The limits of β-plane turbulence

Abstract

The quasigeostrophic shallow-water system on the mid-latitude $\beta$ plane with weak, small-scale turbulent forcing is explored in the limit of large energy. Forcing is weak in the sense that the energy input rate relative to the energy of the flow is very small, of the order of $10^{-5}$ – $10^{-10}$ , and the potential vorticity assumes an approximate staircase structure. The flow has large energy in the sense that the jet spacing is equal to the domain width so that no further jet mergers can occur. Quasi-stationary numerical experiments, in which the energy grows linearly, reveal late-time quasi-steady, translating solutions comprising a single jet and vortex dipole, with details of the jet-vortex configuration depending on the deformation radius. At a smaller deformation radius the jet may traverse the entire domain in the $y$ direction one or more times, giving a jet orientation that is predominantly north–south, rather than the usual east–west orientation characteristic of $\beta$ -plane jets at lower energy. In these meandering cases, a mode number is proposed that quantifies the degree of meandering relative to the vortices. Besides the steadily translating solutions, topological changes in the jet-vortex structure are identified that occur via a transient interaction of a meandering jet with a vortex. At high energy, these give rise to apparently periodic solutions of the system; at low energy, before a single, domain-wide jet is established, they indicate that jet merger may occur through more complicated processes than the simple merging of neighbouring jets.