Nonlinear interaction of small-scale Rossby waves with an intense large-scale zonal flow

Abstract

The system of Rossby waves is unstable with respect to modulations. This instability results in the generation of a large-scale zonal flow with zero mean vorticity. It is shown that in both stable and unstable situations, a finite-time singularity formation takes place in the system. Such a singularity has the form of peaks on the vorticity profile of the zonal flow. The situation is also considered when some zonal flow with nonzero mean vorticity is initially present. Solitary-wave solutions appropriate for the description of nonlinear behavior of such systems are found. In the case of weak mean vorticity of the zonal flow, the solitons break and the singularities develop. If the mean vorticity is strong, then the evolution of the system can be considered as the dynamics of a soliton gas. Soliton dynamics possesses some interesting properties, such as formation of soliton pairs and annihilation of solitons during collision.

A phase-space study of jet formation in planetary-scale fluids

“…A generalisation of the analysis of Manin & Nazarenko [10] was used to predict a finite fastest growing zonal wavenumber. This was then compared with the results of a number of numerical simulations for varying values of eastwest wavenumber k 0 .…”

“…The most relevant work to this section, however, is the previously cited Manin & Nazarenko [10], in which a Vlasov equation similar to (20) was used to study the latitudinal instability of planetary waves in the limit of small β-effect. Here, we begin by summarising their methods and the result of their instability calculation.…”

“…It should be emphasized at this point thatk y = −i∂ y is an operator, as defined in (10), and hence (26) and (27) make no assumption of scale separation.…”

“…Among the first researchers to take this approach were Dyachenko et al [9], who derived an equation for the interaction between waves and largescale vortices. Later, Manin & Nazarenko [10] used a Vlasov equation to study the interaction between scaleseparated zonal flows and planetary waves in the limit of small β-effect. They found that planetary waves in the presence of a zonal flow could become modulationally unstable, which led to singularity formation and soliton propagation in the model they used.…”

Triple Cascade Behavior in Quasigeostrophic and Drift Turbulence and Generation of Zonal Jets

“…Two main zonal jet generation mechanisms considered in the literature are the modulational instability [16,8,17,9,18,19,20,21] and the anisotropic inverse cascade [22,26,23,12]. The inverse cascade mechanism brings the energy from initial small-scale turbulence to the large-scale zonal flows in a step-by-step (local in the scale space) transfer mechanism similar to the inverse cascade in 2D Navier-Stokes turbulence [24,25].…”