## Abstract

This study examines quasigeostrophic Rossby eigenmodes of homogeneous as well as stratified oceans. Analytical studies of Rossby basin modes are usually done by using simple basin geometries. Simple means that solutions are available by applying a separation *ansatz* in looking for basin modes. Here the focus is on half-trapezoidal geometry, in particular on a stratified ocean with a half-trapezoidal (zonal) cross section. In this case, separation is not possible. However, approximate solutions can be found. Different approximations are discussed: one related to linearization of the boundary conditions, one related to a sight change in boundary geometry, and one that uses an asymptotic expansion. It is shown that the widely used “linearized boundary conditions” give wrong results for low-frequency modes. The reason is that weak asymmetries of the basin are neglected and wave energy is therefore distributed too homogeneously in the basin. Although the basin's geometry used deviates only slightly from a square and the eigenvalue spectrum corresponds well with that of a square basin, eigenmodes can still differ greatly.

*Corresponding author address:* Uwe Harlander, The Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, Netherlands. Email: harland@nioz.nl