Copyright 1970. All rights reserved

TIDES MYRL HENDERSHOTT AND WALTER MUNK

Scripps Institution of Oceanography, La Jolla, California

1. INTRODUCTION This review deals with ocean tides; moreover, with that class of long

surface waves that is hardly affected by the internal density stratification (but see section 4). Internal (or baroclinic) tides play an important role in oceanic-tide phenomena, and are crucial with regard to atmospheric tides; they deserve to be separately reviewed. Earth tides have been well observed only in the last decade. They are mentioned only in passing, yet they strongly influence ocean tides and vice versa; so much so that neither subject will be solved without the other.

The literature is vast, and mostly Victorianl; digital techniques are revolutionizing the subject with regard to measurements, analysis, and theory. If a subject should be reviewed when it is rapidly developing (as we hold), and not when it has become stationary, then this is indeed a good time for review.

2. MEASUREMENTS Tides are usually measured by recording the position of a float in a

vertical well. There have been few changes since 1882 when Lord Kelvin (1) argued the case for the pencil before the Institution of Civil Engineers. The relatively high frequencies associated with wind waves are suppressed by an orifice at the bottom of the tide well. But orifices are notoriously nonlinear, and they are subject to siltation and biological fouling (octopi have been known to insert tentacles into the orifice); yet the gauges are hardly ever calibrated. On the other hand, tide records are unique for their continuity, with hourly readings over half a century or longer being available at more than a dozen ports (comprising some 107 observations) .

With a Vibrotron pressure transducer on the sea bottom at 20 meters depth, connected by cable to shore and recording digitally at S-minute intervals, Snodgrass (2) was able to reduce the instrumental noise level by a factor 106• A cable-connected sea-bottom installation by the Lamont Geo-

1 The best sources of the classical literature are Vol. 1 of Sir George Darwin's Scientific Papers (Cambridge 1907); R. A. Harris' Manual of Tides (U.S.C.G.S. Washington 1894-1907); and Vol. 2 of A. Defant's Physical Oceanography (Pergamon 1961).

205

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

206 HENDERSHOTT & MUNK logical Observatory (3) 100 miles (!) west of San Francisco at a depth of 4000 meters has yielded excellent tidal observations. In the meanwhile, Eyries in France (4), Filloux (5) and Snodgrass (6) in the U.S.A. have developed instruments that can be placed almost anywhere on the sea bottom. Snodgrass uses a freely falling capsule with self-contained computer-compatible recording. The instrument is recovered, typically after a month's operation, by acoustic command from a surface vessel. (Work is underway to extend the capsule lifetime to one year. ) On the basis of half a dozen capsule drops, the tidal pattern in the northeast Pacific has begun to take shape (7).

The number of standard tide gauges ever in operation is of the order of 104; typical locations are at the mouths of harbors for which predictions are required. But these are in the midst of the very coastal features most likely to exert anomalous effects on the phase and amplitude of tides; from the point of view of studying the basic problem of tides. there could be no worse situation. It is as if one is recording some process through a series of filters, each with a different time constant than the others, and having in common only that none of them are known. So the deep-sea observations do much more than just add a few tidal constants to the vast existing tabulations.

Apart from this consideration, and perhaps surprisingly, the deep-sea observations even now are of a quality comparable to the best land observavations, with a signal: noise energy in excess of 103: 1 for a month's record. In part this is the result of the favorable temperature environment (and the absence of electronic technicians) in the deep sea; in part it is that the lowfrequency noise continuum associated with variable wind stress (storm tides) is especially pronounced near the coastline.

Tilt meters and gravimeters at coastal land stations measure a significant tide component associated with ocean loading and may provide useful moments of offshore tides. And as this review is being written, there is serious discussion of whether the tides can be measured with satellite altimeters.

3. ANALYSIS AND PREDICTION The connection between Moon and tides is so obvious that long before

the formulation of any theory quite satisfactory rule-of-thumb predictions of tide were made and published. Tide tables constructed by undivulged method were considered as gainful family possessions and passed on from father to son. Some early tide tables for Liverpool published by a clergyman named Holden carried this art to its highest perfection.

Starting in 1831, Sir John Lubbock initiated what has been called the nonharmonic method of tide prediction. Tides at a given port are represented by certain tidal elements: time and height of high water, low water. These are related to astronomical observables: the age of the Moon (reckoned from new Moon), the declination, and the parallax of Moon and Sun. The prediction is made by successive approximations. For example,

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 207 height of high water

=mean height abovc datum+correction for age+correction for declination+correction for parallax+diurnal inequality.

The harmonic method of tide analysis was developed by Lord Kelvin and Sir George Darwin starting in 1867. The tidal elevation at a given port is predicted according to the representation

I;(t) = 1: Ck cos (21Tk. it + Ok). k

The "tidal constants" Ck, Ok are found by harmonic analysis of the tide record I;(t) over a denumerable set of frequencies

(k = 0, ± 1, ± 2, ... ), where f is a five-dimensional vector whose components are the basic frequencies in the motion of Earth, Moon. and Sun; namely:

!I-I = 1 lunar day is the period of the Earth's rotation (relative to Moon) , h-1 = 1 month is the mean period of the Moon's orbital motion, fa-I = 1 year is the mean period of Sun's orbital motion, j4-1 "" 8.85 years is the mean period of lunar perigee, j5-1 "" 18.61 years is the mean period of regression of lunar nodes.

The "Doodson number" k= (k1 k2 ka k4 k6) completely defines the frequency kI The values kJ = 0, 1, 2 refer to frequencies near 0, 1, 2 cycles per lunar day: the long period. diurnal and semidurnal species. In the language of spectroscopy, each tidal species shows three orders of splitting: monthly splitting, a fine structure due to yearly splitting, a hyperfine structure from lunar perigee and regression.

A fictitious tide f(t) called "equilibrium tide" corresponds to a sea surface that is, at any instant, a surface of constant gravitational potential. Doodson (8) has performed the harmonic expansion of the tide potential for the first four decades in amplitude. This involves 400 terms; those in the first decade are shown in Table I. The equilibrium amplitude at latitude 8 is Ck K Pnm(8), where

K = ! (Mmoon/Me.rth)a2/ra = 53. 7 cm is the "general lunar coefficient", with M designating mass, a the Earth's radius and r the mean lunar distance.

To what extent do the observed amplitude factors Ck, resemble the equilibrium factors Ck? At Honolulu (Fig. 1) the admittance CklCk for the semidiurnal species lies between 1 ± t, and the phase lag is about 2 radians. Coastal resonance can produce admittances of order 10. But. even at mid-sea island stations there is no resemblance between observed and equilibrium tides. By accident, the depth D of ocean basins, the radius a, gravity g, and the angular velocity 0 of the Earth are such that the characteristic wave

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

208 HENDERSHOTT & MUNK TABLE I

LEADING EQUILIBRIUM CONSTITUENTS IN THE DOODSON EXPANSION

Designation Darwin Doodson&

M, 0 2 0

Diurnal SPecies, Pzl 01 1 1 0 PI 1 1 -2 KI 1 1 0

Semidiurnal Species, Pz2 Nz 2 -1 0 Mz 2 0 0 S2 2 2 -2 Kz 2 2 0

Period

327h .84

25h .82 24h.07 23h.93

12h .66 12h .42 12h .00 11'.97

Amplitude Factor Gk

0.156

0.377 0.176 0.531

0.174 0.908 0.423 0.115

I

)

Latitude Factor Pnm(o)

Hi-sin2O)

sin I) cos I)

i cos2 0

a First three numbers only. Doodson (8) adds 5 to all numbers except the first. b M, is the fortnightly tide; the monthly (Mm), semiannual (Ssa), annual (Sa);

and lunar nodal tides fall below the limit T1, Mz=.09 of this table.

velocity, v'gD, is of the same order as the velocity Qa of the sublunar point (el = 2Qa/ v' gD is of order 1, see section 7). As a consequence, global tides are broken up into a dozen or so amphidromic cells within which tides are turning about a central stationary point (sections S, 8).

The assumption is implicit in the harmonic method of tide prediction that any degree of precision could be attained if only a sufficient number of spectral lines were included (e.g. , Doodson's 400 terms). This is equivalent to asserting that there is zero noise energy. But noise-free processes do not occur (except in the literature on tidal phenomena). In fact, a glance at Fig. 1 shows that the weak lines are swamped by noise and useless for prediction.

This suggests, as an alternative to the almost universally used harmonic method, the determination of the station response (admittance G/G and relative phase 0-0) to the various spherical harmonic input functions, rather than the evaluation of Gk and Ok themselves. G/C and 0-0 can be expected to be reasonably smooth functions of frequency; G and 0 certainly are not. The equilibrium elevation tnm(t) can be numerically generated directly from the lunar and solar ephemerides (m,n =0,2, 1,2, 2,2 is usually adequate), and the prediction performed according to

m,n lJ

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 209

. :1 3 J-l �.. I II 5,

':' ':> � ':'

: ] 1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIiIIJlllllllillii �lllllllllllllllllllllllllllllllllllIlllmlllllllll Eli

"5 ! [ , [ !

1.95 CYCLES! DAV

9

2.0 2.5

FIG. 1. Honolulu tide spectra at 1 cycle/year resolution for the semidiurnal spherical harmonic P22. The upper panel shows the energy of the gravitational equilibrium tide relative to 10-4 em'; some Darwin designations (Mz, Kz, ... ) and Doodson triplets (2 00,220) are shown. The second panel shows the observed sea spectrum; the height of the filled portions designates the line energy coherent with the equilibrium tide, the unfilled portion the residual energy density. The bottom panel gives the amplitude and phase lead in radians of the observed relative to the equilibrium tide. Adopted from (9).

with the prediction weights w (the Fourier transforms of the complex station admittance) determined by least-square methods (9). Astronomy and oceanography are now separated explicitly. The response method is readily extended to include weak nonlinear effects, and even meteorological effects (Cartwright, 10) Whereas tidal processes, having a discrete spectrum, offer a choice between the harmonic and response methods (with complicated input spectra and simple admittances favoring the response method), there is no such choice in the case of meteorological processes with a continuous spectrum.

4. THE LAPLACE TIDAL EQUATIONS (LTE) Since the work of Laplace in 1775, the integration of Newton's dynamical

equations to predict the response of the oceans to the tide generating forces has been approximated by attempts to solve Laplaces' Tidal Equations (LTE);

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

210

and

HENDERSHOTT & MUNK

au . -g aCt - f) - - (2QsmO)v = -- ---at a cos 0 acJ> av . -g a(t - D -

+ (2QsmO)u = - , at a ao

at + _1_ (aCUD) + a(vD cos 0») = o. ot a cos 0 oq, ao

Here (cJ>, 0) are longitude and latitude, (u, v) are the corresponding eastward and northward components of fluid velocity. The earth's mean spherical radius is denoted by a, the magnitude of the rotation vector by fl, the undisturbed depth of the ocean by D(q" 8), and the deviation of the sea surface from its undisturbed level by r. The potential U of all the tide generating forces is conveniently introduced in terms of the fictitious equilibrium tide r= Ujg, where g is gravity.

The first two equations are formally obtained from the Navier-Stokes equations of momentum conservation for a homogeneous fluid by neglecting (i) all dissipative terms (section 8) , (ii) nonlinear terms (thus excluding tidal bores and other shallow-water distortions), (iii) variations in the effective gravity g and in the geocentric radius of fluid particles, and supposing (iv) that the pressure field is hydrostatic. To the approximate equations of momentum conservation must be adjoined a conservation relationship (the third of LTE), obtained by integrating the statements of continuity and mass conservation, combined in the usual manner for a homogeneous and incompressible fluid, from the sea floor to the free surface.

The hydrostatic assumption makes the horizontal components of fluid velocity depth-independent, a result used explicitly in deriving the conservation relationship. It requires neglect of local vertical fluid accelerations, of buoyancy forces, and of the radial component of Coriolis force (the corresponding zonal component of CorioIis force associated with the tangential component of the earth's rotation must simultaneously be neglected if the approximated equations are to conserve energy). Attempts to justify the hydrostatic assumption have separately invoked either the extreme aspect ratio (of width to depth) of the real oceans or else their relatively great static stability.

Aspect ratio.-Bjerknes et al. (11) realized that the hydrostatic assumption excludes a class of free inertial (

TIDES 211 With a rigid sea surface, Stern (14) finds free low-frequency equatorial waves, whereas Stewartson & Rickard's (15) expansion in essentially Dja (closely related to the inverse aspect ratio) does not yield nonpathological free oscillations. Both results (14, 15) are very different from those obtained with the hydrostatic assumption. The circumstances under which L TE are

useful approximations in the homogeneous case have not been set forth definitively.

Stability.-At first hand, it is surprising that the small variation of seawater density by only a few parts per thousand in about four km of depth could have any appreciable effect on the tides; but the pertinent criterion is the Vaisala (or stability) frequency2 N as compared to the rotation Q, and N2> > Q2 over most of the water column. Hyelleras (16) and Bretherton (17) therefore include the buoyancy force, but otherwise make the "traditional approximation" (Eckart, 18) of neglecting the Coriolis forces associated with the tangential component of the earth's rotation. The baro

tropic solutions to the resulting equations are nearly identical with the corrcsponding solutions of L TE. Thrane (19) does not assent. Moreover, at

very great depths N2jQ2 may no longer be a large number; in fact, N vanishes within a few centimeters of the deep sea bottom (Wimbush & Munk, 20). The consequences of this are not clear.

Boundaries.-In the foregoing discussion, the ocean depth has been taken as constant. The nonhydrostatic equations for each tidal constituent are hyperbolic in space; it is not evident that the effect of sea-floor roughness is properly modeled in LTE. The presence of lateral boundaries introduces additional complications. These questions are all fundamental. Nevertheless, LTE are the point of departure for almost all theoretical studies of tides.

Solutions.-Simplified though they are, L TE are difficult to solve even in the case of fluid of uniform depth covering the globe. Laplace's solutions

were subject to controversy until the case was settled a century later by Lord Kelvin (21). Hough (22, 23) replaced Laplace's power series in sin (J with an expansion in spherical harmonics (each of which is a possible free oscillation in the absence of rotation), thus treating the case of small E (slow rotation or great depth). Longuet-Higgins (24) extended the analysis of the free oscillations to all values of £ (section 7).

Between the works of Laplace and Longuet-Higgins, a very large number of solutions for nonglobal basins bounded by parallels of latitude and longitude were carried out (summarized by Doodson, 25). These studies demonstrate the computational complexity of solving L TE. Improved under

standing of the allowed motions has largely been derived from investigations in which the equations are, ab initio, radically simplified.

2 N = (-gp-1ap/az)112 for the incompressible case, with -ap/az designating the downward increase in density.

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

212 HENDERSHOTT & MUNK 5. THE ,·PLANE ApPROXIMATION

Lord Kelvin (26) considered the oscillations of a horizontal sheet of fluid of uniform depth rotating about its normal. In this case, L TE reduce to

and

au au; -f) --fv= -g ---at ax av a(r -f) -+fu= -g ---at ay at (au av) -+ D -+- = 0, at ax ay

in which (x, y) are Cartesian coordinates in the plane of the fluid with (u, v) the corresponding veloci ty com ponen ts, and f = 212 is the Coriolis parameter.3 When such a plane fluid is confined within a circular basin at whose perimeter the normal component of fluid velocity must vanish, the solutions are simply expressible in terms of Bessel functions. If the radius a of the basin is small relative to that of the earth, these solutions may be expected to approximate solutions of LTE in a small, circular, nonequatorial sea (Lamb, 27).

For free oscillations at frequency !T, the surface elevation r must satisfy

and

r = a, where r is the radial distance from the center of the basin, cjJ is the longitude increasing in the sense of rotation, and 'i/2=r-1ajar(rajar)+a2jacjJ2.

Solutions are of the form

Zz is the Bessel function of real (!T2>f2) or imaginary (!T2

TIDES 213 equation with Neumann boundary conditions applied at the perimeter of a circular region. The solutions represent ordinary shallow-water waves standing in the radial direction with no longitude dependence (l=0) or progressive in either sense around the basin (l = 1.2, . . . ) . The introduction of rotation raises the frequencies of modes n = 1,2 . .. so that 10"1 >1 and splits the formerly identical frequencies of clockwise and counterclockwise traveling modes. The balance between pressure-gradient force and local fluid acceleration (as in gravity waves) is nearly retained, subject to a rotational perturbation. An interesting case is n =0 for the modes traveling in the sense of rotation. Then u O;

t = exp( -iut + iux(gD)-1/2 - fy(gD)-1/2), U = -(igju)(ar/ax), v = O.

For this case, unlike the general curvilinear case, the flow is everywhere parallel to the coast. Waves progress with a speed (gD)i, the coast lying to the right of the direction of propagation in the northern hemisphere. Surface elevation and parallel-to-shore velocity diminish exponentially away from the coast.

An integral number of Kelvin waves fitted around the coast of a sufficiently large rotating plane basin constitute a free oscillation different from familiar gravity-wave seiches. Taylor (28) displayed Kelvin-like modes in a rectangular basin. The analytical novelty is associated with reflection of a Kelvin wave at a normal closure between parallel walls. The conditions of vanishing normal velocity (v = 0) at the two walls y = 0, b of such a canal are satisfied by any linear combination of the two oppositely traveling Kelvin waves

t = exp( -iui + iux(gD)-1/2 -fy(gD)-1/2)

and

t = exp( -iut - iux(gD)-1/2 + fey -b)(gD)-1/2).

An infinite sum of "Poincare (29) modes"

tm = [cos(m1Ty/b) -(j/u) (lb/m1T) sin(m1Ty/b)] exp( -iut±ilx),

1 = [(u2 -F)/gD - (m1T/b)2]1/2, m = 1, 2, ... ,

is needed to satisfy the additional condition of vanishing axial velocity (u=O) at the closure x=O. These Poincare modes satisfy v=O at y=O,b and decay for large J x l if 12

214 HENDERSHOTT & MUNK closure only if (J"2

TIDES 215 representing waves all having a westward component of phase velocity and frequencies that tend to zero as the rotation rate is diminished. The balance of forces on a fluid particle is nearly geostrophic in x and y, and the velocity field tends to be horizontally nondivergent (the above dispers ion relation is readily obtained by setting Bujax+BvIBy=O from the outset.) These waves are commonly called Rossby or planetary waves. They are approximations to solutions of the "second class" (Eckart, 18) of L TE. Haurwitz (33) displayed the divergence-free case for the sphere. Longuet-Higgins (34, 35) developed systematic approximations, and showed how Rossby waves may be combined to give free oscillations of low frequency in closed basins.

Construction of free planetary-wave oscillations in a beta-plane basin is especially simple in the divergenceless case. The horizontal velocity components may then be written in terms of a stream function t/; which satisfies an equation of the same form as that obeyed by the velocity components themsel ves, 4

a2t/; a2tf; i(j at/; - + - + -- = 0, Bx2 ay2 11 ax

and which vanishes at the perimeter of the (simply connected) basin. A transformation

t/; = 1; exp (-i({3/211)X)

results in Helmholtz's equation for the transformed v ariable ,

a2� a2� � - + - + -1;=0, ax2 ay2 4112

with Dirichlet boundary conditions (!;=O) at the basin's perimeter. In a rectangular basin 0 �x �a, 0 �y �b, the free oscillations are repre

sented by

t/I=sin (nZ7rx/a) sin (n-rr'y/b) exp ( -ium,nt -i({3/2um,n)x)

where

As the number of interior nodes increases, the frequency decreases, in marked contrast with the frequencies

Um•n = [gD7r2(m2/a2 + n2/b2)]1/2

of long gravity-wave seiches in a nonrotating basin of the same size and shape.

4 In the low-frequency limit with dissipation, cr becomes positive imaginary, and we have a homogeneous version of the equation solved by Stommel (36) to demonstrate westward intensification for steady circulation.

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

216 HENDERSHOTT & MUNK Such planetary modes have not been clearly identified in tide records, but

this may not be pertinent because the waves tend to be nondivergent and of vanishing amplitude at the coast. What is needed are long time series of currents.

Longuet-Higgins (37), to whom this development is due, has shown that the beta-plane approximation is useful even for an ocean covering one hemisphere. The shortest planetary-wave period is then 1.62 days (1.53 by the beta plane approximation). Major ocean basins are of lesser extent and their planetary-wave periods accordingly longer.

7. SOLUTIONS FOR GLOBAL GEOMETRIES The validity of the f- and beta-plane approximations is systematically

discussed in Longuet-Higgins' (24) definitive study of the free oscillations allowed by L TE in the case of a fluid covered sphere. Asymptotic representations of the free oscillations for either large or small values of the parameter e=4f22a2jgD are connected by numerical integration. For small e, the solutions are either long gravity waves perturbed by rotation ("first class") or divergenceless planetary waves ("second class"). For large e, three distinct types of solutions emerge, the additional type corresponding to an equatodally trapped, eastward-traveling Kelvin wave. The case €»1 also includes the baroclinic solutions for which the surface-wave velocity ygD is replaced by the much smaller phase velocities appropriate to internal modes [Munk & Phillips (38)].Baroclinic solutions are of particular importance in the theory of atmospheric tides.

Longuet-Higgins extended the analysis to include negative values of e. The resulting free oscillations, formally those of an ocean of negative depth, are physically realizable only as components of forced motions but they are then necessary for completeness. This was first pointed out by Lindzen (39).

The insight offered by these asymptotic studies is invaluable but quantitative calculations of the barotropic response of the oceans to the tide generating forces must be based on the unabridged L TE (as El/2 is order 1). The nonuniform depth of the oceans introduces new classes of oscillations (section 8) and profoundly modifies some of the ones already discussed.

8. TIDES IN THE WORLD'S OCEANS The transition problem.-So far we have dealt with oceans of constant

depth or with depth as some artificial function of latitude for analytical convenience. The sloping topography of continental shelf regions gives rise to a family of Stokes-type edge waves (Ursell, 40; Munk, Snodgrass & Gilbert, 41); the introduction of rotation (Reid, 42) modifies them in a manner analogous to the modification of long gravity waves traveling around a circular basin (Section 5). Deep-sea measurements off California (section 2 ) and California coastal measurements can be tolerably fitted by a superposition of free and forced modes, provided allowance is made for the elastic yielding of the sea bed in response to the tide-producing forces [Munk,

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 217 Snodgrass & Wimbush (7)]. This is in line with Proudman's (43) attempt to interpret the central Atlantic M2 tide as a superposition of Kelvin, Poincare, and forced modes, a study not completed because the unfavorable geometry necessitated, for each component wave, a numerical treatment as complex as a direct solution for the composite tide. Sloping topography in a rotating fluid can give rise to yet another class of oscillations. Cross-differentiation of the beta-plane equations (with variable depth D in the continuity relationship) yields the vorticity equation

!-.- (av _�) = � at + �v + L (u aD + v a D) at ax ay D at D ax ay

Changes in the vorticity av/ax-au/ay of a fluid column are the result of (i) stretching of the column as the free surface rises and falls, (ii) meridional motion of the column, and (iii) stretching of the column as it moves about in a region of varying depth. In the particular case aD/ax=O, (f/D) aD/ay = constant, the equivalence of the second and third mechanisms is obvious. The variable topography associated with the continental shelves and with major isolated features of the sea floor is thus capable of trapping an entire family of shelf waves (Robinson, 44; Mysak, 45; Longuet-Higgins, 46, 47; Rhines, 48, 49). Finally, islands are capable of trapping energy (LonguetHiggins, 50, 51), with and without adjacent variations of topography.

Particularly as a result of topographic trapping, modes of widely varying spatial scale may have nearly identical frequencies. Attempts to construct models which concentrate upon the larger, computationally more economical, scales therefore run the risk of omitting or aliasing significant modes.

Empirical Charts.-Global cotidal charts (as in Fig. 2 but usually without corange lines) have been drawn for some hundred years by interpolating coastal and island measurements without direct recourse to theory. The first such attempt (Whewell, 52) is now only of historical significance. Harris (53) divided the oceans into simply shaped regions whose free oscillations were supposed to dominate the tidal regime. With this interpretation of the sparse

observations then available, he constructed a global chart for the semidiurnal tide. Based on the growing observational evidence, Sterneck (54, 55) constructed empirical global charts for M2 and for a composite diurnal tide, Dietrich (56) for M2, S2, Kl and 01• Villain (57) reconsidered the observed global distribution of M2 and 52 in more detail. Other authors have constructed charts for single major ocean basins (Prufer, 58; Bogdanov, 59, 60), or for marginal seas (Defant, 61-bibliography).

These constructions have been influenced to a varying degree by known solutions to the L TE: amphidromic systems, free oscillations, Kelvin waves. They are perhaps most fairly regarded as crude integrations of L TE with tidal elevation specified at the boundaries. The constructions have been valuable and stimulating in the development of the subject, and now provide at the very least a description of global observations sufficiently smoothed to be comparable with approximate theoretical calculations.

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

218 HENDERSHOTT & MUNK

FIG. 2. Cotidal and corange lines for the M. tide obtained by solving L TE with coastal values specified. Cotidal lines radiate from amphidromic points, corresponding to the progression of tidal crests in the sense indicated by the heavy arrows around these points of vanishing tidal range (Section 5). High tide occurs along the cotidal lines labeled 00 just as the moon passes over Greenwich meridian. Successive cotidal lines delineate tidal crests at lunar hourly intervals. (For clarity, only selected cotidal lines are labeled, 300 corresponds to a delay of one lunar hour.) Corange lines (5, 10, 20, 50, 75, 100, 125 em) connect locations of equal tidal amplitude (not double amplitude). They surround amphidromic points (where range vanishes and phases vary rapidly) and range maxima (where range is largest and phases nearly constant).

Semi-empirical charts.-A numerical treatment of any realistic model is mandatory. Defant (62, 63) attempted to use Green's (64) law of propagation for long waves in a canal of slowly varying cross section to describe diurnal and semidiurnal tides in the Atlantic and in marginal seas. Recently, Nekrasov (65) has given a similar discussion of tides in the Arctic Sea. The principle defect in these essentially one-dimensional computations is the neglect of Coriolis forces; the neglect was remedied in part by working with plane rectangular basins (section 5) whose size and orientation were chosen to model small seas crudely. Kelvin and Poincare waves could then be included explicitly in the solutions (Godin 66, Defant 61).

Numerical schemes.-With the advent of automatic methods of computation, application of the method of finite differences to the tidal problem was inevitable. Hansen's (67) calculation of the M2 tide in the North Atlantic was the precursor of sim;ilar st'udies by Accad & Pekeris (68); Bogdanov,

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 219

Kim & Magarik, (69); Bogdanov & Magarik (70); Hendershott (71 and unpublished); and Tiron, Sergeev & Michurin (72).

The difference equations are solved by an elimination procedure [Accad & Pekeris (68) and Hendershott (71)] (if a suitable representation of the inverse of the finite-difference matrix of coefficients is stored on tape, an efficient iterative treatment of mild nonlinearities may be feasible). Bogdanov et al. use Gauss-Seidel iteration (they note that it does not converge completely in the dissipationless case). Hendershott (71) suggests that the usual techniques of sequential overrelaxation' commonly used in finite-difference calculations (Forsythe & Wasow, 73) might be divergent or very slowly convergent for L TE.

In all of these studies, the motion is ab initio taken to be harmonic in time with the frequency of a major tidal constituent. Hansen (74), Gohin (75), and Deno (76, 77) obtain all constituents simultaneously by timestepping. The problem generally posed is to solve L TE, either in their primitive form or after elimination of one or two dependent variables, with

(i) vanishing normal velocity at coastlines (Pekeris, Hansen), (ii) specified (observed) values of the constituent at coastal stations and

at selected islands (Bogandov et al.) or at coastal stations only (Hendershott) ,

(iii) specified (observed) values of the constituent at selected coastal and island stations plus vanishing normal velocity at the remaining coastal boundary points (Tiron et al.), or

(iv) a specified albedo a (normal velocity = a.\") at coastlines or continental shelves (Gohin).

Basin resonance.-At the 1960 General Assembly of the IDGG in Helsinki, Pekeris presented the first global numerical integration of L TE. Subsequent refinement of the computational grid and of the associated representation of the boundary did not result in uniform convergence of the solution. In spite of various improvements5 the semidiurnal solutions displayed by Pekeris (78) at the 1967 Genera l Assembly of the IUGG in Berne remained unacceptably sensitive to seemingly minor details in the discretization of the boundary.

The domain of integration in Pekeris' computations is (as in Hansen, 74) a union of rectangular subregions. Hence the computational boundaries, at which the normal component of fluid velocity is made to vanish, contain many re-entrant corners. One re-entrant corner is known to affect adversely the finite-difference approximation of the eigenvalues of the plane Laplacian (Fox, 79); Pneuli & Pekeris (31) remark on the effect of many re-entrant corners in numerical solutions of L TE. But in this instance, precise approximation of the free periods is crucial. Pekeris' solutions evidence significant dynamic amplification, suggesting that the frequencies of one or more normal

5 Writing two coupled equations in the velocity components rather than a single equation in elevation, thus solving the Dirichlet problem rather than one requiring specification of first derivatives at the boundary.

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

220 HENDERSHOTT & MUNK modes of the world oceans lie close to the driving frequency6 (not surprising in view of the demonstrated diversity of modes) . This means that numerical models of the global tides are much more sensitive to details of discretization, re-entrant corners included, than computational experience with Laplace's potential equation might have led one to expect.

Hendershott's provisional results (Fig. 2 ) for the global M2 tide are realistic in the North and Central Atlantic and in most of the Western Pacific. The distribution of phases in the Indian Ocean is realistic but the computed amplitudes are larger than observed at island stations by factors of two to three. Computed amplitudes depend sensitively on the computational representation of Indian Ocean topography; the situation is clearly dominated by near semi-diurnal resonance. In the Western North Pacific between Japan and New Guinea, where the cotidal maps of Dietrich and Villain show a welldocumented confluence of cotidal lines, Hendershott's calculation yields a pair of amphidromic points. The calculated phases are thus in error by nearly 1800 over part of the Caroline and Mariana Islands. The numerical model is evidently on the "wrong side" of a local normal mode, the erroneous phases may be made nearly correct by small adjustments of the boundary values.

Dissipation.-One might hope to take advantage of the known large energy dissipation to obtain solutions less sensitive to details of discretization.

The role of tidal dissipation was first proposed by Kant in 1754. 7 We here simply recall that the rate of dissipation, 2 .7 X 1019 erg/sec, derived from the observed acceleration of the moon, is so great that all of the energy stored in the tides at any instant is dissipated in less than a day. Independent estimates based on coastal tidal observations have been made by Taylor (80) , Jeffreys (81) , Heiskanen (82 ) , and Miller (83) . Miller estimates 1.7X1019± 50% erg/sec for the dissipation at coasts and in shallow seas. Greater precision is precluded by the quality and distribution of available observations. The crucial question of whether substantially all tidal dissipation occurs at the edges of the great ocean basins is thus left unresolved. If Miller's estimate errs on the small side, neglect of dissipation in the open sea may be permissible but Cox & Sandstrom (84) suggest that significant tidal energy may be scattered into internal waves by sea floor roughness.

A numerical model which allows for dissipation in detail is computationally out of the question (even if the mechanisms of turbulence were understood) . One naturally seeks a parametric representation. Hansen's success in predicting the tides of the North Sea using an empirical bottom-stress

S In principle, Fourier analysis of the "ringing" of a timestepping model after impulsive excitation would permit direct determination of the frequencies of free oscillation.

-

7 'Untersuchung der Frage, ob die Erde in ihrer Umdrehung urn die Achse, wodurch sie die Abwechselung des Tages und der Nacht hervorbringt, eine Veranderung seit den ersten Zeiten ihres Ursprunges crlitten habe, und woraus man sich ihrer versichern konne.'

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 221

representation, 'Ypl ul u ('Y�0.003), suggests that this widely used parameterization is acceptable for one aspect of the dissipative process. But Hansen's work also shows that the size of the computational grid must be to or smaller before this parameterization will lead to acceptable results. Pekeris has introduced an artificial frictional term, proportional to uj D3, a choice intended to limit dissipation to near coastal regions.

Other investigators have used coastal and island observations as boundary values in otherwise dissipationless numerical models. The use of observed tidal elevation makes it convenient to combine L TE into a single equation in the elevation. Hendershott uses coastal tidal elevations augmented by interpolation along sparsely sampled coastlines, and attempts to predict island tides. He finds it essential to include the static earth tide. Tiron et al. (using the equations for a rotating plane (!) fluid) and Bogdanov et al. ignore the earth tide, but still obtain results that are not unrealistic, presumably because island observations are included as specified boundary values. Implicit in these procedures are two (possibly inconsistent) assumptions: (i) that coastal and island values are directly representative of adjoining deep-water values, and (ii) that the regions of dissipation are localized near the coasts or boundaries. Tidal dissipation and gross coastal or continental-shelf albedos may be computed by this method rather than being imposed as adjustable parameters. One may hope also that this implicit parameterization of friction would allow the use of much coarser meshes.

Proudman (85) has given an integral formulation of L TE. To date, it has been utilized only by Fairbairn (86) who computed the K2 tide along the equator of the Indian Ocean using observations along the coast of the northern Indian Ocean as boundary values.

Timestepping.-Hansen (67) has pioneered application of timestepping to tides in the North Sea. He uses a staggered grid system in which velocity components and elevations are computed at adjacent (not at coincident) points. The mesh width 21:J.1 is of order 100 km or smaller. By the Courant criterion for stability,

I:J.t/ I:J.l < (2gDm .. ,,)-1/2, the time step I:J.t is of order ten minutes in the North Sea. The computed fields are virtually periodic in time and independent of the initial values within five tidal periods. This rapid convergence is the result of the strongly dissipative nature of the model; without dissipation initial transients would persist indefinitely as a superposition of the free oscillations of the basin [see Hansen (87) and SCOR Tech. Report No. 25 (88)].

The restrictions imposed upon I:J.l by bottom friction and upon I:J.t by the stability criterion make it difficult to apply timestepping techniques to the deep sea. Gohin (75) has carried out such calculations for M2 in the North Atlantic and in a domain composed of the South Atlantic and the Indian Oceans. He neglects bottom friction and chooses a variable coastal albedo ex to obtain a "satisfactory" configuration of cotidal and corange lines. The resulting boundary dissipation is sufficient to damp initial transients. His

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

222 HENDERSHOTT & MUNK

results in the interior North Atlantic compare favorably with observations at islands and with calculations by Hansen (67) and Boris (89). Gohin includes the static earth tide and suggests an iterative procedure for tidal self gravitation. Veno (76, 77) has carried out calculations excluding bottom friction (he does not discuss how initial transients are damped) with large (2�l= 10°) mesh spacing.

Bottom Reliej.-In all of these computations, it is essential to use some representation of the actual bottom topography. Numerical experiments by Hendershott (unpublished) show that even when coastal tides are forced to agree with observations, the finite-difference representation of the bottom relief significantly influences the deep-sea tides. Bottom relief must be sufficiently smoothed that it is not aliased when sampled at the mesh spacing. Pekeris' experience (private communication) suggests that an appreciably greater degree of smoothing is required for computational stability. It is noteworthy and possibly quite important that such a degree of smoothing must necessarily make the continental shelves far broader and of far more gentle slope than they are in nature. Any parameterization of coastal dissipation may be profoundly affected by this procedure.

Prospects.-To solve the global tide problem one must learn to understand certain small-scale problems in sufficient generality so they can be embedded in a coarse global model: energy trapping along shelves, ridges and around islands,s bottom friction over extended regions, and energy conversion into internal modes. The actual ocean division into distinct topographical provinces favors such an approach.

9. LONG-PEmoD TIDES All of the foregoing calculations are more or less tailored to the response of

the sea to tide-generating forces having periods of the order of the period of rotation. In the past it has been taken for granted that the ocean's response to the long-period tide-generating forces is nearly static. The work of Wunsch (90) suggests strongly that the fortnightly (Mf) and monthly (Mm) tides are not static but rather consist of a superposition of rather short planetary waves most intense in the western parts of ocean basins (section 6). Rhines (48) suggests that such a disturbance will be prevented from reaching the coasts by continental-shelf topography, an apparent explanation of Wunsch's observation that the fortnightly component tends to be insignificant at coastal stations. Similar considerations may apply to tides of even longer period (semiannual, annual, pole tides).

ACKNOWLEDGMENT

The Office of Naval Research has supported our work in tides.

8 This is necessary to compare island observations with global computations, even if the island effect is negligible on a global scale.

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

TIDES 22 3

LITERATURE CITED 1. Thompson, Sir W., Mathematical and 27. Lamb, H., Hydrodynamics, (Cam-

Physical Papers, Vol. VI, (Cam- bridge Univ. Press, 738 pp., 1932) bridge University Press, 1 9 1 1 ) 28. Taylor, G. I., Proc. Land. Math. Soc.,

2. Snodgrass, F. E., Science, 146, 1 98-208 (2), 20, 148-81 (1920) ( 1964) 29. Poincare, H., Theorie des Marees,

3. Nowroozi, A. A., Sutton, G. H . , AUld, Lecons de Mecanique Celeste, Vol. B., Ann. Geophys., 22, 5 1 2-1 7 III, (Gauthier-Villars, Paris, 469 (1 966) pp., 1910)

4. Eyries, M . , Cah. Oceanogr., 20, 355-68 30. Grace, S., Mon. Not. Roy. Astron. Soc. ( 1 968) Geophys. Suppl., 2, 385, (1931)

5. Filloux, J. H . , Trans. Am. Geophys. 3 1 . Pnueli, A., Pekeris, C. L., Phil. Trans. Union, 49, 2 1 1 (1 968) Roy. Soc., London, A, 263, 149-71

6. Snodgrass, F. E., Science, 162, 78-87 ( 1969) ( 1 968) 32. Rossby, C. G., J. Mar. Res., 2, 1, 38-55

7. Munk, W., Snodgrass, F., Wimbush, ( 1 939) M., Geophys. Fluid Dynam. , I, (in 33. Haurwitz, B. , J. Mar. Res., 3, 1, 254-67 press) ( 1970) ( 1 940)

8. Doodson, A. Too Proc. Roy. Soc., Lon- 34. Longuet-Higgins, M. S., Proe. Roy. don, A, 100, 305-29 (1921) Soc., London, A , 279, 446-73 ( 1964)

9. M unk, W. H., Cartwright, D. E., Phil. 35. Longuet-Higgins, M. S., Proc. Roy. Trans. Roy. Soc., London, A , 259, Soc., London, A , 284, 40-54 (1965) 533-81 (1966) 36. Stommel, H . , Trans. Am. Geophys.

10. Cartwright, D. E., Phil. Trans. Roy. Union, 29, 202-06 (1948) Soc., London, A, 263, 1-55 ( 1968) 37. Longuet-Higgins, M. S., Phil. Trans.

1 1 . Bjerknes, V., Bjerknes, J., Solberg, H., Roy. Soc., London, A, 260, 3 1 7-50 Bergeron, T., Physikalisehe Hydro- (1966) dynamik, (Berlin, 797 pp., 1933) 38. Munk, W. H., Phillips, W., Rev. Geo-

12. Solberg, H . , Astrophys. Norv. I, 7, 237- phys., 6, 447-72 ( 1 968) 340 (1936) 39. Lindzen, R. S., Mon. Weather Rev., 94,

1 3 . Proud man, J., Proe. Roy. Soc., Lon- 295-301 (1966) don, A, 179, 261-88 (1942) 40. Ursell, F., Proe. Roy. Soc. London, A ,

1 4 . Stern, M . E. , Tellus, 1 5 , 246-50 ( 1 963) 214, 79-97 ( 1 952) 15. Stewartson, K., Rickard, J. A. , J. 4 1 . Munk, W. H., Snodgrass, F., Gilbert,

Fluid Meeh., 35, 759-73 ( 1 969) F., J. Fluid Meeh., 20, 529-44 16. Hyelleras, E. A., Astrophys. Norv., 3, ( 1964)

1 39-64 (1939) 42. Reid, R. 0., J. Mar. Res., 16, 109-44 1 7. Bretherton, F. P., Tellus, 16, 2, 1 81-85 (1 958)

( 1964) 43. Prondman, J., Mon. Not. Roy. Astron. 1 8. Eckart, C., Hydrodynamics of Oceans Soc., 104, 244-56 ( 1 944)

and A tmospheres, (Pergamon Press, 44. Robinson, A. R., J. Geophys. Res., 79, New York, 290 pp., 1 960) 367-68 (1 964)

19. Thrane, P., Geofys. Publik., 18, 1-36 45. Mysak, L., J. Mar. Res., 25, 205-27 (1951) ( 1 967)

20. Wimbush, A. M . , Munk, W., The ben- 46. Longuet-Higgins, M. S., J. Fluid thic boundary layer, The Sea, (In- Meeh., 3 1, 41 7-34 (1968a) terscience Publishers, New York, in 47. Longuet-Higgins, M. S., J. Fluid press) Meeh., 34, 49-80 (1968b)

2 1 . Thomson, Sir W., Phil. Mag., 4, 227- 48. Rhines, P. B., J. Fluid Meeh., 37, 1 5 1-42, ( 1 845) 89 ( 1 969)

22. Hough, S., Phil. Trans. Roy. Soc., Lon- 49. Rhines. P. B. , J. Fluid Meeh., 37, 191-don, A, 189, 201-57 ( 1 897) 205 (1969)

23. I'Iough, S., Phil. Trans. Roy. Soc., Lon- 50. Longuet-Higgins, 1\1. S., J. Fluid don, A, 191, 1 39-85 ( 1 898) Meeh., 29, 781-831 (1967)

24. Longuet-Higgins, M. S., Phil. Trans. S 1 . Longuet-Higgins, M . S., J. Fluid Mech., Roy. Soc., London, A, 262, 5 1 1-607, 37, 773-84 ( 1 969) ( 1967) 52. Whewell, W., Trans. Roy. Soc., London,

25. Doodson, A. T., A d'llan. Geophys., 5, 147-236 ( 1 833) 1 1 7-52 (1958) 53. Harris, R. A., Manual of Tides IV,

26. Thomson, Sir W., Proe. Roy. Soc. Edin- B. U. S. Coast Survey Rept., App. 5, burgh, 10, 92 ( 1 879) 3 1 3-400 (1904)

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

22 4 HENDERSHOTT & MUNK S4. Sterncck, R., Sitz/;er. d. A kad. Wiss.

Wien, A bt. IIa, 129, 1 31-50 (1920) 55. Sterneck, R., Sitzber. d. A kad. Wiss.

Wien, Abt. IIa, 130, 363-71 (1921) 56. Dietrich, G., Veroffentl. Inst. Meeresk,

Univ. Berlin, A41, 1-68 ( 1944) 5 7. Villain, C., A nn. Hydrog. (Paris) , 3,

269-388 (1952) 58. Prufer, G., Veroffentl. Inst. Meeresk,

Univ. Berlin, A 3 7, 1-56 (1939) 59. Bogdanov, K. T., Doklady A kad. Nauk

SSSR, 138, 2, 441-444 ( 1 961) 60. Bogdanov, K. T., Doklady A kad. Nauk

SSSR, 139, 3. 713-16 (1961) 61. Defant, A., Physical Oceanography, Vol.

II, (Pergamon Press, 598 pp. (1961)

62. Defant, A., A nn. Hydrog. (Berlin) , 52, 153-66; 1 7 7-84 (1 924)

63. Defant, A., Wiss. Ergeb. Deut. Atl. Exped. "Meteor", 1925-1927, 7, 1, 1-318 (1932)

64. Green, G., Trans. Cambridge Phil. Soc., 6, 457-462 (1 838)

65. Nekrasov, A. V., (abstract) Abstracts of papers, Vol. V. , Intern. Assoc. Phys. Oceanogr., XIV General Assembly, IUGG, 80 (1967)

66. Godin, G., Ms. Rep. Ser. No. 2, Mar. Sci. Br., Dept. Mines Tech. Surveys, Ottawa, 156 pp. ( 1966)

67. Hansen, W., Deut. Hydrog. Z. (Ergiinzungskejt), 1, 1-46 (1952)

68. Accad, Y., Pekeris, C. L., Proc. Roy. Soc., London, A , 278, 1 1 0-28 (1964)

69. Bogdanov, K. T. , Kim, K. V., Magarik, V. A., Trudy Inst. Okeanologii, LXXV, 73-97 (1964)

70. Bogdanov, K. T., Magarik, V. A., Doklady A kad. Nauk SSSR, 172, 6, 1 315-17 (1967)

71. Hendershott, M. C., Proc. Symp. Math. Hydrodynam. Invest. Phys. Proc. Sea, Moscow, 8-21 (1966)

72. Tiron, K. D., Sergeev, Y. N., Michurin, A. N., Vest. Leningrad. Univ., 24, 123-35 (1967)

73. Forsythe, G. E., Wasow, W., Finite Difference Methods for Partial Differential Equations. (John Wiley & Sons, New York, 444 pp., 1960)

74. Hansen, W., Symp. Math. Hydrodynam. Meth. Phys. Oceanog. Hamburg, 25-34, (1961)

75. Gohin, F., Symp. Math. Hydrodynam. Meth. Phys. Oceanog., Hamburg, 1 79-97 (1961)

76. Veno, T., Oeeanog. Mag., IS, 2, 99-1 1 1 (1964)

77. Ueno, T., Oceanog. Mag., 16, 1-2, 47-55 (1964)

78. Pekeris, C. L., Accad, Y., (abstract) A bstracts of Papers, Vol. V., Intern. Assoc. Phys. Oceanog., XIV Gen. A ssem. IUGG, 85 (1967)

79. Fox, L. N., Numerical Solution oj Ordinary and Partial Differential Equations, (Addison-Wesley, 509 pp., 1962)

80. Taylor, G. !', Phil. Trans. Roy. Soc., London, A, 220, 1-33 (1919)

81. Jeffreys, M., Phil. Trans. Roy. Soc., London, A , 221, 239-64, (1921)

82. Heiskanen, W., Ann. A cad. Sci. Fenn., A, 18, 1-84 (1921)

83. Miller, G., J. Geophys. Res., 71, 4, 2485-89 (1 966)

84. Cox, C. S., Sandstrom, H., J. Oceanog. Soc. Japan, 499-5 1 3 ( 1 962)

85. Proudman, J., Phil. Mag., 6, 49, 5 70-79 (1925)

86. Fairbairn, L. A., Phil. Trans. Roy. Soc., London, A, 247, 191-2 1 2 (1954)

87. Hansen, W., Tides, The Sea, I, (Interscience Publishers, New York, 864 pp., 1962)

88. SCOR Tech. Report No. 25, Mitteilungen Inst. Meereskundc Univ. Hamburg, Ny. V, 57 pp. (1966)

89. Boris, L. I.. Trudy Leningrad. Meteor% g. Inst., 10, (1961)

90. Wunsch, C., Rev. Geophys., 5, 4, 447-75 ( 1 967)

Ann

u. R

ev. F

luid

Mec

h. 1

970.

2:20

5-22

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

Acc

ess

prov

ided

by

Old

Dom

inio

n U

nive

rsity

on

11/0

8/18

. For

per

sona

l use

onl

y.

Annual Reviews OnlineSearch Annual ReviewsAnnual Review of Fluid Mechanics OnlineMost Downloaded Fluid Mechanics ReviewsMost Cited Fluid Mechanics ReviewsAnnual Review of Fluid Mechanics ErrataView Current Editorial Committee

ar: logo: