The Eigenfunctions of Laplace's Tidal Equations over a Sphere

Abstract

Numerical calculations are presented for the eigenvalues of Laplace’s tidal equations governing a thin layer of fluid on a rotating sphere, for a complete range of the parameter e ( omega = rate of rotation, R = radius, g = gravity, h — depth of fluid layer). The corresponding eigenfunctions or ‘Hough functions’ are shown graphically for the lower modes of oscillation. Negative values of e, which have application in problems involving forced motions, are also considered The calculations reveal many asymptotic forms of the solution for various limiting values of e. The corresponding analytical expressions are derived in the present paper. Thus, as e0 through positive values we have the well-known waves of the first and second class respectively, which were found by Margules and Hough. These can be represented in terms of spherical harmonics. As e-> + oo there are three distinct asymptotic forms. In each of these the energy is concentrated near the equator. In the first type, the kinetic energy is three times the potential energy. In the other two types the kinetic and potential energies are equal. The waves of the second type are all propagated towards the west. The waves of the third type are Kelvin waves propagated eastwards along the equator. All three types are described in terms of Hermite polynomials. As e 0 through negative values there is only one asymptotic form of solution, representing motions which are analytically continuous with Hough’s ‘waves of the second class’. As e —> — oc there are three different asymptotic forms, in each of which the energy tends to be concentrated near the poles of rotation. In the first two types the energy is mainly kinetic and the motion is in inertial circles. In the third type the energy is mainly potential. The modes tend to occur in pairs of almost the same frequency, one being symmetric and the other antisymmetric about the equator. The analytical forms of the solutions involve generalized Laguerre polynomials. In the special case of zonal oscillations, the first two limiting forms as E -> — oo go over into a different form in which the frequency tends to zero as e tends to a finite negative value. In this case the third type does not occur. The way in which the various asymptotic solutions are connected can be traced in figures 1 to 6 (e > 0) and figures 16 to 21 (e 4 4 are tabulated in tables 1 to 10. The eigenfunctions for the lower modes are presented graphically.