IOLong Waves andOcean TidesMyrl C. Hendershott10.1 IntroductionThe main purpose of this chapter is to summarize whatwas generally known to oceanographers about longwaves and ocean tides around 1940, and then to indi-cate how the subject has developed since then, withparticular emphasis upon those aspects that have hadsignificance for oceanography beyond their importancein understanding tides themselves. I have begun witha description of astronomical and radiational tide-gen-erating potentials (section 10.2), but say no more thanis necessary to make this chapter self-contained. Cart-wright (1977) summarizes and documents recent de-velopments, and I have followed his discussion closely.The fundamental dynamic equations governing tidesand long waves, Laplace's tidal equations (LTE), re-mained unchanged and unchallenged from Laplace'sformulation of them in 1776 up to the early twentiethcentury. By 1940 they had been extended to allow fordensity stratification (in the absence of bottom relief)and criticized for their exclusion of half of the Coriolisforces. Without bottom relief this exclusion has re-cently been shown to be a good approximation; thedemonstration unexpectedly requires the strong strat-ification of the ocean. Bottom relief appears able tomake long waves in stratified oceans very differentfrom their flat-bottom counterparts (section 10.4); adefinitive discussion has not yet been provided. Finally,LTE have had to be extended to allow for the gravita-tional self-attraction of the oceans and for effects dueto the tidal yielding of the solid earth. I review thesematters in section 10.3.Laplace's study of the free oscillations of a globalocean governed by LTE was the first study of oceaniclong waves. Subsequent nineteenth- and twentieth-century explorations of the many free waves allowedby these equations, extended to include stratification,have evolved into an indispensible part of geophysicalfluid dynamics. By 1940, most of the flat-bottom so-lutions now known had, at least in principle, beenconstructed. But Rossby's rediscovery and physical in-terpretation, in 1939, of Hough's oscillations of thesecond class began the modem period of studying so-lutions of the long-wave equations by inspired or sys-tematic approximation and of seeking to relate theresults to nontidal as well as tidal motions. Since then,flat-bottom barotropic and baroclinic solutions of LTEhave been obtained in mid-latitude and in equatorialapproximation, and Laplace's original global problemhas been completely solved. The effects of bottom re-lief on barotropic motion are well understood. Signifi-cant progress has been made in understanding the ef-fects of bottom relief on baroclinic motions. I haveattempted to review all those developments in a self-contained manner in section 10.4. In order to treat this292Myrl C. Hendershott 1 _· IC _ __ _vast subject coherently, I have had to impose my ownview of its development upon the discussion. I havecited observations when they appear to illustrate someproperty of the less familiar solutions, but the centraltheme is a description of the properties of theoreticallypossible waves of long period (greater than the buoy-ancy period) and, consequently, of length greater thanthe ocean's mean depth.Although the study of ocean surface tides was theoriginal study of oceanic response to time-dependentforcing, tidal studies have largely proceeded in isolationfrom modem developments in oceanography on ac-count of the strength of the tide-generating forces, theirwell-defined discrete frequencies, and the proximity ofthese to the angular frequency of the earth's rotation.A proper historical discussion of the subject, althoughof great intellectual interest, is beyond the scope ofthis chapter. To my mind the elements of such a dis-cussion, probably reasonably complete through thefirst decade of this century, are given in Darwin's 1911Encyclopedia Britannica article "Tides." Thereafter,with a few notable exceptions, real progress had toawait modem computational techniques both for solv-ing LTE and for making more complete use of tidegauge observations. Cartwright (1977) has recently re-viewed the entire subject, and therefore I have given adiscussion in section 1].0.5 that, although self-con-tained, emphasizes primarily changes of motivationand viewpoint in tidal studies rather than recapitulatesCartwright's or other recent reviews.This discussion of tides as long waves continuouslyforced by lunar and solar gravitation logically could befollowed by a discussion of tsunamis impulsivelyforced by submarine earthquakes. But lack of bothspace and time has forced omission of this topic.Internal tides were first reported at the beginning ofthis century. By 1940 a theoretical framework for theirdiscussion had been supplied by the extension of LTEto include stratification, and their generation was(probably properly) ascribed to scattering of barotropictidal energy from bottom relief. The important devel-opments since then are recognition of the intermittentnarrow-band nature of internal tides (as opposed to thenear-line spectrum of surface tides) plus the beginningsof a statistically reliable characterization of the inter-nal tidal spectrum and its variation in space and time.The subject has recently been reviewed by Wunsch(1975). Motivation for studying internal tides hasshifted from the need for an adequate description ofthem through exploration of their role in global tidaldissipation (now believed to be under 10%) to specu-lation about their importance as energy sources foroceanic mixing. In section 10.6 I have summarizedmodem observational studies and their implicationsfor tidal mixing of the oceans.Many features of the presentday view of ocean cir-culation have some precedent in tidal and long-wavestudies, although often unacknowledged and appar-ently not always recognized. The question of whichparts of the study of tides have in fact influenced thesubsequent development of studies of ocean circulationis a question for the history of
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