Linear Rossby waves in a continuously stratified ocean over a corrugated rough-bottomed topography are investigated by asymptotic methods. The main results are obtained for the case of constant buoyancy frequency. In this case there exist three types of modes: a topographic mode, a barotropic mode, and a countable set of baroclinic modes. The properties of these modes depend on the type of mode, the relative height δ of the bottom bumps, the wave scale L, the topography scale Lb and the Rossby scale Li. For small δ the barotropic and baroclinic modes are transformed into the ‘usual’ Rossby modes in an ocean of constant depth and the topographic mode degenerates. With increasing δ the frequencies of the barotropic and topographic modes increase monotonically and these modes become close to a purely topographic mode for sufficiently large δ. As for the baroclinic modes, their frequencies do not exceed O(βL) for any δ. For large δ the so-called ‘displacement’ effect occurs when the mode velocity becomes small in a near-bottom layer and the baroclinic mode does not ‘feel’ the actual rough bottom relief. At the same time, for some special values of the parameters a sort of resonance arises under which the large- and small-scale components of the baroclinic mode intensify strongly near the bottom.

As in the two-layer model, a so-called ‘screening’ effect takes place here. It implies that for Lb<Li the small-scale component of the mode is confined to a near-bottom boundary layer (Lb/Li)H thick, whereas in the region above the layer the scale L of motion is always larger than or of the order of Li.