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 Lb) 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 Lb.

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 Δ = (Δh/h2)/(L/a) measuring the relative strength of the topography and β-effect. Here Δh/h2 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 Δ and increase monotonically with increasing Δ. Both these modes become close to pure topographic modes for Δ>1.

The dependence of the baroclinic mode on Δ is more non-trivial. The frequency of this mode is of the order of f0L2i/aL (Li is the internal Rossby scale) irrespective of the magnitude of Δ. At the same time the spatial structure of the mode depends strongly on Δ. With increasing Δ the relative magnitude of motion in the lower layer decreases. For Δ>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 Lb<Li the small-scale component of the wave with scale Lb 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, Li. In other words, the stratification prevents the ingress of motion with scale smaller than the internal Rossby scale into the main thermocline.