P 2 f O 2 2 x fo p Pz 2 0 a opx fokp t f v N p z k 0 P2 0 x a upx rop a t fO O N b q x 2 P kfo d l a 1 2p c change i e provided that the relief does not scatter energy from one mode into others Constancy of n is indeed a feature of Wunsch s solutions but it cannot be expected to hold for more abrupt relief especially if the relief slope exceeds the characteristic slope If the relief couples modes efficiently then scattering into higher modes allows I to remain real even in deep water far from shore so that energy is not refractively trapped near the coast In principle scattering into internal modes thus even destroys the perfect trapping of long surface gravity waves predicted by LSW theory over a step shelf but in practice appreciable trapping is often observed The efficiency of mode coupling depends both on the relief and on the vertical profile N z of the buoyancy frequency so that a general result for internal waves is difficult to formulate d 10 4 8 Free Oscillations of Ocean Basins P1 e Stratified Problem f Figure Io 2o The deep water surface wave analog d f of two shelf problems involving topographic Rossby waves in uniformly stratified rotating fluid a stratified problem b result of affine transformation c result of rotation d equivalent deep water problem velocity potential e stratified problem f equivalent deep water problem atmospheric pressure P must be maintained lower than P2 for physical realizability 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 refracted just like surface gravity waves by the shoaling relief and that refractively trapped edge modes occur From the dispersion relation o0 2 12 k 2 for plane internal waves of the form w sin D exp i t ilx iky over a uniform bottom Do I must ultimately become imaginary if Do is allowed to grow parametrically offshore while n and k are held fixed One would therefore expect a WKB treatment of internal waves over gently shoaling relief to result in refraction and refractive trapping provided that the mode number n does not Finding the free oscillations allowed by LTE in rotating ocean basins is difficult even in the f plane section 10 4 2 Platzman 1975 1978 has developed powerful numerical techniques for finding the natural frequencies and associated flow fields of free oscillations allowed by LTE in basins of realistic shape and bottom relief The general classification of free oscillations into first and second class modes characteristic of the idealized 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 Atlantic and Indian Oceans there are 14 free oscillations with periods between 10 and 25 hours Some of these are very close to the diurnal and semidiurnal tidal periods and all of them being within a few percentage points of equipartition of kinetic and potential energies are first class modes There are also free oscillations of much longer period for which potential energy is only about 10 or even less of kinetic energy they are second class modes I know of no extratidal peaks in open ocean sea level records that correspond to these free oscillations There is some evidence in tidal admittances for the excitation of free modes but the resonances are evidently not very sharp see section 10 5 1 Munk Bryan and Zetler private communication have searched without success for the intertidal coherence of sea level across the Atlantic that the broad spatial scale of these modes implies The modes are evidently very highly damped 10 5 The Ocean Surface Tide 10 5 1 Why Ocean Tides Are of Scientific Interest The physical motivation for studying and augmenting the global ensemble of ocean tide records has expanded enormously since Laplace s time In this section I have tried to sketch the motivating ideas without getting 317 Long Waves and Ocean Tides involved in the details of theoretical models some of these 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 observations made explicitly for predictive purposes may be the table of flod at london brigge due to Wallingford who died as Abbot of St Alban s in 1213 Sager 1955 Mak ing practical tide predictions was probably the preoccupation of observers for the next 500 years In 1683 Flamsteed Sager 1955 produced a table of high waters for London Bridge as well as in the following year corrections making it applicable to other English ports Darwin 1911a quotes Whewell s description written in 1837 of how successors to Flamsteed s tables were produced The course would have been to ascertain by an analysis of long series of observations the effects of changes in the time of transit parallax and the declination of the moon and thus to obtain the laws of phenomena Though this was not the course followed by mathematical theorists it was really pursued by those who practically calculated tide tables Liverpool London and other places had their tables constructed by undivulged methods handed down from father to son The Liverpool tide tables were deduced by a clergyman named Holden from observations made at that port for above twenty years day and night Holden s tables founded on four years of these observations were remarkably accurate At length men of science began to perceive that such Mr Lubcalculations were part of their business bock finding that regular tide observations had been made at the London docks from 1795 took nineteen years of these and caused them to be In a very few years the tables thus proanalyzed duced by an open and scientific process were more exact than those which resulted from any of the secrets Quite aside from its proprietary aspects Darwin 1911b explicitly notes the synthetic nature of this process it at least conceptually represents the oscillation of the sea by a single mathematical expression provided by Bernoulli in 1738 for an inertialess ocean the equilibrium tide by Laplace for a global ocean obeying Newton s laws of motion and assumed to exist for actual oceans even if too complex to represent in simple form Kelvin in about 1870 Darwin 1911b introduced the harmonic method which Darwin 191lb calls analytic because synthesis
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