A scaling theory for jets in beta-plane turbulence

An idealized model of jets in a forced barotropic beta-plane turbulence is constructed. In the statistically steady zonal-mean momentum equation, meridional fux of eddy vorticity is balanced by linear damping of zonal-mean wind. The vorticity flux is represented by eddy diffusivity parameterization. Inspired by the “potential vorticity staircase” model, the eddy diffusivity is assumed constant except that it drops to zero at a regular interval ( y = … -2L, -L, 0, L, 2L, … etc.). These equally spaced, infinitesimally thin “gaps” in diffusivity represent transport barriers. Although the diffusivity drops to zero, the mean vortictity gradient is infinite (i.e., mean vorticity jumps) there, so the vorticity flux across remains finite at the barriers.

This analytical model is solved for the zonal-mean zonal velocity. The obtained zonal-mean flow has westerly cusps at the barriers, connected by a cosh profile in between. With additional constraints, we derive an expression of L in terms of powers of beta, damping rate, and (externally specified) energy input rate (“I”). The constraints are: the global integral of zonal-mean momentum vanishes; the mean energy dissipation of zonal-mean flow due to damping equals the energy input rate; the eddy diffusivity equals u*L, where u is the rms velocity of energy-containing eddy; and I = u*u*u/L. This theoretical prediction of the jet scale will be tested with direct numerical simulations of beta-plane turbulence in a range of parameter space.