Do Rossby-wave critical layers absorb, reflect, or over-reflect?

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Abstract
The StewartsonWarn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat's eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale eddy-viscosity−1 [double less-than sign] 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen.

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