State of the art

An extensive numerical investigation of the coherent asymptotic states obtained at relaxation from turbulent states in the non-dissipative incompressible (Euler) fluid have been carried out by David Montgomery and his collaborators, in the years 1970 - 1990. It has been inferred from numerical studies that the streamfunction obeys the sinh-Poisson equation. Therefore the subspace of final states is only a tiny part of the space of all possible coherent flows allowed by the conservation equations. The explanation has been given in terms of the maximum entropy of a discrete set of interacting point-like vortices (equivalent with the Euler equation), viewed as a statistical system. Experiments in the rotating water-tank have shown stable vortical structures. When the geometry is quasi-two-dimensional (there is an axis of main rotation) the process of vorticity concentration is observed, collecting into few filaments the vorticity initially spread in the volume. There is no qualitative and quantitative theory to predict the final 2D profile of the filaments and their volume distribution. In a similar case, crystal-type disposition of tornadoes is observed. For the plasma in strong magnetic field and for planetary atmosphere the loss of scale invariance is the difficulty that has prevented an unequivocal identification of the equation governing the ordered final states. Several formulas have been proposed to fit the scatterplots (Ψ,ω) (streamfunction, vorticity) obtained numerically but none is fully acceptable at present. It is not known the criterion that may decide which is the natural one. In the physics of atmosphere the tropical cyclone is usually examined in a unitary approach, i.e. including the cyclogenesis and the landfall, with extensive implication of the thermal processes. This requires numerical simulation. However the quasi-stationary phase of the cyclone (frequently observed) has not been examined analytically as a system evolving on a low dimensional manifold, with the vorticity balance being the dominant factor that determines the spatial characteristics of the vortex. In non-neutral plasma there have been observed quasi-stable crystal-symmetric structures of vortices. The breaking up of a ring of concentrated vorticity in electron plasma is seen as a manifestation of the Kelvin-Helmholtz instability. Since this is just a mechanism and not an unequivocal characterization of an intrinsic trend of evolution toward a preferred state of the plasma, we still need to know why the crystal of vortices is an attractor and why is a metastable state. At present no theory exists for this. In linear plasma machines large scale stable vortices have been observed.

Finally, for the two-dimensional magnetohydrodynamics the numerical studies and experimental observations have clearly indicated the evolution to highly organized states, from turbulent initial conditions. But an equation for the asymptotic stationary states, equivalent of the sinh-Poisson equation, is not known. The statistical approach has not been able to provide any hint.

We conclude that at present it is fully recognized that the evolution by self-organization belongs to the fundamental properties of fluids and plasmas. However we do not have a physical model, we do not know how to accede to the elaboration of a theory for this, we do not know what are the adequate concepts to formulate a description of this evolution.

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