A uniformly sheared basic state is characterized by three non-dimensional numbers: s the stratification within the layer relative to the buoyancy jump across the free boundary, ν the slope of this surface relative to the inclination of the isopycnals, and b/ν the planetary gradient of potential vorticity relative to the topographic one. The integrals of motion are used to derive the instability region in (s,ν,b) space, as well as to find bounds on the growth of a wave (difference of the state from the longitudinal mean) and of a perturbation (idem, from the basic state). A reduced model is developed, in which two waves interact with two components of the mean flow. This system is Hamiltonian an integrable. The solutions are quasi-periodic and satisfy the bounds derived from the integrals of motion. Optimum perturbations are important only for the earlier evolution; most initial conditions compatible with given values of the Casimirs reach the maximum allowed growth.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics