III. Development of the numerical instruments for the investigation of
IV. Possible extension to 2D Magnetohydrodynamics (MHD)
As explained, there are unequivocal proofs that the asymptotic stationary states of MHD plasma are highly organized. There is at this moment no model for the evolution to, and for the selection of- these states by the system. It is missing the intermediate model: for MHD there is no model of 2D discrete set of point-like objects (vortices and currents) whose dynamics would be given by Kirchhoff-type equations. One possible starting model would be the Ashkin-Teller generalization of the Ising model, where two kinds of spins are interacting with distinct energies. The most attractive possibility is however rather different: the Born-Infeld Lagrangian for a field theory in ten dimensions allows to obtain at one limit the Nambu-Goto Lagrangian for the world-surface of a vortex string (also known to be equivalent with the dynamics of an ideal fluid with Chaplygin polytropic law) and at the opposite limit the Lagrangian of a the free Maxwell electromagnetic field. Investigation of this field theory is worthwhile, although difficult. On the other hand we will look to correlations that are apparent in numerical simulations: the maximum of current density appears to coincide with the maximum of the vorticity, and the vorticity exhibits a multiple concentric ring structure. This type of results, even if they will be obtained from separate field theoretical models (for current: Liouville and for vorticity: the Abelian dominance model) have a considerable impact on researches in tokamak plasma: filaments of vorticity and of current may represent a stable structure generated by a synergetic evolutions of the two fields, explaining periodic destruction of the Internal Transport Barriers or the periodic breaking-up by filamentation of the H-mode velocity shear layer.
V. Possible extension of the field-theoretical approach to three dimensional fluid evolution
We see this as a tentative since the problem appears to be highly non-trivial. Curiously enough, the field theory is sensibly more advanced in the analysis of three dimensional vortex lines than the classical fluid theory. In field theory the instruments are prepared for the description of the self-interacting vortex line in a perfect fluid (Lund and Regge, Sato and Yashikozawa, the string theory, etc.) and can be taken as a starting point for more extended investigations. Although rather vague at this moment, the main difficulty appears to us to be the impossibility of the self-duality in three spatial dimensions: the self-duality, as a geometric-algebraic property of a fiber space exists when a differential form is equal to its Hodge dual, or in three dimensions (at fluid stationarity) this is impossible. The Chern-Simons term simply cannot be written in 3D (which is 4D at stationarity). We will focus, instead, on topological properties of physical strings i.e. vorticity lines and similarly of magnetic field lines, where reconnection is suppressed by very low resistivity. For this problem the field theory offers a formalism in which the physical string is replaced by a continuous field whose topological Hopf invariant reflects the linking with a reference direction. The applications are the statistical properties of the stochastic magnetic lines in magnetized plasmas, which are known to influence decisively the energy and particle transport properties in a confinement structure.
VI. Extension into a new class of fundamental problems: the connection between statistical stationarity and self-duality.
The self-duality (probably the most important conclusion we have obtained as a common aspect of the FT models we have developed so far) is known to be the source of exact integrability of the equations like sinh-Poisson, KdV, nonlinear Schrodinger, sine-Gordon, Painleve Transcendents, Yang-Mills, etc. The exact integrability (e.g. of the Liouville and sinh-Poisson eq.) involves a high genus Riemann surface generated by the compactification of the plane of the spectral variable (the parameter of the Lax pair), after exclusion of the points of the main spectrum, and the solution (after Abel map and Jacobi inversion) consists of ratios of Riemann theta functions having as arguments the phases on the cycles of the Riemann surface. The connection may come from the property of the theta function of being a solution of the diffusion equation, which possibly may be represented as a statistical limit of instantons on cycles. The idea is complex and requires much work, still complicated by the fact that our equations for the planetary atmosphere and magnetized plasma apparently are only close to exact integrability.Return to the main page