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.