K. C. Shaing

Bootstrap current and neoclassical transport in tokamaks of arbitrary collisionality and aspect ratio

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 Fusi...

Neoclassical flows and transport in nonaxisymmetric toroidal plasmas

The moment equation approach to neoclassical transport theory has been generalized to nonaxisymmetric toroidal systems under the assumption of the existence of magnetic surfaces. In particular, the parallel plasma flows and bootstrap current are calculated in both the Pfirsch–Schluter and banana regimes. It is found that both parallel plasma flows and the bootstrap current can be reduced as the toroidal bumpiness increases in an otherwise axisymmetric system.

Plasma transport coefficients for nonsymmetric toroidal confinement systems

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.

Magnetohydrodynamic-activity-induced toroidal momentum dissipation in collisionless regimes in tokamaks

It is shown that in the presence of magnetohydrodynamic (MHD) activities, such as resistive wall mode, the toroidal symmetry in tokamaks is broken. This results from the modification on |B|, and the variation of |B| on the distortion of the magnetic surface due to the MHD activities. Here, B is the magnetic field. The toroidal plasma viscosity in the collisionless regime for the nonresonant MHD modes is calculated. It scales like (δB/|B|)2 with δB, the typical magnitude of the perturbed magnetic field strength. The theory is applied to determine the toroidal flow speed in the presence of these modes.

Transport associated with the collisionless detrapping/retrapping orbits in a nonaxisymmetric torus

The differences among the various scalings in the transport calculations associated with the collisionless detrapping/retrapping orbits in a nonaxisymmetric torus are clarified. A simple random walk argument indicates that these differences are mainly due to an implicit assumption made about the retrapping rate of the toroidally trapped particles.

Neoclassical quasilinear transport theory of fluctuations in toroidal plasmas

A transport theory of fluctuations with frequencies less than the gyrofrequency in toroidal plasmas is developed to calculate particle and heat fluxes, bootstrap current, Ware pinch flux, and modification to plasma conductivity. It is found that the particle and heat fluxes are proportional to ∑m,n,ωm2(eΦmnω/Te)2, the bootstrap current and Ware pinch flux are proportional to ∑m,n,ωm[(m−nq)/‖m−nq‖] (eΦmnω/Te)2, and the modificationto plasma conductivity is proportional to ∑m,n,ω‖m−nq‖(eΦmnω /Te)2, where m (n) is the poloidal (toridal) mode number for the (m,n) mode with frequency ω, Φmnω is its mode amplitude, e is the electric charge, Te is the electron temperature, and q is the safety factor. Alternatively, in terms of the poloidal wave vector kθ and the parallel wave vector k∥, the particle and heat fluxes are proportional to 〈k2θ〉〈(eΦ/Te)2〉, the bootstrap current and Ware pinch flux are proportional to 〈kθk∥/‖k∥‖〉〈(eΦ/ Te)2〉, and the modification to plasma conductivity is proportional to 〈‖k∥‖〉〈(...

Neoclassical transport in a multiple‐helicity torsatron in the low‐collisionality (1/ν) regime

For a sufficiently high number of toroidal field periods (m/ι>l), the magnetic field of a multiple‐helicity torsatron can be reduced to a simple form such that the second adiabatic invariant J can be calculated. It is found that the particle and the heat fluxes for a multiple‐helicity torsatron in the low‐collisionality (1/ν) regime have the same geometric dependences. An optimization of both quantities is carried out for a given equilibrium constraint. It is shown that the transport fluxes can be smaller than those of the conventional stellarator by an order of magnitude. The effect of finite plasma beta on the neoclassical fluxes is also studied.

Neoclassical transport fluxes in the plateau regime in nonaxisymmetric toroidal plasmas

The neoclassical fluxes in the plateau regime in an arbitrary large aspect ratio nonaxisymmetric toroidal system are calculated using fluid equations. The results are evaluated explicitly for a stellarator and for a rippled tokamak. For typical stellarator parameters, the helical magnetic field contribution to the particle and heat fluxes is comparable to the toroidicity contribution, and therefore, cannot be neglected. In addition, the helical magnetic field contribution to the parallel viscosity is capable of reversing the directions of the bootstrap current and Ware pinch flux in the plateau regime. In contrast, the neoclassical particle and heat fluxes in a rippled tokamak are comparable to those in an axisymmetric tokamak after the radial electric field and parallel flow velocity are determined (without external sources).

THE AMBIPOLAR ELECTRIC FIELD IN STELLARATORS

In a three-dimensional device like a stellarator, the ambipolar electric field must be determined self-consistently from the ambipolarity constraint and can have a significant effect on transport through the diffusion coefficients. A differential formulation and an algebraic formulation for the electric field are solved, together with the density and temperature equations. The results are compared, and in both cases multiple electric field solutions can exist, with bifurcations occurring between different solutions. It is shown that heating of the electrons encourages bifurcation to the more favourable positive electric field root.

An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks

An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities is presented. The expression smoothly joins transport fluxes or plasma viscosity in all the known collisionality regimes derived from the solution of the bounce averaged drift kinetic equation and should be useful in modelling results of existing and future tokamak experiments. It also incorporates some of the extensions of the known expressions to include the effects of finite ∇B drift in the non-resonant transport processes. Here, B is the magnitude of the magnetic field. The toroidal momentum balance equation is a nonlinear function of the radial electric field when the neoclassical plasma viscosity is dominant. It can have bifurcated solutions for the radial electric field and may lead to better plasma confinement as a result.