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Once Again: Once Againmtidal Friction

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Once Again: Once Againmtidal Friction Prog. Oceanog. Vol. 40~ pp. 7-35, 1997 © 1998 Elsevier Science Ltd. All rights reserved Pergamon Printed in Great Britain PII: S0079-6611( 97)00021-9 0079-6611/98 $32.0(I Once again: once againmtidal friction WALTER MUNK Scripps Institution of Oceanography, University of California-San Diego, La Jolht, CA 92093-0225, USA Abstract - Topex/Poseidon (T/P) altimetry has reopened the problem of how tidal dissipation is to be allocated. There is now general agreement of a M2 dissipation by 2.5 Terawatts ( 1 TW = 10 ~2 W), based on four quite separate astronomic observational programs. Allowing for the bodily tide dissipation of 0.1 TW leaves 2.4 TW for ocean dissipation. The traditional disposal sites since JEFFREYS (1920) have been in the turbulent bottom boundary layer (BBL) of marginal seas, and the modern estimate of about 2.1 TW is in this tradition (but the distribution among the shallow seas has changed radically from time to time). Independent estimates of energy flux into the mar- ginal seas are not in good agreement with the BBL estimates. T/P altimetry has contributed to the tidal problem in two important ways. The assimilation of global altimetry into Laplace tidal solutions has led to accurate representations of the global tides, as evidenced by the very close agreement between the astronomic measurements and the computed 2.4 TW working of the Moon on the global ocean. Second, the detection by RAY and MITCHUM (1996) of small surface manifestation of internal tides radiating away from the Hawaiian chain has led to global estimates of 0.2 to 0.4 TW of conversion of surface tides to internal tides. Measure- ments of ocean microstrncture yields 0.2 TW of global dissipation by pelagic turbulence (away from topography). We propose that pelagic turbulence is maintained by topographic scattering of barotropic into baroclinic tidal energy, via internal tides and internal waves. Previous estimates by BAINES (1974); BAINES, (1982)) of this conversion along 150,000km of continental coastlines gave a negligible 0.02 TW; evidently the important conversion takes place along mid-ocean ridges. The maintenance of the abyssal global stratification requires a much larger expenditure of power, 2 TW versus 0.2 TW. This is usually attributed to wind forcing. If tidal power is to play a significant role here, then the BBL estimates need to be reduced. The challenge is to estimate dissipation from the energy flux divergence in the T/P adjusted tidal models, without prior assumptions concerning the dissipation processes. © 1998 Elsevier Science Ltd. All rights reserved "In 1920 it appeared that Jeffreys had solved the problem of tidal friction. We have gone back- wards ever since." These are the opening sentences of the 1968 Harold Jeffreys Lecture "Once again--tidal friction" (MuNK, 1968). At a 1982 symposium, CARTWRIGHT (1984) spoke to the subject: "Finally, I cannot leave the subject.., without at least a paragraph on a problem which has always been dear to Waiter's heart.., the dissipation rate. Twenty years ago, one of the main reasons for striving for a global M2 map was to provide lower bounds to the total loss of rotational energy in the Earth-Moon system... As my group's study of the northeast Atlantic showed, direct measurements of localized dissipation can be quite different from that assumed by numerical models. Trying "to quantify the localised distribution of energy sinks" (my quotes).., seems to be the last remaining obstacle to progress in our two centuries of effort to produce a set of correct tidal maps." Cartwright's assessment was made in 1982 at a symposium held on my 65th birthday. The following essay is dedicated to David Cartwright on his 70th. 8 WAklER MUNK I. INTRODUCTION A unique feature of this subject is that the total dissipation of tidal energy can be inferred accurately from astronomic observations. Independent estimates come from ancient eclipses, ~ from modem measurements of the increase in the length of day, length of month and lunar distance, and the tidal perturbation of satellite orbits. 2 After a rather stormy history, all methods now give 2.50 + 0.05 TW (1 Terawatt = 1012 W) for the dissipation associated with the lunar M2 component. (For orientation, 3.0 TW is the total lunar dissipation, and 4.0 TW is the dissi- pation from the Moon and Sun.) The agreement is one of the triumphs of 20th century science. Tides of the solid Earth 3 dissipate 0.1 TW, leaving 2.4 TW ocean dissipation. This is tiny com- pared to the solar radiation of 2 x 105 TW, and small even compared to the heat flow from the Earth's interior of 30 TW. It is comparable to the 2.9 TW of installed global electric capacity in 1995. 2. SURFACE TIDES In this section we review the generation of surface tides and various dissipating processes with which the generation is in balance. 2.1. The working of the Moon on the water In addition to the astronomic determinations referred to in the Introduction, the working of the Moon on the ocean surface tide can be computed directly (the Sun is ignored in this essay). Figure 1 illustrates how a lagging tidal bulge transfers spin momentum (the length of day) to orbital momentum (length of month). The display, which goes back to Kelvin, is a good rep- resentation of the solid earth tide which is almost of equilibrium shape with a small lag of 0.21 ° associated with 0.1 TW of tidal energy dissipation? The ocean tide has large local positive and negative phases relative to the equilibrium tide. The net oceanic phase lag of 130 ° (corresponding to an angle ½ 130 ° in Fig. 1) is associated with 2.4 TW of M2 dissipation. Let u and ~" designate horizontal velocity and vertical displacement, respectively, of a surface particle in a tidal motion, and let qb designate the tidal force. The working of the Moon on the ocean is proportional to (~.d~/dt) integrated over the world's oceans (CARTWRmHT, 1993, 127-- 8; HENDERSHOTT, 1981). There are positive and negative areas in each ocean basin (we are dealing with a small difference between large numbers), and the pre-satellite results varied between wide limits, 1.6 to 7.3 TW! The Topex/Poseidon-adjusted tide models of RAY et al. (1996) and KANTHA et al. (1995) are in excellent agreement with the astronomically derived 2.4 TW. A thorough review of the entire subject is given by KAGAN and SONDERMANN (1996). 2.2. The search for dissipators In a steady state situation, the net work done by the Moon on the water is balanced entirely by net dissipation of oceanic tides. With a global tide energy of 4 × 105 TJ, this implies that once every 48 hours all of the tide energy is renewed! During this time interval, tides travel 34,000 kin, far enough to place the entire ocean tide in contact with the marginal seas and continental shelves at the ocean boundaries. The global balance does not tell us anything about the geographic distribution of the dissi- Once again: once again--tidal friction 9 Fig. 1. Effect of tidal friction on the length of day and length of month. Centrifugal force and gravitational attraction are in balance at the mass centers of Earth and Moon, but the gravitational pull is greater than average on the side of the Earth facing the Moon (pulling the Earth away from its center), and less than average on the opposite side (pulling the center away from the Earth). Tidal friction retards the two bulges relative to the sub-lunar and anti-lunar points. The gravitational force exerted by the Moon on the two bulges retards the Earth's spin; the gravitational force exerted by the bulges on the Moon accelerates the orbital motion of the Moon pation. The modem treatment was initiated by the paper of G.I. TAYLOR (1919) on tidal dissi- pation in the Irish Sea. Taking d = Cl)p(u 3) W/m 2, CD = 0.0025, ( 1 ) tor the dissipation in the turbulent bottom boundary layer (BBL), Taylor estimated d = 1.04 W/m< Integrating over the area of the Irish Sea yielded D= f d da = 0.041TW. area Taylor 4 independently confirmed the result by computing the flux in and out of the Irish Sea, F=j'fd/-ffdl=O.O6OTW, (2) entr exit where 10 WALLER MUNK ( (,i, ~¢(t) Fig. 2. A simplified sketch of the energy flux balance in a shallow sea. ~(t) is the tidal force, p(t) is pressure, u(t) is the coastward component of the barotropic tidal current, st(t) is the tidal elevation, and ~ = d~/dt. The horizontal influx (there can also be a horizontal outflux) is h(p.u) = pgh(~.u) per unit width of coastline, and this vanishes if high tide lags flood current by 90 ° f = pg(~u)h W/m. Here p g ~" u -- pressure × velocity is the instantaneous horizontal energy flux per unit depth per unit width of entrance or exit (Fig. 2). One year after Taylor reported his results, JEFFREYS (1920) was looking for 28 Irish Seas to account for the global dissipation, and he found all he needed, and then some. (Jeffreys' estimate of 1.4 TW for the global dissipation is too low, he should have looked for 48 Irish Seas. z) Table 1 sketches some milestones. Jeffreys looked mainly at shallow seas by the BBL method. He took an unrealistic u -- 2.5 knots -- 125 cm/s in the Bering Sea which alone accounted for Table 1.
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