Stable and Unstable Eigensolutions of Laplace’s Tidal Equations for Zonal Wavenumber Zero

Laplace's tidal equations are of great importance in various fields of geophysics. Here, the special case of zonal symmetry (zonal wavenumber m = 0) is investigated, where degenerate sets of eigensolutions appear. New results are presented for the inclusion of dissipative processes and the case of unstable conditions. In both instances the (nonzero) eigenfrequencies are complex. In the latter case, additional stable (i.e. real) eigenfrequencies appear in the numerical results for the absolute value of the Lambparameter ε being larger than a critical value εc. Further, it is shown that any degeneracies are removed through the inclusion of dissipation. Moreover, asymptotic relations are derived employing the relation of Laplace's tidal equations for m = 0 to the spheroidal differential equation. The implications of these findings to numerical techniques are demonstrated and results of computations are presented.