Asymptotic analysis of long-time behaviour of zonal flows in two-dimensional turbulence on a β plane

In forced two-dimensional turbulence on a rotating sphere, it is well known that a multiple zonal-band structure, i.e. a structure with alternating eastward and westward jets, emerges in the course of time development. The multiple zonal-band structure then experiences intermittent mergers and disappearances of zonal jets, and a structure with only a few large-scale zonal jets is realised as an asymptotic state (Obuse et al., 2010). With the view of understanding the long-time behaviour of the zonal jets in two-dimensional turbulence in rotating systems, we consider large-scale zonal flows superposed upon a homogeneous zonal flow and a small-scale sinusoidal transversal flow on a β plane, which is the model originally introduced by Manfroi and Young (1999), and investigate the merging and disappearing processes of zonal jets. First, we analytically obtain solutions of steady isolated zonal jet of the evolution equation of such zonal flows. Then it is shown that these steady zonal jet solutions are all linearly unstable. The numerical time integration of the evolution equation also confirms that the final state of a perturbed unstable steady solution is a uniform flow. These results suggest that mergers and disappearances of zonal jets in two-dimensional turbulence on β plane and on a rotating sphere might be due to the instability of the zonal jets caused by the effect of turbulence. Utilising the analytical solution of steady isolated zonal jet, the weak interaction between two neighbouring zonal jets is also studied. The time derivative of the distance between two identical zonal jets is estimated by a perturbation method, confirming that two zonal jets placed apart attract each other, and the attraction becomes stronger as the distance between them gets shorter. The estimated time derivative of the distance between two zonal jets is in agreement with that obtained from the numerical time integration of the evolution equation. It is also found by numerical simulation that the two zonal jets then merge to a new steady isolated zonal jet of different parameters. Because of the linear instability of the new steady zonal jet, the final state is expected to be a uniform flow. These results are consistent with gradual mergers and disappearances of zonal jets seen in forced two-dimensional turbulence on β plane and on a rotating sphere, and implies the importance of the weak interaction between neighbouring zonal jets for the long-time behaviour of zonal jets in forced two-dimensional turbulence in rotating systems. Finally, we modify the Manfroi-Young model by taking account of the spatial variation of the disturbance in the zonal direction, and the surface variation of the fluid layer, in order to make the model a little more realistic. The linear stability analysis of analytical solutions of steady isolated zonal jet in these models suggests that the instability of zonal jets is widely common in turbulence on β plane.