G. A. CORBY 551.511.32 : 551.513.1 : 525.624

Laplace’s tidal equations - an application of solutions for negative depth

By G. A. CORBY Meteorological Ofice, Bracknell

(Manuscript received 27 February 1967)

SUMMARY

It is shown that solutions of Laplace’s tidal equations in the presence of a non-zero zonal flow may be obtained from the usual solutions for perturbations about a state of rest relative to the Earth, provided one uses an appropriate modification of the equivalent depth. Typically the modification leads one to the solutions which for zero zonal flow would be applicable to negative depths.

1, INTRODUCTION

Laplace’s tidal equations have recently received further attention from a number of workers, notably Lindzen (1966; 1967) and Longuet-Higgins (1967). The former has drawn attention to the use of negative depth solutions for cases where there is a forcing function and has exploited this in computations of the thermally driven diurnal tide in the atmosphere. Longuet-Higgins has carried out extensive computation of the solutions and essentially has tabulated the eigenfunctions over the complete range of depths from - co to + CO. These treatments are concerned with the form and frequency of perturbations about a state of rest relative to the Earth. For some purposes, however, solutions appropriate to the case where there is a finite zonal flow are needed and the purpose of the present note is to show that such solutions may be obtained by suitable interpreta- tion of the negative depth solutions of the simpler case.

2. THEORY The linearized equations for the perturbations of an incompressible fluid about a state of rest

relative to the Earth are

’ (1)

In these equations the Earth’s radius is the unit of length, and

0 = colatitude 4 = longitude

u, v = perturbation velocity components D = Earth‘s angular velocity 5 = displacement of free surface about mean depth H.

Longuet-Higgins (1967) introduces a representation of the velocity perturbation in terms of potential and stream functions @, Y and derives the following equations for them

368

SHORTER CONTRIBUTIONS 369

where a trial solution like exp i (3 - at) has already been inserted and

s = zonal wave number u = frequency h = non-dimensional frequency = u/ZQ

E = 4Q2/gH p = C O S B

D = (1 - p2) 3/3 p.

His paper includes comprehensive tabulations of the eigenfunctions over the full range - ca < E < + co, the results covering Rossby type waves as well as gravity waves. Although not asserted specifically in the literature it is sometimes implied that the Rossby type solutions are equally applicable to perturbations about a constant angular wind, say a, provided one then uses (Q + a) instead of Q in the definition of E . That this is not so can be seen from the following analysis.

The linearized equations of motion in the presence of a constant angular wind a are

3u 3u 1 3 - + a - - ~ v ( S Z + a ) c o s B + - - ( g o = 0 bt b+ s i n e 34

. . (6)

3v 3v b - + a - + Z U ( Q + a) COSB - - (g t ) = O . 3t 34 3e

(7)

where 5 is now the perturbation of depth relative to the zonal mean depth, and the a part of the flow is geostrophic. Referred to axes moving with the zonal wind these equations would differ from Eqs. (1) and (2) only in the replacement of Q by (SZ + a). However, similar considera- tions do not apply to the continuity equation. If we represent the fluid depth by H + h + 5 where H is the global mean, h is the departure of a zonal mean from H and 5 is the local departure, then the full continuity equation would be

= 0. . (8)

If we are interested in the Rossby type solutions rather than the gravity wave solutions we may argue that as the flow is then quasi-geostrophic the whole of the advective terms in Eq. (8) may be neglected. Furthermore, if H is chosen for simulation of the atmosphere (H .- 10 km) and h accords through the geostrophic equation with a plausible a, then Ihl is small compared with H. Hence the appropriate form of continuity for our purpose is

1 35 + (a + A) 31 - v3 (h + 5) ( H + h + 5 ) bu 3 (us in@ 3t slnB 34 be + s i n 0 [g- 38

It is to be noted that, in contrast to the motion equations, Eq. (9) does not have the same form as its counterpart Eq. (3) when it is referred to axes moving with the zonal flow. This is simply a consequence of the presence of a mean meridional gradient of the free surface depth.

However, we may still proceed by applying to Eqs. (6), (7) and (9) the analysis which Longuet- Higgins used to derive Eqs. (4) and (5 ) . Instead of the latter we obtain

If we now put

u - as = A’

2 (Q + a)

4 (Q + a)2 u and

___- - € gH (u - as)

370 G. A. CORBY

then Eqs. (10) and (11) assume precisely the same form as Eqs. (4) and (5) and we may expect to obtain solutions of Eqs. (10) and (11) by suitable interpretation of those of Eqs. (4) and (5). It is clear that if the disturbance moves eastwards at a rate less than that of the mean zonal flow, i.e. 0 < u < as, then solutions of Eqs. (10) and (11) for positive depth will correspond to solutions of Eqs. (4) and (5) for negative depth. (There will of course be other cases when the magnitude but not the sign of the depth is modified). This additional application of the negative depth solutions of Laplace’s tidal equations appears to be of interest and some value especially in applica- tions concerning the atmosphere when a non-zero zonal flow is normal.

3. AN APPLICATION OF NEGATIVE DEPTH SOLUTIONS

The full primitive equations for the free surface model, viz. the complete non-linear forms of Eqs. (I), (2) and (3, have been widely used for numerical integration experiments, and especially so for experiments designed to test finite difference schemes and techniques of integration for the primitive equations. For such purposes one really requires data with a known analytical solution so that the accuracy and properties of any particular numerical scheme can be appraised. Phillips (1959) dealt with this problem by generating analytically balanced data from a Haurwitz type solution of the non-divergent vorticity equation. Such a solution is not of course also a solution of the equations in primitive form because these provide for divergence, but it must be a fair approximation to a solution. Indeed, it can be shown that when used as initial data for an integra- tion of the primitive equations the main difference between the computed evolution and the analytical solution must be some reduction in the phase speed of the disturbance. Several workers have been satisfied to achieve qualitative agreement of this sort. However, by making use of negative depth solutions for the simpler case with no zonal flow we can be more precise as to the reduction in phase speed to be expected.

As an example, the writer has been associated with some work+ involving numerical integra- tions of the primitive equations for the free surface model and, following the Phillips approach, the initial wind field comprised a disturbance in wave number four having a meridional variation like the Associated Legendre Polynomial P54 (p) and superposed on a constant angular flow of 0.1 52. In the non-divergent (or infinite depth) case the disturbance moves eastward at 9.6 deg/day without change of form. In a 20 day integration of the primitive equations with a global mean free surface depth of 9.418 km it moved with negligible change of form at a little over 8.8 deg/day. If we interpret the tabulated solutions of Longuet-Higgins (1967) in the manner proposed in the foregoing section, we obtain a speed of movement of 8.84 deg/day, in very good agreement with the numerical integration. This solution of Eqs. (10) and (11) corresponds to a solution of Eqs. (4) and (5) appropriate to a mean fluid depth of - 24 km.

REFERENCES

‘On the theory of the diurnal tide,’ Mon. Wea. Rev., 94,

Quart. 1. R. Met. Soc., 93, p. 18. ‘The eigenfunctions of Laplace’s tidal equations over a

‘Numerical integration of the primitive equations on the

Lindzen, R. S. 1966

1967 1967

1959

p. 295.

Longuet-Higgins, M. S.

Phillips, N. A. sphere,’ Phil. Trans. (in course of publication).

hemisphere,’ Mon. Wea. Rev., 87, p. 333.

The work referred to is described in a paper by Grimmer and Shaw on p. 337 in this issue.