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Late states of incompressible 2D decaying vorticity fields

Two-dimensional decaying turbulent flow is known to approach apparently stable states after a long time evolution. A few theories and models have been so far proposed to account for this relaxation. In this paper, we compare results of numerical experiments with the predictions of these theories to assess their applicability. We study the long time decay of initially multilevel vorticity fields on the periodic box, and characterize the outcoming final states. Our final states do not match the predictions of the theories; a broader variety of dipole profiles, as well as nonstationary final states are found. The problem of the robustness of the relaxational state with respect to variations of the Reynolds number and different numerical resolution is addressed. The observed configurations also do not necessarily possess the maximal energy, in contrast to what is anticipated by some of the theories. We are led to conclude that the mixing of the vorticity is generally not ergodic, and that some metastable configurations may inhibit the attainment of an equilibrium state.