IO 10.1 Introduction Long Waves and The main purpose of this chapter is to summarize what Ocean Tides was generally known to oceanographers about long waves and ocean tides around 1940, and then to indi- cate how the subject has developed since then, with Myrl C. Hendershott particular emphasis upon those aspects that have had significance for oceanography beyond their importance in understanding tides themselves. I have begun with a description of astronomical and radiational tide-gen- erating potentials (section 10.2), but say no more than is 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 tides and long waves, Laplace's tidal equations (LTE), re- mained unchanged and unchallenged from Laplace's formulation of them in 1776 up to the early twentieth century. By 1940 they had been extended to allow for density stratification (in the absence of bottom relief) and criticized for their exclusion of half of the Coriolis forces. Without bottom relief this exclusion has re- cently been shown to be a good approximation; the demonstration unexpectedly requires the strong strat- ification of the ocean. Bottom relief appears able to make long waves in stratified oceans very different from their flat-bottom counterparts (section 10.4); a definitive 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 due to the tidal yielding of the solid earth. I review these matters in section 10.3. Laplace's study of the free oscillations of a global ocean governed by LTE was the first study of oceanic long waves. Subsequent nineteenth- and twentieth- century explorations of the many free waves allowed by these equations, extended to include stratification, have evolved into an indispensible part of geophysical fluid dynamics. By 1940, most of the flat-bottom so- lutions now known had, at least in principle, been constructed. But Rossby's rediscovery and physical in- terpretation, in 1939, of Hough's oscillations of the second 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 the results to nontidal as well as tidal motions. Since then, flat-bottom barotropic and baroclinic solutions of LTE have been obtained in mid-latitude and in equatorial approximation, and Laplace's original global problem has 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 have attempted to review all those developments in a self- contained manner in section 10.4. In order to treat this 292 Myrl C. Hendershott 1 _· IC _ ___ vast subject coherently, I have had to impose my own Many features of the presentday view of ocean cir- view of its development upon the discussion. I have culation have some precedent in tidal and long-wave cited observations when they appear to illustrate some studies, although often unacknowledged and appar- property of the less familiar solutions, but the central ently not always recognized. The question of which theme is a description of the properties of theoretically parts of the study of tides have in fact influenced the possible waves of long period (greater than the buoy- subsequent development of studies of ocean circulation ancy period) and, consequently, of length greater than is a question for the history of science. In some cases, the ocean's mean depth. developments in the study of ocean circulation subse- Although the study of ocean surface tides was the quently have been applied to ocean tides. In section original study of oceanic response to time-dependent 10.7 I have pointed out some of the connections of forcing, tidal studies have largely proceeded in isolation which I am aware. from modem developments in oceanography on ac- count of the strength of the tide-generating forces, their 10.2 Astronomical Tide-Generating Forces well-defined discrete frequencies, and the proximity of these to the angular frequency of the earth's rotation. Although correlations between ocean tides and the po- A proper historical discussion of the subject, although sition and phase of the moon have been recognized and of great intellectual interest, is beyond the scope of utilized since ancient times, the astronomical tide-gen- this chapter. To my mind the elements of such a dis- erating force (ATGF) was first explained by Newton in cussion, probably reasonably complete through the the Principia in 1687. Viewed in an accelerated coor- first decade of this century, are given in Darwin's 1911 dinate frame that moves with the center of the earth Encyclopedia Britannica article "Tides." Thereafter, but that does not rotate with respect to the fixed stars, with a few notable exceptions, real progress had to the lunar (solar) ATGF at any point on the earth's await modem computational techniques both for solv- surface is the difference between lunar (solar) gravita- ing LTE and for making more complete use of tide tional attraction at that point and at the earth's center. gauge observations. Cartwright (1977) has recently re- The daily rotation of the earth about its axis carries a viewed the entire subject, and therefore I have given a terrestrial observer successively through the longitude discussion in section 1].0.5that, although self-con- of the sublunar or subsolar point [at which the lunar tained, emphasizes primarily changes of motivation (solar) ATGF is toward the moon (sun)] and then half and viewpoint in tidal studies rather than recapitulates a day later through the longitude of the antipodal point Cartwright's or other recent reviews. [at which the ATGF is away from the moon (sun)]. In This discussion of tides as long waves continuously Newton's words, "It appears that the waters of the sea forced by lunar and solar gravitation logically could be ought twice to rise and twice to fall every day, as well followed by a discussion of tsunamis impulsively lunar or solar" [Newton, 1687, proposition 24, theorem forced by submarine earthquakes. But lack of both 19]. space and time has forced omission of this topic. The ATGF is thus predominantly semidiurnal with Internal tides were first reported at the beginning of respect to both the solar and the lunar day. But it is this century. By 1940 a theoretical framework for their not entirely so. Because the tide-generating bodies are discussion had been supplied by the extension of LTE not always in the earth's equatorial plane, the terres- to include stratification, and their generation was trial observer [who does not change latitude while (probably properly) ascribed to scattering of barotropic being carried through the longitude of the sublunar tidal energy from bottom relief. The important devel- (solar) point or its antipode] sees a difference in ampli- opments since then are recognition of the intermittent tude between the successive semidiumal maxima of narrow-band nature of internal tides (as opposed to the the ATGF at his location. This difference or "daily near-line spectrum of surface tides) plus the beginnings inequality" means that the ATGF must be thought of of a statistically reliable characterization of the inter- as having diurnal as well as semidiurnal time variation. nal tidal spectrum and its variation in space and time. Longer-period variations are associated with period- The subject has recently been reviewed by Wunsch icities in the orbital motion of earth and moon. The (1975). Motivation for studying internal tides has astronomical variables displaying these long-period shifted from the need for an adequate description of variations appear nonlinearly in the ATGF. The long- them through exploration of their role in global tidal period orbital variations thus interact nonlinearly both dissipation (now believed to be under 10%) to specu- with themselves and with the short-period diurnal and lation about their importance as energy sources for semidiurnal variation of the ATGF to make the local oceanic mixing. In section 10.6 I have summarized ATGF a sum of three narrow-band processes centered modem observational studies and their implications about 0, 1, and 2 cycles per day (cpd), each process for tidal mixing of the oceans. being a sum of motions harmonic at multiples 0,1,2 of the frequencies corresponding to a lunar or a solar day 293 Long Waves and Ocean Tides plus sums of multiples of the frequencies of long-period the N}'1 are sets of small integers (effectively the Dood- orbital variations. son numbers); and the S,(t) are secular arguments that A complete derivation of the ATGF is beyond the increase almost linearly in time with the associated scope of this discussion. Cartwright (1977) reviews the periodicity of a lunar day, a sidereal month, a tropical subject and supplies references documenting its mod- year, 8.847 yr (period of lunar perigee), 18.61 yr (period ern development. For a discussion that concentrates of lunar node), 2.1 x 104yr (period of perihelion), re- upon ocean dynamics (but not necessarily for a prac- spectively. tical tide prediction), the most convenient representa- The frequencies of the arguments XNi' Sj(t) fall into tion of the ATGF is as a harmonic decomposition of the three "species"-long period, diurnal, and semidi- the tide-generating potential whose spatial gradient is urnal-which are centered, respectively, about 0, 1, and the ATGF. Because only the horizontal components of 2 cpd (N1 = 0,1,2). Each species is split into "groups" the ATGF are of dynamic importance, it is convenient separated by about 1 cycle per month, groups are split to represent the tide-generating potential by its hori- into "constituents" separated by one cyle per year, etc.