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Do Rossby-wave critical layers absorb, reflect, or over-reflect ? (English) Zbl 0676.76040

The Stewartson-Warn-Warn (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-reflectig 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 \(\delta\) q, i.e. the initial absolute vorticity gradient \(dq_ 0/dy\) times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate \(du_ 0/dy\) of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat’s-eye pattern, probably leadig to chaotic motion there, with very large ‘eddy-viscosity’ values. Those values correspond to critical-layer Reynolds numbers \(\lambda^{-1}\ll 1\), suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76E99 Hydrodynamic stability
76B65 Rossby waves (MSC2010)
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