (7) is the geocentric(ocean + Earth) tide. and kinetic (KE)) advectedinto the volume Equations(5) and (6) give expressionsfor the local bal- anceof momentumand mass. These equations may be com- P.fidl = (pu).fidzdl+ H binedin anynumber of differentways to form an energybal- anceequation that describesthe trade-offbetween terms that may be identifiedwith work, flux, anddissipation at a given $((PE+KE)u).fidl.(10) location [e.g., Hendershott, 1972; Zahel, 1980; Le Provost In (10) fi is the outwarddirected normal to the surface,_q, and Lyard, 1997]. The different forms of energy balance and PE and KE are verticallyintegrated potential and ki- that are found in the literaturereflect omission of supposed neticenergy densities. The advectedkinetic energy term can secondaryterms like tidal loading and self-attraction,and be neglectedbecause it involvesthird powers in the current different definitions for work and flux, which do not have an velocity[Taylor, 1919]. The meanpressure throughout the obviousform when allowanceis madefor a nonrigidEarth. watercolumn is« pg(H + •), sothe first term on the right Of course,all correctbalance equations must arrive at equiv- sideof (10) is «pg(H + •)2u. fl. Combiningthiswith the alent dissipations. potentialenergy from (8), the meanenergy flux is A derivationof an energybalance equation from firstprin- cipleswas given by Taylor [1919], who wrote down expres- sionsfor potentialand kinetic energyin a fixed Eulerianvol- e = ,og(u(•'2 +(H +((s +(sH)) (11) ume of oceanand then used the principalof energyconserva- • pg (U(( + _Cs)), tion to derivethe balancebetween work doneby bodyforces and forcesacting on the boundary,advection of energyinto assuming,as usual,that H >> •. This too agreeswith Hen- the volume,and dissipation. In the nextsection we usea sim- dershott.The flux P computedusing (11) is plottedbelow ilar approachto clarify the properdefinition of work andflux in Figure 1. termson a nonrigidEarth. After droppingsmall terms,con- 3.2. Energy Balance Equation sistentwith the approximationsleading to (5) and (6), equa- tions identicalto thosegiven by Hendershott[1972, 1977] Equations(6) and(7) arereadily combined to form a mean are obtained. Because several of the terms take an unfamiliar energybalance of the form [Hendershott,1972, 1977] form (and becauseof sometypographical errors in the origi- Gravitationalwork Bottom flux Horizontal flux Dissipation nal papers),it is worth consideringthis derivationin further W - F - V. P = D. (12) detail. An expressionfor P is givenin (11). The meanrate of work- 3.1. Potential Energy and Flux on an Elastic Earth ing of tidal gravitationalforces (including the self-attraction forces)is The followingdiscussion briefly repeatsthe analysisof Taylor[1919] for potentialenergy and energy fluxes, but al- w = p (u. vr) (13) lowingfor elastictidal deformationsof the Earth. = pV. (FU) +p (ra•/at), The usualexpression for the potentialenergy density is derivedby integratingpgz overthe watercolumn [e.g., Gill, the lastline from substituting(6). The term -F represents 1982, p. 80]. When the seabedis also movingvertically, mechanicalworking by the solidtide againstthe ocean,or •'s = •'b -Jr-(l must be addedto z to accountfor the addi- equivalently[Platzman, 1985] F is a flux of energyfrom the tionalpotential energy associated with verticaldisplacement oceaninto the solid Earth. This flux is givenby thepressure of the entire water column. FollowingTaylor, we thus take fluctuationpg• timesthe downwardvelocity of the ocean- the extendedpotential energy density to be solid interface

PE-- pg (z+ (s)dz = «pg((2_ H2 + 2((s+ 2H(s). F = -{pg• O•s/Ot) = (Pg•s O•/Ot). (15) H (8) Note that our useof F and W followsPlatzman [1985] (ex- 22,480 EGBERT AND RAY: TIDAL ENERGY DISSIPATION ceptthat his variables denote global integrals); our (W - F) leastin Taylor'scase it wasprobably permissible because is equivalentto the (Wt) of Hendershott[1972]; andthe in theIrish Sea is muchgreater than the equilibrium tide. global integral of our (W - F) is equivalentto the W of Cartwright and Ray [ 1991]. 3.4. Energy Balance Over the Globe The work termsin (12) can be decomposedas W = Wa + As is well known[Hendershott, 1972; Platzman,1985], Wb and F = Fa + Fb, where furthersimplification of the balanceequation occurs when the spatialintegration is takenover the entire globe. All di- Wa = p (U. V(1 +k2)cI)) (16) vergenceterms of form V. (UX), whenintegrated to im- Wb = p (U. V En(1+ k;)gan•'n) (17) permeablecoasts, drop out. In addition,all termsinvolving Fa = pg (•'bOr/at) (•8) k• or h• dropout because they lead to factorsof theform (•'nO•'n/Ot) = 0. We are left with Fb = pg (•'t 3•'/3t). (19)

Thus Wa and Fa arisefrom the large-scaleastronomical po- • =Pg f•obe (•'m(O•'/Ot)) dS. (21) tential and body tide, while Wb and Fb arise from the load- ing and self-attractioneffects. Theseterms are individually Of all thesurface harmonics •n •'nforming •', onlythe n - evaluatedand plottedbelow in Plate 1. 2 termin quadraturewith the equilibrium tide •'• contributes to thisintegral. The result, along with the Saito-Molodensky In principle, evaluationof the quantitiesW, F, and P ! in (12) providesa method for mappingthe dissipationD relationk2 = k2 - h2,leads directly back to (3). throughoutthe ocean,regardless of the physicalmechanism of that dissipation. The complicationsarise from the un- 4. Estimatesof Tidal Volume Transports avoidableerrors in estimatingthese quantities, which depend 4.1. General Considerations on tidal volumetransports throughout the ocean. To map barotropictidal energydissipation using (11)- 3.3. Energy Balance Over Patches (15), we require the oceanand solid Earth tidal elevations •' and •'s,the completetidal potentialF, and volumetrans- One cananticipate that the spatialresolution of dissipation estimates derived from T/P tidal elevations will be limited, portsU. T/P altimetrydata provide direct constraints on the elevations,and a numberof nearlyglobal maps of •' arenow both by noise and by the incompletedata coverage. How- available.Given •', calculationof the tidal loadingand self- ever, the integral of D over a large oceanpatch might be attractionparts of F and •'sare straightforward[Ray, 1998]. reliably estimatedeven when the small-scaledetails of D in Theprimary challenge is estimationof thevolume transports the samearea are poorlydetermined. There are severalways U in the openocean. in which thesepatch integralscan be calculated,and some With elevationsalready specified (from the altimetry data) care is warranted.To make this explicit, let an overbarde- the momentumequations essentially involve two unknown note area integration,so that the dissipationin someclosed fields:U and.•'. Given.•', or strongenough assumptions regionis givenby D = W - F - V. P. By expressingW aboutthis dissipationterm, it would be straightforwardto via (14), ratherthan as p (U ßVI'), all termsin D involving directlysolve for U. For example,assuming that U are surfaceintegrals of divergences,and may thusbe re- placedby line integralsin which knowledgeof U is required .•' = rU/H, (22) only along open boundariesof the patch. Theseboundaries with the lineardrag coefficient r known,volume transports may be convenientlyplaced to avoid shallow seasand re- can be estimatedby substituting•' into (5) and solvingthe gions of complextopography where the currentsare more resulting2 x 2 linearsystem for U at eachpoint in the do- likely to be poorly determined. main.This is theapproach (with r = 0) usedby Cartwright Substitutingexplicit expressions from (11), (14), and(15), andRay [ 1989]in theirestimate of dissipationon the Patago- and simplifyingyields nianshelf from Geosat altimeter data. This simple approach requiresonly a local calculationand is thuseasily imple- • = Pgff ((ro+rs,) or/at) dS- mented,but the explicitassumptions about energy dissipa- tionrequired might be expectedto biasestimates of dissipa- Pgf ((•' - •'m - •'s^L) U.fi) dg, (20) tion computedfrom the resultingU. More importantly,this simple approachcompletely ig- wherefi is a unit vectordirected normal to the openbound- noresthe continuityequation, so the estimated volume trans- ary and away from the region of interest. These integrals portswill not in generalconserve mass [e.g., Cartwright et may be interpretedas a direct work integralminus a bound- al., 1980]. In fact, (6) providesa powerful constrainton ary flux integral (althoughthe terms do not correspondto U, independentof any assumptionsabout dissipation. It ef- our W and P). The flux-like integralincludes the so-called fectivelyprovides the additionalequations needed to extract equilibriumflux of Garrett [1975], extendedto includethe meaningfulinformation about both .•' and U from knowl- self-attractionand loading effects. Garrettpoints out that edgeof •' and henceto map dissipation.Enforcing (6) also Taylor[1919], and many others since Taylor, left outthe •'m leadsto smootherestimates of volumetransports which are term from the flux; some authorscontinue to do so, but at less affectedby noise in the tidal elevationfields. As we EGBERTAND RAY: TIDAL ENERGY DISSIPATION 22,481 shall showbelow, estimatesof dissipationcomputed with- tionsand volume transports Simultaneously and already al- out explicitlyenforcing mass conservation are too noisyto lowsfor misfitbetween the estimated•' andthe alfimetry be useful. data. Ourstrategy isto estimate the currents by fitting both (5) To allowfor nonlinearityin theshallow water equations, and (6) in a leastsquares sense. Since (6) is a simplestate- (23)is minimized with a two-stepprocedure. In thefirst step ment of massconservation that doesnot dependon any un- a priormodel is calculatedby time steppinga finitediffer- knownparameters, fit to thisequation should be emphasized. ence approximation tothe nonlinear shallow water equations Basedon comparison of a numberof globaltidal solutions to on a 1/4ø nearlyglobal (80øS-80øN) grid. At thenorthern pelagicand island tide gauges, Shum et al. [1997]estimate limit, elevations derived from the FES94.1 model of LeP- errorsin T/P constrainedopen oceanelevation fields to be rovostet al. [1994] are usedfor boundaryconditions. The of the order of 1 cm or so. Fit to (6) should be consistent forcingincludes the four dominantconstituents, M2, S2,K1, with this level of accuracy.If misfit to the continuityequa- O1, with tidal loadingand ocean self-attraction (i.e., •s^Lin tion is significantlylarger in the openocean, the estimated ($)) calculatedas described by Ray [1998]using elevations transportsare not truly consistent with the altimetrically con- fromthe TPXO.3 globalsolution of Egbert[1997]. Bottom strained elevations. On the other hand, the momentum con- frictionis assumedto be quadraticin velocity servationequations (5) entailseveral approximations and de- pendon imperfectly known bathymetry aswell as the dissi- .•' = coUv/H, (24) pativeterm •c. Mostof themisfit between the dynamics and the altimeter data shouldthus be accommodatedby these wherev is the total waterspeed (including all tidal con- equations. stituents),and the nondimensionalparameter co = 0.003. The leastsquares fitting approach still requiresus to make Advection,the nonlinearterm in the continuityequation, a priori assumptionsabout dissipation in orderto complete anda horizontaleddy viscosity term HAh V2u, withAh = the definitionof the momentumequations. To verify that 103m 2 are alsoincluded in the prior modelcalculation, theseassumptions do not undulyinfluence the resultswe althoughthe effect of theseadditional terms appears to be usea largesuite of tidalelevation models (both T/P andsyn- minimal.In thesecond step we linearizethe dissipation us- thetic)with a wide rangeof assumptionsabout •c. We also ing thespatially varying velocities from the prior solution, usetwo distinctapproaches for estimatingU: thevariational omitother nonlinear terms from the shallow water equa- dataassimilation scheme of Egbertet al. [1994] (hereinafter tions,and use the reduced basis representer approach of EBF referredto as EBF) and a schemebased on least squaresfit- to approximatelyminimize (23). SeeEBF andEgbert and tingof theshallow water equations to griddedtidal elevation Erofeeva[2001] for furtherdetails on the computational ap- •'•o t,,,•y, 2001]ß For completenessand to •'•,•c,,t,te ;'; • sub• proach sequentdiscussion, we briefly summarizethe two methods For the inverse solutions described here we fit T/P data here. for232 orbit cycles, including data from all crossover points andone point between each crossover. Because of computa- 4.2. Data Assimilation tionallimitations the final inverse solution was computed on By assimilatingthe T/P altimetry data into the shallow a 3/4ø grid,instead ofthe 1/4 øgrid used for the purely hydro- waterequations, tidal elevationand volume transport fields dynamicprior model. To assess the sensitivity of dissipation can be estimatedsimultaneously. For this study, new M2 estimatesto assumptionsabout the dynamical errors, inverse tidal solutionshave been computed using a refinementof the solutionsare computedwith threedifferent assumed forms variational assimilationscheme described in EBF [see Eg- for El. In thefollowing we refer to theseinverse solutions as bert and Bennett,1996; Egbert, 1997; Egbert and Erofeeva, TPXO4a, b, and c. Further details on the error covariances 2001]. Briefly, the methodrequires minimizing a quadratic usedare givenin EBF and in section5.3. penaltyfunctional formally expressed as The assimilationapproach allows us to calculateformal error bars on the tidal fields and on the estimatesof tidal en- if[w] = (Lw- d)*E•1 (Lw - d) + ergyflux and dissipation. These error bars of coursedepend on theassumed a priorierror structure (i.e., El) andshould (Sw- a)*E•-l(Sw- a), (23) be interpretedin conjunctionwith othermeasures of solu- where w representsthe tidal fields (•' and U), S repre- tionstability and validation data. The posterior errors for the sentsthe shallowwater equations (including boundary con- inversesolutions are computed by a MonteCarlo method. ditions),a representsthe forcing,and d = Lw q- ea repre- First,a realizationof therandom forcing error is generated sentsthe altimetrydata. Ee and Ef are covarianceswhich with covariance•f. The correspondingerror in the tidal expressa priori beliefs about the magnitude and correlation model •W i is thenfound by solvingthe linearized shallow structureof errorsin thedata (ca) andthe assumed dynami- waterequations. The realization of therandom tidal fields is calequations, respectively. As in EBFwe assume a nonlocal then given by wi = w0 q-tSwi, where w0 is thesame prior dynamicalerror covariance Ef with spatially varying am- modelused for the inverse solution (i.e., the solution tothe plitudes,and a globallyconstant decorrelation length scale, astronomicallyforced nonlinear shallow water equations). generally5 degrees. The continuity equation (6) is assumedThe solution is sampledwith the spatial and temporal pat- exact,since the assimilation approach estimates the eleva- ternof thealtimeter, random data errors are added, and the 22,482 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

resultingsynthetic data vector •s invertedfor x•i. The differ- linearparameterization of bottomdrag (22) is usedin (5), encewi -•9i is thena realizationof theerror in theestimated with r variedover a wide range. tidal fields. If a number of realizations wi, i = 1,..., I are calculated,actual and estimated dissipation maps Di, l•i can 5. Results be computedfor eachof wi and•9i, anderror bars for D (or areaintegrals of D) canbe calculated.By adjustingthe scale Both the data assimilationand the least squaresproce- of the assumeddynamical error covariance Ef we canensure duresresult in elevationand volumetransport fields which thatthe dynamicalresiduals for the synthetictidal fieldsused can be substitutedinto (11)-(19) to yield estimatesof time- in the errorcalculation have amplitudes and spatial structure averagedfluxes, work terms,and dissipation. Some numer- consistentwith the actualdynamical residuals required to fit ical detailsof thesecalculations are givenin the Appendix. the data. Further details on the posteriorerror calculation E-R presenteddissipation maps derived in thisway for two are given by Dushaw et al. [1997] and Egbert and Erofeeva differenttidal solutions:GOT99, an empiricalcorrection [2OOl]. by Ray [1999] to the FES95.2 solutionof Le Provostet al. [1998], with currentscomputed using the weightedleast 4.3. Least Squares Inversion of Elevation Solutionsfor squaresapproach of section4.3, andthe TPXO.4 inverseso- Volume Transports lutiondescribed in section4.2. Basedon thesemaps, E-R concludedthat about25-30% of theM2 tidal dissipationoc- Using the weightedleast squaresprocedure described by curredin the open ocean,mostly in areasof roughbottom Ray [2001], volume transports,and henceestimates of en- topography.Here we considerthese results in muchgreater ergy dissipation,can be computedfrom any of the global detail, and we demonstratethe robustnessof theseempiri- T/P tidal elevationsolutions. Given a griddedtidal elevation cal dissipationestimates to detailsin the T/P tidal solutions, field •, U is estimatedby minimizingthe weightedmisfit to prior dynamicalassumptions and weightings,and errorsin the equations(5)-(6), bathymetry. (25) A4u[U, r] + wrA4r[U, r]. 5.1. Energy Fluxes and Rates of Working Here.A4u and A//r givethe squared misfits to thetwo equa- We beginwith a brief examinationof the individualterms tions,and the weightwr controlsthe relativedegree of fit in the energy balance equation (12). As these are almost to eachequation. In the limit of largewr, continuityis en- indistinguishablefor the full range of solutionsconsidered, forcedexactly, while in thelimit of smallwr themomentum resultsare shownfor only TPXO.4a. The mean barotro- equationswill be satisfiedexactly. As Ray [2001]shows, in pic energyflux for the M2 constituentis shownin Figure 1. theopen ocean currents estimated by solvingthis large least Thesefluxes immediately suggest that certainshallow seas squaresproblem are quitesimilar to thoseobtained by the are importantenergy sinks. Large fluxes are observeden- assimilationmethod discussed above and showgood agree- tering the Europeanshelf, the Norwegian, Greenland,and mentwith reciprocalacoustic tomography and current meter Labrador Seas, the Yellow, East China, and Timor Seas, and data. Furtherdetails, including treatment of coastalbound- the PatagonianShelf. The very substantialflux of energy ary conditionsand computationalprocedures are givenby passingfrom the SouthAtlantic into the North Atlantic is Ray [20011. astonishing,amounting to 420 GW (althoughthis is about In additionto wc, theleast squares solution will depend 200 GW lessthan the hydrodynamicestimate of Le Provost on the assumedform for the bottomdrag term in themomen- and Lyard [1997], even after accountingfor differencesin tumequations. To keepthe least squares problem linear, the the definition of flux). Other features to note include the

60'

30'

0 ø

30'

60'

Figure1. Energyflux vectors P for M2 derivedusing elevations and currents from the assimilation solution TPXO.4a. EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,483 pronouncedcounterclockwise flux of energy aroundNew body forcesand the moving seafloor(W - F) and the di- Zealand, and the flux parallelingthe west coastof North vergenceof the energyflux V ßP. Thus,for example,most America. The patternof energyflux inferred from the T/P of the energyinput to the oceanin areasof positivenet work data is generallysimilar to that obtainedby Le Provostand (red areasin Plate 1e) propagatesaway as barotropicwaves. Lyard [ 1997] from the finite elementhydrodynamic solution Plate 1 underscoresthe difficulty of estimatingdissipation, FES94.1. which is the differencebetween the two very similar terms Plate 1 showsthe cycle averagesof the work terms Wa, D = (W - F) - V. P. Evidently, great care must be ex- Wb, -F a, and -Fb defined in (16)-(19), their sum, and the ercisedin all phasesof thesecalculations. In particular,al- energyflux divergenceV. P. Note the differencein color thoughtidal loadingand self attractionare small compared scales: the terms associatedwith the large-scalepotential with the total work W - F, they cannotbe neglectedand, in and body tides (Wa and -Fa) are an order of magnitude fact, mustbe calculatedas accuratelyas possible. largerthan the termsassociated with loading and self attrac- tion (Wb and -Fb). Wa (the work done by the body forces 5.2. Localized DissipationEstimates associatedwith the large-scalepotential) is large only in the deepopen ocean where volume transports are significant.It Before discussingdissipation maps estimatedfrom the is generallypositive, with large areasof energyinput in the T/P data, we brieflyconsider dissipation in the purelyhy- South Atlantic and Southwest Indian Oceans, but there are drodynamicprior solutionused as the prior or firstguess for also areaswhere the oceanlocally does work on the Moon all of the assimilationsolutions. The prior was computed (andthe solidEarth). The globalintegral of Wa for TPXO.4a by time steppingthe nonlinearshallow water equations on a is 4.566 TW. 1/4o grid, and then averaging the elevations (0 andtransports The work term associatedwith the body tide (-Fa) has U0 ontothe 3/4ø grid usedfor assimilationof the T/P data. a magnitudecomparable to Wa but a very differentappear- Dissipationwas then estimatedusing (11)-(15). The result ance. There is significantwork done by the oceanon the is plottedin Plate2a. Note the occurrenceof bothnegative solidEarth in a seriesof large (blue) spotsaround the equa- (blue) and positive(red) dissipation.Negative dissipation tor. At midlatitudesthe situationis reversed,with a tendency is, of course,physically implausible and is indicativeof the for the body tide to cio work on the oceans. The body tide level of noisein the empiricaldissipation maps. Consistent alsodoes significant work on the oceanaround Australia and with theparameterization of bottomdrag with the quadratic in the northIndian Ocean. Fa nearlybalances the part of Wa law of (24), for whichdissipation is cubicin currentspeed, due to work by the body tide gravitationalpotential on the significantenergy sinks in the prior solutionare restricted to ocean.In contrastto Wa, Fa is not obviouslyaffected by the shallow seas and broad continental shelves where tidal ve- bathymetry(e.g., note the smoothnessof this term) and can locities are greatest. be quite large in shallowwater. In additionto showingwhere dissipation by bottomdrag The loading and self-attractionterms Wb and--Fb have is expectedto be large, Plate 2a illustratessome of the arti- smalleramplitude and tend to be dominatedby smallerscale factsthat can contaminateempirical dissipation maps. The features(especially Wb). Although Wb is also integrated plot has a slightly noisy appearancewith numeroussmall over the water column and obviouslyaffected by bathym- spotsof negativeand positive dissipation in the openocean etry, it is still large in some shallow seaswhere large tidal (e.g., in the North Atlantic, the westernPacific aroundNew elevationsand shortspatial scales can resultin large ampli- Zealand,the westernIndian Ocean near Madagascar, around tudesfor thegradients of •n(1 + k'n)gOtn•'n.For example, the Hawaiian Islands,and the KerguelanPlateau). Noise in notethe large amplitudeof Wb on the European,North Aus- Plate 2a (which is derivedfrom a purely numericalmodel) tralian, and Patagonianshelves, and near the Gulf of Maine resultsprimarily from averagingof the 1/4ø solutiononto a on the eastcoast of North America. The work done by the coarsergrid in whichsome bathymetric details are no longer load tide on the ocean (-F•,) is the smallest of the terms resolved.Other contributingfactors include the inexactcan- considered.Magnitudes peak in shallowseas and along the cellationin the energyequation of the Coriolis termson the edgesof continents.The global integralsof Wt, and Ft, are C grid and noisein the loadingand self attractionterms due 0.009 and -0.001 TW, respectively.The two termsshould in to truncationof the sphericalharmonic expansion used for principleintegrate to zero globally,since we have assumed their computation.In general,the smallblue spotsof neg- real Love numbers.The small discrepanciesarise from sev- ativedissipation occur in tandemwith red spotsof positive eral sources:(1) combiningloading and self-attractioncal- dissipation.The clearestexample is providedby the Hawai- culated from TPXO.3 tidal elevations with tidal elevations ian Ridge, which appearsred on the northside and blue on and currents from a different inverse tidal solution, TPXO.4, the south.The Azores,and Canary and Cape Verde Islands (2) nonvanishingof currentson the coastal boundariesin in the easternnorth Atlantic, Somoaand Fiji in the south TPXO.4a, and (3) incompleteglobal coverageof the tidal Pacific, the GalapagosIslands, and the MascereneIslands solution. These factorsalso lead to small discrepanciesbe- in the Indian Oceanhave a similar appearance.Compari- tween the global totals for work (Table 1) and dissipation sonwith Figure 1 revealsthat in all casesthe blue patches (Table 2) for each of the solutions. occur on the "downstream" side of the unresolved islands, The most strikingaspect of Plate 1 is the almostidenti- relativeto the directionof tidal energyflux. Thesefeatures cal appearanceof the net work done on the ocean by all result primarily from residualsin the dynamicalequations 22,484 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

' • E t ' E EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,485

-20

-40

-60 0 50 100 150 200 250 300 350 Figure2. Shallowseas and deep ocean areas used for integrated dissipation computations. Areas outlined with solidlines, numbered 1-28, includeall of the shallowseas where significant dissipation due to bottom boundarylayer drag would be expected. In particular,essentially all of thedissipation in theprior model (Plate2a) occurswithin these areas. Areas outlined with dashedlines and labeled A-I are deep-water areaswhich showenhanced dissipation in the T/P estimatesof Plate2. whichsimulate the interactionof thebarotropic tide with to- -V. U/iro agreeswith the tidal elevation•' to within about pographicfeatures present in the fine grid but not resolved 1 cm overmost of the openocean [Ray, 2001 ]. by thecoarser grid. The pairingof red andblue spotsaround The dissipationmap from TPXO.4a has the cleanestap- this unresolvedtopography implies that theseresiduals on pearance,while the map from the purely empirical DW95 averagedo no net work aroundany particularfeature. We solutionis noisiest,with significantareas of negativedissi- demonstratethis more explicitly below by computingarea pation throughoutthe ocean. However, all three maps have integralsof dissipationfor the prior solution. Becausethe many featuresin common. The areasof intensedissipation real oceanhas topographythat will not be resolvedin the expecteddue to bottom boundarylayer drag in the shallow dynamical equationsused to estimatecurrents, we should seas (e.g., Patagonian Shelf, Yellow Sea, northwest Aus- expectsimilar sortsof noise(and areasof negativedissipa- tralianShelf, European Shelf) are clearlyevident in all cases tion) in all of our estimatesof D. (compareto the prior solution of Plate 2a). The areas of The remaining three panels of Plate 2 show dissipation enhancedopen oceandissipation discussed in E-R are also mapsbased on threetidal solutionsrepresentative of the dif- seenin mapsfor all threeT/P-constrained solutions (but not ferent approachesused to estimatetides from T/P data: for- for the prior solution). For example, dissipationis clearly mal assimilationmethods, empirical corrections to a hydro- enhancedin the Pacificover the Hawaiian Ridge, the Tu- dynamicmodel, and purely empirical solutions. The first amotu archipelago,and over the back arc islandchains ex- two of thesemaps (Plates2b and 2c) are the TPXO.4a and tendingfrom Japansouthward to New Zealand. The west- GOT99hf estimatespresented in E-R. The third approach, ern IndianOcean around the MascereneRidge and south of a purelyempirical solution with no explicitreliance on any Madagascaralso exhibitsenhanced dissipation in all three dynamicalassumptions, is representedby the DW95 solu- of themaps. The Mid-Atlantic Ridge shows up mostclearly tion of Desai and Wahr[ 1995] (Plate 2d). For the GOT99hf in the assimilation estimate but is evident also in the other and DW95 solutions,volume transports were computedus- two estimates. In all three cases there is substantial dissi- ing the least squaresapproach of section4.2, with the lin- pation throughoutmuch of the North Atlantic. All of these ear friction parameterof (22) set to a relatively high value areaswhere dissipation is consistentlyenhanced are charac- (r = 0.03), and the weight wc of (25) chosenso that terizedby significantbathymetric variation, generally with

Plate 1. Termsin the time averagedenergy balance equation (12), derivedfrom the assimilationsolution TPXO.4a. In Plates la throughld the work termsare subdividedas in (16)-(19). (a) The work done on the oceanby the large-scalegravitational potential, and (b) the work doneby potentialterms associated with loadingand self-attraction.(c) and (d) Work done by the moving bottomon the ocean,associated with thebody tide •'b,and load tide •'t,respectively. (e) The sumof thefour work termsplotted in Platesla throughld, and(f) the divergenceof the energyflux V ßP. Note that Platesle and I f nearlycancel; the differenceis usedto estimatethe oceanictidal energydissipation rate. 22,486 EGBERT AND RAY: TIDAL ENERGY DISSIPATION elongatedfeatures such as ridgesand islandchains oriented are aroundHudson Bay (includingthe LabradorSea, Baffin perpendicularto tidal flows. Bay, and CanadianArctic straits),the Europeanshelf (in- Dissipationmaps computed for otherT/P-constrained tid- cluding the North Sea), the Yellow and East China Seas, al solutionsare grosslysimilar. Ratherthan plot all of these the northwestcoast of Australia,the Patagonianshelf, and maps, we computeintegrals of D over discretepatches of the northeast coast of Brazil in the area around the Ama- oceanto comparemore quantitativelythe large-scalepat- zon Cone. Other shallowseas with significantdissipation terns of dissipation. E-R presentedestimates of M2 tidal (greaterthan about 50 GW each) include the South China dissipationfor major shallowsea sinksand selecteddeep Sea,the St. LawrenceSeaway and Gulf of Maine (including ocean areas for TPXO.4a, GOT99hf, and several variants on the Bay of Fundy), the AndamanSea, East Africa (includ- these solutionswhich we considerin greaterdetail in the ing the MozambiqueChannel), New Guineaand northeast- nextsection: TPXO.4b,c (computedwith a differentdynam- ern Australia,and Antarctica.The dissipationin the Arctic ical error covarianceEl') and GOT99nf (basedon currents Oceanand Norwegian Sea is alsosignificant (together nearly estimatedwith "no friction," i.e., r = 0 ). Here we present 100 GW), but becausethe T/P data do not extendbeyond more completeresults for a larger set of tidal solutions,in- 66øN latitude,we do not attemptto divide dissipationbe- cludingintegrated dissipation estimates for all shallowseas tweenthese seas. The total, whichis determinedprimarily and continental shelves. These are divided into 28 areas with by the energyflux out of the northeastAtlantic (in an area boundariesindicated by the solid lines in Figure2. The ar- of goodT/P datacoverage), is similarfor all of theT/P solu- easchosen roughly follow the compilationof Miller [ 1966], tionsand thus appears to be reasonablywell constrained. thoughwe havemerged some adjacent small or low dissipa- For almostall of thesemajor shallowsea sinks the TPXO tion shelfareas (e.g., the narrowshelf along the west coast of and GOT99 dissipationestimates (red and blue symbolsin North America is treatedas one area for our computations). Plate3) agreewithin approximately 10-15%. Betteragree- We alsocompute the area-integrateddissipation for 11 open mentis obtainedfor isolatedshallow seas that can be cleanly oceanpatches, outlined by dashedlines and labeledA-I in separatedfrom other possiblesinks (e.g., the Patagonian Figure 2. Thesehave been chosen to correspondto the gen- Shelf (5), the St. LawrenceSeaway/Gulf of Maine (9), and eral areasof enhanceddissipation seen in Plate2. In contrast the Bering and OkhotskSeas (15) and (17)). Note that for to the shallow seas, none of these areas should contribute theseareas the error barscomputed for the TPXO.4a solu- significantlyto dissipationby bottomdrag (see Plate 2a). tion are small. Dissipationestimates based on the otherT/P Tidal currentsare unlikely to be well constrainedby al- solutions(green symbols in Plate 3) showmore scatter,but timeter data in shallow seas or in areas with small islands exceptfor a few outliersmost estimates are still within about or other topographiccomplications. However, by comput- 20% of the average.It shouldbe notedthat theseother tidal ing the dissipationintegrals using (20) we requirevolume solutionsare basedon muchless T/P dataand are generally transportsonly on the open oceanboundaries. By drawing of lower qualitythan the morerecent TPXO and GOT99 so- theseboundaries in deep water and avoidingareas of com- lutions[e.g., Ray, 1999]. There is alsogenerally excellent plex topography,reasonable estimates of dissipationcan be agreementamong all solutionsin areasof minimaldissipa- computedeven for shallowseas. This is also the approach tion (e.g., the westcoast of SouthAmerica, the eastcoast of usedby Miller [1966], who relied on a very sparseand only NorthAmerica (south of the Gulf of Maine), the westcoast partly reliable set of tidal elevationand currentobservations of Africa,and the coastof Australia(excluding the north- in his estimatesof shallowseas dissipation. west coastand Coral Sea). Resultsare given in Table 2 and Plate 3 for the five so- The area of greatestdisagreement between solutions is lutionsconsidered by E-R (TPXO.4a,b,c, GOT99hf,nf), for aroundIndonesia. Division of dissipationbetween the var- TPXO.3 (an older version of the inverse solution, described iouspatches in theseareas is problematic,since boundaries by Egbert [1997]), and for a selectionof additionaltidal so- mustbe drawnin shallowareas with complexbathymetry lutions derived from T/P altimeter data: DW95, CSR3.0, whereour estimates of volumetransports are most question- SR96, and FES95.2. For all of these additional solutions able. Agreementbetween all estimatesis muchbetter for the (which are summarizedby Shum et al. [1997]), volume total dissipationin the threeareas which shareboundaries in transportswere computedusing the least squaresscheme thisarea of complextopography (4, 8 and 19, the northwest with r = 0.03, andwc chosento emphasizefit to thecon- Australian shelf, the South China Sea, and Indonesia,re- tinuity equation.Table 2 also givesthe area-integrateddis- spectively).For this total the TPXO and GOT99 solutions sipationcomputed for the regriddedprior modelof Plate2a, againagree to within about15% (seeTable 2). anderror bars for the TPXO.4a solution,computed using the Summedover all shallowseas, all of the T/P dissipation Monte Carlo procedureoutlined above. For comparisonwe estimatescome up well shortof therequired total of 2.4 TW. also includetwo previouslypublished estimates of shallow With the exceptionof DW95 (which is a bit of an outlier, sea dissipationin Table 2; thesewill be discussedin sec- evenfor the globaltotal), all of the additionalT/P solutions tion 6. considered here are consistent with the TPXO and GOT99 For all of the T/P estimatesthe dominantsinks of energy resultspresented by E-R, with totalshallow sea dissipation are in a small number of shallow seas and broad continen- around1.6-1.8 TW. The remainingdissipation, required to tal shelves.The largest(greater than about 100 GW each) match the well-determinedglobal total must occur in the EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,487 22,488 EGBERT AND RAY: TIDAL ENERGY DISSIPATION deeperocean. For the T/P baseddissipation estimates this lar to thatobtained with the largefriction parameter r = 0.03 amounts to about 0.6-0.8 TW, or about 25-30% of the total. (Plate2c), althoughthere is a slighttendency toward smaller The breakdown into some of the major areas of deep openocean dissipation in someareas (such as the North At- oceandissipation is given in the secondpart of Table 2. The lantic) for the no-frictioncase. The similar appearanceis most significantareas (each accountingfor approximately confirmedmore quantitativelyby the area-integrateddissi- 100 GW in all estimates)are Micronesia and Melanesia in pationestimates of Table2. Basedon thiscomparison, and the western Pacific, the western Indian Ocean, and the Mid- furtherexperiments with intermediatevalues of r, we con- Atlantic Ridge. The first of these areas containsa num- cludethat estimatesof dissipationare relativelyinsensitive ber of significantelongated topographic features including to the assumedlinear friction coefficientover a wide range. the Kermedec, Tonga, Lau, and Norfolk Ridges north of In fact,the two valuesexplicitly considered here are extreme New Zealand. All of these are prominentin the dissipa- andcertainly bracket any plausiblevalue of r. tion maps of Plate 2. The secondarea includesthe Ma- The similarityof the GOT99 dissipationmaps to thoseob- scereneand SouthwestIndian Ridges and the Madagascar tainedwith the assimilationapproach provides further ev- Plateau.These specific topographic features again generally idencefor the insensitivityof our conclusionsto prior as- showup as areasof enhanceddissipation in Plate 2. Other sumptionsabout 3 v. For TPXO.4a we have assumedthe deep-oceanareas of note include the Hawaiian Ridge (con- quadraticdrag law of (24). With this quadraticparameter- sistentlyestimated to accountfor about 18-20 GW of dissi- ization,energy dissipation by bottomdrag can be large in pation), Polynesia(including the TuamotuRidge; about40 shallowseas where current speeds are large,but is smallin GW), and the South Honshu Ridge (about 50 GW). There the openocean (Plate 2a). Althoughthe spatialdistribution is also a significantamount of dissipation(generally around of dissipationresulting from this parameterizationis quite 200-300 GW) spreadaround the remainingopen ocean. In differentfrom that implied by a linear drag law with any fact, some of this remaining area (e.g., the North Atlantic valueof r, essentiallythe samepattern of tidal energydissi- between the northwest coast of Africa and the Mid-Atlantic pationis obtainedwhen the T/P dataare fit adequately. Ridge) has moderatelyrough bottom topographyand ex- 5.3.2. Work done by errors in f'. For the TPXO.3 as- hibits consistentlyelevated dissipation levels. AreasA-I of similationsolution used by Egbert[1997], purely linear dy- Figure 2 clearly do not containall of the roughtopography namicswere used with the bottomdrag again parameterized in the deep ocean. Note that the openocean dissipation for as (22) with r = 0.03. This relativelycrude treatment of the prior model is indeedvery near zero, demonstratingthat bottomdrag was found to be the dominantsource of error the noise associatedwith topographicfeatures has little net in the dynamicsassumed by EBF, especiallyin openocean effecton dissipationintegrated over larger patches of ocean. areaswhere the linear drag law resultedin excessivedissi- We concludefrom the comparisonsof Table 2 and Plate 3 pation[Egbert, 1997]. To correctfor this deficiencyin the that the large-scalepattern of tidal dissipationdoes not de- priormodel, and bring the elevations into agreement with the pend strongly on details in the estimatedtidal elevations, T/P data,significant misfit to the momentumbalance equa- providedthe fit to the T/P data is sufficientlygood. tion (5) wasrequired. The residuals•u to thisequation can be viewedformally as an extraforcing term, whichdepends 5.3. Sensitivity to Assumed Dynamics on the explicitassumed form for f'. If we evaluatedissipa- To demonstratethe insensitivityof dissipationto prior dy- tion directly using the estimatedcurrents and the assumed namicalassumptions, E-R comparedresults from five of the dissipationparameterization estimates summarized in Table 2 and Plate 3, TPXO.4a,b,c Do = pu..?, (26) and GOT99hf,nf. These comparisonsshowed that the re- sults are only weakly dependenton the assumedform for thenthe work doneby theseresiduals the friction3 v andthe prior dynamicalerror covariance. We expandon theseissues here, providingfurther details and WE = pu.•u (27) discussion.We also addressmore explicitly the importance of properweighting of the continuityequations for estima- mustbe includedin the energybalance equation (12). If misspecificationof 3v is the dominanterror in the energy tion of volume transportsand considerpossible effects of balanceequation, then the true dissipationis approximately errorsin the assumedbathymetry. 5.3.1. Sensitivityto assumed3 v. The mostobvious area D = Do- WE. (28) of concernis that both the assimilationand weightedleast squaresmethods require some sort of prior assumptionabout Equation(28) clarifieshow the dissipationestimates can be the dissipativeterm f' in (5). To keep the problemlinear soinsensitive to prior assumptionsabout the frictionalstress: for the weightedleast squares approach, we restricted3v to the dynamicalresiduals do work to effectivelycorrect the the simple linear drag law (22). By varying r one can eas- assumedparameterized dissipation and bring the modeled ily test sensitivityof resultsto the assumeddrag coefficient. elevationsinto agreementwith the T/P data. Plate 4a showsdissipation estimates for the GOT99nf solu- For TPXO.3. where the dissipationcomputed with the tion, computedby applyingthe least squaresapproach with assumedparameterization (Do) was too large in the open r = 0.0 (i.e., no friction). The dissipationmap is very simi- ocean,the work doneby the dynamicalresiduals (WE) was EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,489

8

E

E

i

.. . t .. J-_ ...... ! ....

I i I

,..q [

I 22,490 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

! I (1)Hudson Bay (2)European Shelf (3)Yellow Sea (4)NW Australian Shelf (5)Patagonian Shelf (6)NE Brazil (7)Arctic/Norweg. Sea (8)China Sea (9)St. Lawrence/Fundy (10)Andaman Sea - (11)E Africa/Arabia - (12)E Austr/N Guinea - (14)WC(13)Anta N. Americarctica (15)Bering Sea - (16)New Zealand - El (17)Okhotsk Sea - (18)India - (19)Indonesia-' 'n (20)Australia - (21)E. N. America - ,' (22)Carribean - (23)W. Africa+Med.- (24)Java-Sumatra - (25)Kerguelen Plat- (26)WC SA - (27)Japan - O TPXO4a (28)WC S. America - • TPXO4b 1:3 TPXO4c O GOT99hf (A)Micronesia/Melanesia • GOT99nf (B)W. Indian Ocean O DW95 (C)Mid-Atlantic Ridge • CSR3.0 (D)W Pac Trench [3 FES95.2 (E)MidPac Mt •, SR96 (F)Polynesia • TPXO.3 (G)E. Pacific (H)Hawaii (I)Galapagos

0 50 100 150 200 250 300 35O

Plate 3. Plot of area-integrateddissipation estimates computed for the 10 tidal solutionsof Table2. Red symbolsare usedfor the TPXO assimilationestimates, computed with threedifferent dynamical error covariances.Blue symbolsare usedfor the GOT99hf and GOT99nf estimates,computed with linear frictioncoefficients of r -- 0.03 and r -- 0, respectively.Green symbols are usedfor the othertidal solutions.The weightedleast squares procedure with r -- 0.03 was usedto estimatevolume transports for theseother solutions (except for TPXO.3). Error barswere computedfor the assimilationsolution TPXO.4a usingthe Monte Carlo proceduredescribed in the text. EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,491 22,492 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

søF(a) ;...-

20

-20

-40

-60

,40 ,-' '.| t "' ½ "' 20 - .•

o '", -,. •,

-60 ...... • ...,..- 0 50 100 150 200 250 300 350

_. -0.02 -0.01 0 0.01 0.02 mW/m• Plate5. Workdone by thedynamical residuals in theassimilation solutions. (a) For TPXO.3 a linear parameterizationof bottom drag • = rU/H wasused with r = 0.03,making the assumed energy dissipationin thedynamical model too large in thedeep ocean. To fit the altimeterdata, residuals are requiredin themomentum equations. In thiscase these do workin thedeep ocean to overcomethe excessivedrag. (b) ForTPXO.4a a quadraticparameterization of • wasused, so a priorithere is little dissipationin the deep ocean. Now the residuals tothe momentum equation do work in theopposite sense to increasedissipation almost everywhere in theopen ocean. EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,493 generallypositive, as illustratedin Plate5a. In contrast,with dynamicalerror covariance from 5 to 2.5 degrees.We also the quadraticdrag law usedfor the new TPXO.4a solution, slightlymodify the dynamical error variances from that used Do in the open oceanis now very small and W• is almost for TPXO.4a to increaseerror variancesfor individual grid alwaysnegative in deepwater, especially over roughtopog- cells containingsignificant fine scalevariations in bottom raphy (Plate 5b). Startingfrom two extremesfor Do (min- topography.This modified error covariance Ef tendsto put imal and excessiveopen oceandissipation), very different largeerrors (and thus potentially larger deviations of dissipa- dynamicalresiduals, which do work in oppositesenses, are tionfrom the prior) in areaswith topographic complications. requiredto fit the altimeterdata. However,very similar es- The result(Plate 4c) is as expected,with topographicfea- timatesof dissipationD are obtainedfor both cases,pro- turessuch as ridges and island chains more sharply resolved viding very strongevidence that significantopen ocean dis- (and also somewhatnoisier). However,the generalpattern sipationis requiredby the T/P altimeterdata. Note that a remainsessentially the same as for theother two cases.More similar analysiscould be appliedto the two weightedleast quantitatively,Table 2 and Plate 3 showthat detailsin the as- squaresdissipation estimates discussed in section5.3.1. For sumeddynamical error covariance Zf havevery little effect the caser = 0, Do = 0 and all dissipationin the final esti- on the large-scalepattern of dissipation. mate arisesfrom work done by the dynamicalresiduals. For 5.3.4. Importance of mass conservation. We next con- the r = 0.03 case,the dynamicalresiduals in the openocean siderthe effectof relaxingthe fit to the continuityequation. generallydo work in the oppositesense. As notedin section3, if the continuityequation is ignored Betweenthe assimilationand least squares approaches we and a linear dragparameterization is assumed,U can be es- have considereda very broad range of prior assumptions timatedlocally by solvingtwo linear equationsin two un- aboutthe frictionaldissipation •, includingboth quadratic knowns(i.e., (5)) at each point in the domain. In the least and linear drag laws with a wide rangeof drag coefficients. squaresapproach this entails setting the weightw c -- 0 in The generallygood agreementbetween all of the dissipa- (25). The resultingdissipation estimates are extremelynoisy tion mapsdemonstrates convincingly that our resultsare not (Plate 4d). The reason is not hard to understand. The flux unduly biasedby prior assumptionsabout the natureof dis- divergencemay be written sipation. 5.3.3. Sensitivity to assumed covariance. A second V. P = pgV. (U(( + (29) area of possibleconcern is the effect of the assumedprior error covarianceon the estimateddissipation. This is most = pg ((• + Cs)v. u) + pg (u. v(c + Cx)). (30) easilyexplored with the assimilationapproach where the co- variancesare explicit. Since computationof U alreadyrequires differentiation of For TPXO.4a a prior estimateof the magnitudeof errors the estimatedelevation fields (, direct evaluationof V. U in the momentumequations (5) was obtainedfollowing the requirescomputing second derivatives of a measuredfield, generalanalysis of EBF, but with allowancefor the improve- which necessarilymagnifies noise. However, if the continu- ment we have made to the dynamics.For the solutionscon- ity equationis enforcedexactly, then sideredhere the dynamical error variance is dominatedby errorsin the bathymetryand the effectsof unresolvedtopog- V. U = -iw(, raphy. As a result we have assumeda priori that errorsin the dynamicsare largestin placeslike the Hawaiian ridge or andthe divergenceof U neednot be computedby actualdif- the westernPacific, where many islandsand seamountsare ferentiation.Second derivatives are thuscompletely avoided not resolvedby our coarsenumerical grid. Although these when (6) is fit exactly(as for the assimilationestimates) and prior assumptionsabout momentum equation errors may be significantlystabilized when only a small misfit is allowed reasonable,it is also possiblethat they bias our dissipation (as for the leastsquares estimates). estimates.Away from topographiccomplications dynamical As we have arguedabove, there are good a priori argu- errors are assumedto be smaller, so that dynamical errors, ments for taking the continuityequation as a very strong and hence any required excessdissipation, will tend to be constraintin deriving estimatesof volume transports. The tile larger over rough topography.To assessthis possibilitywe re•u,t• o• t-,ate•ct, and" argumentof (29)-(30) showsthat considertwo variantson the preferreddynamical error co- thereare alsogood practical numerical reasons for enforcing mass conservation. variance. For the first case (TPXO.4b) we assumea spatiallyuni- 5.4. Inversion of Synthetic Data form dynamicalerror variance.As for TPXO.4a, decorrela- tion length scalesare assumedto be 5 degrees. The result- As a definitive test of the ability of the T/P elevation ing dissipationmap is given in Plate 4b. Comparedwith datato constraintidal dissipation,we appliedthe assimila- TPXO.4a the dissipationestimates are slightly smoother, tion procedureto datagenerated from syntheticmodel runs. with featureslike the Hawaiian Ridge less distinct. How- First, the nonlinearshallow water equationswere solvedby ever, the overall pattern is very similar, and all significant time-steppingon a 1/4ø grid,with severaldifferent input as- areasof enhanceddissipation remain. For the secondcase sumptionsabout friction and bathymetry.The tidal eleva- (TPXO.4c) we reducethe decorrelationlength scalefor the tion fields were then sampledwith a spatialand temporal 22,494 EGBERT AND RAY: TIDAL ENERGY DISSIPATION patternequivalent to thealtimeter, noise was added, and the and inversesolutions for all of the shallowand deep sea inverseapproach was usedto computetransports and esti- areasof Figure 2. For referencewe also showthe dissipa- matedissipation. Since the synthetictidal currentsat the tion computedfor the prior solution(which was usedas the original1/4 ø resolutionare available,we cancompare the "first guess"for inversionof the syntheticdata). As can be actualdissipation in thenumerical model with the inversion seen,the modifiedlinear drag coefficientsresult in signif- results.The procedure is essentiallyidentical to thatused for icant changesin dissipationin some seas. After inversion the Monte Carlo errorbar calculation,except that insteadof of the syntheticdata the estimatestrack the actualdistribu- specifyingrandom forcing and boundaryconditions for the tion of dissipationclosely, with typicalerrors of the orderof syntheticcalculations, we ran the forwardmodel with mod- 5-10%. Fits are somewhatpoorer for the North Australian ified dynamics. In all casesfor inversionof the synthetic Shelf, the China Sea, and Indonesia, which share boundaries data,we usedthe same3/4 ø grid, with the bathymetry,dis- in an areaof complexbottom topography. However, the sum sipation,and dynamical error covariance used for TPXO.4a. of dissipationin theseareas for the syntheticrun (490 GW) By computingthe synthetic"truth" with differentbathyme- is quite closeto the estimatefrom the inversion(497 GW). try we can assessthe importanceof this additionalsource of This behaviorof the dissipationestimates is consistentwith uncertainty. what we have seenwith actual data for this complex area: For thefirst set of syntheticruns the bathymetry was taken althoughthe total dissipationis well constrained,the parti- from the GTOPO30 database of Smith and Sandwell [1997] tion betweenindividual seasis not. Figure 3a confirmsthe averagedonto a 1/4ø grid. The bathymetryfor thesesyn- impressionof Plate 6, that thereis little deep-oceandissipa- theticruns was thusof higherresolution but otherwisecon- tion in the inverse estimate for the first case. The totals for sistentwith that usedfor the inversion.Dissipation was mod- all deep areasin Figure 3a are 38 GW for the actual syn- ified from the quadraticdrag law usedfor the prior inverse thetic solution, and -1 GW for the inverse estimate. For the solutionin severalways. Startingfrom the prior solution, secondcase of Figure 3b (wherewe haveincreased the drag we computeda spatiallyvarying linear drag coefficient that coefficientin deepwater) the deep-oceantotals are 484 GW wouldproduce the samedissipation as the quadratic law (for for the synthetic,and 505 GW for the inverseestimate. the prior solutioncurrents). We thenmodified the dragco- To testthe effectof errorsin bathymetryon dissipationes- efficient in selected areas and reran the forward model with timates,we modifiedthe bathymetryused for the synthetic this modifiedlinear friction. Plate 6 comparesthe actualsyn- calculations,while still using the standardbathymetry for theticand estimated dissipation maps for two variantson this the inversion. Resultsfor two variantson this experiment procedure.For the first case(Plate 6a) all dissipationwas are shownin Figure 3c and 3d. The spatiallyvarying drag confinedto shallow seas,but the drag coefficientin some coefficient used for these runs is identical to that used for seaswas modified (e.g., by settingit to 10% or 200% of Figure 3a. For Figure 3c uncorrelatedGaussian errors with the level calculatedfrom the prior solution).For the second standarddeviation 20% of the water depth were addedto case (Plate 6b) dissipationwas addedto deeperocean ar- the standard1/4 ø bathymetry.This makesthe syntheticba- eas.To do this we followedthe theoryof Sjiibergand Stige- thymetryquite rough, with a substantialnumber of extra brandt[1992] to estimatean approximatelinear drag coeffi- seamountseverywhere in the ocean.For Figure 3d smooth cient that accountsfor conversionof barotropictidal energy random errors (with a decorrelationlength scale of 5 de- to baroclinicmodes. The requiredbuoyancy frequency was grees)were addedto the standardbathymetry. For this ex- computedfrom the climatologyof Levitus[1999] and ba- perimentthe relativeperturbations to the bathymetrywere a thymetry(on a 5t resolutiongrid) was taken from Smith and functionof depthas follows: 25% for H < 100 m; 15% for Sandwell [ 1997]. Details of the calculation(which are some- 100 < H < 200 m; 10% for 200 < H < 1000 m; 6% for what involved but follow the developmentof Sjiiberg and 1000 < H < 3000 m; 3% for H > 3000. These relative er- Stigebrandt[ 1992] closely)are not importantfor ourdiscus- ror levelsapproximate the depthdependence of the statistics sionhere. For this syntheticrun thereis significantdissipa- of differences between the older ETOPO5 [National Geo- tion overareas of roughtopography in the openocean where physicalData Center,1992] databaseand GTOPO30 [Smith the theorypredicts larger linear drag coefficients (Plate 6b). and Sandwell,1997], and probablyprovide an upperbound Estimatesof dissipationfrom inversionof the synthetic on the magnitudeof large-scaleerrors in the bathymetrywe data are shown for the two cases in Plates 6c and 6d. The in- haveused for the T/P dissipationestimates. versionresults appear somewhat blurred, with someareas of Uncorrelatederrors in the bathymetry(i.e., with length negativedissipation, and a tendencyto be noisiestin areasof scalesbelow the grid resolutionused for the inversion)have roughtopography. However, the overallagreement between very little effecton the dissipationestimates (Figure 3c). In actual(synthetic) and estimated dissipation is quitegood for fact,results for thiscase are slightlybetter than when the in- both cases.In particular,there is no significantdissipation versionis donewith the correctbathymetry (i.e., Figure3a). in the deep oceanfor the first case(Plates 6a and 6c). For The major improvementoccurs for the three adjacentareas the secondcase (Plates 6b and 6d) the areasof assumeddeep notedabove (North Australian Shelf, China Sea, and Indone- oceandissipation are reasonablycaptured in the estimates. sia) and hasmost likely arisenby chance.The largerscale Figure 3 comparesdissipation in the syntheticforward errorsin thebathymetry used for Figure3d leadto somewhat EGBERTAND RAY:TIDAL ENERGYDISSIPATION 22,495

(1)HudsonBey (2)EuropeanShelf -t (3)YellowSea -I (4)NWAustralian Shelf (5)PatagonlanShelf (6)NEBrazil (7)N. Sea/Arctic (8)ChineSea (9)St.Lawrence/Fundy '1 (10)AndamanSee (11)EAfrica/Arabia. -1 (12)EAustr/N Guinea -1 (13)Antarctica -1 (14)WCN. America -1 (15)Bering Sea -I (16)NewZealand, (17)OkhotskSee •%._ -1 (18)IndiaI- '••.. (19)IndonesiaI-. • '1 (20)AustraliaI;•"• (21)E.N. America• (22)Carribean• -1 (23)W.Africa+Med.l• '1 (24)Java-Sumatra-:• (25)KerguelenPlat• -I (26)wc SA• (27)Japan• -I (28)WCS.America l (A)Micronesla/IdelaneslaI• '1 (B)W.Indian OceanS. (C)Mid-AtlanticRidge !• 't (D)WPac Trencl• -1 (E)MidPacMt• -I (F)PolyneslelG -I (G)E. Pacific• -I (H)Hawaii• -I

(I)Galapigo• I I I 0 lOO 2oo 3oo 0 100 200 300 0 100 200 300 0 100 200 300 GW Figure3. Comparisonofactual (solid lines) and estimated (dashed lines) dissipation in shallow seas and deepocean areas of Figure 2 for four synthetic runs. Symbols give dissipation forthe prior hydrodynamic solution.(a) Modified linear drag, restricted toshallow seas (corresponding tothe dissipation maps of Plates6a and 6c). (b) Modified linear drag, with drag over rough topography inthe deep ocean estimated followingSj•iberg and Stigebrandt [1992] (Plates 6b and 6d). (c) As in Figure3(a), but with 20% random errorsin bathymetry (nospatial correlation). (d)As in Figure 3(a), but with spatially correlated (5 degree lengthscale) errors in bathymetry(amplitudes as given in text). largererrors in the inverseestimates of dissipation.How- from our hydrodynamicprior solution, and from the finite ever,agreement between actual and estimated dissipation re- elementhydrodynamic solution FES94.1 [Le Provostand mainsexcellent. Lyard,1997]. Note thatfor FES94.1only the majorsinks We haverun experimentswith additionalvariations in andthe global integral were given in theoriginal reference. draglaws and errors in bathymetryand boundary conditions Dissipationin shallowseas computed from the purely hy- in variouscombinations. The results presented here are typi- drodynamicprior model tends to be largerthan in theT/P cal. Weconclude that the sampling pattern of theT/P datais estimates.The mostsignificant differences are for the Eu- sufficientto extractaccurate estimates of thespatial localiza- ropeanShelf and in theArctic/Norwegian Sea, where dissi- tionof dissipation,with a resolutionof 5ø or so,even if the pationin the priorsolution is large,and about 50% above truebathymetry is only imperfectlyknown. The synthetic the empiricalestimates. More generally,shallow sea dissi- modelingresults also suggest that small-scale unresolved to- pationin theprior and T/P solutionsis similar,with a slight pographicfeatures do notsubstantially affect dissipation in tendencyto largervalues in theprior. Agreementbetween eitherthe synthetic runs or theinverse solutions. the T/P estimatesand the hydrodynamiccalculations of Le Provostand Lyard [1997] is similar,also with a tendency 6. Comparisonto Other DissipationEstimates towardlarger shallowsea dissipationin the finite element model.The largest difference between the TPXO.4 prior so- In Table2 we includeestimates of energyflux intoshal- lutionand FES94.1 is in the Arctic/NorwegianSea, which low seasfrom Miller [1966](regrouped where appropriate is 157 GW in the prior and only 21 GW for FES94.1. At to correspondto thedivision of shallowseas of Figure2), leastsome of thedifference between the hydrodynamic mod- 22,496 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

40

20

.. -20

-40

I I I 50 100 150 200 250 300 350 0 0.5 I 1.5 2 x 10-2 m25-2

Figure 4. Magnitudeof the verticallyintegrated body force resulting from flow of the barotropictide over oceanbathymetry, estimated using the linear theoryof Baines[1982]. This givesa qualitativeindication of where forcing of internaltides is likely to be largest,which indeedis similar to areasof open-ocean energydissipation in Plates2 and4. els probablyreflects differences in the shallow sea bound- this sort of modelingstudy, Glorioso and Flather [1997] ariesused for our calculations(Figure 2), andthose used usedopen boundary conditions extracted from Schwiderski by Le Provostand Lyard [1997]. The total shallowsea dis- [1978]to modeltides on thePatagonian shelf. For M2 they sipationfor thetwo hydrodynamic models is in remarkably estimatedthe dissipationto be 228 GW, in goodagreement goodagreement, ataround 2.01 TW, well below the required with the 245 GW estimateby Cartwrightand Ray [1989] global total of 2.4 TW. basedon Geosataltimeter data, but more than twice the T/P Agreementbetween the T/P dissipationestimates and the estimates,all of whichare near115 GW. We suggestthat energyflux estimates of Miller [1966]is poorer.Some areas theT/P estimatesare preferable. Cartwright and Ray [1989] thatMiller estimatedto be majorsinks (the Bering Sea and estimateddissipation using an approachsimilar to ours,but Seaof Okhotsk,each estimated to be over200 GW) arerel- with currentscomputed without the extraconstraint of mass ativelyinsignificant (only 30-40 GW each)in all of ouresti- conservation(6). As a result, theseestimates should be ex- mates.Some other significant sinks in Miller'scompilation pectedto be very noisy,as indicatedby Plate 4d. Simi- are alsoconsistently reduced by a factorof 2 or morein the larly,there are alsoreasons to questionthe hydrodynamic T/P estimates(the west coast of North America,shallow seas results. Gloriosoand Flather [1997] increasedboundary aroundIndia, Japan,the AndamanSea). At the sametime, amplitudesfrom the Schwiderski[1978] solutionby 60% Miller's estimatesof flux into HudsonBay andthe Yellow to obtaina betterfit to coastaltide gauges,an amountthat Sea,two major sinks in all of theT/P estimates,are low by a appearsexcessive given known crrorsin Schwiderski'sso- factorof 2. Somewhatremarkably, given some of thelarge lution[Schrama and Ray, 1994]. Basedon experiencewith disagreementsin specificareas, Miller's [1966] estimatefor localfinite element calculations for theYellow Sea, Lefevre thetotal shallow sea dissipation falls neatly within the range et al. [2000] concludedthat regionaldissipation estimates 1.6-1.8TW whichbrackets most of theshallow sea dissipa- arevery sensitive to the assumedopen boundary conditions, tion estimates in Table 2. whichstrongly influence energy flux intothe model domain. SinceMiller's [1966] compilation,dissipation estimates In the Yellow Sea studyopen boundaryconditions were have been publishedfor a number of shallowseas. Most takenfrom the FES94.2solution. Tuning of the quadratic of thesehave been based on localhydrodynamic modeling, bottomdrag coefficient used in the hydrodynamic model had ratherthan actualin situ currentand elevationdata, as Miller very little effecton totaldissipation, which remained close used.For example, Sandermann [1977] used tide gauge data to the 182 GW foundby Le Provostand Lyard [1997] for from the Aleutianarc to provideboundary conditions for a this area (Table 2; note that this is also close to the T/P numericalmodel of theBering Sea. His dissipationestimate estimates).Since boundary conditions for mostpublished for thisarea (29 GW) wassignificantly less than earlier esti- regionalscale hydrodynamic models have been taken from mates[Jeffreys, 1920; Miller, 1966]. The T/P estimates(for olderglobal solutions of relativelylimited accuracy, conclu- a slightlylarger area) are in goodagreement (29-47 GW for sionsfrom these numerical studies should be treatedvery the TPXO and GOT99 solutions). cautiously.For the PatagonianShelf the T/P dissipation Unfortunately,there is rarelysufficient in situdata to ade- estimatesare probably more reliable, especially given the quatelydefine open boundary conditions for the sort of large extremelygood agreement between different tidal solutions shallowsea or shelf areasconsidered here, so recoursemust andestimation approaches, and the smallerror bars. be madeto a largerscale model. As a recentexample of By far the mostcareful empirical estimates of tidalener- EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,497 22,498 EGBERT AND RAY: TIDAL ENERGY DISSIPATION

geticsfor a major si.nkare by Cartwrightet al. [1980] for andsurface forces. Large-scale features in thesemaps de- the Europeanshelf area, includingthe North Sea and En- pend only weakly on the specificT/P tidal elevationmodel glishChannel. Basedon measurementsfrom bottompres- used,prior assumptionsabout the natureof dissipation,de- suregauges and currentmoorings, this studyestimated the tailsof priorcovariances, and errors in theassumed bathym- flux onto the Europeanshelf to be 250 GW. The part of etry. All of the T/P dissipationestimates considered are the shelfconsidered roughly coincides with our shallowarea in closeagreement on the distributionof dissipationamong 2. Our dissipationestimates are all somewhatlower, around shallowseas, and all exhibitsimilar large-scale patterns in 200 GW for the TPXO.4 and GOT99 solutions,but theseal- the openocean. Experimentswith syntheticdata reinforce low for work doneby the Moon andby the solidEarth. For the conclusionthat the coverageand accuracyof the al- TPXO.4a these work terms sum to -11 GW, and the actual timeter data are sufficient to allow accurate estimation of flux into area2 is 219 GW. More generally,about 10 GW smootheddissipation fields, even if the true bathymetryis shouldbe addedto thearea 2 dissipationestimates of Table2 notknown or completelyresolved. We stressagain that mass for comparisonto the energyflux estimatesof Cartwrightet conservationmust be stronglyenforced when estimating U al. [1980]. Allowing for the differencesin the work terms, to obtainthese stable and reproducible results. the altimetricestimates are thusconsistently about 15% less Resultsfrom all of our experimentsstrongly support the than those obtained from the in situ data. The error bars for conclusionof E-R thatapproximately 25-30% of theM2 en- the TPXO.4a dissipationestimates are fairly largefor this ergydissipation (0.7 4- 0.15 TW) occursin the deepocean. area(31 GW), andthe uncertaintyassociated with the in situ In theseareas, tidal velocitiesare too low for energyloss estimatesmust be comparable,given the sparseness(espe- by bottomdrag to be significant,at leastwith the traditional cially in depth)of the currentmeasurements. Allowing for quadraticparameterization. Zahel [1980] hassuggested that these error bars, we conclude that the in situ and satellite turbulenthorizontal viscosity may also play an important estimatesof energyflux are reasonablyconsistent. role in tidal energetics.We havenot explicitlyincluded this Cartwright et al. [1980] also provideddissipation esti- termin our energybalance equation, but dissipationby this matesof thisarea with finergeographic resolution. Flux into mechanismis implicitlyaccounted for in our genericstress the southernpart of theEuropean Shelf, between Ireland and f'. Syntheticcalculations of the sort consideredin section the Brittany coast, was estimatedto be 190 GW. The flux 5.4 showthat even with the verylarge (and perhaps implau- throughthe English Channel into the North Sea was esti- sible)values for the coefficientof horizontalviscosity as- mated to be 17 GW (with minimal flux north out of the Irish sumedby Zahel [1980] (Ah = 2x 105 m 2 s-l), littledissipa- Sea),for a net flux into the EnglishChannel and Irish Seaof tion wouldbe expectedin the deepocean, so this mechanism 173 GW. The comparablenet flux estimatesobtained with alsocannot account for theobserved deep ocean dissipation. TPXO solutionsare in reasonablygood agreement,rang- While our estimatesof barotropicenergy balance can- ing from 153-156 GW for the three covariances.However, notdirectly constrain dissipation mechanisms, the consistent agreementfor the GOT99 solutionsis poorer: 107 GW for spatialpattern in the deepocean is highly suggestive.Much GOT99hf and 138 for GOT99nf. The dissipationmaps in of the deep ocean dissipationis concentratedin areas of this area are quite noisy (see Plate 2) and even coarsede- roughbottom topography, particularly over ridges and island tails in the bathymetryof this areaare very poorlyresolved arcsoriented perpendicular to barotropictidal flows. This is in theseglobal solutions. For example,England and Ire- exactlywhat would be expectedif the energyloss from the landare not evenresolved as separate islands on thesegrids. barotropictide is from generationof baroclinicmodes as the The relativelypoorer consistency of theT/P solutionsamong stratifiedocean is carriedover steeptopography. themselves,and with the in situdata, at thishigher resolution It is instructiveto compareour dissipationmaps to a sim- is thusnot surprising.The resolutionof our globaldissipa- ple linear theory for generationof internal tides. Baines tion mapsis really quite coarse,probably no betterthan 5 ø [ 1982] assumeslinear equationsfor a rotatingstratified in- or so. This presentsus with difficultiesin comparingour viscidfluid, andwrites the tidal velocityas the sumof a baro- dissipationestimates to local data-basedstudies of tidal en- tropiccomponent and a baroclinicperturbation u = Ul q-ui, ergetics.With the exceptionof the Cartwright et al. [1980] and the perturbationof densityfrom the local staticprofile study,these have been restrictedto small geographicareas (/5) as p = Pl q- Pi, where Pl is the densityperturbation which are not reasonablyresolved by our globalscale study. causedby the barotropicmotion. Linear equationsfor the internalwave motionsui, Pi are thenreadily derived: 7. Discussion OtUiq- f X Ui q-p0 -1 (Vpi q-PigS) = F, (31) Throughexperiments with real and syntheticdata we have shown that T/P altimeter data provide sufficientlystrong wherethe depth-dependentbody force F can be computed constraintson M2 elevationsto yield useful information from the barotropictransport field (U = Hul) andbathym- aboutthe globaldistribution of tidal energydissipation. Pro- etry H as vided the altimeter data and the continuityequation are fit F = U. [ito-lN2zV(1/H)]•, (32) sufficientlywell when estimatingvolume transports, mean- ingful dissipationmaps can be computedas the balanceof whereN = (-gOzlS/po)1/2 is thebuoyancy frequency. barotropicenergy flux divergenceand local workingby body Equation(32) givesa simpleexpression for thebarotropic EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,499 forcingof the baroclinictide, which we have calculatedus- edge (e.g., Plates2b, 2c, 4a, and 4c), just where Figure 4 ing bathymetryfrom Smith and Sandwell [1997] averaged predictslarge forcing of internaltides. ontoa 5• grid,with N 2 estimatedfrom the Levitus [1999] Many of theareas where we havemapped significant open climatology,and volumetransports from the TPXO.3 solu- oceandissipation are alsoknown generators of internaltides. tion [Egbert, 1997]. Low-modeinternal tides phase-locked to the barotropictide The body force dependson the positionin the water col- can be seenas small modulationsin amplitudeand phase umn z but hasa constantphase at any location.To get some in along-trackestimates of M2 harmonicconstants from T/P idea where the body force is large, we computethe inte- data [Ray and Mitchum, 1996, 1997]. Particularlyclear ex- gral over depth of IF'l. The resultingmap of body force amplesare seen around the Hawaiian Ridge and the Tuamotu amplitude,plotted in Figure 4, is only a qualitativeindica- archipelagoin the Pacific.Numerical modeling of the inter- tor of whereconversion from barotropicto baroclinicmodes nal tide near Hawaii by Merrifield and Holloway [2000] is shouldbe energeticallysignificant, since the actualconver- able to reproducethe surfacepattern of low-modeinternal sion efficiencyat any point will dependon the shapeof the tidesquite well. Their estimatedbaroclinic energy flux is topographyin a largerarea. In particular,conversion should 9.7 GW. For the first baroclinicmode alone they obtain6.0 be mostefficient where bottom slopes are tangentto the rays TW, in reasonableagreement with 5.4 GW obtainedby Kang or characteristicsof the internalwaves [e.g., Baines, 1982]. et al. [2000] with a simpletwo-layer model. Our estimates Nonetheless,there is a good qualitativeagreement between of totalbarotropic dissipation in this areaare exceptionally maps of dissipationand internaltide body force magnitude well constrained(presumably due to the isolationfrom other over mostof the openocean. Almost all of the areaswhere sinksand the lack of nearbytopographic complications) to significantforcing of internaltides are predictedby the sim- around 18-20 GW in all cases(Plate 3 and Table 2). Al- ple linear model indeed exhibit enhanceddissipation, and though larger than the estimatedbaroclinic radiation, the conversely. T/P estimatesare not inconsistent,as the additional 10 GW The spatialpattern of dissipationmapped from the T/P of barotropicdissipation may generatelocal turbulenceand data is also reasonablyconsistent with the calculationsof mixing, rather than low-moderadiated internal waves. conversionof barotropictidal energy to baroclinic modes Our resultshave obviousimplications for how energy computedby Sjdibergand Stigebrandt[1992] using a sim- shouldbe dissipatedin numericaltidal models. More im- ple theoreticalmodel. We used formulae derived from this portantly,our findingthat a significantfraction of tidal dissi- modelto estimatelinear drag coefficients due to topographic pationoccurs over rough topography in theopen ocean could interactionsin the openocean for the syntheticmodel run of haveprofound implications for verticalmixing in theabyssal Plate6b. The resultingpattern of dissipationis quitesimilar oceanand possiblyeven for long-termvariations in climate to the empiricalT/P mapsof Plates2 and4. The agreement [Munk and Wunsch,1998]. of thesemaps with the syntheticinversion result of Plate6d Munk [1966] and Munk and Wunsch[1998] estimatethat is evenmore striking. Morozov [ 1995] alsoestimated energy a globallyaveraged diapycnal diffusivity of 10-4 m2 s-1 conversionto internalwaves in the deepocean, using a com- is requiredto maintainthe observedabyssal stratification. binationof the theoryof Baines[1982] and directmeasure- Withoutthis degreeof verticalmixing they arguethat the ments of internal tidal currentsand vertical displacements. meridionaloverturn circulation would shut down. Typical His mapsare lessdetailed than those of Sjdibergand Stige- backgrounddiffusivities in thepelagic ocean estimated from brandt [1992], but most of the areaswhere Morozov [1995] ocean microstructure[Gregg, 1989] or tracer release data inferssignificant conversion also show up in the empirical [Ledwellet al., 1993]are of theorder of only10 -5 m2 s-1, dissipationmaps. butmuch of the requiredmixing across isopycnals may oc- The globaltotal for the energeticsof M2 barocliniccon- cur in localizedhot spotswhich have been poorly sampled versionestimated by Sjdibergand Stigebrandt[1992] was [Munk, 1966;Armi, 1978]. Recentin situobservations sup- 1.3 TW and by Morozov [1995] was 1.1 TW. This is rather port this hypothesis[e.g., Polzin et al., 1997; Lueck and higherthan the roughly0.7 TW estimatewe obtainfrom the Mudge, 1997; Ledwell et al., 2000], with diffusivities as T/P data. We shouldnote, however,that our estimatesmay largeas 10 -3 m2 s -1 observedin specific areas over rough be somewhatconservative. To avoid topographiccomplica- topography. Ledwell et al. [2000] have further shown that tions, we have drawn boundariesfor shallow seas well out measuredturbulent energy dissipation rates are positively in the deepocean. At leastsome conversion to baroclinic correlatedwith spring/neapvariation in barotropictidal ve- modesmay occurin the areaswe have classifiedas shal- locities,suggesting that tidesmay provideat leastsome of low. This may particularlyoccur along continental shelves the neededforcing. where submarinecanyons and other along-shelfvariations Munk and Wunsch[1998] calculatethat approximately2 in bathymetrymay significantlyenhance local generation of TW of mechanicalenergy must be providedto stirthe ocean internaltides [e.g., Cumminsand Oey, 1997; Petruncioet andmaintain the observedstratification. As the workingof al., 1998],relative to thepredictions of the two-dimensional the winds on the oceansurface is estimatedto accountfor modelof Baines[1982]. The AndamanSea (area 10 in Fig- only about1 TW [Wunsch,1998], theyfurther suggest that ure2) providesan exampleof a shallowsea where there ap- tidesmay provideabout half the neededpower. Extrapolat- pearsto be significantdissipation specifically near the outer ing ourresults for M2 to all otherlunar and solar constituents 22,500 EGBERT AND RAY: TIDAL ENERGY DISSIPATION doesyield about1 TW, sothe tides could indeed conceivably With these conventions for calculation of the discrete dot providethe additionalpower. product,and of productsbetween scalar (½) andvector (U) Calculations of Samelson [1998] and Marotzke [1998] fields,scalars are alwaysdefined at the centersof the grid suggestthat the large-scaleocean circulation may be very cells,and vector fields at theedges. It is readilyverified that sensitiveto thespatial distribution of verticalmixing. An un- theusual vector identities such as V. ((pu) = (pv.u+u.v(p derstandingof whereand how thismixing occurs may thus holdexactly with theseconventions. This guaranteesexact be critical to realisticnumerical modeling of long-termpro- equivalencebetween the surface and line integrals discussed cessessuch as climatechange. A moredetailed understand- in thetext when these are evaluated numerically using the ing of the processesby which tidal energyis transferedto obvious metric terms. turbulentfluctuations in the deep oceanmay thusturn out to be necessaryfor sensiblemodeling of climate.More specu- Notation latively,maintenance of stratificationmay depend on tidally Unlessotherwise noted, the followingvalues for various forcedmixing througha processthat in turn dependson the geophysicalconstants are employedthroughout this paper stratification. Such an interaction between tides and strati- (seemain text for definitions). ficationmay be an important,but neglected,factor in long- GM 3.9860 x 10TM m 3 s-2. term climatefluctuations. Further experimental and theoret- R 6.371 x 106m. ical studieswill be requiredto clarify the role of the tides, H 0.63194m (M2). long seen as too periodic and regular to be interesting,in co 1.4052x 10-4 s-1 (M2). drivingor modifyingslower and more irregularocean pro- f2 7.2921 x 10-5 s-1. cesses. p 1035kg m -3. Re 5515kg m -3. Appendix' Discrete Energy Balance h2 0.609. Calculations k2 0.302. Harmonic constantsof volume transportand elevation h•, k• arefrom Farrell [1972]. fields are defined on a discreteC grid, with • defined at Acknowledgments. We thankBrian Beckley at NASA God- the center of each cell, and U and V defined on the verti- dardSpace Flight Centerfor his helpwith theNASA Pathfinderal- cal and horizontalcell edges,respectively. Because the indi- timeterdata. GDE thanksProf. Andrew Bennett for encouragement vidual field componentsare definedat differentgrid nodes, and manydiscussions that ultimatelyled to this work. This work and becausewe work in sphericalcoordinates, some care is waspartially supported by theNational Science Foundation (grants requiredto estimatedissipation using (11)-(19). OCE-9633527and OCE-9819518to GDE), and by the National Aeronauticsand Space Administration through the Topex/Poseidon The discretegradient and divergenceoperators in spheri- andJason-1 projects. cal coordinatesare definedin the usualway for the C grid. Thus the gradientoperator maps naturally from the • nodes (cell centers)to the U/V nodes(cell edges).The energyflux References vectorP = gp• H u involvesthe productof fieldsdefined at Armi, L., Someevidence for boundarymixing in the deepsea, J. bothedge and center nodes. 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