Jet formation by potential vorticity mixing at large and small scales

We examine the formation of zonal jets in geostrophic turbulence with an emphasis on the inhomogeneous potential vorticity mixing caused by Rossby waves and turbulent eddy motions, and the control of the wave and eddy motions by the emerging potential vorticity distribution itself. Through the diagnostic relation between eddy fluxes of potential vorticity and the convergence of the eddy flux of zonal momentum, generation and dissipation of these wave and eddy motions give rise directly to accelerations in the zonal direction and the emergence, under suitable conditions, of strong zonal jets. Jet regimes are found to depend in a simple way upon two non-dimensional parameters, which may be related to three natural length scales of the system: the Rhines scale, the forcing scale, and a length scale relating the strength of the forcing to the background potential vorticity gradient. Three key results will be discussed: (i) the emergence of strong jet motions when dynamical forcings are weak; (ii) the independence of the jet formation process from the two-dimensional inverse energy cascade; (iii) the identification of two distinct types of potential vorticity mixing, one dominated by turbulent eddies, the other by the action of localized Rossby wave critical layers.

To permit a clear analysis we consider the simple system of geostrophic turbulence on a mid-latitude beta-plane, making use of high-resolution, long-time numerical integrations. Two separate cases are considered, in which forcing scales are either (i) much smaller than, or (ii) comparable to the scale of the emerging jets. In the first case, the late-time distribution of potential vorticity is found to depend in a simple way on a single non-dimensional parameter, which may be conveniently expressed as the ratio of the traditional Rhines scale and a length scale relating forcing strength and planetary potential vorticity gradient. It is shown here that jet strength increases as the value of this ratio increases, with the limiting case of the potential vorticity staircase, comprising a monotonic, piecewise-constant profile in the north-south direction, being approached for values around ten. At lower values, eddies created by the forcing become sufficiently intense to continually disrupt the steepening of potential vorticity gradients in the jet cores, thus preventing strong jets from developing.

In the second case, in which the forcing scale is comparable to the scale of the emerging jets, the flow evolution depends on the ratios of all three length scales. The potential vorticity is again found to organize into a piecewise constant staircase-like profile, monotonic in latitude, provided only that the Rhines scale is at least of the same order as the forcing scale. That strong jets are observed even when these two scales are similar indicates, in particular, that jet formation may be considered completely independently from dynamical processes associated with the two-dimensional turbulent inverse energy cascade. More generally, the character of potential vorticity mixing is shown to depend on a parameter involving forcing strength, forcing scale, and planetary vorticity gradient, occurring either predominantly in localized critical layers, or through the more uniform small-scale turbulent eddy mixing. In the former case, care must be taken with the form of the dynamical forcing to ensure that the material advection of potential vorticity is not obscured. A combined condition for the formation of strong zonal jets may be expressed, summarizing the overall dependence on all three length scales.