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Dynamical pattern formation in two-dimensional fluids and Landau pole bifurcation

A phenomenological theory is proposed to analyze the asymptotic dynamics of perturbed inviscid Kolmogorov shear flows in two dimensions. The phase diagram provided by the theory is in qualitative agreement with numerical observations, which include three phases depending on the aspect ratio of the domain and the size of the perturbation: a steady shear flow, a stationary dipole, and four traveling vortices. The theory is based on a precise study of the inviscid damping of the linearized equation and on an analysis of nonlinear effects. In particular, we show that the dominant Landau pole controlling the inviscid damping undergoes a bifurcation, which has important consequences on the asymptotic fate of the perturbation.

Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere

The large-scale circulation of planetary atmospheres such as that of the Earth is traditionally thought of in a dynamical framework. Here we apply the statistical mechanics theory of turbulent flows to a simplified model of the global atmosphere, the quasigeostrophic model, leading to nontrivial equilibria. Depending on a few global parameters, the structure of the flow may be either a solid-body rotation (zonal flow) or a dipole. A second-order phase transition occurs between these two phases, with associated spontaneous symmetry breaking in the dipole phase. This model allows us to go beyond the general theory of marginal ensemble equivalence through the notion of Goldstone modes.