On the Application of the Spheroidal Wave Equation to the Dynamical Theory of the Long-Period Zonal Tides in a Global Ocean
The frictionless solution for the fortnightly and monthly long-period zonal tide heights in a constant-depth global ocean has been known for a century. An equation for the tide height is obtained from Laplace's tidal equations. Darwin (1886) gave a solution to this equation in a power series expansion, and Hough (1897) gave one in terms of Legendre polynomials. In this article we derive a meridional velocity equation, which is an inhomogeneous form of the spheroidal wave equation. This allows a more straightforward approach to a solution. The forcing term is the associated Legendre function of degree two and order one, and so the meridional velocity is conveniently represented as a series of associated Legendre functions of order one. This expansion is substituted into the spheroidal wave equation, use is made of Legendre recurrence relations, and a three-term recurrence relation is derived for the coefficients. This is solved by a continued fraction expansion in a similar manner to the classical solution for the tide height. Series expansions for the tide height, zonal velocity, stream function, and velocity potential are then easily derived by Legendre recurrence relations.
Proceedings of the Royal Society of London Series A
- Pub Date:
- October 1992