On Laplace’s differential equations for the...

On Laplace’s differential equations for the tides

By J. P roudman, F.R.S,

[Received 22 May 1941)

1. Introduction

This paper is an investigation of the validity of the assumptions made by Laplace in framing his well-known differential equations for the tides of the ocean. Since his time these equations have been used by all the principal workers on the dynamical theory of the tides, but in recent years it has been claimed, mainly in Norway, tha t they are quite inadequate.

Let (o denote the angular speed of the earth’s rotation and let t denote the time; let r denote the distance of a point in the ocean from the earth’s centre, and 6 the co-latitude of this point; let u, v and w denote the components of the velocity of a particle of water in the directions respectively of increasing 6, east longitude and r. Then the corresponding components of the acceleration of the particle are respectively:

du uv 2 d t ^ r r cot 6 — 2o) cos dv—co2r sin 6 cos 6,

-7--I------1----cot 6 + 2(o cos 6u -f 2co sin 6w,dt r r (M )

dw u2 + dt r 2oj sin 6v — sin2 6.

In forming his differential equations Laplace neglected all terms of the first two of these expressions involving u, v and w, except those on the left- hand side of the equations (1-2) below; and he dispensed with the use of the third by neglecting the vertical component of the acceleration. By interpreting u and v in the Eulerian sense he neglected product terms and thus restricted his consideration to small motion. He also neglected the vertical component of the tide-generating forces and the vertical variations in the horizontal components of these forces. He assumed the mean surface of the ocean to be the sphere r = a.He took the centrifugal terms of (1*1) to be merged in the earth’s effective gravity, and assumed the acceleration g

[ 261 ]

on June 13, 2018http://rspa.royalsocietypublishing.org/Downloaded from

http://rspa.royalsocietypublishing.org/

262 J. Proudman

of this effective gravity at the earth’s surface to be uniform. His equations of motion thus take the form

dudtdv

2(o cos dv a d d ’

+ 2o) cos du — — 9a sin d dx

( 1-2 )

where x denotes the east longitude, denoting the elevation ofthe free surface, and — the variable gravitational potential on

I t is a consequence of the equations (1*2) that there is no vertical variation in uand v,and then the equation of continuity takes the form

^ ( l ( A s i n 0M) + |< t o ) + ' (1-3)

where h denotes the mean depth of the ocean at any place. I t is also a consequence of Laplace’s equations that the vertical component of current varies linearly between the ocean bottom and the ocean surface.

In 1933 V. Bjerknes, J. Bjerknes, H. Solberg and T. Bergeron stated tha t the neglect of the vertical acceleration was a serious error, which might give totally erroneous results for a tidal constituent of period 2 such tha t ct2 < 4oA They emphasized the possibility of ‘ cellular oscillations ’ associated with negative values of the factor 1 — 4&>2/<r2, a possibility which was indicated by the presence of this factor in the equations obtained in 1880 by Sir W. Thomson (Lord Kelvin) on the vibrations of a columnar vortex. The main force of their criticism referred to the diurnal constituents, for which cr2 is near oj2; they stated that Laplace’s equations were not seriously in error for the semi-diurnal constituents, for which is near 4 I t may be mentioned that the vertical acceleration had been retained by R.O. Street in 1930 in his study of the surface oscillations of water in a rotating cylindrical vessel, but he did not examine the case of or2 < 4

In 1933 M. Brillouin and J. Coulomb published a detailed study of the oscillations, both free and forced, of the water in a flat circular basin of uniform depth, in which they retained the vertical acceleration. They emphasized the fundamental way in which the distribution of the motions depends on the sign of 1 — 4<y2/<r2, and they paid particular attention to possible applications to the tides. Their problem of forced tides, when cr2 < 4 (o2,led to certain difficult questions of convergence. They pointed out that for the semi-diurnal constituent K 2, in which cr2 = 4o2, Laplace’s equations did not appear to be valid.

In 1936 H. Solberg published the first part of a monograph on the free



On Laplace’s differential equations for the tides 263

small oscillations of a homogeneous liquid on a rotating earth, with special reference to the dynamical theory of the tides in the oceans and atmosphere. For the equations of motion he took

- — 2o) cos Ova ti b p

‘ r 0 0 ’

— + 2o)cos Ou + 2(o sin Owdtdw — 2(0 sin 6v

1r sin 6 ’a pdr ’

(1-4)

where P (1-5)

p denoting the pressure, p the density of the water, and V the potential of the gravitational and centrifugal forces. For the equation of continuity he took the general equation

l a . . . . . i di3 /9 xi^dde{sm0u)+^redx+rd-r(rhv) = o- (1'6)

He took account of the spheroidal nature of the mean surface of the ocean, and he considered it an ‘internal contradiction’ to retain the dynamical effect of the earth’s rotation on the tides and yet to neglect it on the shape of the mean surface of the ocean.

For the free oscillations of an ocean of uniform depth covering the whole earth, Solberg showed that the analytical character of the solution completely changed as 1 — 4w2/<r2 passed through zero. For free oscillations which are the same in all longitudes, and in which tr2 > 4 he obtained results which are not very different from those obtained by S. S. Hough (1897) on the basis of Laplace’s equations. He also gave some attention to the cases in which <x2 = 4 (o2. The detailed development of the cases in which a2 < 4 (o2,where Solberg expected results differing greatly from those

obtained by Hough, he reserved for a second part of his monograph.

2. Summary of the present paper

In the present paper consideration is restricted to the tides of the ocean and the water is taken to be homogeneous. Forced oscillations are considered, as the actual tides belong to this class. The types of motion can thus be restricted to those corresponding to the actual tides. A direct comparison can also be made between the results of solving Laplace’s equations



264 J. Proudman

and the results of a more general theory. For this more general theory Solberg’s differential equations, viz. (1-4) and (1*6), are used.

First it is shown that it is reasonable to simplify the problem by neglecting the ellipticity of the meridians (§3) and the gravitational attraction of the disturbed water (§4). With these simplifications the problem is still mathematically self-consistent and the question of cellular oscillations remains.

In §§ 5, 6 the general mathematical formulation is given, and in § 7 it is shown that cellular oscillations are only liable to occur for semi-diurnal constituents near the poles and for long-period constituents near the equator.

Certain cases provided by the vanishing, in turn, of one of the components of current are then considered. I t is shown (§ 8) that Laplace’s celebrated particular integral for the constituent K x in an ocean of uniform depth may be immediately and without approximation extended to Solberg’s equations. For a narrow ocean of uniform depth bounded by meridians (§9) and for an endless canal of uniform depth along a parallel of latitude (§10), it is shown that Laplace’s equations always provide a high degree of approximation.

Two basins for which Laplace’s equations do not always give valid solutions are next considered. These are a circular sea of uniform depth near the North Pole (§11) and a broad channel of uniform depth near the equator (§ 12), and in both cases the curvature of the meridians is neglected. As already indicated, the cases of failure of Laplace’s equations are those of semi-diurnal constituents in the first basin and of long-period constituents in the second basin.

Finally, an ocean over which the depth is a function of the latitude only is considered. For the K % constituent (§13) and for the limiting form of long-period constituents (§ 14) general differential equations in which there is only one independent variable, viz. the latitude, are obtained. When the depth is a very small fraction of the earth’s radius, these differential equations reduce to the corresponding ones of Laplace except near the poles for the K% constituent and near the equator for the limiting form of long-period constituents. I t appears that in §§11, 12 the neglect of the curvature of meridians has tended to exaggerate the degree of invalidity of Laplace’s equations.

For the diurnal constituents, Laplace’s equations appear always to give a good approximation in problems which really resemble those of the actual tides.

In the present paper only such problems as may be solved without the use of expansions in infinite series are considered.




3. N eglect of the ellipticity of meridians

The ellipticity of a meridian section of the mean surface of the ocean depends on cao2jg, while the gyrational effects on the tides depend on o>/<r. From a mathematical point of view, these are independent parameters. If in the solution of the equations (1*4) and (1*6) the latter is retained but the former neglected, and the mean free surface of the ocean is taken as a mathematical problem without any mathematical inconsistency is formulated. And as the solution of this simplified problem will have the same general character as that of the more complete problem, consideration wdll be given only to the simplified problem.

The contribution to the potential V from the constant part of the earth’s gravitation and from the centrifugal force of the earth’s rotation is taken to be g(r — a), where g is constant.

4. N eglect of the gravitational attractionOF THE DISTURBED WATER

By using spherical harmonics, Hough, in his solution of Laplace’s equations for an ocean covering the whole earth, was able to allow for the disturbance of the gravitational field produced by the tides themselves. In a similar way and for the same ocean, Solberg, in the solution of his equations, was able to make the same allowance. But when the ocean does not cover the whole earth, the allowance for the gravitational attraction of the disturbed water is a difficult problem, even in the equilibrium theory of the tides (H. Poincare 1896, 1910; M. Brillouin 1928). No special cases of the equilibrium theory appear yet to have been worked out, though Brillouin’s method makes this possible.

As this matter is not one of those which is the subject of controversy, in the present paper the disturbances in the gravitational field produced by the tides themselves are neglected.

5. Conditions at free surface

From (1*5) and the results of §§ 3, 4,

p = ^ + 9(r-a)-gf(r) ,(5-1)

where f(r)is a given function of r such tha t/(a) = 1.

Vol 179. A. 18



266 J. Proudman

At the free surface of the ocean

so that, from (5-1)

r = a + £, = 0 ,

P = g{Z-f{r) a

but on working to the first order of small quantities only

(5-2)

on r = a ,since £— £ = £ •In the differential equations (1*4) and the boundary conditions, P enters

as a whole. Hence the function/(r) does not affect the determination of P ; it only affects the determination of the pressure p.

For the semi-diurnal, diurnal and long-period constituents of the actual tides

respectively, H denoting a constant length appropriate to each case and cr being positive. By taking

a and s being positive, more than one of these constituents may be considered at the same time, and, when F(6) does not include the functions of 6 in (5*3), it will be a general member of a sequence of functions specially convenient for the basin in question. The assumption is made tha t the wavelength of the variations of £ is large compared with the depth of the ocean. This assumption is important, as it is the basis of our contention that Laplace’s equations are valid for constituents other than those of semidiurnal and long periods. I t corresponds to the usual criterion for long waves, and it is only for such motion that Laplace’s equations have ever been Assumed to hold.

r— = sin2 0 cos + 2y),

r— = sin d cos 6 cos (at + x), (5-3)

= |(3 cos2 1) cos at,

= F (0) cos (at+ sx),



On Laplace's differential equations for the tides 267

6. Harmonic motion

For harmonic oscillations of period 2 7 in which t enters only through ei<rt Laplace’s equations (1*2) yield

ior - 1 /0£' 2a>eos#0£'\\ dd icr sin

icr - 1asin# dxK - ~ c o S02(0

icrd£w \

( 6 - 1 )

and then on substituting into the equation of continuity (1-3),

1 [—sin d {I

A sin#

1 — —7T COS2 0 ^

0 .

( 6-2 )

When o'2 ^ 4 co2,a similar procedure may be followed with the equations (1*4) and (1-6). The equations (1-4) can be solved algebraically so as to give u, v and w in terms of dP/dd, dP/dx and dP/dr. The conditions at the ocean bottom, a t the surface, and at the coasts can then be expressed in terms of P and of these same derivatives. On substituting into the equation of continuity (1-6), a differential equation of the second order is obtained for P, to be satisfied through the body of the whole ocean. These conditions are sufficient to determine P, u, v, w, £ and p.

When o’2 = 4o>2, there is a linear relation between dP/dd, dP/dx and dP/dr. To determine u, v and w use must be made of two of the equations (1-4) and the equation (1-6).

The procedure may be illustrated more concisely in cylindrical coordinates, and in fact much use will be made of these co-ordinates. Denoting them by vr, x and z with the axis that of the earth, the equations of motion may be written as

n i l .2o)v dP

dm ’du dtdv „ 1

or + 2mu =---- ,ot m dxdw dP

dz '

(6-3)



268 J. Proudman

Using the time factor ei,rl, it is deduced that, when # 4

icr IdP 2w j.0P \U a2 — i(o

icr / 1 dP 2a> 0P\! — 4(o2 \U7 dx icr dm)'1 dw — —icr *

The equation of continuity may be written as

1 8 . , 1 -— (U7W) + — 5-+-V- = o, m dm m ox

and on substituting from (6-4)

m dml dP\ 1 d2P(. 4 w2\ \ dm/ m2 dx2 + \ cr2 J

When cr = 2(o, it follows from (6-3) or from (6-4), that

dP .dP----- 1 = °»dm dx

and the equation (6-6) may then be written

0 . d \ l d P > dx) \ dm

Also, from (6*3) and (6-5),

. du„ , x 1 dm mU '>tldx <r\w 2 m

(6-4)

(6-5)

( 6 -6 )

(6-7)

( 6 -8 )

7. The possibility of cellular oscillations

Since the depth of the ocean is very small compared with its lateral dimensions, cellular oscillations can only occur when the wave-length of the undulations of P in the vertical direction is very much smaller than the wave-length of these undulations in the horizontal direction. For the possibility of this we examine the equation (6-6), and notice that it depends on very small or very large values of 1 — 4

Near the poles the vertical co-ordinate is z and d2P/dz2 must be large compared with

1 0 / dP\m dm \ dm / (7-1)




This will be the case when 1 — 4&>2/<r2 is very small, unless

(7-2)

is also very small. The m atter is examined in detail in §§ 11 and 13.Near the equator the vertical co-ordinate is m, and for cellular oscilla

tions (7-1) must be very large compared with This may be the casewhen 4w2/(r2 is very large, so that the tides are of long period. The matter is examined in detail in §§ 12 and 14.

For other latitudes and other constituents cellular oscillations do not appear to be possible.

8. Laplace’s particular integral for the constituent K 1

When the depth is uniform and a = o) the equations (1-4), (1-6) and the conditions at the bottom and at the free surface of the ocean are satisfied by

As there is zero vertical component of current at all levels, it is clear that the above expressions, with r — a, P = <?£', also satisfy Laplace’s equations; and in fact this is the solution of his equations given by Laplace himself, in which there is no rise and fall of the ocean surface.

Of course the satisfaction of coastal conditions requires the addition of complementary functions which, in general, do involve a rise and fall of the ocean surface.

2sin 0 cos 0 cos ( + y),

w = 0, £ = 0 .



270 J. Proudman

9. N arrow ocean bounded by meridians

Consider now a basin of uniform depth h bounded by two meridians which are so near together that there are no transverse currents. Take

l Pn(COS 6) COS (Tt, (9-1)

where Pn( ) denotes Legendre’s polynomial of integral order n.On writing v = 0 and dv/dx = 0 in the equations (1-4) and (1-6), and then

on eliminating u and w, it follows that

1 0 / • n dP\^ r e w { s m e de)4 (4 ) o, (9-2)

an equation which does not contain co. The solution of all the conditions is given by

P

UcrAw

crA

[ r \n n / M 2n+1/a \n+1) _ .(a) + r e + l ( 1~ a ) (?) « » «*•

~ — lcr2a\

- 4 *cr2a

where HA n il (■-n - i -ar a n+ 1

When h/a is small, (9-3) gives, to the first order, on and in viewof (5-2),

£_A 'u

aA'w

VA'

Pn{cos 6) cos art,

9 dcr2a

ghcr2a2

where HA 7

Pn(cos 6) sin

n(n + 1) Pn(cos 6) sin

gh

(9-4)

n{n-\-1) cr2a2 1.

On writing v = 0 in Laplace’s equations (1-2) and (1-3), and eliminating u, the equation

1a2a sin 6 dd (sin4 ) +£'=-«> (9-5)



is obtained. The solution of this equation and the corresponding expressions for u and w are given by (9*4), so th a t Laplace’s equations are valid.

I t follows from the first of (9-3) on r = that


rH

J^((cos 6) cos at

where qis a known function of A/a. Denoting by q0 the corresponding part of (9-4), viz. n(n + 1) A/a, the magnitude of q/q0 — 1 may be used as a measure of the degree of approximation afforded by the solution of Laplace’s equations. Now

H)2n+1| a__________ ! i

/ M2*+ln + 1 + ny 1 — - 1

- 1,

and when A/a is small, the principal part of this is

- \n ( n + l ) - 2.

10. Canal along parallel of latitude

Next consider a canal of uniform rectangular section along the parallel of latitude, for which 6 = a, and suppose that it is so narrow that there are no transverse currents. Then

IH cos {at + sx), ( 10- 1)

and there is no point in including the case of = 0 , as there would then be zero tide-generating forces along the canal. As this rules out cases of long- period constituents no large values of 2 coccur.

On writing u — 0 and du/dd = 0 in the equations (1*4) and (1-6), and on eliminating v and w, the equation

v l /2 scofr(r S:7 + ( v ~ ~ S V P = 0’ ( 10-2 )

is obtained, provided that a2 # 4oj2 sin2 a. On writing

2so)(T

s2sin2 a = -n (n + 1), (10-3)



272 J. Proudman

the solution takes the formPgA

n2 sin2

g sin a cr2a s

w a A

where

- J - fcr2a \

x cos(<rtf + s%),

r - i - m * - * *(10-4)

H L / A\2w+1| gj (w + l)2sin2a / w2sin2a \ /A ~ \ \ a) j a-2a \In obtaining the expressions for v and w in (10-4), the factor or2 — 4&>2 sin2 a

has been removed from both numerators and denominators, so tha t these expressions are determinate when tr = 2o> sin a. On starting with = 2wsina, and applying the special method indicated in § 6, the same expressions are obtained directly.

When hja is small, (10-4) becomes, on and in view of (5-2),r

- j r = cos (crt + sx),

voA '

_j___ g_sin a <r2a cos (o-t + sx),

woA '

s2 gh Sin2 cl cr2a2sin (crt + sx),

(10-5)

where H sj* gjiA ' sin2 a ar2a’

and these expressions are identical with those obtained by solving Laplace’s equations (1*2) and (1*3) after writing = 0.

I t follows from the first of (10-4), on that

rH

cos (<rt + sy)

and on writing the first of (10-5) in the same form with for q,

H ri - * 'l a,

^ n + l j | sin2 cl a 1 s2 h 1

- i + ( . + * " - + (' n2 sin2 j

( - 3

|2 n + l




When hja is small, the principal part of this is

q ( s2 , , |A- 2— - 1 = -~2 n(n 1) - = -,q0 (sin2a )a a a

and this gives a measure of the degree of approximation afforded by the solution of Laplace’s equations.

11. Flat circular sea

Consider next a sea of uniform depth bounded by a parallel of latitude which is so near to the North Pole that the earth’s curvature may be neglected. Using the cylindrical co-ordinates m, and z the coast is given by m = c, the mean sea surface by z = 0, and the sea bottom by z

The equation (6*6) has a solution of the form

PgA

J8(ktd) cos A k{z + h) cos (art + sx), (1M )

where A is a constant length, k a constant reciprocal length and

. /4w2\

ko2 \~*O’2 *

Then from (6-4) it follows that

2 Js(K7D)u _ Kg A2 crA J'8{kw) + cos Ak(z + h) sin {crt + sx),a“ {”' ' a kvt

£ - ■ + cos cos (crt + 3 ;),KTUKgX

= L i*™)sin + h) sin {at + ax)•

(H-2)

From (11-2) it is seen that the condition 0 a t the sea bottom — is satisfied; also that the condition u = 0 at the coast m — c will be satisfied if

kcJ's{kc)+ Js{kc) — 0 . (11*3)

This equation determines k, and the simplest distribution for a prescribed value of s will be given by the lowest root of (11*3). The conditions (5-2) at the free surface z — 0 give

2 = — sin A Kh + cos J8{kvj) cos {at + sx),



274 J. Proudman

so that iff

= Js(k w ) c o s (at + sx), (11-4)

then — = gin XkU + cos AA a2(11-5)

When A Khis real and sufficiently large, the expressions (11*2) and (11-5) show that cellular oscillations will occur.

When A Kh is small, it follows that

where

P (4oj2 — a2ct2k «, ,~gA' = ~ Js(KVT){ — " — + ) J C O S (a t + ) >

V^A ' {— J's(k w ) + scos

| (7 KJU )W

aA ' — Js(kvj) k(z + h sin (at +

H_A 7

7 4o2 — a-2Kh-\----------- ,xg

( 11-6 )

and when z = 0 these expressions are identical with those obtained by solving Laplace’s equations.

From (11*1) on z = 0, and (11-5)

rH

Js(kvj) c o s (at + sx)

and on writing the first of (11-6) in the same form with for q, we have

q j tan A q0 AkJi

as a measure of the degree of approximation afforded by the solution of Laplace’s equations. When A Kh is small, the principal part of q/q0 — 1 is i(AAc/i)2.

For the long-period constituents a/2oj is small, so that A is small; also 5 = 0, and then the lowest value of is 3-83. Taking cjh = 100, then Kh = 0*0383, and Laplace’s equations give a very high degree of approximation. For the limiting case in which cr-> 0 and A->0 , the solution given by (11*1), (11*2) and (11*5) becomes identical with that given by (11*6).



For the constituent K x, cr = a>so that A = 0-5774, and s — 1, so that the lowest value of kc is 1-84. Again taking cjh — 100, then A — 0-0106, and again Laplace’s equations give a high degree of approximation.

For the constituent K 2, s = 2, and A->co. Then from (11-1), (11*2) and (11-5) i t is deduced that P tends to vanish while u and v tend to become infinite; but of course a t z = 0,

^ = = - ,72(OT)8in(<rt+2* )-

On starting with a = 2m, (6-7) gives

dm ’and as the solution of this is

p - ( £ ) * ' « •


it follows that, for finiteness at m = 0, F(z) = 0 . Then 0 and = 0, sothat the conditions (5-2) a t the free surface cannot be satisfied. The situation is analogous to that of resonance, in that a boundary condition involving finite elements can only be satisfied by a distribution involving infinite elements. This impossibility of providing, by a finite distribution, the necessary vertical currents at the free surface when A is infinite, throws a light on the part played by the vertical currents of the cellular oscillations when A is large.

From the solution (11*6) with s = 2, it is seen that, as er-> and kTiremains finite

PgH

J ^ kvj) p (£z2 + zh). cos (at + 2x),

- fh + '2 ^ i r } 8in (at+ 2*>>

- i { j ^ ) + 2 ^ - )}c°S<<r( + 2A:),

VO / z \-a f j = - hr + 1) sin (<rt + 2x).

(11-7)

In order that these expressions may satisfy the fundamental equations and conditions, it is necessary, according to the different orders of magnitude involved, sometimes to retain P and sometimes to neglect it.



276 J. Proudman

On starting with a = 2coin Laplace’s equations the same solution with z = 0 is obtained. The expressions (11*7) are the result of first taking A small with A finite and then of making A -> oo, and this corresponds to the procedure offered by Laplace’s equations. But the conditions of the physical problem require that A->oo with kJi finite though small, and to this it has been seen that there is no definite solution. Hence for the K 2 constituent, Laplace’s equations give a definite solution when the more general equations show that one does not exist. If the curvature of the meridians had been neglected in (5*3) as well as in (1*4) it would have followed that £ = 0 and not (11*4); both Laplace’s equations and the more general ones would then have given the same zero solution.

12. Broad equatorial channel

Consider now a channel of uniform rectangular section along the equator, and suppose that it is sufficiently broad to admit of transverse currents. The equations will be simplified by supposing tha t in the neighbourhood of the channel the earth is cylindrical with gravity perpendicular to its axis. Cylindrical co-ordinates will be used and also

?H cos kz cos + sy), ( 12- 1)

where s is an integer and & is a constant such that at the sides of the channel sin kz — 0. The case of k = 0 is not included, as this would exclude transverse currents and merely reproduce the problem of § 10 with a = \ir. I f the coasts of the channel are given by z — 0, z = b,it follows tha t kb = where n isan integer. But in order to prevent the wave-length of £ from being of the order of the depth h, n must not be too large. The assumption tha t kh is small, a consequence of tha t made in § 5, is important, because it rules out the possibility of cellular oscillations except for long-period constituents.

The appropriate solution of the equation (6*6) is given by

— F(w)cos kz cos (at + sy),

where F(m) satisfies the Bessel’s equation:

( 12*2 )




Then, from (6-4),

u —g ldF2s(o F \ 1 . . . . 1*H = < ra -4^ U + J « > 6 faam <rt + <X).

*H = - L * ( a d w + Sw ) COakz C0S (<r‘ + **>>

F sin kz sin {at + sx),

(12-3)

with dF 2so)F— 1-------- = 0 (m = a — h),dm a m

and g (dF 2so) F \_ ,2 A A s + ) = 1 = a),a2 — 4a)2 \dm a m

in order to satisfy the conditions at the bottom and the free surface. At the bottom m = a — h ,it follows that u — 0, and

v so F(a — h) 1— = ----*------=— cos kz cos (crt + sv),crH a2 a — h (12-31)

while at the free surface

- {F(a) + 1} cos kz sin ( + sx),aaHv

(tH \ ^ a + 7T (« ) + !] j 008 kz co8 (<rt +(12-32)

When h/a is so small that it is only necessary to retain the first term of the Taylor expansion of

dF 2F,vJj--- I-----dm <r

this function may be taken as

,d/ .dF \i + 7 f )

orh[ l - ijf ) {s* + kV)F{a~

at m = a. Then (12-4)

but it is clear that this can only be valid in the absence of cellular oscillations. Under the same condition and to the same order, may be



278 J. Proudman

replaced by F(a)/a in (12-31) and the term involving F(a)+ 1 omitted in the expression for v in (12-32), so tha t the expressions for v in (12-31) and (12-32) become the same. On substituting (12-4) in (12-2), in the third equation of (12-3) and in (12-32), expressions identical with those obtained by solving Laplace’s equations are obtained.

The Bessel’s equation may be written in the form

d2dm2

and on neglecting the variations of j ) /^ 2, it follows that

d2dm2 (ruiF) + ( vj^F) — 0 , (12-5)

where

Then

F(ru) ( a/ru)* (x(a — h) cos k(tu — a + h) — y sin k(vj + h)}

{yKh cos kJi + [y2 + K2a(a — /&)] sin } 91*4w2 — a2 K(a — h) cos Kh + y sin kK

where

the form of which indicates the possibility of cellular oscillations. When Kh is small, F(a), as given by (12-6), reduces to Laplace’s solution (12-4), but again it is clear that this excludes cellular oscillations.

For the constituent K 2, 5 = 2, and when in (12-2) and (12*3), cr->2w, it is found, after some reduction, that

P / a \ 2 gA ~W

u = crA 4

v _ g I

cos kz cos ( + 2x),

(rA 4

( 1 ) + i W ( ^ [ ( ‘ ' « ) +p2a2( 1 _ i ) 2] f } ^ f c s in ( < T ( + 2x ),

{” (£ ) + p 2“2(£)_ [(1- 5) + iW (1-s ) ]j}cosfacos(< rt + 2

w a A a^ (S ) sin ^2 sin(<rt + 2y),

where HA

( 12-6 )

(12-7)



On starting with s = 2 and cr = 2 io,and on using the equations (6-7) and (6-8) the result (12-7) is obtained directly. I t is clear tha t (12-7) does not admit of cellular oscillations, and when h/a is small it reduces to Laplace’s solution (12-4). From the first of (12-7) on to = a


rH

cos kz cos

9 94<o2a - 1

and on writing Laplace’s solution in the same form with q0 for q, we have

(4 -f k2 -a

as a measure of the degree of approximation afforded by Laplace’s solution. When hja is small, the principal part is given by

q _ 20 + 3 q0 8 + a '

For the constituent K x, s = 1 and cr = so that

\ k2cl2 = k2a2 — \, y = f .

As kh is small, xh is also small, and cellular oscillations do not occur. Treating (12-4) and (12*6) as before,

{\kK cos Kh + [f + %K2a(a — /&)] sin Kh}

{/c(a — h) cos Kh — § sin kK] (1 + k2a2)

the principal part of which is

q__ _ 11 + 6P a 2q0 4 + 4 a ’

For long-period constituents, s = 0, so th a t y = — and 2 is large,



280 J. Proudman

I t is then seen from (12-6) that, to the first order,

PgAu

a A

2 0)(Tkg

-C)

ei cos (m — a + h) cos cos

sin (w — a + h) cos kz sin at,a

sin (tu— aaA a \uj) aw

aA2a>/a\i 2o)k . , , . 7— I —I cos---- (m — a +a \m a

( 12-8 )

. H .vvhere -r = s in ------ .A a

The motion given by (12*8) involves cellular oscillations, and when h/a is small the expressions do not reduce to those obtained from Laplace’s equations.

When in (12-8), a->0, no definite limiting motion is obtained. On starting with a = 0, the equations (14-1) of a later section are found, but the presence of the factor cos kz requires = 0 and hence = 0 and £ = 0 , £ denoting the radial displacement. I t is then impossible to satisfy the condition £ = £ at the free surface. On making <r-^0 in Laplace’s solution, it follows that £ '->0, u->0, v->0, w-+ 0 which is the equilibrium solution, and the same result is obtained by putting 0 in Laplace’s differential equations. Hence Laplace’s equations give a definite limiting form for the constituents of long period when the more general equations show that one does not exist.

13. The constituent K 2

Consider now an ocean over which the depth is a function of the latitude only. On taking a = 2w and supposing tha t t and y enter P only through the factor e ^ + 2x> the equations (6-3) take the form

.. . dPa (iu -v ) = - — ,

a(iv + u) = — 2 >w (13-1)

dptaw = — -X—. dz



On Laplace's differential equations for the tides

From the first two of these equations it is deduced th a t

281

ei(a-t+2x)}

where F( ) is a function to be determined. For finiteness a t the pole it is necessary tha t

for z ^ a — hr

h0 being the depth of the ocean at the pole. Write, for shortness,

j = 4 )> f , =4 A o) ’ rFrom the third of (13’1) it is seen tha t

dz2 1D-iaaw er-F 'eH<rt+2x)} (13-2)

and on substituting for v and w in the equation of continuity (6-5),

SO that + + (13-3)

where f(z/a) is another function to be determined.Let tjr denote the downward slope of the ocean bottom in the southerly

direction, so tha t tan \]r — dhjadd. The equation of the bottom may be written as

m2 + z2 = (a —

and on the bottom both h and may be regarded as functions of z. Take vj1 such that

w\ = ( a — h) 2.

Now the component of current normal to the ocean bottom is

u sin (6 + ifr) + wcos (6 + i/r),

and on substituting for u and w in terms of z, the vanishing of this normal component gives

- 1 f e F - f e )1 J + c o t + f e ) ’ < 1 3 4 >Vol. 179. A. 19



282 J. Proudman

for 0 < 2 < a - h0. At the free surfaceu sin 0 + wcos 6 = i<r£, (13*5)

so that, from (13*2) and (13-3),

( s i n { - i ( £ ) * " + ( £ ) * ' + © ' ] - 008 j ^ + = ° -(13-6)

Take cosa = 1 —h0/a. When hja = 1/2000, a = 1°*8.

For d ^ a , F = 0, so that from (5-2), (13-2) and (13-3), = 0 , = 0,

icrau

and hence, from (13*6),sin2 6f(cos 6) ei<~(Tt+2 + <x2a£ = 0 .

This defines f{z/a) from of (5-3),

a — h0 to z = a, and, in fact, on using the first

— cr2aH.

I t is interesting to compare these relationships near the pole with those for the same constituent in the flat sea of § 11. There the vanishing of w made it impossible to satisfy the condition w = iar£, while here the curvature of the earth does make it possible to satisfy the corresponding condition (13-5) through the non-vanishing component u.

Now on r = a it is seen that

F9

IT9

F"

ei(<rt+2X) = sin20£',

ei(crt+2x) sin 6~ — 2 cos 0%, cu

g e ^ ) = g + 3 c o t e | - 2 S',

so that, from (13-4) and (13-6)

1 f/M 12 l\roi/

sin2 6 —1

( , n a cos 6 cos ilr — td, sinxjf ( a\ 3 . „ ,+ {00t 6 - c o s f+ a c o s e sin jr fe ) S,“ °\ \H§ + 2 COt * )

- f e )♦ I 1 4sin46»jr + ~ t I 9

(13-7)




In (13*7) h,which enters through mv and rjr, regarded as functions of d, correspond to that circle of the ocean bottom for which = cos 6. As 6-» a, £'->(), and the remaining boundary condition is obtained from the coastal condition or that a t the other pole. I f the ocean does not extend to either pole, then both boundary conditions are obtained from the coastal conditions. As d->ot,it is seen from (13-7) tha t it is possible to have

so that, as

„ 3 % r.———> 0 —— -> 0do ’ ’

z->a

F ^ 0 , F'->0, 0,

and from (13-3) it is seen th a t the conditions for the finiteness of a t 0 are satisfied.

Thus there are two distributions of tides, one on either side of the plane z = a — h0 which is perpendicular to the earth’s axis and which touches the ocean bottom immediately under the pole. Across this plane there is continuity of the motion, since on both sides

P = 0, — - e i <rt+2x\w = 0 .

When the depth is uniform, h is constant, i/r = 0, and on

(?)’ sin2 d — sin2 a.

The equation (13-7) may then be written in the form

(sin2 d — sin2 a) — (sin2 d -f sin2 a) cot d ott* oal s in 2

+ l2sin20 — 8 + 4 ^ 2 ^ K , + (sin2(9 — sin2a)2/?£ = 0, (13-8)

whereft 4H‘)-

When h/a is small compared with sin2 0 and also is small, the equation (13*7) takes the form

1 d ( h 0£'\ ( 2 / cosd\sin d dd \sin d dd] \sin 0 dd \ sin2 d)

4 h\ sin40) r+ 4o)2a2 = 0, (13-9)

which is identical with Laplace’s equation (6-2) when cr 2a>, and y enters through the factor e2ix.

19-2



284 J. Proudman

In Laplace’s solution for an ocean of uniform depth, = 0 and d^'jdO = 0 at 6 = 0, whereas in the present solution these conditions obtain at As 6 increases beyond a the equation (13-8) rapidly approximates to Laplace’s equation for uniform depth. From these facts it can be seen that, for an ocean of uniform depth, the solution of the present equations will differ very little from that of Laplace’s equations.

When the depth follows the law h = sin2 6, being constant, so tha t h0 = 0, and when hx/a is small, Laplace’s equation (13*9) is valid everywhere. In this case the solution takes a very simple form, as noted by Sir G. B. Airy in 1842.

14. Limiting form of long-period constituents

Again, consider an ocean over which the depth is a function of the latitude only, but now suppose that P is independent of x and t.

Let £ and tj denote displacement-components in the directions of w and z respectively, so tha t

0£ ... dVdt ’ W dt *

Then the equations (6*3) give

2 (ov = —dm ’dv a 0£

0,

0 = -0z ’

(14-1)

so that, in the limiting form of periodic motion,

P = F(m), v + 2(o£, = 0, 4(0% F', (14-2)

where F( ) is a function to be determined. From the equation of continuity (6-5) it is seen that

1 d

in the limiting form of periodic motion, and hence

4oj2dy 1 d

1so that 4o)2t] = (w F ) + f ( m ) ,

where j( ) is another function to be determined.

(14-3)



Take z\ = { a - -

and ftas in § 13.Now the component of displacement normal to the ocean bottom is

£ sin (0 + ft) + cos ( + and from the vanishing of this a t the bottom it is found tha t

f(m) = - - 14~ (w*") + tan (0 ft) (14-4)tu dm

The equation (14*4) defines/( ) in terms of F( ) for 0 < — wheredenotes the depth of the ocean a t the equator.

At the surface of the ocean£ sin 0 + 1] cos = £,

so that, from (14*2) and (14*3),

sin0F ' — cos# j ^ ^ ( t i 7.F ')+ /j + 4 a — 0, (14*5)

while F(vj) = g£',

F, _ 0 Ka cos 0 d0 ’


9 0 / B i t t e ndm K ’ a cos 0 d0\cos

Take sin a = 1 — h ja .When hx/a — 1/2000, a = 88°-2. For 0 < cc,f{w) is given by (14*4), and on substituting into (14-5)

I)\ zx(so&ft — a8m .0sm ftjd0 g

(14-6)

in which h and ft, regarded as functions of 0, correspond to the circle of the ocean bottom for which m — a sin 0. At the equator, where for h and ft, 0 = cl, it will be supposed tha t ft — 0. As 0->a, it is then deduced from (14*6) that j

and it follows, from (14*4) and (14*5), that, as m-+a — hx,

0- aF " - ^ e ^ °£/cos20 being evaluated on r = a



286 J. Proudman

To illustrate the case in which the ocean extends over the equator, it will be taken as symmetrical about the equator. Then, a t the equator, z = 0, d£'/dd = 0, 7/ = 0, and hence/(tt7) = 0, for < The equation(14-5) then becomes

1 0 sin 6 W

0 (14-7)

for d ^ a \ and at 6 = cl and 6 = \n , d^'/dd = 0 . Using the third of (5*3),

C CTt = + P2(cos #) + constant,Jut t l

P2( ) denoting Legendre’s polynomial of the second order, and hence the solution of (14*7) may be written

where

^ = - P 2(cos0) +P 2(cosa) P;(cosa) w

(cos 6) + constant,

HA n(n+ 1) = —4 o)2a

Rn(cos 6) denoting the Legendre’s function

F {-b n , \ + \n , i , cos20).

This distribution gives, at w =

F' — 0 — o* ’ a cos20 “ U’

£/cos 2 6 being evaluated on r = a ,and the value of £ must be the same as in (14-6). On comparing with §12, where a definite limiting form did not exist, it is again seen tha t it is the curvature of the earth which has enabled all the conditions to be satisfied.

Thus there are two distributions of tides, one on either side of the cylinder m = a — which is parallel to the earth’s axis and which touches the ocean bottom along the equator. Across this cylinder there is continuity of

/ , F, F ', F'and hence of P , £, ij and v.

When the depth is uniform, h is constant, ^ 0, and on

cos2 6 — cos2 a.




The equation (14*6) may then be written in the form

! 1 a !f'sin# 3£'\

(sin # 3#'i cos # 3# /

+sin# K \ - 4(i)2a

cos #[cos2 6 — cos2 a]* 3#j g£ = 0. (14-8)

When A/a is small compared with cos2#, and \}r also is small, the equation (14-6) may be written as

1 3 /Asin#3£'\ 4a>2a2sin#3#\cos2# 3#/ g

(14-9)

which is identical with the form taken by Laplace’s equation (6-2) when <t—> 0 and £' is independent of y.

In Laplace’s solution for an ocean of uniform depth covering the whole earth, d^'jdd — 0 a t 6 = 90°, whereas in the present solution this condition obtains at # = a. As 6decreases from a the equation (14*8) rapidly approximates to Laplace’s equation for uniform depth. From these facts it can be seen that the solution of the present equations will differ very little from that of Laplace’s equations.

15. Comparison with previous work

The chief result of the paper is to modify very considerably the impression created by V. Bjerknes and his co-authors. The emphatic statements of these authors appear to be based on the possibility of the existence of cellular oscillations of diurnal period, and on the possibility of resonance caused by the tide-generating forces of equal period. Now it has been seen that diurnal cellular oscillations are only possible when the horizontal wave-lengths are of the same order of magnitude as the depth. This indeed occurs in the examples given by Bjerknes and his co-authors, but it does not occur in the actual tides of the ocean. The part of the present paper which most nearly corresponds to the examples given by these authors is § 12.

The motion of § 11 may be regarded as one of the elemental motions used by Brillouin and Coulomb in their expansions. There again it is the dynamically possible oscillations of very short horizontal wave-length which cause their difficulties of convergence.

Comparison with Solberg’s work is possible in connexion with §§13 and 14, as the basins here considered include an ocean of uniform depth covering the whole earth. The two problems here considered occur at the two extreme ends of the range of negative values of 1 — 4w2/<x2. I t has been



288 J. Proudman

shown that though Laplace’s equations apply over the greater part of such an ocean, they do not apply near the poles for the constituent i£2ior near the equator for the limiting form of long-period constituents. But for the ocean taken as a whole, the necessary correction to the solution of Laplace’s equations is only very slight. Though Solberg does not use cylindrical coordinates explicitly, he does obtain equation equivalent to those of § 13 as far as (13*3). But the distribution of tides which has been taken here to apply only near the poles, Solberg took to apply to the whole ocean, so tha t he did not obtain the equation (13*7). The problem of § 14 is a limiting case of those to be considered in Solberg’s future paper.

R eferences

Airy, Sir G. B. 1842 Tides and waves. London.Bjerknes, V., Bjerknes, J., Solberg, H. and Bergeron, T. 1933 Physikalische -

dynamik, pp. 450-452. Berlin.Brillouin, M. 1928 C.R. Acad. Sci., Paris, 186, 1665-1669. Brillouin, 1VL and Coulomb, J. 1933 Oscillations d'un liquid pesant dans un

cylindrique en rotation. Inst. M6can. Fluides Univ. Paris.Hough, S. S. 1897 Phil. Trans. A, 189, 201-257.Poincar6, H. 1896 J. Math, pures appl. (5), 2, 57—102.Poincare, H. 1910 Lemons de Mecanique celeste, 3. Paris.Solberg, H. 1936 Astrophys. Norveg. 1 , 237—340.Street, R. O. 1930 Proc. Camb. Phil. &oc. 26, 446-452.Thomson, Sir W. 1880 Phil. Mag. 10, 155-168. Math. Phys. Papers, 4, 152-165.



Date post:	30-Apr-2018
Category:	Documents
Upload:	dominh
View:	220 times
Download:	3 times

On Laplace’s differential equations for the...

Documents