This zonostrophic instability is controlled by two independent parameters, β* = β kf-5/3ε-1/3 and μ*=μ kf-2/3ε-1/3. For a fixed value of β*, the formation of jets is completely determined by the magnitude of the drag coefficient, μ*, which, when sufficiently large, results in a flow that is laminar and jetless. As the drag is gradually decreased the flow becomes turbulent, as indicated by an inverse cascade, but it remains isotropic and jetless. Eventually at a small enough value of the drag, the flow becomes zonostrophically unstable so that jets emerge spontaneously and remain quasi-steady in time.
We analyze this zonostrophic instability by neglecting the eddy-eddy nonlinearity, but retaining the eddy-mean flow nonlinearity. Taking advantage of this "quasi-linear" formulation and the rapid temporal decorrelation of the applied forcing, one can construct a closed evolution equation for the spatial correlation function of the vorticity. Zonostrophic instability is studied within the quasi-linear approximation by obtaining the marginal stability curve both numerically and analytically. The jetless isotropic basic-state flow is linearly stable if μ* > 0.2462. For large values of β* the marginal stability curve takes the form, μ*neut ~ 2/β*2 while the most unstable wavenumber on this curve has the scaling, mneut ~ 31/3/β*. The inverse dependence of mneut on β* flatly contradicts Rhines scaling, and is confirmed by numerical simulation of the quasi-linear equations. (For the full nonlinear system there is no such clear contradiction of Rhines scaling.) Further, the value of μ* required for fully nonlinear zonostrophic instability is less that of quasi-linear, by more than a factor of four. On the other hand, far away from the neutral curve, in the small-drag regime where full non-linear dynamics results in robust steady jets, the wavelength of the quasi-linear jets agrees with that of the full non-linear jets to a remarkable degree, and both are consistent with Rhines scaling. This offers hope that quasi-linear theory can be used to understand upgradient momentum transfers in the robust-jet regime.
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