Onsager relaxation of toroidal plasmas

The slow relaxation of isolated toroidal plasmas towards their thermodynamical equilibrium is studied in an Onsager framework based on the entropy metric. The basic tool is a variational principle, equivalent to the kinetic equation, involving the profiles of density, temperature, electric potential and electric current. These profiles enter two functionals reflecting the entropy of the field plasma - plasma system and the entropy production rate. These functionals are symmetrical. By themselves, they would drive an Onsager evolution of the system. However, the variational principle also contains an antisymmetrical functional reflecting the trajectory effects. These effects are eliminated, so that the Onsager relaxation is automatically established, in situations of low collisionality where the trajectories are integrable and close to the magnetic surfaces (e.g. in axisymmetric tokamaks). In such situations the Onsager character of the slow relaxation is a mere consequence of the Hamiltonian nature of the field - plasma system. In the collisional or non-integrable cases, an Onsager evolution may be still derived from the variational principle, but the plasma layers around successive magnetic surfaces must be independent enough, in the sense that unconfined trapped particles are forbidden unless they are detrapped long before they depart significantly from the magnetic surfaces. New minimization procedures are proposed to obtain entropy and entropy production rate functionals expressed in terms of the profiles of density etc, which drive the Onsager relaxation of the profiles. Onsager relaxations are possible in the presence of a turbulent field, either in an integrable situation (e.g. well separated magnetic islands) or in a non-integrable case (overlapping islands). The variational principle then involves the characteristic frequencies of the turbulent field, on the same grounds as the profiles of density, etc.