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

Linear Rossby waves in a two-layer ocean with a corrugated bottom relief(the isobaths are straight parallel lines) are investigated. The case of a rough bottom relief (the wave scale L is much greater than the bottom relief scale L-b) is Studied analytically by the method of multiple scales. A special numerical technique is developed to investigate the waves over a periodic bottom relief for arbitrary relationships between L and L-b. There are three types of modes in the two-layer case: barotropic, topographic, and baroclinic. The structure and frequencies of the modes depend substantially on the ratio Delta = (Delta h/h(2))/(L/a) measuring the relative strength 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 stratification for small and large Delta and increase monotonically with increasing 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 frequency 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 Delta the relative magnitude of motion in the lower layer decreases. For Delta much greater than 1 the motion in the mode is confined mainly to the upper layer and is very weak in the lower one. A similar concentration of mesoscale motion in an upper layer over an abrupt bottom topography has been observed 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 motion 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 internal Rossby scale into the main thermocline.