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.