The five-dimensional (5D) drift kinetic Fokker–Planck equation for fast charged particles confined in a tokamak with a toroidal field (TF) ripple magnitude below the Goldston–White–Boozer stochasticity threshold is averaged over the banana and superbanana timescales. As a result, a three-dimensional (3D) Fokker–Planck equation in the constants of motion (COM) space describing the collisional transport of charged high-energy particles is obtained. Toroidally trapped particles with the toroidal precession being in resonance with the ripple perturbations are shown to yield the main contribution to the ripple induced transport. It is found that the rates of ripple superbanana diffusion and convection in the radial coordinate significantly exceed the corresponding rates of the bananas in the axisymmetric limit. The superbanana diffusion and convection shown to be dominant in the MeV energy range may be responsible for the loss of partially thermalized fusion products observed in the Tokamak fusion test reactor (TFTR) [S. J. Zweben, R. L. Boivin, C.-S. Chang et al., Nucl. Fusion 31, 2219 (1991); H. W. Herrmann, S. J. Zweben, D. S. Darrow et al., ibid. 37, 1437 (1997)].
Under the assumption of a general symmetry (dependency on two space variables only), a generalized Grad–Shafranov equilibrium equation is derived and discussed. An elementary formulation of the boundary conditions is given and the existence of solutions is investigated. It emerges that from the equilibrium requirements almost no restrictions follow for the two arbitrary functions appearing in the equilibrium equation.
A multiple time-scale derivative expansion scheme is applied to the dimensionless Fokker–Planck equation and to Maxwell’s equations, where the parameter range of a typical fusion plasma was assumed. Within kinetic theory, the four time scales considered are those of Larmor gyration, particle transit, collisions, and classical transport. The corresponding magnetohydrodynamic (MHD) time scales are those of ion Larmor gyration, Alfvén, MHD collision, and resistive diffusion. The solution of the zeroth-order equations results in the force-free equilibria and ideal Ohm’s law. The solution of the first-order equations leads under the assumption of a weak collisional plasma to the ideal MHD equations. On the MHD-collision time scale, not only the full set of the MHD transport equations is obtained, but also turbulent terms, where the related transport quantities are one order in the expansion parameter larger than those of classical transport. Finally, at the resistive diffusion time scale the known transport equations are arrived at including, however, also turbulent contributions.
The results of a Fokker-Planck simulation of the ripple induced loss of charged fusion products in TFTR are presented. It is shown that the main features of the measured "delayed loss" of partially thermalized fusion products, such as the differences between DD and DT discharges, the plasma current and major radius dependencies etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR.
Ideal MHD equilibria with an ignorable space variable are investigated. It is shown that only three classes of these symmetric equilibria exist: the systems with a straight, a (cylindric) helical, and a circular magnetic axis.
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