Equilibrium states of eddy-driven jets in a generalized potential vorticity staircase model

Possible equilibrium states of eddy-driven jets are studied using an idealized 1D (equivalent latitude) model on the barotropic beta plane with a generalized potential vorticity (PV) staircase geometry and a parameterized PV (absolute vorticity) flux. The basic premises are that jets emerge as a result of inhomogeneous mixing of PV, which is represented by piecewise constant effective diffusivity and PV gradients, and that the jets in turn limit mixing by reducing effective diffusivity locally. Specifically, the domain is divided into “mixer” (a small PV gradient and a large effective diffusivity) and “barrier” (a large PV gradient and a small effective diffusivity) regions with regular intervals. In the former the effective diffusivity is assumed to be a function of eddy kinetic energy alone, whereas in the latter it is assumed to depend also on the peak velocity of the jet. This flow feedback on mixing, together with the required continuity in the flux at the barrier-mixer boundaries, gives rise to a cubic equation for the peak velocity of the jet for a given combination of beta, widths of the barrier and mixer regions, eddy kinetic energy, and a micro-scale diffusion coefficient for PV. It is found that multiple equilibria are possible for a sufficiently narrow and deep barrier, although only one of them is physically realizable. For a given geometry of the barrier, there exists a critical eddy kinetic energy beyond which no steady jet is possible. This may explain the threshold behavior of the jet at large forcing in the dynamically consistent model of chaotic PV mixing documented in the literature and have bearing on the rapid breakdown of the polar vortex during the Southern Hemisphere final warming.