On computation of Hough functions

Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

On computation of Hough functions

Hough functions are the eigenfunctions of Laplace’s tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly-used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact Matlab code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wavenumbers.

Asymptotic eigensolutions of Laplace’s tidal equation

Asymptotic approximations to the eigenfunctions of Laplace’s tidal equation (Hough functions) are obtained for prescribed λ = σ /2 ω ( σ = angular frequency, ω = angular velocity of planet) and large values of Lamb’s parameter, β = 4 ω 2 a 2 / gh ( a is the planetary radius, and h the equivalent depth for a particular vertical structure), qua eigenvalue. Both positive and negative eigenvalues are considered. The results are validated by comparison with the extensive numerical results of Flattery (1967) and Longuet-Higgins (1968). They should be useful in atmospheric tidal studies, especially for a rapidly rotating planet, and may be useful for studies of equatorial motions in the oceans.