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Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. (English) Zbl 0636.76120
Summary: It is shown that, for a sufficiently large value of \(\beta\), two- dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate \(\Omega\). This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Lyapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute- vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity can be derived. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which \(\epsilon\) decreases in time) seem to evolve into a non-ergodic regime at large scales.

MSC:
76U05 General theory of rotating fluids
76F99 Turbulence
76E99 Hydrodynamic stability
76B47 Vortex flows for incompressible inviscid fluids
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