Nonlinear stability of a zonal shear flow

Within the framework of a weakly nonlinear theory, this paper considers, 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 (the zeroth harmonic) and increases with increasing Reynolds number, R, as R-cubed. At the short-wave boundary of the instability region, the nonlinearity plays a stabilizing role irrespective of the value of the parameter beta, and this includes the region in which the neutral curve has a maximum. At the long-wave boundary, there exists a range of wave numbers for which the nonlinearity has a destabilizing effect.