C. L. Hedrick

Orbits in asymmetric toroidal magnetic fields

Orbits in an asymmetric toroidal magnetic field are studied for the case in which the local variation of the field strength due to ripple is rapid compared with that due to toroidicity. In this case, to lowest order the poloidal variables are constant and particles move primarily in the toroidal direction. Invariants and averaged equations of motion for the locally passing and locally trapped particles are derived based on this approximation. The equations imply that transitions between the locally trapped and locally passing states occur. The probabilities for these transitions are calculated.

Stability of magnetically-confined, high-beta plasma

Plasma waves propagating perpendicular to the magnetic field in a high‐beta, magnetically confined plasma can be unstable in configurations for which the guiding‐center drifts are in the ’’favorable’’ direction. The basic instability mechanism is illustrated and the conditions for growth are obtained using the Maxwell and Vlasov equations in a slab model of the plasma.

Variational corrections to ELMO Bumpy Torus neoclassical ion plateau transport

Low collisionality corrections to ELMO Bumpy Torus (EBT) ion plateau transport coefficients are calculated using a variational method. Using a bounce‐averaged drift kinetic equation and a drift orbit model appropriate to collisionless EBT ion transport, a solution is obtained that is accurate to first order in e(Ω0/ν0)2/3 (e is the inverse aspect ratio, ν0 is the 90° deflection frequency, and Ω0 is the average poloidal gradient B drift frequency). This is substituted into a variational principle (valid over both the plateau and banana regimes) from which particle and energy fluxes are obtained. The sensitivity of the transport rates with respect to variations in the radial and poloidal components of the ambipolar electric field and with respect to variations in the poloidal drift frequency scale length is examined. The transition from plateau to crescent orbit scaling occurs at approximately ν0/Ω0=e3/2.

Resonant ion transport in the ELMO Bumpy Torus

Modeling the ELMO Bumpy Torus as a bumpy cylinder with toroidally induced vertical drift, neoclassical transport coefficients are obtained for resonant ions as integrals over the energy‐dependent flux. A continuous approximation to this energy‐dependent flux reduces to the correct results in the banana and plateau regimes and yields simple analytic formulas for the diffusion coefficients which agree well with numerical results.

Stochastic particle diffusion in velocity space for a bumpy torus

Nonadiabatic changes of the magnetic moment μ in the ELMO Bumpy Torus‐Scale (EBT‐S) have been studied both analytically and numerically. Simple forms of Δ μ and gyrophase change were obtained, permitting the changes in these quantities to be studied using an iteration mapping. The mapping results show stochastic behavior for particles having high energy and low initial μ. Otherwise, superadiabatic motion appears. The stochastic diffusion coefficient for the variation of μ was measured numerically by mapping and was also calculated from quasilinear theory. The results are shown to agree well in the stochastic region. For high‐energy particles, the diffusion in μ caused by nonadiabaticity can be comparable to collisional diffusion when stochastic motion occurs for EBT‐S.

Monte Carlo estimates of particle and energy confinement times in a bumpy torus and a bumpy square with poloidal electric fields

Since significant poloidal structure in the electrostatic potential in Elmo Bumpy Torus (EBT) has been observed experimentally [Phys. Fluids 2 8, 2848 (1985)] and predicted theoretically (C. L. Hedrick, submitted to Phys. Fluids), a Monte Carlo calculation has been used to make estimates of the particle and energy confinement times in EBT with varying degrees of asymmetry in the electric field. The code is applicable to the bulk ion population and the ‘‘cool’’ electron population in EBT, but not to the intermediate‐energy electrons believed to be responsible for the formation of the potential. A similar calculation is possible for an alternate magnetic field configuration known as a bumpy square, which is expected to have more symmetric potential profiles because of much better centering of the particle orbits. The calculations indicate that the confinement time in a bumpy square would be two to three orders of magnitude better than in EBT.

Diamagnetic wells in axisymmetric mirrors

Here the question of whether hot‐electron rings have reversed gradients in B and ∮ dl/B in symmetric mirrors is reexamined. The focal point for this reexamination is a particular data set from a symmetric mirror device. Using standard least‐squares fitting techniques, this and three other cases (having comparable stored energy) have been fitted with a five‐parameter function for the perpendicular pressure. Three of the four cases show reversal of the gradients. All cases pass standard tests for goodness of fit. The probable errors, deduced by fitting a total of ten experimintal data sets, were found to be consistent with the single‐measurement accuracy. Sensitivity tests for random and systematic error were performed. The errors would have to be implausibly high in order to change the conclusions regarding gradient reversal. More complex functions for the perpendicular pressure, involving more parameters, have been examined to see if radial asymmetries or pressure pedestals extending to large radius are i...

Poloidally asymmetric electrostatic potentials in closed line bumpy toroids

Here an analytic expression is developed for the poloidal asymmetry in the electrostatic potential observed experimentally in the T mode of operation in the Elmo Bumpy Torus (EBT) [Phys. Fluids 28, 2848 (1985)]. A multiple fluid treatment for the ‘‘cool,’’ ‘‘warm,’’ and ‘‘hot’’ electrons is used. The central idea is that the ‘‘warm’’ electrons dominate both the radial and poloidal structure of the electrostatic potential in the T mode of operation. The expression for the poloidal asymmetry in the electrostatic potential, which agrees reasonably well with experiment, is also applied to a modification of the EBT magnetic configuration. It is found that a substantial reduction in the asymmetry of the electrostatic potential below that of the Elmo Bumpy Torus is possible in the modification of the EBT magnetic configuration.

Nonresonant ELMO Bumpy Torus transport coefficients in the small electric field regime

A conservative BGK collision operator is used to obtain nonresonant neoclassical transport coefficients for a bumpy torus when the ∇B drift dominates over the E×B drift. Previous nonresonant coefficients have considered the opposite limit. For large collisionalities, the diffusion coefficients are only weakly dependent on the E×B drift, while for small collisionalities, plateau diffusion coefficients are obtained which have an exponential dependence on the E×B drift.

Monte Carlo calculation of resonant diffusion coefficients in the ELMO Bumpy Torus and the applicability of local diffusion theory

A Monte Carlo calculation of resonant diffusion coefficients in the ELMO bumpy torus (EBT) is described and is shown to agree well with analytic theory in a regime where the assumptions of local, diffusive behavior are valid. Since present and future EBT devices do not fall into this regime, local diffusion theory may not be entirely applicable.