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.