The plasma energy W p = j [\B 2-h p)dV is minimized over a toroidal domain ft p using an inverse representation for the cylindrical coordinates R = ^ R mn (s) cos(m6-nf) and Z = VJ Z mn (s) sin(m#-n<;) , where (.s,0,f) are radial, poloidal, and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when tt p is varied to make the total pressure-B 2 + p continuous at the plasma surface E p and when tie vacuum magnetic field B^ satisfies the Neumann condition B w • dS p = 0. The vacuum field is decomposed as B v = Bo I V$, where Bo is the field arising from plasma currents and external coils and $ is a single-valued potential necessary to satisfy B v • d*£, p = 0 when p == 0. A Green's function method is used to obtain an integral equation over S p for the scalar magnetic potential $ = YJ$ mrl sin(mfl-n<;). A linear matrix equation is solved for $ mn to determine ^B% on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to I-2 stellarator equilibria are presented.
A variational principle is developed for computing accurate values of local plasma transport coefficients in nonsymmetric toroidal confinement configurations. Numerical solutions of the linearized drift Fokker–Planck equation are used to obtain the thermodynamic fluxes as functions of collision frequency and the radial electric field. Effects resulting from the variation of the longitudinal adiabatic invariant J along an orbit (resulting from particle transitions from helically trapped to toroidally trapped orbits) are treated. The velocity-space distribution resulting from trapped, circulating, and transition particle orbits is well represented by a Legendre polynomial expansion in the pitch angle coordinate. The computational effort is significantly reduced from that required with Monte Carlo methods through use of an efficient treatment of the disparity between the time scales of collisionless and collisional particle dynamics. Numerical computations for a stellarator configuration are presented.
A variational principle is developed for the linearized drift-kinetic, Fokker–Planck equation, from which both upper and lower bounds for neoclassical transport coefficients can be calculated for plasmas in three-dimensional toroidal confinement geometries. These bounds converge monotonically with the increasing phase-space dimensionality of the assumed trial function. This property may be used to identify those portions of phase space that make dominant contributions to the transport process. A computer code based on this principle has been developed that uses Fourier–Legendre expansions for the poloidal, toroidal, and pitch-angle dependences of the distribution function. Numerical calculations of transport coefficients for a plasma in the TJ-II flexible heliac [Nucl. Fusion 28, 157 (1988)] are used to demonstrate the application of this procedure.
Bootstrap currents in the low collisionality regime in nonaxisymmetric toroidal plasmas are calculated analytically using a model drift kinetic equation. The results obtained using the analytic expression are compared with those calculated from the numerical solutions of the drift kinetic equation. Qualitatively and quantitively, the results are in reasonable agreement.
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