Tokamak plasmas are inherently comprised of multiple ion species. This is due to wall-bred impurities and, in future reactors, will result from fusion-born alpha particles. Relatively small densities nI, of highly charged non-hydrogenic impurities can strongly influence plasma transport properties whenever . The determination of the complete neoclassical Onsager matrix for a toroidally confined multispecies plasma, which provides the linear relation between the surface averaged radial fluxes and the thermodynamic forces (i.e. gradients of density and temperature, and the parallel electric field), is reviewed. A closed set of one-dimensional moment equations is presented for the time evolution of thermodynamic and magnetic field quantities which results from collisional transport of the plasma and two-dimensional motion of the magnetic flux surface geometry. The effects of neutral-beam injection on the equilibrium and transport properties of a toroidal plasma are consistently included.

An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest-descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive-definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self-consistent value for λ.

A multi-species fluid model is described for the steady state parallel and radial force balance equations in axisymmetric tokamak plasmas. The bootstrap current, electrical resistivity, and particle and heat fluxes are evaluated in terms of the rotation velocities and friction and viscosity coefficients. A recent formulation of the neoclassical plasma viscosity for arbitrary shape and aspect ratio (including the unity aspect ratio limit), arbitrary collisionality, and orbit squeezing from strong radial electric fields is used to illustrate features of the model. The bootstrap current for the very low aspect ratio National Spherical Torus Experiment [J. Spitzer et al., Fusion Technol. 30, 1337 (1996)] is compared with other models; the largest differences occur near the plasma edge from treatment of the collisional contributions. The effects of orbit squeezing on bootstrap current, thermal and particle transport, and poloidal rotation are illustrated for an enhanced reverse shear plasma in the Tokamak Fusion Test Reactor [D. Meade and the TFTR Group, Plasma Physics and Controlled Nuclear Fusion Research, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. I, p. 9]. Multiple charge states of impurities are incorporated using the reduced ion charge state formalism for computational efficiency. Because the force balance equations allow for inclusion of external momentum and heat sources and sinks they can be used for general plasma rotation studies while retaining the multi-species neoclassical effects.

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.