Potential vorticity mixing by marginally unstable baroclinic disturbances

Abstract

The weakly nonlinear dynamics of a baroclinic wave in a two-layer model near minimum critical shear is described in terms of a nonlinear critical layer problem which is completely integrable in the absence of dissipation. Sufficiently small amplitude supercritical inviscid disturbances are found to equilibrate to leading order, even though the absolute potential enstrophy of the lowest layer is always transient and transferred irreversibly to smaller and smaller scales. The inviscid equilibrium amplitude of the fundamental is found to be larger by a factor of √2 than the weakly dissipative value found by Pedlosky, implying that the limits t → ∞ and vanishing dissipation are not interchangeable. The fundamental equilibrates when the mixing in the lowest layer results in the streamwise homogenization of the coarse-grained potential vorticity. It is also noted that coarse-grain homogenization can be established on faster time scales than the fine-grain versions discussed by Rhines and Young. With weak or vanishing supercriticality or larger initial disturbance amplitudes, the fundamental may either equilibrate or oscillate periodically, depending on the form and strength of the initial disturbance. In the oscillating case, the baroclinic vortex reversibly wraps up and un-wraps and there is no tendency to mix potential vorticity. Finally, the possibility of further instabilities similar to those discussed by Killworth and McIntyre, and Haynes for the Rossby wave critical layer problem is also noted.