The Laplace Tidal Equation:
Its Geophysical Significance and Mathematical Difficulty.
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.