Once Again: Once Againmtidal Friction

Home , Earth tide, Tidal force, Walter Munk

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 marginal 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 dissipation 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 compared 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 representation 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 dissipation 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. Estimates of lunar surface tide dissipation in TW (1012 watts). For each of the columns we include only the largest reported dissipation areas.(Detailed configurations of the shallow seas for the three columns differ, and so the numerical values in Table 1 cannot be readily compared.) The numbers 2.7 and 3.5-4.3 in the line labelled "Astronomy" were not due to the named authors but from other contemporary authorities

JEFFREYS (1920) MILLER (1966) CARTWRIGHT ( 1977-- KANTHAet al. 1980) (1995) M2 only

Astronomy 1.4 2.7 3.5-4.3 2.4 Bering Sea 1.5 0.24 0.07 Yellow Sea 0. l 0. l 1 European Shelf 0.12 0.16 0.25 0.30 Okhotsk Sea 0.04 0.2 l Patagonian Shelf 0.13 0.24 0.16 Other shallow 0.5 0.9 1.4 Total shallow 2 2 1.7 2.1 Deep ocean 0.0 0.02 0.01 Total ocean 2.2 1.7 2. I Once again: once again--tidal friction I1

all his required dissipation. MmLER (1964, 1966) looked at shallow seas and shelves largely by the flux method, based on 50 pairs of published harmonic constants for tidal elevation ~ and velocity u. Let

~'=A cos(~ot- ~bl), u = Ucos(wt- ~b2).

If these are out of phase, then (~-u) = 0 and the flux f = 0; if on the other hand, +~ = 0b2, then the flux is at a maximum,

f= ½pgAuh W/m, as is nearly the case in the Irish Sea where flood current occurs just before high tide. In the latter case we are dealing with a progressive wave in shallow water for which the energy flux equals the wave energy per unit area times the shallow water group velocity ~/g-h, with U = "~g-hA/h, so that

F=ec~,= ½ pgA2"~-~= ½gAUh (4) is in agreement with the previous expression.

2.2.1. Pelagic tidal measurements There was a brief window (between the development of a deep-sea tide gauge in 1965 and the GEOSAT altimetry launch in March 1985) during which the observational program was dominated by the pelagic measurements. 5 The first deep station was occupied off California in August 1968 (CARTWRI6HT et al., 1969). Eventually about 348 stations have been occupied, many under Cartwright's initiative (SMITHSON, 1992). The data are of very high quality, with better signal/noise ratios than the traditional coastal records. They provide an excellent test for the validity of both models and altimetry. But the necessarily sparse deep-sea network is no competition to today's global satellite coverage. Still there remains a place for bottom pressure recorders to provide information for small areas, and to fill special needs such as long period tidal constants. CARTWRIGHT et al. (1980) have estimated tidal power fluxes using measured tidal currents and pressures. By a judicious placement of deep-sea instruments augmented by island and coastal stations, Cartwright and his collaborators were able to obtain the fluxes onto the European shelf (Fig. 3). The flux values are greater than those found by Miller (see Table 1). For the North Sea, Cartwright obtained 38 GW for BBL, far short of the 76 GW flux value. (But see also DAVIES et al., 1985). For the Irish Sea, Cartwright refers to Taylor's classical BBL estimate of 41 GW (compared to the 45 GW flux value) "... as the only plausible balance between the two methods", but then gives reasons why Taylor's BBL estimate should be cut in two. (There are other problems; see footnote 4.) For the combined area (Fig. 4), Cartwright's flux estimate of 266 GW is in good agreement with the BBL estimates of 272GW (Kantha, personal communication) and 271 GW (Le Provost and Lyard, this volume). Still I conclude that the BBL estimates must be used with caution. CARTWRI6HT et al. (1980) also calculated fluxes into a large off-shore area (Fig. 3). Large positive and negative fluxes are nearly in balance, and the total from the four sides plus the work done by the Moon is given by 3 + 26 = 29 GW. The area (3.3 × 106km 2) is about 1% ] 2 WAI/[ER MUNK

60 ° 334 -- N I I I I Y~ +77 I ( -1 I 76 I BBL 38 I • v I I I 47

"l +29 io ~BBL (K) 3 Irish Sea

I a °o Z +45 I oo ( o |~ 170 45 ° BBL 41 8 J- 196

Io °

30°W 15 ° 0 o

Fig. 3. The dissipation over and adjacent to the European shelf (adapted from CARTWRIGHT et al., 1980). Dots show location of tide gauges. Arrows give flux estimates in GW. For the North Sea, the combined flux Y = 60 + 17 = 77 GW and the working of the Moon of - 1 GW compare to a bottom boundary layer dissipation estimated at BBL = 38 GW. For the Irish Sea Z = 45- 0.1 compared to Taylor's (1919) estimate of 41 GW (now considered to be too high; the Moon is here negligible). For an offshore box the flux plus lunar estimates of 29 GW greatly exceed the BBL estimate of 3 GW (Kantha, personal communication)

of the global ocean, and so the computed dissipation rate, if representative, would account for the total global dissipation. The conclusion is that open sea flux estimates are not yet reliable. Once again: once again--tidal friction 13

62°N r ..... --~------"

+334

Open Ocean Box

+196

3SON ~ ...... 28.75° W

-- ..... "~ 35%

+61 Open Ocean Box

_( -21 /

+95

Fig. 4. Lunar tidal energy flux into the European and Patagonian shelf areas. The former estimate is based largely on measurements with deep-sea tide gauges and current meters, the latter estimate derives from satellite altimetry. For the combined European waters (North Sea plus English Chan- nel plus Irish Sea) the influx plus lunar working is 266 GW, about 10% of the global dissipation, and compares to a BBL estimate of 272GW for the combined area (Kantha, personal communication). Patagonian Shelf magnitudes are comparable. Together, the two shallow seas account for about 0.5 TW, 20% of the global dissipation. For the two open-sea boxes, the flux estimates greatly exceed the BBL estimates (Patagonia flux convergence is negative) 14 WA1.TER MUNK

2.2.2. Satellite altimetry The launching of GEOSAT introduced a new dimension into the tidal problem (see CART- WRmHT (1993) for a general discussion). But most of the effort has gone into "correcting" for tides as an unwanted noise so that the altimetry data can be used for other problems. With regard to tidal dissipation, the velocities v are now computed from the horizontal gradients of the measured elevations (so that the flux divergences are differences of differences). Unlike the calculation of the lunar working on the water, which is robust and not unduly weighted over the shallow sea areas, the complex situation over shelves and banks is crucial. The procedure is iterative, almost circular; one needs a fairly good understanding of the dissipation physics to develop a reasonably realistic model for deducing the dissipation. CARTWRIGHT and RAY (1989, 1990, 1991 ) applied the flux method to the Patagonian Shelf (Fig. 4). Here the work done by the Moon is negative, and the computed value 245-17 = 228 GW is nearly twice the estimate of Miller and considerably larger than the BBL value of 161 GW. For an offshore box the flux method yields + 2 - 21 = - 19 GW as compared to a BBL estimate of + 0.3 GW. The Patagonian results have a resemblance to those for the European shelf. The areas are comparable. The flux estimates into the shelves are similar, amounting together to 0.5 TW, 20% of the global dissipation. They are within 30% of the BBL estimates. For the adjoining offshore areas, the flux divergences do not significantly differ from zero (in our view), and are not useful for identifying dissipation processes. Altimetry estimates greatly improved with the August 1992 launch of Topex-Poseidon (TP) satellite. The fourth column in Table 1 is based on the global tide model of KANTHA et al. (1995), tuned by TP altimetry. (I am indebted to Lakshmi Kantha for providing some detailed numbers.) In Kantha's model, BBL dissipation (Eq. (1)) is included in the dynamic equations, and dissipation is computed accordingly. The dissipation excludes any processes other than the turbulent bottom boundary layer. Eleven major centers of shallow water dissipation are identified, adding up to 1.7 TW. Altogether the Kantha model yields 2.1 TW of shallow water dissipation. For the European shelf, Kantha obtains 272 GW compared to Cartwright's 266 + 16 = 282 GW. For the Patagonian shelf, Cartwright's 245 - 17 = 228 GW flux estimate 6 compares to Kantha's 161 GW and Lyard and LeProvost's 185 GW BBL estimates.

2.3. Deep bottom boundary layer

The hypothesis that dissipation is a matter entirely of turbulent bottom boundary layers with an u 3 dissipation leads to the conclusion that 99% of the ocean area accounts for less than 1% of the dissipation. Typical tidal currents are of order 1 cm/s in the deep sea, and of order 1 knot = 50 cm/s in shallow marginal seas, consistent with the foregoing result. The numbers also set the stage for a "catastrophe" point of view: a few singular situations provide much of the answer. Whether or not the deep sea situation is important for solving the dissipation problem, we need to understand the numbers. For orientation, we use

A = 0.5 m, U = "~g-h.A/h = 2.5 cm/s (5) for the amplitudes of elevation and current velocity of the surface tide in h = 4 km of water. Allowing for (Icos31) = 4/37r, the mean BBL dissipation is computed according to

d = 0-.u) =

for a global total of less than 0.01 TW. But there are many problems with this estimate. Co is not a constant but a (weak) function of the stress. Equation (6) pertains to a steady current in a non-rotating reference frame. GILL ( 1982, 328--332) discusses the complex issues associated with a variable Ekman boundary layer. At the time "Once again--tidal friction" was written, the oceanographic community was not aware of the persistent presence of mesoscale activity. Let Urns ---- 5 to 10 cm/s designate typical mesoscale velocities and assume u,,~. to be in the same direction as the tidal u(t). Then since the tidal (u) = 0, ((Ums "l- U)3) ~. Ums3 -I- 3Ums(U2). Presumably the second term is pertinent to tidal dissipation, and this term is in the ratio of (97r/8) Urns /U to the term in (6). This argument does not apply to the marginal seas where the tidal flow dominates. Direct observations of BBL dissipation are few. CHRISS and CALDWELL (1982) have measured ~- = pv.d u/dz within acm thick viscous sub-layer on a sandy sea bottom with sparse rough- ness elements. They find that the measured bed stress may be several times smaller than ~- = 0.0025 p u 2 (with u measured above the turbulent BBL). GRoss et al. (1986) have measured Reynolds stresses in a deep boundary layer using BASS, the Benthic Acoustic Stress Sensor designed by WILLIAMS (1985). The drag coefficient lies generally within the limits 2 to 3 × 10-~, but occasionally deviates widely from this range of values (Williams, personal communication).

2.4. Perforations in the coastal boundaries

Bays, inlets, fjords and harbours offer some of the most dramatic evidence of tidal power, yet their total contribution to the global dissipation budget is almost negligible. Dissipation in the Bay of Fundy is 20 GW (GREENBERC, 1979; MILLER, 1966 estimated 23 GW); including the Gulf of Maine yields 50 GW. Here high tide (amplitude 3.2 m) follows flood current ( 1.8 knots) by only 25 °, the entrance is 60 km wide with a maximum depth of 90 m, yielding an incoming flux of almost 1 MW/m. For comparison, the M2 dissipation in the Straits of Juan de Fuca is only 3.3 GW (FOREMAN et al., 1995). Perhaps the most dramatic manifestations of tidal dissipation are the tidal bores. They occur along funnel-shaped fiver estuaries, where the incoming tide is transformed into a steep front gushing upstream. In the estuary of the Amazon River, tidal bores (locally called Pororoca) reach heights of 8 m and travel upstream with velocities of 6 m/s, producing a roar that can be heard at distances of 20 km (DEFANX, 1961, p. 470). But the contribution to the global dissipation is very small; I estimate less than 1 GW when spread over a tidal cycle. Many harbours and bays are analogous to "reactive impedances", delaying high water and flood current about equally, but with the two components remaining roughly in quadrature. A tide wave travelling parallel to the coast is delayed but not dissipated. Other harbours and bays correspond to resistive impedances, with current and elevation almost in phase, corresponding to high dissipation. There are some subtle points (GARRETT, 1975). FREELAr~D and FARMER (1980) and FARMER and SMIXH (1980) have given a detailed account of tidal dissipation in Knight Inlet, a fjord in British Columbia, Canada. A strong seasonal variation, from almost 10 megawatts in June to 5 megawatts in January, gives evidence of the decisive role played by stratification in the dissipation process. The turbulent BBL accounts for only 0.3% of the Knight Inlet dissipation. The dominant process is the extraction of baroclinic energy: tidal energy is converted into steep internal waves of soliton character and subsequently into the formation of an internal hydraulic jump. (Rapid surface convergence causes capillaries to break and emit hissing sounds which can be heard on quiet days.) At other times the formation of internal lee waves is dominant. Knight Inlet is a veritable laboratory of different processes 16 WAI.TI-R MUNK

involved in the conversion from barotropic to baroclinic energy. Some of these processes have been observed in the open sea, others will be. For the total dissipation in bays, inlets and other such perforations in the global coastlines I very roughly estimate the equivalent of five Bay of Fundy's, or 0.1 TW. This corresponds to a flux density of 4 × 104 W/m (10% of the Bay of Fundy) into 2500 km of coastal perforations (2.5% of the global coastline). Dissipation by the flexing of the Antarctic ice shelves, as suggested by DOAKE (1978), contrib- utes less than 25 GW (RAY and EGBERT, 1997).

2.5. Scattering into internal tides and waves

In our discussion of Knight Inlet, we referred to a loss of tidal energy into an internal hydraulic jump. Ocean processes are predominately baroclinic, and such a conversion into baroclinic energy is of intense oceanographic interest (whether or not it dominates the global tidal budget). We shall discuss the loss of surface tide energy into internal tides (§3) and internal waves (§4) as part of the generation process. The conversion comes in many different faces, and the processes involved are summarized in §5. We refer the reader to Fig. 5 for orientation.

3. INTERNAL TIDES

Internal waves of tidal frequencies (internal tides, baroclinic tides) have long been recognized as an undesired source of noise in hydrographic casts. Semidiurnal tides with up to 40 m double amplitudes were measured in Indonesian waters on the "Snellius" expedition in 1930 (LEK, 1938). They are much too large to have anything to do with the equilibrium tide potential. Surface displacements are of order Ap/p = 10 -3 times the interior displacements, a few centimeters. Horizontal currents associated with the internal tides are of the same order as the barotropic tidal currents, making an interpretation of tidal current records difficult. Internal tide energies are typically half of those of the barotropic tides. See WUNSCH (1975) and HENDERSHOTT ( 1981 ) for reviews. Until very recently, the discussion of the role played by baroclinic tides was influenced by the belief that they were incoherent with the neighbouring barotropic tides and indeed incoherent among themselves at small horizontal and vertical separation. Tidal "cusps" (a sharp rise in the continuum spectrum adjoining the strong tidal lines) found in twenty years of Honolulu records (MuNK and CARTWRmrtT, 1966) gave some early evidence of incoherent baroclinic energy. Observations in the last few decades have shown a much more coherent structure. (Phases relative to the tide-producing forces remain constant within a quarter cycle for the duration of the record.) SCIJOTT (1977), a member of the Deep-sea Tides Working Group, was one of the earliest to claim that internal tides have a significant coherence with the surface tide, but estimated that they contributed only about 1 GW to the global dissipation. HENDRY (1977) found that "about 50% of the main thermocline temperature variance in the M2 band is coherent with astronomic forcing". The subsequent detection of modulations in acoustic transmissions caused by internal tides (DUSHAW et al., 1995) and of related surface manifestations in satellite altimetry (RAY and MITCHUM, 1996) is responsible for the present attempt of a quantitative assessment of the role played by internal tides in the dissipation of surface tide energy. Once again: once again--tidal friction 17

t,.~ "~

~<~

-.E m~ .... e,,! ©

i 2xN2 m~ i =~

71- ~ ~~

~"u oF" o._~

~,- II "- .~

~-~ ..= >,_o ~

o0 ~ [.,..~ ~-~ ~,~ 18 WALTER MUNK

3.1. Generation over shelf breaks, banks and ridges

Following the work 7 by RATTRAY (1960), BAINES (1974, 1982) has developed a model for converting surface to internal tides over bottom topography (see Fig. 6). The theory differs for "flat" or "steep" slopes, depending upon whether the bottom slopes are small or large compared to the internal tide characteristic slope

tan ~ = qto2 _f2)l(Aa _ to2). (7) where to, f, N are the tidal, inertial and buoyancy frequencies, respectively. At low latitudes, = to/N = 0.03 radians. Baines' energy convergence consists of 3 terms of comparable magnitude: (i) interfacial waves at depth d moving seaward, (ii) interfacial waves moving shoreward, (iii) waves generated in the deep continuum (seaward only for steep slopes). All terms depend upon the volume flux across the shelf break, which is computed according to

Q = toAl (8) where to is the tidal frequency, A is the local amplitude, and 1 the local shelf width. Baines applied his model to 230 coastal sections of approximately uniform geometry and tidal amplitude, comprising 155,000km of coastline. The total conversion is 15 GW, an average of 100 W/m. It is instructive to consider the 12 regions (some of them including several coastal sections) of largest conversion, accounting for 12 out of the total of 15 GW:

Region 1: 5.7 GW over 1800 km, 3156 W/m

Region 1-12: 11.8 GW over 17,700km, 670W/m

Region 2-12: 6.1 GW over 15,900 km, 385 W/m.

Ap 0 g -~" = 10 -2 ms-2

d __~ 100 m

N(z) 200 m

N 300 m 0 3 cph 400 m

Fig. 6. The Baines model for barotropic to baroclinic tide conversion. A mixed layer of depth d = 100 m overlies a stratified ocean with buoyancy frequency 3 cph, separated by a density jump Ap/p = 10 -3. A shelf of constant depth h = 150 m extends to the shelf break at distance l, beyond which the bottom drops with slope a Once again: once again--tidal friction 19

Region 1, stretching from the North British Isles to the Bay of Biscay, accounting for half of the 12 GW, is associated with a conversion density of 3000 W/m. The next largest density, off Cape Cod, is only 800W/m. For comparison, RATa'RAY (1960) measured 500W/m of internal tide generation at Blake Plateau, and WuNscn (1975) reports similar values. This corresponds to internal tide amplitudes of 10 m. Baines suggests that his global conversion of 0.016 TW is subject to a 50% uncertainty. It is in all events negligible as compared to 2.5 TW global dissipation. This result has been widely referred to as the basis for neglecting the generation of internal tides as an important consideration in the dissipation budget. But all these estimates are proportional to Q2, and the volume flux Q~ across the continental shelf edge is generally small as compared to the flux parallel to the shelf edge. For the extreme case of the Kelvin wave travelling northward along the California coast with velocity c = x/g-H ~ 200 m/s, the components are

Q+ = to l A, QII = HU = Hc(A/H) = cA, with a ratio QJQII = 27rl/Atides of order 10 ~. Baines himself notes that "... along-shore tidal velocities near the shelf break are generally larger than those on-shore...". This has two important consequences: (i) off-shore ridges which do not constrain the flux of barotropic tidal energy are more favorably situated with regard to energy conversion. A glance at global co-tidal charts indicates that this is generally the case. (ii) Transverse canyons, 8 rills, gullies and other irregu- larities in the shelf edge, which have been neglected in the 2-dimensional shelf representation, may be important scatterers (THORPE, 1992, 1996; PETRUNCIO, 1996), and in fact CUMMINS and OEY (1997) with a 3-dimensional model of the coast of British Columbia get 5 times the mode conversion obtained by Baines. MoRozov (1995) has estimated the generation of internal tides over subsurface ridges, based on the Baines model, but computing Q = HU cos0, where 0 is the angle between the direction of the tide and the ridge normal. In the spirit of the previous discussion, I consider Morozov's five largest generating regions:

Region l S. Atl 0-25S 95 GW over 2700 km, 35,100 W/m

Region 2 S. Pac Kusu-Palau 41 GW over 2316 km, 17,700 W/m

Regions 1-5 to one side 237 GW over 19,400 km, 12,200 W/m

Regions 1-5 to both sides 473 GW over 19,400 kin, 24,400 W/m

All of these regions have flux density exceeding 10,000 W/m on both sides requiring mode 1 wave amplitudes of 50 m. Morozov estimates about 400 GW for the Atlantic, 400 GW for the Pacific, and 300 GW for the Indian Ocean. His total of 1.1 TW is then about half of the total dissipation of 2.4 TW. He interprets these calculations in terms of the measurements in many locations in the Atlantic and Pacific Oceans taken during the Polygon deployment in 1970 and the Mesopolygon deployment in 1985. Intensive spectral peaks (at M 2 frequency) were regis- tered by instruments located on the trajectories of internal wave rays commencing from the top of ridges whereas off-ray peaks were weak. Slant coherences were strong between time series at two points on a ray but weak otherwise. Measurements along a profile at fight angle to the 20 WALTER MUNK

ridge with distances up to 500 km suggest decay distances of order 1000 km and decay times of order 4 days. LEVINE et al. (1983) have measured internal tides and waves with an array of moored and towed instruments during the JASIN experiment. They identify Rockall Bank about 100 km to the southwest as the source of an energetic beam of internal tides; the associated flux of order 1000 W/m is associated with low (but not just the lowest) internal modes. Some new results have come from rather unexpected sources. DUSHAW et al. (1995) found internal tide components in acoustic travel times recorded in a 1000 km tomographic triangle deployed 2000 km north-north west of the Hawaiian islands. The angular resolution of such an array is very high, and indicated an arrival from the Hawaiian Ridge, and coherent with the local Hawaiian barotropic tide. Dushaw estimates a flux of 180 W/m at a range of 2000 kin. RAY and MITCHUM (1996) convincingly confirm the coherent conversion of barotropic into baroclinic tide energy by the Hawaiian Ridge from Topex-Poseidon altimetry. The M2 barotropic tide propagates into the region from the north northeast and impinges upon the ridge roughly perpendicularly, favorable to the internal tide generation. The surface manifestation of 5 cm is easily detected, and found to be phase-locked to the local surface tide. A power spectrum along the path reveals peaks at 150 and 85 km, corresponding to the horizontal wavelengths of M2 internal modes 1 and 2, and permitting estimates of group velocities and energy fluxes. The northward flux is about 3000 W/m near the ridge, with an e ~ decay of 1000 km, in rough agreement with the acoustic measurements. The decay distance of 1000 km with a group speed of 2 m/s corresponds to a decay time of 5 days. The total conversion of the Hawaiian Ridge is estimated at 15 GW. The detection of internal tides from satellite altimetry came as a surprise to the oceanography community who had been accustomed to subject internal waves to a "solid lid" surface boundary condition. On retrospect, the known internal amplitudes of tens of meters multiplied by a density contrast Ap/p of order 10-3 give the measured surface manifestation of several centimeters. Our conclusion is that the generation of internal tides over ridges and other offshore features is indeed significant, but that Morozov's global estimates are too high. Quite arbitrarily (and with an eye on the budget), I take 4000 W/m (2000 W/m to each side) as representative, yielding 0.2 TW (or 14 Hawaiian Ridges) for 50,000 km of global submarine ridges. This is consistent with a time constant of 5 days for a global internal tide energy of 105 TJ (250 j/m2). [Author's note: Since this paper was written Kantha (personal communication) has made the following global estimates from TP altimetry: 0.20TW (M2), 0.24TW (all semidiurnals), 0.32 TW (semidiurnals plus diurnals).] We assume that the dissipation of internal tides takes place by transfer of energy to the internal wave continuum, as will be discussed in §4.

3.2. Scattering on a bumpy sea floor

Internal tides generated by the flow of a stratified fluid over an irregular bottom have been considered by various authors. Cox and SANDSTROM (1962) considered a bottom spectrum k -3 for scales between 1 and 10 km with 15 m rms vertical displacement. BELL (1975) considered a distribution of bottom bumps. SJOBER6 and STIGEBRANDT (1992) considered a bathymetry made up of a large number of densely packed rectangular vertical pillars. The results for the three investigations are (1, 1, 4) × 10 -3 watts/m 2, of order 1 TW of global dissipation. All these theories are very sensitive to underlying assumptions. There is an urgent need for experimental testing of the foregoing numerical estimates. Once again: once again--tidal friction 21

ARMI and D'ASARO (1980) found a bottom boundary layer above the Hatteras abyssal plane of typically 20 m thickness. Temperatures within this layer varied by less than 1 m°C. Any bottom bumps embedded within this layer (such as those considered by Cox and SANDSTROM, 1962) will be ineffective as internal tide generators. Seamounts and islands are a different matter, and someone should do a global sum. (For the time being I include them in the ridge conversion. )

4. INTERNAL WAVES

Gravity waves in the ocean's interior are as common as waves at the sea surface--perhaps even more so, for no one has ever reported an interior calm. There have been many reviews but no clear-cut decisions about the generation and dissipation of internal waves (see THORPE, 1975; GARRETT and MUNK, 1979; GARRETT, 1991; D'ASARO, 1991 ). Thorpe's frequently repro- duced schematic (Fig. 7) takes a decidedly solarian view of internal wave generation; our specu- lation (Fig. 5) admittedly shows a lunarian prejudice. We review some of the evidence. Internal waves and tides are both low in the Arctic (LEVINE and PAULSON, 1985) and Mediterranean Sea (Uwe Send, personal communication). We have previously referred to the JASIN experiment (LEVINE et al., 1983); internal tides and waves were measured with an array of moored and towed instruments. Spectra of internal waves show a strong peak at M2 frequency, constituting a significant fraction of the total vari-

If. .&.

' ,~',z.0 112,'~

. j J

Fig. 7. A solar view of physical processes affecting internal waves. "As a challenge, the reader is left to decipher the symbols himself' (THORPE, 1975) 22 WALTER MUNK

ance. Rockall Bank about 100 km to the southwest was identified as the source of an energetic beam of the internal tides. During the 1.5 months of measurements, the tidal variance and continuum variance both fluctuated by an order of magnitude, but the ratio between the two variances remained almost constant. There was no correlation with local wind, even at 50 m depth. A spectrum of internal waves taken over a period of two months off California (Fig. 8) is remarkably stationary even though the wind stress varies by an order of magnitude. This could be the result of very long time scales for internal wave generation and dissipation. Alternately, it could be the result of a rapid replenishment by horizontal internal wave fluxes from outside the wind area. GARREXT (1991) presents the following argument in favour of wind generation. Let F = Fo(l + r cos o~t) designate a wind related forcing function, so that the internal wave energy level E(t) is governed by

dE/dt+E/r=F(t), E=Eo[I +scosog(t-to)],

s = r(1 + ~2I"2)-1/2, Ogto = tan Io~1-].

From his table 1, ~- = 81 days, yielding to = 55 days, s/r = 0.58 for seasonal forcing. BRISCOE and WELLER (1984) have measured the seasonal variation in internal wave energy and find a lag of 2 to 3 months, and a relative modulation s/r of about 0.4. On this basis winds cannot be ruled out as a source of internal waves.

spectrum of vertical displacement 0067 .067 mZ/cph.67 6.7 67 i , ~ i i i i i ~ I I i L i i i i I

4 "~ N- -N o 3

f_ ...... i. - ' _f

100 wind stress

~-~ 0 I I I I I I I I I I I 17 27 19 25 June July 1974

Fig. 8. The spectrum of internal waves as measured by a yo-yoing capsule remains stationary for two months (from CAIRNS and WILLIAMS, 1976) Once again: once again--tidal friction 23

In regions with strong inertial peaks and weak tidal lines the excitation of inertial oscillations by winds must play an important role. In other regions tidal components dominate the spectrum and suggest a tidal origin. Regardless of whether internal waves are of tidal or inertial origin, the cascade of energy towards higher frequencies determines the form of the spectrum.

4.1. Influx from internal tides

Measurements of internal tides show definite evidence of strong non-linear interactions, such as the development of harmonics. It is not unreasonable to suppose that the tidal line spectrum "diffuses" into the internal wave continuum. Interacting resonant triads obey the rule

o)t = o)2 + o)3; kt = k2 + k3. (9)

It used to be thought that non-linear resonant interactions were much too weak to account for the internal wave generation (HoLLowAv, 1980; OLBERS and POMPHREV, 1981 ), but new estimates by HIRST (1996) are much enhanced relative to previous estimates. Among several possible resonances governed by (9), "parametric subharmonic instability" transfers energy from low to high wave numbers, but frequencies to half the tidal frequencies. In previous work the tide was treated as a small amplitude wave (like any other component of the internal wave continuum) whose phase is only weakly correlated with that of the background waves. Hirst finds that because of their coherence and large amplitude, subharmonic modes to which the tide is unstable grow to large amplitude. The computed interaction times of 3 to 10 days are in general accord with the previously cited observations. But there is a problem. There are two critical latitudes:

f = wM2 at 75°; f= ½ o)M2 at 28.8 °.

Internal semidiurnal tides are evanescent at latitudes higher than 75.1 °. Above 28.8 ° the resonant condition (9) cannot be satisfied (since o) > fand hence ~o~ + o)2 > 2f), and the preceding instability analysis does not apply. Only 50% of the world ocean lies between 28.8°N and 28.8°S. There may be other ways out. GARRETT (1991) suggests that bottom scatter of low internal tide modes into high mode internal tides and waves may provide an adequate mechanism. In all events, it would be interesting to generate a precise internal wave climatology to search for an indication, if any, of critical latitudes?

4.2. Impulsive generation (solitons)

"It has been known over a century that.., there are occasionally seen on the surface of the sea long, isolated stripes of highly agitated features that are defined by audibly breaking waves and white water... These features propagate past vessels at speeds that are at times in excess of two knots..." (APEI. et al., 1995). The events repeat at tidal intervals, and are referred in the nautical literature as "tide rips". Such features were recognized in early satellite images, and were soon thereafter associated with internal solitons. By now they have been seen at hundreds of locations around the globe. Solitons in the Sulu Sea have been studied by coordinated satellite and surface measurements (LIU et al., 1985). As the semidiurnal tidal current flows southward out of the Sulu Sea across 24 WALTER MUNK

a shallow sill, an isopycnal trough forms in the lee. Six hours later, as the tidal flow reverses, the trough is carried northward over the sill into the Sulu Sea. This acts as a localized source which develops into a packet of solitons. The packet typically consists of 4 solitons, with the leading wave being associated with a 70 m deep trough of the 200 m isopycnal, diminishing to a 20 m trough for the fourth wave. Distances between solitons are of order 6 km (40 minutes). The sequence is repeated after 12.4 hours. (Peak-to-trough displacements up to 150 m have been reported elsewhere.) We estimate a power density of 30,000 W/m, or 1.5 GW along the 50 km crest. The dissipation averaged over the tidal cycle is less than 0.3 GW. As an example of open sea occurrence we cite the measurements in the Western Equatorial Pacific during the TOGA COARE experiment (PINKEL et al., 1997). A surprise was the detection of 50 m high internal solitons propagating at 5 knots from an apparent generation site at the Nugarba Islands 200 km to the southwest. Evidence consists of clutter images on a doppler radar, backscatter measured with high-frequency sonar, and 500 m to 1000 m offsets in the ship's navigational records. (In the words of SANDSTROM and OAKEY (1995), the ship is "surfing on the internal solitary waves.") The mean dissipation is again estimated at 0.1 GW. For comparison, Fu and HOLT (1984) place an upper bound of 0.5 GW for the rate of loss of tidal energy to soliton-like internal waves in the Gulf of California, representing 10% of the total Gulf dissipation. For one hundred Sulu Seas the total is 0.03 TW. We surmise that tidal solitons are major factors in local mixing, but they are not significant actors in the global dissipation budget, at least not in their most virulent appearances. HENYEY and HOERING (1997) emphasizes that the passage of solitons (as ordinarily defined) leaves the stratification unchanged from its initial shape. In the cases of Knight Inlet and the Strait of Gibraltar, the boundary between a thin upper and a thick lower layer is deepened by the passage, resembling bore-like dynamics. (They suggest using the term "solibore", implying that both sets of properties are important.) From our view the important consideration introduced by Henyey and Hoering is that the propagating baroclinic disturbance can extract energy from the stratification and can lose energy directly into turbulence. We have ignored these processes in interpreting the soliton-like radiation as a flux of tidal energy into internal wave energy.

4.3. Generation of pelagic turbulence

Measurements of internal waves indicate that the spectral distribution of energy is close to instability (MLrNK, 1981). For example, ((du/dz)2)/N 2 is of order (1) indicating shear (or Richardson) instability. This suggests that the flow of energy into internal waves is balanced by an equal loss of energy to pelagic turbulence. GREGC and SANFORD (1988) have measured the turbulent dissipation in a region of the eastern Pacific without strong currents and fronts, where they consider internal wave breakdown as the only known source of turbulence. The procedure involves measuring the velocity microstructure down to 1 cm scales to ascertain realistic estimates of the total mean-square-shear. The dissipation e W/kg equals the mean-square-shear times the kinematic molecular viscosity. They find e(z) ~ N2(z) with the proportionality constant interpreted as follows. Define the diapycnal diffusivity kv (the eddy coefficient of buoyancy flux F) such that

F = (w'p') = k~dp/dz = (p/g)ep, e r = eR/( 1 - Rt~), where t3p is the fraction of e going into potential energy, and Ry ~ 1/5 is the "flux Richardson number". This yields Once again: once again--tidal friction 25

~'(Z) ~" ( 1 -- Rf)RTIk~f2(Z) W/kg.

The measured e(z) corresponds to kv = 10-5 m2/s. GREGG and SANFORD (1988) found an internal wave spectrum corresponding roughly to a Garrett-Munk spectrum, with

N=Noexp(-z/h), No=5.2x 10 -3s -~ (3cph), h=lkm,

beneath the surface-mixed layer. The dissipation per unit area is then

d = fdz p e (z) = ½ Rf' p k, N~o h = 6.8 x 10-4 W/m 2, (10)

or 0.2 TW globally. An interpretation of MODE and POLYMODE profiles in terms of Gregg scaling is consistent with kv = 10-5 m 2 s -l (KUNZE and SANVORO, 1996). The result is consistent with the milliwatt/m 2 downward flux of internal wave energy measured by Pinkel (personal communication). Internal wave dissipation in the benthic bottom boundary layer over a flat bottom is relatively small, of order 10-5 W/m 2 (D'ASARO, 1982). We use the value of d given by (10) as a benchmark for the flux of internal wave energy into pelagic turbulence. For an internal wave energy of 3800 J/m e, the decay time is then about 60 days but strongly depth dependent, varying from 30 days at 100 m to 130 days at 1 kin. A diapycnal diffusivity kv = 10-5 m 2 s -j determined by the microstructure measurements of Gregg and Sanford is in agreement with values obtained from tracer release experiments (LEDWELL et al., 1993). This value is 70 times the molecular conductivity of 1.4 × 10 -v m 2 s ~, but only 1/10th the canonical value of 10-4 mZs J (or l cgs) required for a global balance between downward diffusion and upward advection (MUNK, 1966). An overall dependence of on N 2 (shear) 4 (GRErG, 1989) gives much higher values in regions of enhanced shear. Gregg scaling is consistent with recent measurements over Cobb Seamount (LUECK and MU~E, 1997) and over the Mid-Atlantic Ridge in the Brazil Basin (PoLzIN et al., 1997). Diffusivities as high as k~ = 10-3 m 2 s -~ were measured locally over rough topography. Perhaps the most remarkable aspect of these experiments is that the diffusivities settle down to the pelagic value kv -- 10-5 m 2 s -~ within a few km of the topography. One is reminded of the astounding universality of the internal wave spectrum (GARRETT and MUNK, 1979) where intensities remain within a factor of two over a broad range of conditions. This suggests a tidal excitation for both internal waves (with a 100 day time constant) and pelagic turbulence, with severe paleo implications: one hundred million years ago the basin structure differed from what it is now, and five hundred million years ago the moon was much closer.

5. THE BAROTROPIC-BAROCLINIC CONVERSION

In the preceding two sections, we have come across a number of processes converting surface tides to internal tides and waves. The processes fall into three broad classes, according to whether the internal Froude Number is subcritical (F < 1), transcritical (F ~ 1 ) or supercritical (F > 1 ). The latter two cases are strongly influenced by non-linearities (MELVILLE and HELVR~CH, 1987; HELFmCH and MELVILLE, 1990) and have been documented in Knight Inlet. We refer the reader to Fig. 5 for orientation. Weak (resonant) interactions (F<

tol = 0)2 -I" 0)3; kl = k2 + k3. ( 11 )

We distinguish two cases. In the first case, the conversion is from a surface tide of frequency to! = 2 cycles per lunar day to an internal tide of the same frequency, to2 = to~, by interaction with bottom topography having a k 3 wave number spectrum. The bottom does not move, to3 = 0. This interaction includes the linearized scattering models of Rattray and Baines. In the second case, the surface tide, or the internal tides once generated, scatters into internal waves of non-tidal frequency. Here the foregoing equations have been widely applied to three interacting wave packets within an internal wave continuum and found to be too weak to be relevant, but HmST (1996) has shown that the situation is different if one component of the triad is a discrete high-amplitude wave (such as a tide). Strong interactions. This is the case when the orbital velocity of the surface tide equals the oppositely directed phase velocity of the internal wave:

Usr = - Clw( to,kj); to = to(kj) (12) or a Froude Number F = u/c or order 1. With c(w, k, j) being a given function of frequency, wave number and vertical mode number, the Froude resonance then determines the components of the internal wave spectrum that are excited. The generation of lee-waves in the atmosphere is a case in point. Winds blowing over a mountain ridge generate lee waves travelling upwind at phase velocity equal to the wind and thus remaining stationary. Fig. 9 illustrates the situation for a Baines-like density distribution in a deep ocean (Fig. 6). Tidal currents are generally less than 0.5 m/s. Internal tides of low mode number are too fast for Froude resonance; internal waves are generally too short to be effectively generated by ocean ridges with dimensions exceeding several kilometers. The situation is far more complex when allowance is made for a continental shelf cut by transverse canyons, and there appear to be many opportunities for Froude resonance (THORPE, 1992, 1996). CARTWRIGHT (1959) has identified tidal lee-waves generated at a shelf break. Impulsive generation. Evidently the situation is more favorable for the generation of an evanescent baroclinic lee trough which is swept across the ridge when the tidal current reverses and becomes the starting point for the impulsive generation of an internal wave train. We consider the linear case. At the forward edge, the wavelength is determined by the dispersive character of the wave guide independent of the initial disturbance scale. For a large class of solutions for which maximum group velocity V is associated with maximum wavelength, the linear solutions can be written in terms of the Airy function (for example, JEFFREYS and JEFFREYS, 1950, 17.08-9)

Ai[x - Vot)lV ], V = - (½xVo"/Vo) '/3 (13) where Vo" = d2g]dk 2 evaluated at k = 0. The interval between the first two peaks of the Airy function (wave period) at a range x is given by 3.80V~V. The period increases slowly like x ~/3 with increasing distance from the source. The amplitude decreases like x -1/3 (instead of x -~/2 in the interior) and so becomes more and more the most prominent feature of the disturbance. For the case of a two layer system with a thin upper layer h, Eq. (13) yields an interval between the first and second crest of

2WCol( ½xh2) I/3, Co = ~l(g"Vp/p)h. Once again: once again--tidal friction 27

i I I u I/ I I / / // I / / / / N l ' / N

/ /

/ /; / f f f

dzl

mode #

/ . M2 I I I I I 5 10 c/km

0 -~ ~ h = 100 rn 0.10 3 cph

/ M 2 ," / / / / / /

0 • 05 ! ' / " / / H=4km

i t// / /

0 /z/ I I 0 0.1 o12 c/km

Fig. 9. Dispersion of internal waves in a Baines-like ocean. Beneath a mixed layer (N = 0) of thickness h = 100 m, the buoyancy frequency decreases exponentially with depth according to N = No exp - (dz,) to the bottom at 4 km, with No = 3 cycles per hour, and Zo = 1 km. The dispersion is drawn for vertical modes 1 to 10. The lower left plot shows the shaded rectangle on an enlarged scale 28 WALTER MUNK

For x -- 100 km, h = 100 m, Vp/p = 10 -3, this gives an interval of 50 minutes or 3 km, not too much out of line with the Sulu Sea observations. For high amplitude disturbances, the radiation will include internal solitons, and so the foregoing linear analysis may not be pertinent. With reference to Fig. 5, the important conversion processes appear to be the resonant generation of internal tides and the impulsive generation of internal waves (including solitons) at tidal intervals. In both cases we assume a subsequent transformation into an internal wave continuum. Both cases have line spectra at tidal frequencies and their harmonics (Fig. 10), and so the soliton-like flux in Fig. 5 could have been included in the generation of internal tides.

6. THE FLUX BUDGET

Figure 5 is an attempt to pull together the diverse evidence into a coherent picture of tidal dissipation. The numbers refer to the lunar tidal dissipation, which is the dominant factor. Solar dissipation adds something like 40% to the lunar values. The conclusions are not so different from earlier attempts and by no means definitive. In this essay I have been guided more by observation than by theory. The dissipation physics appears to be dominated be a few, almost singular, situations. Looking for average situations won't solve the problem. In the past, there has been difficulty in identifying possible sources of dissipation. We now find ourselves in a situation where there is too much dissipation (as is characteristic of our time): 2.1 TW (Kantha's bottom boundary layer) + 1.1 TW (Sj6berg and Stigebrandt's conversion into

-~- T+~

T~.1

Ts t ' f 0 T T 2TT Ts ~ ~ -1

Fig. 10. Conversion of barotropic to baroclinic tidal energy (schematic). The internal tide (with non-linear distortion) is generated at the shelf edge according to the Rattray-Baines theory (upper left), with spectrum (upper right). The impulsive generation of a few soliton-like waves at tidal intervals TT is shown in lower panels. The interval between crests (soliton period 7~.) is determined by the group velocity Vo and its derivative Vo" = d2V/dk2 evaluated at the wave number k = 0 Once again: once again--tidal friction 29

internal tides) + 0.1 TW (coastal perforations) for a total of 3.4 TW, compared to the allowable 2.4 TW of M2 dissipation. In Fig. 5, I have ignored the conversion over a bumpy bottom (there being no observational evidence) and have reduced the conversion over ridges to 0.2 TW, in line with the recent direct measurements of radiating internal tides. The proposed budget follows 75 years of tradition in attributing most of the dissipation to turbulence in the BBL of marginal seas, but I am concerned that the BBL estimate is high and the ridge conversion is low (see E6BERT, 1997). There are two conclusions: (i) scattering of surface tides into internal tides is a minor (10%) source of surface tide dissipation, but the dominant source of open-sea internal tides and possibly of internal waves, and hence of pelagic turbulent processes; (ii) tidal dissipation may contribute significantly to the much larger power required for maintaining the global abyssal stratification. With regard to (i), STICEBRANDT (1979) has long ago demonstrated that vertical diffusion in Fjords is driven by coastal breaking of progressive internal tides generated at the sills '° (STIGEBRANDT and AURE, 1989). The concept was extended by SJOBERG and STIGEBRANDT (1992) to a global mixing model with internal tides being generated at the sea bottom, islands and ridges (portrayed as a large number of densely packed vertical pillars) and locally dissipated into internal turbulence. With regard to (ii), this calls for a massive reduction in the BBL allocation. Many of the models postulate the BBL dissipation as the only process balancing the working of the Moon, and there is concern here about a self-fulfilling prophesy. Coming out with the 'right' diapycnal diffusivity kv = l0 _5 m 2 s 1 as a measure of pelagic turbulence is a strong consideration in our budgetary allocations. But there can be no doubt that the conversion of lunar tidal energy over deep ocean ridges is a significant source of baroclinic- ity. Oceanographers had better pay more attention to the Moon, as urged by HENDERSHOTT ( 1981 ) in his review of long waves and ocean tides in the Stommel anniversary volume. When asked which is more important, the Moon or the Sun, Kozma Prutkov (of Russian literary fame) replies: '...the Moon of course, because the Sun is shining only in daytime when it is bright anyway...'. Tidal modelers have made a remarkable contribution towards the subtraction of 'tidal noise' from the altimetry measurements, and so exposing other processes to be studied. To make an equivalent major contribution to the understanding of the tide problem--rather than its elimin- ation-requires, in the words of CARTWRIGHT (1984) '... to quantify the localized distribution of energy sinks... (as) the last remaining obstacle to progress in our two centuries of effort...'. Until this is done, I doubt whether modelers will make such a major contribution--unless they get their feet wet.

7. ACKNOWLEDGEMENTS

1 am indebted to many people: D. Agnew, J. Apel, L. Armi, D. Caldwell, D. Cartwright, G. Egbert, D. Farmer, C. Garrett, M. Gregg, M. Hendershott, L. Kantha, K. Melville, R. Pinkel, R. Ray, A. Stigebrandt, S. Thorpe and C. Wunsch. I thank the reviewers (C.G. and C.W.) for many constructive comments, including 'three strikes and you're out' (by C.G.). This work is supported under the terms of the Secretary of the Navy Chair in Oceanography.

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9. ASTRONOMICAL AND HISTORICAL APPENDIX

Whe cumulative tidal deceleration of the diurnal spin over a few thousand years is associated with a significant displacement in the longitude of where an eclipse is observed. The angular apertures of Moon and Sun happen to be very nearly the same, and the report that at some identifiable historical time '... an eclipse was total in Babylon' (rather than in London, say) is a precise and fairly unambiguous measure of the perturbation in the angular positions of Moon, Sun and Earth. 2Changes in the length of day are due also to changes in the Earth's moment of inertia. Lunar tidal dissipation and the acceleration of the lunar orbital velocity (a negative quantity) are related by (see, for example, MUNK and MACDONALD (1960), 201-203; LAMBECK (1980))

r2MML. D = - h(~ - n) -- 3(M + ML) where n, M, r are the Moon's orbital velocity, mass and distance, h = dn/dt, and where ~, ME are the Earth's angular velocity and mass. In this essay we take the numerical values

h = - 25" century -2 = - 1.24 × 10-23 rad sec -2

D = 2.5 TW (M2), 3.0 TW (total Moon), 4.0 TW (Moon + Sun).

Spencer-Jones' value of- 22.4"/c 2 was accepted for many years, and is used in MUNK and MACDONALD (1960). But by the early 1970s there was an inexplicable increase in n, leading to Cartwright's use of 40"/c 2 in a 1977 SCOR report (Douglas Inman, personal communication). Most 'aficionados' (Cartwright's expression) now prefer something close to - 25"/c 2 (CARTWRIGHT, 1993, 133). The above values refer to telescopic data. Values based on Babylonian eclipse records range from - 27 to - 79"/c 2, but seem to have settled down to the modem values. Since the study of ancient observations requires competency in both astronomy and antiquities, the field has never been overcrowded. The low value of 1.4 TW used by JEFFREYS (1920) was based on the then available eclipse observations. A better way is to use the telescopic data. What is required is to disentangle the Earth's variable rotation (which includes the effects of winds and moments of inertia) from the tidal deceleration (MUNK and MACDONALD, 1960, 182-5; CARTWR1GHT, 1993, 132). The most precise information now comes from lunar laser ranging using the retro reflector placed on the Moon in 1969 during the Apollo mission (DICKEY et al., 1994). The semimajor axis of the Moon's orbit is increasing at a rate 3.82 -+ 0.07 cm/year. The secular acceleration of the orbit is deduced as - 22.2 + 0.6 and - 4.0 _+ 0.4 arcsec/century 2 associated with semidiurnal and diurnal tides, respectively. ~The bodily tide dissipation is small and difficult to measure. ZSCHAU (1986) estimate is 34 WALTER MUNK

based on an interpolation to tidal frequencies of the Qs associated with the decays of the Earth's normal modes and of the Eulerian nutation (Chandler wobble, MUNK and MACDONALD, 1960, 144-74). Writing

1 1 1( 1 dE Q - 27r E J Ddr-270, D= --dt cycle for the tidal-effective 'specific dissipation function' 1/Q, the associated phase-lag of Me is 0.21 °, and the dissipation is

D= 120+20GW

An independent estimate by RAY et al. (1996) is based essentially on subtracting the effect of the much larger ocean term taking advantage of the high precision of the Topex-Poseidon measurements. The amplitude and phase lag of the degree-2 order-2 M2 ocean tide are 3.23 cm and 129.4 °, respectively, leaving a residual solid Earth dissipation of D = 83 + 45 gigawatts. 4The distortion of the Earth by tidal loading and the self-attraction of the tidal bulge is here ignored. This is taken into account by the appropriate 'Love numbers'. We refer to the discussion on pages 106-7 of CARTWRIGHT et al. (1980) and by LE PROVOST and LYARD (1997). Taylor has somewhat overestimated the flux values by writing ff instead of ff--ffequilibrium in the flux calculation (GARRETT, 1975). The nonlinear BBL stress leads to a complex coupling of lunar and solar tidal dissipations by semidiurnal and diurnal components. We refer to a careful discussion by LE PROVOST and LYARD (1997). 5Measurements of surface waves had convinced us that the deployment of deep-sea tide gauges away from the complex coastal environment was feasible (MUNK and ZETLER, 1967). SCOR Working Group 27 was formed in Paris in 1965. All members of WG27 participated in a sea trial in 1967 off San Diego under the leadership of Frank Snodgrass. The members were taken aboard the E.B. Scripps immediately upon their arrival at San Diego airport, to participate in one of the earliest attempts to measure tidal pressure changes on the deep sea bottom. The experiment failed, but the committee lived up to its designation as a 'working group'. 6The Patagonian value is very sensitive to the northward flux over a narrow segment which was not well resolved by GEOSAT (Cartwright, personal communication). 7The idea goes back to ZEILON (1912). 8WuNSCH and WEBB (1979) report enhanced internal tide activity in submarine canyons. Bore-like internal tides have been observed moving up Monterey Canyon off California, accompanied by 'tidal pumping' of dense water up onto the shelf (PETRUNCIO, 1996). 9The global energy of surface tides is 4.0 × l0 ~7 J (KANTHA and TIERNEY, 1997), or 1100 J/m 2. The GM internal wave spectrum has 3800 J/m e. Internal tidal energy is highly variable, depending (we think) on the distance from topographic features suitable for surface-to- internal conversion. Taking internal tide currents at half the magnitude of those associated with surface tides, the ratios 4: (1/4): l may be representative of the energies of internal waves: internal tides: surface tides. It would be interesting to study the geographic variation of these ratios. Is there an indication of critical latitudes at 75.1 ° and 28.8°? WUNSCH and WEaR (1979) have studied the climatology of deep ocean internal waves and find higher intensities near critical topographic features. Once again: once again--tidal friction 35

1°The indicated mixing efficiency is only 5%, as compared to the traditional 20% assumed in this paper. It is possible (but has not been demonstrated) that the mixing efficiency is a function of the Prandtl Number, and so different in a salt-stabilized fjord than in a temperature- stabilized ocean (Stewart Turner, personal communication).