C. C. Hegna

Threshold for neoclassical magnetic islands in a low collision frequency tokamak

A kinetic theory for magnetic islands in a low collision frequency tokamak plasma is presented. Self‐consistent equations for the islands’ width, w, and propagation frequency, ω, are derived. These include contributions from the perturbed bootstrap current and the toroidally enhanced ion polarization drift. The bootstrap current is independent of the island propagation frequency and provides a drive for the island in tokamak plasmas when the pressure decreases with an increasing safety factor. The polarization drift is frequency dependent, and therefore its effect on the island stability cannot be deduced unless ω is known. This frequency is determined by the dominant dissipation mechanism, which for low effective collision frequency, νeff=ν/e<ω, is governed by the electrons close to the trapped/passing boundary. The islands are found to propagate in the electron diamagnetic direction in which case the polarization drift is stabilizing and results in a threshold width for island growth, which is of the or...

On the stabilization of neoclassical magnetohydrodynamic tearing modes using localized current drive or heating

The effectiveness of using localized current drive or heating to suppress the formation and growth of neoclassical magnetohydrodynamic (MHD) tearing modes is addressed. The most efficient way to use an auxiliary current source is to cause current to flow in the same direction as the equilibrium bootstrap current and phase the current relative to the magnetic island such that the current is deposited on the O-point of the island. Theoretical estimates for the amount of required current to suppress the formation of a large magnetic island is of order a few percent of the equilibrium current. If the suppression is successful, the magnetic island will saturate at a width of order the radial localization width of the current source. Localized heating at the O-point of the magnetic island can also produce stabilizing effects relative to magnetic island growth. The effects of the driven current or heating can be illustrated by using a phase diagram of the island growth.

Computational modeling of fully ionized magnetized plasmas using the fluid approximation

Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. High dimensionality renders this approach impractical for computations for long time scales. Fluid models are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and∕or temporal scales to be explored. Computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. Several ordering and closure schemes are reviewed and discussed, as are their normal modes, and algorithms that can be applied to obtain a numerical solution.

Two-fluid tearing instability in force-free magnetic configuration

In general, the linear two-fluid tearing instabilities are driven by shear Alfven (SA), compressional Alfven (CA), and slow magnetoacoustic (MA) modes modified on short scales by two-fluid effects. Previous two-fluid theories were devoted to either the hot plasma case where coupling of the SA and the MA waves dominates, or to the cold plasma limit, β=0, where the instability is driven by the SA and the CA waves. Taking into account plasma compressibility and the Hall term, we derive general tearing equations that cover the two limiting cases and the transition between them. In particular, in the hot plasma case, equations are derived that depend on the factor β/(1+β) and span the validity of resistive to electron MHD. The important effect of resistive diffusion of the “out-of-plane” component of the magnetic field perturbation B∥(1) is also included. Two new solutions where this effect dominates are obtained within the scope of the hot plasma model. Whistler scaling γ∝Δ′2 is found for the collisionless te...

Effect of a resistive vacuum vessel on dynamo mode rotation in reversed field pinches

Locked (i.e., nonrotating) dynamo modes give rise to a serious edge loading problem during the operation of high current reversed field pinches. Rotating dynamo modes generally have a far more benign effect. A simple analytic model is developed in order to investigate the slowing down effect of electromagnetic torques due to eddy currents excited in the vacuum vessel on the rotation of dynamo modes in both the Madison Symmetric Torus (MST) [Fusion Technol. 19, 131 (1991)] and the Reversed Field Experiment (RFX) [Fusion Eng. Des. 25, 335 (1995)]. This model strongly suggests that vacuum vessel eddy currents are the primary cause of the observed lack of mode rotation in RFX. The eddy currents in MST are found to be too weak to cause a similar problem. The crucial difference between RFX and MST is the presence of a thin, highly resistive vacuum vessel in the former device. The MST vacuum vessel is thick and highly conducting. Various locked mode alleviation methods are discussed.

Stability of tearing modes in tokamak plasmas

The stability properties of m≥2 tearing instabilities in tokamak plasmas are analyzed. A boundary layer theory is used to find asymptotic solutions to the ideal external kink equation, which are used to obtain a simple analytic expression for the tearing instability parameter Δ’. This calculation generalizes previous work on this topic by considering more general toroidal equilibria (however, toroidal coupling effects are ignored.) Constructions of Δ’ are obtained for plasmas with finite beta and for islands that have nonzero width. A simple heuristic estimate is given for the value of the saturated island width when the instability criterion is violated. A connection is made between the calculation of the asymptotic matching parameter in the finite beta and island width case to the nonlinear analog of the Glasser effect [Phys. Fluids 18, 875 (1975)].

Toroidal coupling of ideal magnetohydrodynamic instabilities in tokamak plasmas

A theoretical framework is developed to describe the ideal magnetohydrodynamic (MHD) stability properties of axisymmetric toroidal plasmas. The mode structure is described by a set of poloidal harmonics in configuration space. The energy functional, δW, is then determined by a set of matrix elements that are computed from the interaction integrals between these harmonics. In particular, the formalism may be used to study the stability of finite‐n ballooning modes. Using for illustration the s‐α equilibrium, salient features of the n■∞ stability boundary can be deduced from an appropriate choice of test function for these harmonics. The analysis can be extended to include the toroidal coupling of a free‐boundary kink eigenfunction to the finite‐n ideal ballooning mode. A unified stability condition is derived that describes the external kink mode, a finite‐n ballooning mode, and their interaction. The interaction term plays a destabilizing role that lowers the instability threshold of the toroidally couple...

Theory of pressure‐induced islands and self‐healing in three‐dimensional toroidal magnetohydrodynamic equilibria

The role of singular currents in three‐dimensional toroidal equilibria and their resolution by magnetic island formation is discussed from both analytical and computational points of view. Earlier analytical results are extended to include small vacuum islands, which may, in general, have different phases with respect to pressure‐induced islands. In currentless stellarators, the formation of islands is shown to depend on the resistive parameter DR, as well as the integrated effect of global Pfirsch–Schluter currents. It is demonstrated that the pressure‐induced ‘‘self‐healing’’ effect, recently discovered computationally, is also predicted by analytical theory.

Generalized reduced magnetohydrodynamic equations

A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Ad...

Nonlinear tearing mode interactions and mode locking in reversed‐field pinches

The nonlinear interaction of a set of tearing instabilities and plasma flow is studied in a cylindrical plasma. An analytic theory of mode locking is developed, which includes the effects of the localized electromagnetic torques, plasma inertia, and cross‐field viscosity. The calculation is specialized for the case of mode locking on the Madison Symmetric Torus (MST) reversed‐field pinch [R. N. Dexter et al., Fusion Technol. 19, 131 (1991)]. In MST plasmas, a set of m=1 tearing instabilities become phase locked and form a toroidally localized, rotating magnetic disturbance. An evolution equation for the phase velocity of this magnetic disturbance is derived that accounts for two types of electromagnetic torques. The external torques describe the interaction of the tearing modes with static magnetic perturbations located outside the plasma region. The interior torques describe the nonlinear interaction of three tearing modes that satisfy a wave number resonance condition. For conditions typical of MST, the...