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 integralsof 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 distributionof tidal energydissipation in the world 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