F. L. Hinton

PLASMA TRANSPORT IN TOROIDAL CONFINEMENT SYSTEMS.

The neoclassical theory of plasma transport in axisymmetric, toroidal confinement systems, is developed by means of a variational principle for the rate of irreversible entropy production. The variational principle derived here employs the full Fokker‐Planck collision operator, including both like and unlike species collisions. Using the variational principle, all the relevant neoclassical transport coefficients are systematically evaluated in the “banana” regime of small collisional frequency, to lowest order in the inverse aspect ratio. These results include both the “diagonal” and “cross” coefficients for the particle fluxes, ion and electron heat flux, and electric current. By combining the transport coefficients with appropriate moments of the drift equation, a closed set of equations which accurately summarize the predictions of neoclassical theory in the banana regime is obtained. The significance of these equations, in particular with regard to recent tokamak experiments, is discussed briefly.

Transport properties of a toroidal plasma at low‐to‐intermediate collision frequencies

The neoclassical plasma transport coefficients for axisymmetric toroidal magnetic confinement systems are calculated in the regime of low‐to‐intermediate collision frequency. The problem of solving the linearized drift kinetic equations, and calculating the transport coefficients, is formulated as a variational principle. A maximal form of the variational principle, which is valid in the low‐to‐intermediate collision frequency regime, is used in a numerical solution of the finite dfiference equations, to obtain the distribution function at points on a mesh in phase space. A new analytical result is used to check the accuracy of the numerical calculation for small collision frequencies; this is a correction to the asymptotic banana regime result, obtained from a Wiener‐Hopf analysis of the boundary layer in phase space between the trapped and untrapped particle regions. An analytical correction to the plateau regime analtyical result is also obtained. The transport coefficients are found to be monotonic fu...

Amplitude Limitation of a Collisional Drift Wave Instability

A nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude. The two‐fluid equations are used, including the effects of resistivity, viscosity, and thermal transport. The result, for the wave amplitude as a function of magnetic field in the linearly unstable region close to marginal stability, agrees reasonably well with experiment.

Collision-dominated plasma transport in toroidal confinement systems

The collisional regime of neoclassical transport theory is investigated, using a moment equation approach as well as a method based on the drift kinetic equation. Allowing for both density and temperature gradients, and an externally induced toroidal electric field, the transport coefficients describing particle and energy flux perpendicular to the magnetic field of an axisymmetric confinement system are derived. Relations between the parallel and perpendicular fluxes, which are exact in the collisional regime, are also derived. Charge neutrality is used to obtain the electrostatic potential variation on a magnetic surface.

Plasma transport in a torus of arbitrary aspect ratio

Finite aspect ratio modifications to neoclassical transport theory are considered. In the general case of arbitrary flux surface geometry, a closed set of macroscopic equations is derived. These include the equation which determines the time evolution of the flux surfaces. In the large aspect ratio case, the O (r /R ) corrections to the order O (r )1/2)R (1/2)) transport coefficients, calculated previously, are obtained. In particular, an expression for the electrical conductivity is obtained which may be regarded as exact for all experimentally interesting aspect ratios.

Kinetic theory of plasma scrape‐off in a divertor tokamak

The structure of the plasma boundary, in a tokamak with a poloidal divertor, is considered. The assumption, of hot, collisionless ions, and cold, collisional electrons is shown to be self‐consistent. The density gradient length is shown to be of the order of ρpi, the ion poloidal gyroradius, at typical poloidal angles. Plasma diffusion across the separatrix, and particle motion along the field lines to the collector plates, are described by a kinetic equation. From the solution of the boundary‐layer problem, the ion flux and ion energy flux, to the plates, are calculated. The ratio of these fluxes, and the corresponding ratio for the electrons, complete the specification of divertor boundary conditions, to be used in tokamak transport codes.

Steady‐state response of the electron distribution function to an applied electric field

Steady‐state solutions to the linearized Fokker–Planck equation have been numerically investigated using two models. The Kulsrud model, in which the electron‐electron collision term is simplified by evaluating the integrals using a Maxwellian distribution, is considered first. It is shown that steady‐state solutions of the Kulsrud model, obtained by long time integrations of the time dependent equation, can be obtained more easily by considering a separable solution. A more physically reasonable steady‐state model, which consistently describes both the thermal and runaway regimes and is readily solved numerically, is developed. The resistivity, in agreement with Spitzer and Harm, and runaway production rates, in agreement with the Kulsrud model, are obtained.

Convective amplification of universal drift modes

An analytical solution of the wave equation for universal drift modes is derived. More exact analytical expressions for the dispersion relation and the damping rate are obtained than have been obtained previously. The spatial amplification of an incident wave is derived from the solution of the scattering problem. The dependence of the convective amplification factor on the shear is then determined.

Numerical solution of the drift kinetic equation

The nonadiabatic part of the electron distribution function in toroidal geometry is investigated through numerical solution of a drift kinetic equation. A Lorentz collision operator is used for both trapped and untrapped electrons. No special conditions are invoked at the trapped‐untrapped electron boundary. Poloidal and pitch angle dependence of the electron guiding center drifts are included without bounce averaging. Radially local growth rates are found for the trapped electron mode with ion inertia and Landau damping retained. Comparison is made with simplified descriptions of collisions and drifts, and the dependence of growth rate and distribution function on collision frequency is examined.

Bremsstrahlung spectra associated with anomalous electron thermal conduction

A steady‐state model of the linearized Fokker–Planck equation has been solved numerically for two types of anomalous electron transport. Electron distributon functions which consistently describe both the thermal and runaway regimes and the soft x‐ray spectra computed from them are given. The effect of the anomalous transport on the x‐ray spectra is compared with the effect of an electric field. Due to the simplicity of the analytic model and the significant enhancement of the soft x‐ray spectra predicted, this type of calculation provides a useful tool for analysis of the anomalous electron transport from observed x‐ray spectra.