ABSTRACT
Conclusion The multiregion-relaxation variational approach described in Sec. 3 holds strong promise of being the most satisfactory mathematical foundation on which to base the solution of the MHD toroidal equilibrium problem posed by Grad [44] over forty years ago. The numerical program we are developing will not only provide a practical tool for design and analysis of fusion experiments, but will allow numerical investigation of such fundamental issues as the critical point at which a KAM barrier ceases to be able sustain a pressure jump. One hopes this will stimulate further mathematical developments beyond KAM theory (see e.g., [45]) on the existence of KAM barriers and their breakup. While dissipationless MHD is a standard fi rst-cut model in fusion plasma physics because of its (relative) mathematical tractability, it is clearly inadequate to describe much of the physics of the complex selforganizing system that is a hot, toroidally confi ned plasma. In particular, the lack of diffusive transport in the model allows the unphysically strong (infi nite) gradients we have postulated to occur at a KAM barrier, and also more or less dictates our assumption of complete relaxation between the barriers. A fi rst step away from this oversimplifi cation has recently been taken by Hudson and Breslau [46], who used a simple anisotropic thermal diffusion to resolve the structure of the temperature profi le in a chaotic magnetic fi eld, revealing a much more complex structure than our current relaxation model can represent. Once dissipation is present a simple energy minimization variational principle is no longer appropriate, but it may still be possible to construct a variational relaxation model of plasma steady states by using the thermodynamic MEP principle with a phenomenological Ziegler entropy production function [36]. The Onsager MaxEnt approach has been partially explored in plasma physics, but its utility in climate modeling [30] suggests that it should be developed further. The use of Jaynesian MaxEnt approaches would appear to be an entirely open fi eld in plasma physics, as is the use of statistical-mechanical MEP principles. Given the need for robust variational principles for predicting the overall behavior of fusion we plasmas, we expect entropy-based methods to be increasing importance in this fi eld.
