Planetary waves in a stratified ocean of variable depth. Part 1. Two-layermodel

Linear Rossby waves in a two-layer ocean with a corrugated bottom relief(th e isobaths are straight parallel lines) are investigated. The case of a rou gh bottom relief (the wave scale L is much greater than the bottom relief s cale L-b) is Studied analytically by the method of multiple scales. A speci al numerical technique is developed to investigate the waves over a periodi c bottom relief for arbitrary relationships between L and L-b. There are three types of modes in the two-layer case: barotropic, topograph ic, and baroclinic. The structure and frequencies of the modes depend subst antially on the ratio Delta = (Delta h/h(2))/(L/a) measuring the relative s trength of the topography and beta-effect. Here Delta h/h(2) is the typical relative height of topographic inhomogeneity and a is the Earth's radius. The topographic and barotropic mode frequencies depend weakly on the strati fication for small and large Delta and increase monotonically with increasi ng Delta. Both these modes become close to pure topographic modes for Delta much greater than 1. The dependence of the baroclinic mode on Delta is more non-trivial. The fre quency of this mode is of the order of f(o)L(i)(2)/aL (L-i is the internal Rossby scale) irrespective of the magnitude of Delta. At the same time the spatial structure of the mode depends strongly on Delta. With increasing De lta the relative magnitude of motion in the lower layer decreases. For Delt a much greater than 1 the motion in the mode is confined mainly to the uppe r layer and is very weak in the lower one. A similar concentration of mesos cale motion in an upper layer over an abrupt bottom topography has been obs erved in the real ocean many times. Another important physical effect is the so-called 'screening'. It implies that for L-b < L-i the small-scale component of the wave with scale L-b is confined to the lower layer, whereas in the upper layer the scale of the mo tion L is always greater than or of the order of, L-i. In other words, the stratification prevents the ingress of motion with scale smaller than the i nternal Rossby scale into the main thermocline.