The Laplace Tidal Equation:

Its Geophysical Significance and Mathematical Difficulty.

Peter Lynch

Second Meeting on Differential Equations,

N.I.H.E., Glasnevin, Dublin, 22-25 May, 1989.


The natural oscillations of the atmosphere may be studied using the shallow water equations. The meridional (North-South) structure of these normal modes is determined by the eigensolutions of a second order o.d.e., the Laplace Tidal Equation (LTE). Boundary conditions of regularity at the geographic poles yield a Sturm-Liouville-like problem. However, for low frequency solutions the LTE has singularities within the physical domain. Although the solutions are analytic, the singularities of the equation complicate its numerical solution, and also make asymptotic analysis more difficult.

In this paper, I will outline the origin of the LTE and its geophysical significance. The results of some numerical studies will be discussed; these results indicate a rich variety of asymptotic forms. Some elementary asymptotic solutions will be described, and the desirability of a more comprehensive asymptotic analysis will be stressed.

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