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MIT 12 000 - LECTURE NOTES

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P,2 O 2 .,... . , + f -fo)} {p -kPz -0a{opx-fokp} +t(f2-v?)/N2pz- 0(a)q xP ' (kfo/)d/(l+a2) 1/2p(c)\ P2 0x'(b)(d)change, i.e., provided that the relief does not scatterenergy from one mode into others. Constancy of n isindeed a feature of Wunsch's solutions but it cannotbe expected to hold for more abrupt relief, especiallyif the relief slope exceeds the characteristic slope. Ifthe relief couples modes efficiently, then scatteringinto higher modes allows I to remain real even in deepwater far from shore so that energy is not refractivelytrapped near the coast. In principle, scattering into in-ternal modes thus even destroys the perfect trapping oflong surface gravity waves predicted by LSW theoryover a step shelf, but in practice appreciable trappingis often observed. The efficiency of mode coupling de-pends both on the relief and on the vertical profile N(z)of the buoyancy frequency, so that a general result forinternal waves is difficult to formulate.(e) Stratified ProblemP1(f)Figure Io.2o The deep-water surface-wave analog (d,f) of twoshelf problems involving topographic Rossby waves in uni-formly stratified rotating fluid: (a) stratified problem; (b) resultof affine transformation; (c) result of rotation; (d) equivalentdeep-water problem (velocity potential ); (e) stratified prob-lem; (f) equivalent deep-water problem (atmospheric pressureP, must be maintained lower than P2for physical realizabil-ity).tion. This is the stratified analog of Eckart's (1951)nonrotating LSW study of waves over a sloping beach(section 10.4.6).For beach slopes much smaller than the slope (or/N)of (low-frequency) internal wave characteristics (10.43),Wunsch thus found that internal waves are refractedjust like surface gravity waves by the shoaling reliefand that refractively trapped edge modes occur. Fromthe dispersion relationo0) 2 (1 2 + k2)for plane internal waves of the formw = sin( D) exp(-i t + ilx + iky)over a uniform bottom Do, I must ultimately becomeimaginary if Do is allowed to grow parametrically off-shore while n and k are held fixed. One would thereforeexpect a WKB treatment of internal waves over gentlyshoaling relief to result in refraction and refractivetrapping provided that the mode number n does not10.4.8 Free Oscillations of Ocean BasinsFinding the free oscillations allowed by LTE in rotatingocean basins is difficult even in the f-plane (section10.4.2). Platzman (1975, 1978) has developed powerfulnumerical techniques for finding the natural frequen-cies and associated flow fields of free oscillations al-lowed by LTE in basins of realistic shape and bottomrelief. The general classification of free oscillations intofirst- and second-class modes characteristic of theidealized cases discussed in sections 10.4.2 and 10.4.5(effectively for a global basin) persists in Platzman's(1975) calculations. For a basin composed of Atlanticand Indian Oceans, there are 14 free oscillations withperiods between 10 and 25 hours. Some of these arevery close to the diurnal and semidiurnal tidal periods,and all of them, being within a few percentage pointsof equipartition of kinetic and potential energies, arefirst-class modes. There are also free oscillations ofmuch longer period, for which potential energy is onlyabout 10% or even less of kinetic energy; they aresecond-class modes.I know of no extratidal peaks in open-ocean sea-levelrecords that correspond to these free oscillations. Thereis some evidence in tidal admittances for the excitationof free modes but the resonances are evidently not verysharp (see section 10.5.1). Munk, Bryan, and Zetler(private communication) have searched without suc-cess for the intertidal coherence of sea level across theAtlantic that the broad spatial scale of these modesimplies. The modes are evidently very highly damped.10.5 The Ocean Surface Tide10.5.1 Why Ocean Tides Are of Scientific InterestThe physical motivation for studying and augmentingthe global ensemble of ocean-tide records has expandedenormously since Laplace's time. In this section I havetried to sketch the motivating ideas without getting317Long Waves and Ocean Tides,xa-upx-rop ' t (fO-O-)/N- } 'involved in the details of theoretical models; some ofthese receive attention in subsequent sections.Certain of the ancients knew a great deal about tides[see, e.g., Darwin's (1911a) summary of classical ref-erences], but the first extant reduction of observationsmade explicitly for predictive purposes may be the ta-ble of "flod at london brigge" due to Wallingford whodied as Abbot of St. Alban's in 1213 (Sager, 1955). Mak-ing practical tide predictions was probably the preoc-cupation of observers for the next 500 years.In 1683, Flamsteed (Sager, 1955) produced a table ofhigh waters for London Bridge as well as, in the follow-ing year, corrections making it applicable to other Eng-lish ports. Darwin (1911a) quotes Whewell's descrip-tion, written in 1837, of how successors to Flamsteed'stables were produced:The course . . . would have been to ascertain by ananalysis of long series of observations, the effects ofchanges in the time of transit, parallax, and the decli-nation of the moon and thus to obtain the laws ofphenomena.. . .Though this was not the course followed bymathematical theorists, it was really pursued by thosewho practically calculated tide tables. .... LiverpoolLondon, and other places had their tables, constructedby undivulged methods ... handed down from fatherto son.... The Liverpool tide tables ... were deduced by aclergyman named Holden, from observations made atthat port .. . for above twenty years, day and night.Holden's tables, founded on four years of these obser-vations, were remarkably accurate.At length men of science began to perceive that suchcalculations were part of their business.... Mr. Lub-bock . . . , finding that regular tide observations hadbeen made at the London docks from 1795, . . .tooknineteen years of these . . . and caused them to beanalyzed.... In a very few years the tables thus pro-duced by an open and scientific process were moreexact than those which resulted from any of the se-crets.Quite aside from its proprietary aspects, Darwin(1911b) explicitly notes the


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