Zonostrophic instability

We show that the spontaneous generation of zonal jets on a β-plane is an instability of a jetless basic-state flow. The flow is driven by an external body force and damped by bottom drag. A standard model forcing is isotropic, rapidly decorrelating, homogeneous and acting on a narrow band of wavenumbers centered on k_f; the amplitude of the forcing is controlled by its power input, ε.

This zonostrophic instability is controlled by two independent parameters, β_* = β k_f^-5/3ε^-1/3 and μ_*=μ k_f^-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, m_neut ~ 3^1/3/β_*. The inverse dependence of m_neut 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.