Nonlinear stability of a zonal shear flow

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Abstract
Within the framework of a weakly nonlinear theory, we consider, in the beta-plane approximation, the nonlinear stability of a zonal shear flow. It is shown that in the regime with a viscous critical layer, the Landau constant (the coefficient of the nonlinear term of the evolution equation) is determined mainly by the interaction between the fundamental harmonic and the wave-induced mean flow distortion ( he zeroth harmonic) and increases with increasing Reynolds number, R, as R-1/3. At the short-wave boundary of the instability region, the nonlinearity plays a stabilizing role irrespective of the value of the parameter p, and this includes the region in which the neutral curve has a maximum (p= 4.3(-3/2) for the flow u = tanh y). At the long-wave boundary, there exists a range of wave numbers for which the nonlinearity has a destabilizing effect.

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