R. D. Hazeltine

Island bootstrap current modification of the nonlinear dynamics of the tearing mode

A kinetic theory for the nonlinear evolution of a magnetic island in a collisionless plasma confined in a toroidal magnetic system is presented. An asymptotic analysis of a Grad–Shafranov equation including neoclassical effects such as island bootstrap current defines an equation for the time dependence of the island width. Initially, the island bootstrap current strongly influences the island evolution. As the island surpasses a certain critical width the effect of the island bootstrap current diminishes and the island grows at the Rutherford rate. For current profiles such that Δ’<0 the island bootstrap current saturates the island.

Theory of anomalous tearing mode growth and the major tokamak disruption

An analytic theory of turbulence in reduced resistive magnetohydrodynamics is developed and applied to the major disruption in tokamaks. The renormalized equations for a long‐wavelength tearing instability are derived. The theory predicts two principal nonlinear effects: an anomalous flux diffusivity due to turbulent fluid convection in Ohm’s law and a vorticity damping term due to turbulent magnetic stresses in the equation of motion. In the final phase of the disruption, when fine‐scale fluid turbulence has been generated, detailed considerations show that anomalous diffusivity has the dominant effect at long wavelengths. For a low‐m tearing mode, the solution of the renormalized equations during the turbulent phase yields a growth rate analogous to the classical case but increased by turbulent resistivity: γ∼(∑k′ k′2θ‖φk′‖2)3/8 ×(Δ′)1/2. This analytical prediction is in good accord with computational results.

Reduced magnetohydrodynamics and the Hasegawa–Mima equation

Reduced magnetohydrodynamics consists of a set of simplified fluid equations which has become a principal tool in the interpretation of plasma fluid motions in tokamak experiments. The Hasegawa–Mima equation is applied to the study of electrostatic fluctuations in turbulent plasmas. The relations between these two nonlinear models is elucidated. It is shown that both models can be obtained from appropriate limits of a third, inclusive, nonlinear system. The inclusive system is remarkably simple.

Quasi-linear diffusion and radial transport in tokamaks

An important difference between the classical collision operator, C(f), and Kaufman’s quasi‐linear diffusion operator in action space, D(f), is that only the former conserves particles, momentum, and energy at each spatial point. The nonlocal character of action‐space diffusion allows all transport fluxes to be expressed directly in terms of appropriate moments of D. Thus, a general description of quasi‐linear diffusion and convection across flux surfaces in any collisionless, axisymmetric toroidal system is obtained, in terms of scale factors relating the invariant surfaces to the flux surfaces. Several analogues to the neoclassical problem are apparent, including the collisionless version of the Ware–Galeev pinch effect, whose derivation in action space is especially straightforward. Toroidicity modifies not only particle orbits but also the spatial structure of the fluctuations, and both modifications affect the resulting transport in an important way. The use of appropriate action‐angle variables auto...

Stability of low‐shear tokamaks

It has been suggested that the recently observed fast sawtooth crashes are caused by a low‐shear, pressure‐driven ideal instability. This hypothesis is investigated, using asymptotic methods to solve the toroidal mode equations for a class of equilibria characterized by a low‐shear central region in which q−1 is small, separated from the wall by a region with finite shear. A dispersion relation that differs significantly from previous results is obtained. An explicit expression for the growth rate is given for a model q profile.

Bumpy torus transport in the low collision frequency limit

A variational formulation of transport for a closed field line, large aspect ratio bumpy torus in the low collision frequency limit is presented. The relatively large radial excursions made by those particles with canceling magnetic and electric poloidal drifts requires generalization of previous neoclassical techniques. General expressions for the transport coefficients of both charge species are given. Explicit numerical coefficients are evaluated under certain simplifying assumptions for both the banana and nonbanana species.

Unified theory of tearing modes

A unified theory of tearing modes in slab geometry is presented. All known results are readily derived from our general dispersion relations. Several new results are derived and discussed, and the inter‐relatonship between various tearing modes is established.

Sheared-flow generalization of the Harris sheet

A novel, exact class of solutions to the Vlasov–Maxwell system, with self-generated magnetic fields and nonuniform plasma flows, are constructed. It is shown that a gyrotropic distribution function (independent of gyrophase) does not allow equilibrium shear flow; introduction of agyrotropy is essential for the maintenance of spatially nonuniform velocity fields. The new self-consistent sheared-flow solutions include the shearless Harris sheet [E. G. Harris, Nuovo Cimento 23, 117 (1962)] solution as a special case. These equilibria are likely to be relevant to a variety of astrophysical flows (most natural flows are sheared) and to a better understanding of the laboratory phenomena observed, for example, in the device MRX [Magnetic Reconnection Experiment, M. Yamada, H. Ji, S. Hsu, T. Carter, R. Kulsrud, N. Bretz, F. Jodes, Y. Ono, and F. Perkins, Phys. Plasmas 4, 1936 (1997)] designed to study magnetic reconnection.

Impurity transport in the collisional regime for large poloidal variations

Impurity transport is studied in a toroidal plasma column with large poloidal variations of the plasma parameters (comparable to the variation of the magnetic field) near the edge. It is found that the highly collisional nature of the edge plasma can lead to such variations of density and electrostatic potential. The impurity transport coefficients are significantly modified by this effect, becoming non-linear in the gradients.

Plasma equilibria in dipolar magnetic configurations

The effects of finite plasma pressure and pressure anisotropy, toroidal rotation and gravity on the equilibrium, flow, and stability of plasma in dipolar magnetic configurations are considered. Dipolar equilibria are of interest for magnetic confinement experiments in the laboratory and understanding the physics of magnetospheric and astrophysical plasmas. It is demonstrated that realistic solutions of the appropriate ideal magnetohydrodynamics (MHD) equations can be found in a separable form which drastically simplifies the equations and even allows us to analytically obtain some limiting forms of the nonlinear solutions. The MHD stability of these equilibria is explicitly evaluated in some cases.