How and where the ocean tides dissipate
their energy are longstanding questions with both
oceanographic and astronomical implications. Two
decades ago, Doake suggested that flexing of Antarctic
ice shelves by the underlying ocean tide is an important
energy sink, perhaps accounting for over half the global
dissipation rate. Observational constraints on Antarctic
dissipation have been scarce. Here two new and
complementary ocean-tide models, both derived from
Topex/Poseidon satellite altimeter measurements are
used to determine the flux of tidal energy across 60øS
toward the Antarctic coastline. Our results show relatively
small fluxes and they therefore rule out Doake's
suggestion: Antarctica is an insignificant sink in the
global tidal energy budget.Copyrighted by American Geophysical Union

Egbert, Gary D.

Ray, Richard D.

Undetermined

ScholarsArchive@OSU

GEOPHYSICAL RESEARCH LETTERS, VOL. 24, NO. 5, PAGES 543-546, MARCH 1, 1997 The flux of tidal energy across latitude 60øS Richard D. Ray Hughes STX, NASA Goddard Space Flight Center, Greenbelt, Maryland Gary D. Egbert College of Oceanic &: Atmospheric Sciences, Oregon State University Abstract. How and where the ocean tides dissi- pate their energy are longstanding questions with both oceanographic and astronomical implications. Two decades ago, Doake suggested that flexing of Antarctic ice shelves by the underlying ocean tide is an important energy sink, perhaps accounting for over half the global dissipation rate. Observational constraints on Antarc- tic dissipation have been scarce. Here two new and complementary ocean-tide models, both derived from Topex/Poseidon satellite altimeter measurements, are used to determine the flux of tidal energy across 60øS toward the Antarctic coastline. Our results show rela- tively small fluxes and they therefore rule out Doake's suggestion: Antarctica is an insignificant sink in the global tidal energy budget. Introduction For well over a century, the secular slowing of the earth's rotation and the corresponding acceleration of the moon's mean motion have been attributed to "tidal friction" [Munk and MacDonald, 1960]. Clarifying ex- actly what that term entails--what physical mechanism is at work and where it occurs--contributes not only to our knowledge of ocean dynamics but also to our ability to decipher the history of the lunar orbit. An impor- tant constraint in resolving these questions is the total global rate of the tidal energy loss, and this is now well established: 2.5 + 0.1 terawatts (TW) for the principal lunar semidiurnal tide (M2) [e.g., Ray, 1994]. This rate is deduced from observations of the secular acceleration of the moon [Dickey et al., 1994], from observations of long-period perturbations in the orbits of artificial satel- lites [e.g., Cheng et al., 1992], and from direct calcula- tions of the rate of working of tidal forces on the oceans [Cartwright and Ray, 1991]. Note that the latter tells us the rate at which energy is input to the oceans, but it tells us nothing about how or where that energy is consumed. The leading candidate for the dissipation mechanism has long been bottom friction in the shallow seas, partly Copyright 1997 by the American Geophysical Union. Paper number 97GL50316. 0094-8534/97/97GL-50316505.00 because friction in the deep ocean and the solid earth are apparently one or more orders of magnitude too small [Munk and MacDonald, 1960; Ray et al., 1996]. But the question remains open: a rough estimate of the total dissipation in all shallow seas by Miller [1966] falls well short of the required 2.5 TW, while conversion of energy into internal tides may [$jSberg and Stigebrandt, 1992; Morozov, 1995] or may not [Bell, 1975] be impor- tant. It is with this background that Doake [1978] sug- gested that flexing of Antarctic ice shelves by the un- derlying ocean. tide may alone consume 2 TW (q-50O-/o) of tidal power, The observational evidence to support Doake's hypothesis is meagre and ambiguous. Support- ing it are a few spot tidal observations near the ice shelves which display large tidal "ages" (large phase differences between principal lunar and solar tides) and abnormal nonlinear tides [Cartwright, 1979; Pedley et al., 1986]. Both, particularly the latter, suggest large frictional dissipation in the region. Discounting the hy- pothesis are recent geodetic measurements on Ronne Ice Shelf by Vaughan [1995] who finds that measured ice flexures can be modeled by an elastic, as opposed to Doake's viscous or creep, theology, with reduced energy consumption. If dissipation near Antarctica is important, then there must exist a large flux of tidal energy crossing, say, the 60øS parallel. This is because the greatest part of the (semidiurnal) tidal energy enters the ocean in the lowest latitudes where the astronomical tidal potential reaches its maximum and where large expanses of the ocean have high tides in quadrature with the potential [e.g., Cartwright and Ray, 1991]. In contrast, the (semidiur- nal) potential drops to zero at the poles and the direct working on the ocean in polar regions must be rela- tively small. If tidal energy is consumed at ice shelves, it must cross the 60øS parallel to get there. We here es- timate the flux across 60øS from two recent global tidal models, both derived from Topex/Poseidon satellite al- timeter measurements. Satellites provide the only com- prehensive source of data in this remote location; the Topex data extend to 66 ø S, providing a wide band of ac; curate elevation measurements around 60øS, mostly in deep water where energy fluxes can be reliably inferred. In Miller's global compilation (referred to above) the Antarctic coastline was omitted, presumably because of scarcity of reliable data. 543 544 RAY AND EGBERT: TIDAL ENERGY FLUX ACROSS 60øS Although diurnal tides are strongly enhanced around Antarctica, their known global dissipation rates [Cart- wright and Ray, .1991] are too small to affect he argu- ß ment of this paper. It suffices to 'examine 'M2 only. We do, never[heless, give some figures for O1 and K1. Two Topex/Poseidon Models Topex/Poseidon [Fu et al., 1994] is the most accu- rate satellite altimeter mission yet flown. Owing to its dual-frequency altimeter, multi-channel radiometer, and multiple tracking systems, the satellite acts as an accurate tide gauge, yielding a precise time history of the height of the ocean. Harmonic decompositions of these measurements have resulted in a number of new global tidal elevation models [Le Provost, Bennett, a nd Cartwright, 1994], which have been shown to surpass any pre-Topex global model in accuracy [e.g. Shum et al., 1997]'. The two models used here are from an em- pirical response analysis by Ray et al. [1994] (version 941230; hereinafter denoted Model A) and from a gen- eralized inverse analysis by Egbert et al. [1994] (version TPXO.3; hereinafter Model B). Model B is an {assim- ilation model,' in which altimetry is incorporated into a numerical hydrodynamic model in a way that explic- itly accounts for errors in both the satellite observations .. and the assumed model dynamics. The grid resolution of Model A is 1ø; Model B slightly smaller. Compar- isons to tide gauges suggest that model accuracies in the Southern Ocean are comparable with the global ac- curacies reported by Shum et al. [1997]. For both models tidal current velocities must be de- .. termined by assumed dynamical relationships, but be- . cause these relationships and their implementations are so different, the two models are quite complementary in the present application. Currents for Model A are determine d (in deep OCean only) from tidal height gra- dients by solving the momentum equations of Laplace's tidal equations, With rigorous accounting for ocean load- ing and self-attraction but with no dissipation terms and no exp!ici, t enforcement of mass conservation (f rfurther details, see Cartwright et at. [1992]). Currents for Model B are determined simultaneously with the heights through the assimilation method; this method also invokes the Laplace equations, including the conti- nuity equation, but with simple scaling approximations for ocean loading and self-attraction and with frictional dissipation parameterized linearly in the current veloC, ..• ity. Conceivably, energy fluxes from Model B might be biased by prior assumptions about the nature of dissi- pation at the Antarctic coastline, so that there could be some circularity in our deductions from this model. But such circularity is manifestly absent for Model A' those currents are estimated, at each desired location, strictly from the local elevation gradients; the only rel- evant dynaxnical ssumption isthat friction is zero at such location. Results obta.ined from the two models are quite similar, sugg•esting that in practice the accurate el- evation measurements near 60øS tightly constrain tidal currents and fluxes there. We are thus confident that , our final conclusion. s ar e not dependent on the dynami- cal assumptions of Mod91 B at the Antarctic coastline. Energy Balance , Below we shall ignore the relatively small effects of ocean loading and self-attraction. Then at any location the tidal energy balance is given by D=W-V.P where the mean (integrating over a tidal cycle) oceanic dissipation D is balanced by the rate of working W on the ocean, less the flux divergence X7 ß P. The work rate W includes both gravitational tidal forces from the moon and mechanical forces from the body tide [e.g., Garrett, 1975]: w = a(u. P = gp where g is gravitational acceleration, p the density of seawater, • the tidal height, U'the tidal transport vec- tor (current velocity times ocean depth), 72 a combina- tion of Love numbers that accounts for the body tide's influence on the tidal potential (I), and brackets (.) de- note averaging over one tidal cycle. (Note that Gar- rett discusses a fourth term in the balance, an "equi- librium flux," that must be invoked when the work integral is computed in the more traditional manner: W = p (72(• O•/Ot). This addit:ional f ux term is unnec- ess .ary when W is computed as above.) The mean horizontal energy flux across any section $ (oriented by unit vector fi) may be calculated as a line integral' F = •sP'fidS = •s pg <•U.fi> do e This formula was originally derived by Taylor [1919] in his study of the flux into the Irish Sea. For our two adopted models, integration along all longitudes at 60øS yields -42 GW (Model A)'and -1 GW (Model B), With the negative signs implying atotal flux away from Antarctica. That the total integrated fluxes should be northward clearly implies that direct working by the .moon pn the Antarctic tides, although expected to be small, !s nonetheless nonzero. In fact, Model B may be used to estimate f W dS integrated ove r the complete area south of 60øS (Model A canrmt be used since it does not extend past the Topex limits of 66øS); this estimate yields 25 GW for the rate of working on the ocean in that area, giving a total dissipation south of 60øS of-!7 GW (Model A) and 24 GW (Model B), the former being physically implausible. Although there are differences between models, the RAY AND EGBERT: TIDAL ENERGY FLUX ACROSS 60øS $45 / / ß ( . .•,6• 15 Gigawattsl ß Figure 1. Mean meridional fluxes of M•. tidal energy crossing latitude 60øS, integrated along 2ø-wide longitude sections for Model A (left) and Model B (right). The noiselike behavior of Model A between 55øW and 30øW is suspect and may be due to inadequate spatial resolution ear rugged topography. important point is that they both rule out Doake's very large +2000 GW (+ 50%) flux. Detailed examination of the model flux estimates (Figure 1) shows good agreement except between 80øW and 50øW where Model A indicates larger flux toward the Patagonian Shelf off Argentina; this accounts for most of the 40 GW difference in the flux totals. Model A also appears somewhat noisy in character over the 55ø-30øW band and is probably suspect there owing to inadequate spatial resolution near the rugged topogra- phy between the South Shetland and South Sandwich Island chains, and also perhaps to the assumption of no friction, which is unlikely in this region. On the other hand, Dushaw et al [1997] have shown that cur- rent vectors of Model B are, in general, overly smooth. Cartwright and Ray[1989], using methods similar to 1• 1111• .. ... ••A ':'" 6•66AA •AAAA • AAAAAA• •AAA AAAAAA•A .•; •AAA AAAA•AA •" '-.;. .. .. •AAAA AAA•• •AAAA AA••• :'% .... ••AAAAAA 44 •PP '. ;••AAA• (• ........ ' ( 200 kilowatts/meter Figure 2. Mean M2 tidal energy flux vectors from Model B (an update of the work described by Egbert et al., 1994), evaluated on a stereographic grid. The faint circle denotes latitude 60øS. those of Model A, previously inferred large northward fluxes onto the Patagonian Shelf, although their section was across 55øS. In keeping with their relative amplitudes near Antarc- tica, the diurnal tides are found to exhibit comparable dissipation rates. Across 60øS Model B yields energy fluxes of 13 GW (O1) and 19 GW (K1). The work rates integrated over the area south of 60øS give 18 GW (O1) and 16 GW (El), implying dissipation rates of 31 GW (O1) and 35 GW (K1), comparable to M•.. A more complete picture of the M•. energy flux vec- tors P throughout this region of the globe is given in Figure 2, based on Model B. Here we see that most of the patterns across 60øS evident in Figure I are part of larger circulation features, particularly the fluxes across 160øW-140øE, which capture part of the large circu- lation of energy flux rounding New Zealand, and the fluxes across 90øW-70øW, which capture part of the large flux that runs along the coast of Chile and then turns through the Drake Passage. Energy is seen to enter the outer Weddell Sea from the east along the Antarctic coast, to circulate through toward the west, and to leave near the Antarctic Peninsula where the flux merges with the Drake Passage flow and enters, and presumably dissipates on, the Patagonian Shelf. To the eye, Figure 2 suggests that there is little dis- sipation occurring near Antarctica and our detailed in- tegrated flux estimates confirm this. Compared to the total global M•. power consumption of 2500 GW, it ap- pears that dissipation near Antarctica plays a very mi- nor role in the global tidal energy budget. It is now of some importance to determine the flux of tidal energy entering all coastal and shallow-water regions of the globe. After 30 years, Miller's rough in situ estimates are in dire need of teevaluation. This task will be more difficult than the present determina- tion, where our evaluation of heights • and transports U could be confined (except near 50øW) to deep-ocean areas. As the shallows are approached, both elevations and currents grow larger and spatially complex, mak- ing mapping more difficult. Moreover, in lower latitudes where the tidal forces are larger, direct tidal working on the shallow-water tide must be carefully separated from the deep-ocean working, because some significant frac- tion of the flux across any section may be due (as here at 60øS) to energy flowing away from the shallows. Im- proved accuracies also require accounting for the fluxes and work rates associated with the self-attraction po- tential and with ocean loading. Finally, a careful eval- uation of uncertainties in all estimated terms of the en- ergy balance is required, including an analysis of dy- namical errors from faulty or overly simplistic hydrody- namical assumptions, all of which call for, in our opin- ion, a rigorous use of inverse methods. We anticipate important results along these lines in the coming few years, stimulated in no small measure by the accurate tidal observations from Topex/Poseidon. References Bell, T. H., Topographically generated internal waves in the open ocean, J. Geophys. Res., 80, 320-327, 1975. Cartwright, D. E., Analyses of British Antarctic Survey tidal records, Br. Antarc. $urv. Bull., 49, 167-179, 1979. Cartwright, D. E. and R. D. Ray, New estimates of oceanic tidal energy dissipation from satellite altimetry, Geophys. Res. Left., 16, 73-76, 1989. Cartwright, D. E. and R. D. Ray, Energetics of global ocean tides from Geosat altimetry, J. Geophys. Res., 96, 16897- 16912, 1991. Cartwright, D. E., R. D. Ray, B. V. Sanchez, A computer program for predicting oceanic tidal currents, NASA Tech. Memo. 104578, Goddard Space Flight Center, 22 pp., 1992. Cheng, M. K., R. J. Eanes, B. D. Tapley, Tidal deceleration of the Moon's mean motion, Geophys. J. Int., 108, 401- 409, 1992; Erratum, 110, 218, 1992. Dickey, J. O., P. L. Bender, J. E. Faller, X X Newhall, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, Lunar laser ranging: a continuing legacy of the Apollo program, Science, 265, 482-490, 1994. Doake, C. S.M., Dissipation of tidal energy by Antarctic shelves, Nature, 275, 304-305, 1978. Dushaw, B. D., G. D. Egbert, P. F. Worcester, B. D. Cor- nuelle, B. M. Howe, K. Metzger, Validation of a Topex/ Poseidon global tidal model using barotropic tidal cur- rents determined from long-range acoustic tranmissions, preprint, 1997. Egbert, G. D., A. F. Bennett, and M. G. Foreman, Topex/ Poseidon tides estimated using a global inverse model, J. Geophys. Res., 99, 24821-24852, 1994. Fu, L.-L., E. J. Christensen, C. A. Yamarone, M. Lefebvre, Y. M•nard, M. Dorrer, P. Escudier, Topex/Poseidon mis- sion overview, J. Geophys. Res., 99, 24369-24381, 1994. Garrett, C., Tides in gulfs, Deep-Sea Res., 22, 23-35, 1975. Le Provost, C., A. F. Bennett, D. E. Cartwright, Ocean tides for and from Topex/Poseidon, Science, 267, 639- 642, 1995. Miller, G. R., The flux of tidal energy out of the deep ocean, J. Geophys. Res., 71, 2485-2489, 1966. Morozov, E.G., Semidiurnal internal wave global field, Deep-Sea Res., 42, 135-148, 1995. Munk, W. H. and G. J. F. MacDonald, The Rotation of the Earth, Cambridge Univ. Press, 1960. Pedley, M., J. G. Paren, and J. Potter, The tidal spectrum underneath Antarctic ice shelves, J. Geophys. Res., 91, 13001-13009, 1986. Ray, R. D., Tidal energy dissipation: observations from as- tronomy, geodesy, and oceanography, In: The Oceans (ed: S. Majumdar et al.), 171-185, Penn. Acad. Science, 1994. Ray, R. D., B. Sanchez, and D. E. Cartwright', Some exten- sions to the response method of tidal analysis applied to Topex/Poseidon (abstract), Eos, 75, Spring suppl., 108, 1994. Ray, R. D., R. J. Eanes, and B. F. Chao, Detection of tidal dissipation in the solid Earth by satellite tracking and altimetry, Nature, 381, 595-597, 1996. Shum, C. K. et al., Accuracy assessments of recent global ocean tide models, J. Geophys. Res., in press, 1997. SjSberg, B. and A. Stigebrand, Computations of the geo- graphical distrbution of the energy flux to mixing pro- cesses via internal tides and the associated vertical circu- lation in the ocean, Deep-Sea Res., 39, 269-291, 1992. Taylor, G.I., Tidal friction in the Irish Sea, Phil. Trans. R. $oc. Lond., A220, 1-33, 1919. Vaughan, D. G., Tidal flexure at ice shelf margins, J. Geo- phys. Res., 100, 6213-6224, 1995. G. D. Egbert, College of Oceanic & Atmospheric Sci., Oregon State University, Corvallis, OR 97331-5503 R. D. Ray, HSTX, NASA/GSFC, Code 926, Greenbelt, MD 20771; richard.ray@gsfc.nasa.gov (Received December 11, 1996; revised January 24, 1997; accepted February 3, 1997.) 

The flux of tidal energy across latitude 60°S

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The flux of tidal energy across latitude 60°S

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