L. A. Charlton

Stability of ideal and resistive internal kink modes in toroidal geometry

The stability of the ideal and resistive m=1 internal modes is investigated for tokamak equilibria having a variety of different q(r) profiles, including nonmonotonic q(r) with multiple q=1 surfaces. Detailed comparisons between analytic theory and numerical results from a linear toroidal magnetohydrodynamic code are presented. Particular attention is paid to the study of equilibria that are near marginal stability.

Numerical calculations using the full MHD equations in toroidal geometry

A computer code has been constructed that solves the full magnetohydronamic (MHD) equations in toroidal geometry. The code is applicable to toroidal devices, including tokamaks, stellarators, and reversed field pinches. A fully implicit numerical technique is used that allows linear eigenvalues and eigenfunctions to be found in a very few computational steps. Althought the present work describes the solution of the linearized equations, generalization of the numerical method to the solution of the nonlinear problem is straightforward. Use of the code is illustrated by calculating the n = 1 instability for a tokamak configuration. The results show the structural changes in the eigenfunctions as the plasma pressure is increased.

3D nonlinear MHD calculations using implicit and explicit time integration schemes

Abstract A computer code (KITE) that solves a reduced set of magnetohydrodynamic (MHD) equations with diamagnetic and thermal force effects included has been constructed. It can use two different time integration schemes. A mostly-explicit time integration scheme is shown to be efficient for the nonlinear phase of tearing mode turbulence where numerical stability demands a small timestep. However, for linear calculations, or nonlinear ones in which the level of turbulence is low, a mostly-implicit approach is seen to be more efficient. The two numerical schemes yield the same solutions. Nonlinear MHD calculations in which the solutions are represented by a finite Fourier series allow one to study the dependence of the nonlinear solution on its permitted harmonic content. It is seen that for the 3-dimensional (3D), nonlinear, tearing mode disruption problem, the refinement of the solution by the addition of more modes leads to further destabilization. This contrasts to the rippling mode problem, in which the addition of modes is stabilizing. For the latter, a finite number of modes yields a converged solution that is a saturated state. This contrasting behavior reflects the basic physical mechanism of the nonlinear interaction in each case and not the particular numerical scheme used for the calculations.

Compressible linear and nonlinear resistive MHD calculations in toroidal geometry

Abstract A formalism has been developed and incorporated in the computer code FAR to solve the magnetohydrodynamic (MHD) equations compressibly or incompressibly for either ideal or resistive modes. A linear subset or the full nonlinear set of equations can be solved, in toroidal geometry, with no ordering assumptions. Significant features of the formalism include (1) the addition of compressibility by adding two equations to a basic incompressible set, (2) the ability of the code to converge very rapidly for linear calculations, and (3) the use of a diffusive term in the evaluation of the compressible part of the velocity. This term damps the short-wavelength waves and allows a time step size which is comparable to that needed for incompressible simulations.

Second stability in the ATF torsatron—Experiment and theory

Access to the magnetohydrodynamic (MHD) second stability regime has been achieved in the Advanced Toroidal Facility (ATF) torsatron [Fusion Technol. 10, 179 (1986)]. Operation with a field error that reduced the plasma radius and edge rotational transform resulted in peaked pressure profiles and increased Shafranov shift that lowered the theoretical transition to ideal MHD second stability to β0≊1.3%; the experimental β values (β0≤3%) are well above this transition. The measured magnetic fluctuations decrease with increasing β, and the pressure profile broadens, consistent with the theoretical expectations for self‐stabilization of resistive interchange modes. Initial results from experiments with the field error removed show that the pressure profile is now broader. These later discharges are characterized by a transition to improved (×2–3) confinement and a marked change in the edge density fluctuation spectrum, but the causal relationship of these changes is not yet clear.

Tokamak m=1 magnetohydrodynamic calculations in toroidal geometry using a full set of nonlinear resistive magnetohydrodynamic equations

The linear stability and nonlinear evolution of the resistive m=1 mode in tokamaks is studied using a full set of resistive magnetohydrodynamic (MHD) equations in toroidal geometry. The modification of the linear and nonlinear properties of the mode by a combination of strong toroidal effects and low resistivity is the focus of this work. Linearly there is a transition from resistive kink to resistive tearing behavior as the aspect ratio and resistivity are reduced, and there is a corresponding modification of the nonlinear behavior, including a slowing of the island growth and development of a Rutherford regime, as the tearing regime is approached. In order to study the sensitivity of the stability and evolution to assumptions concerning the equation of state, two sets of full nonlinear resistive MHD equations (a pressure convection set and an incompressible set) are used. Both sets give more stable nonlinear behavior as the aspect ratio is reduced. The pressure convection set shows a transition from a K...

Bifurcations and modulational interaction in negative compressibility turbulence

The results of detailed numerical studies of phenomena in negative compressibility turbulence with sheared perpendicular (i.e., poloidal) flow are presented. The turbulence model is based on the parallel ion flow gradient instability, a representative paradigm for ion drift waves. Studies of coupled turbulence and mean flow evolution indicate the existence of two distinct nonlinear states. In the first state, saturation occurs via nonlinear transfer to damped high‐k modes and sheared flow is heavily damped. In the second state, the turbulence level is controlled by the self‐consistently generated sheared flow. Transition between these states is determined by the competition between instability growth and damping of rotation. The dynamics of the observed transition is well described and consistent with a simple set of coupled envelope equations. Modulational interaction between small scale turbulence and large scale m≠0 shear flows is observed, as well.

Shear flow effects on the nonlinear evolution of thermal instabilities

In the weak radiation drive regime, the coupling between the thermal instability driven by impurity radiation and the self‐consistent flow profile modification leads to a simple dynamical system that can be approximated by the Volterra–Lotka equations. In this system the shear flow acts as a predator and the temperature fluctuations act as prey. The solutions are oscillatory, and their behavior resembles that of edge‐localized modes (ELM’s). The solutions of the simplified model are compared with the three‐dimensional and two‐dimensional nonlinear numerical results for this instability.

Resistive ‘‘infernal’’ modes

Resistive ‘‘infernal’’ modes are considered that are just below the ideal internal mode threshold. Nearest the ideal threshold, a resistive continuation of the ideal infernal mode, whose growth rate scales as η3/13 (η being the resistivity), is seen. Farther away is the conventional tearing mode with a η3/5 scaling. These scalings are found both numerically and analytically.

Magnetohydrodynamic stability and nonlinear evolution of the m = 1 mode in toroidal geometry for safety factor profiles with an inflection point

Magnetohydrodynamic (MHD) stability and nonlinear evolution of the m=1 mode are studied in toroidal geometry for safety factor profiles having an inflection point at the q=1 surface. For ideally stable cases, linear growth rates of the resistive m=1 mode are decreased by any combination of the following variations: decreasing the aspect ratio, decreasing shear at the q=1 surface, increasing S, decreasing q0, and/or decreasing beta. Transitions from resistive kink to tearing mode to complete stability are accompanied by increasing localization as the mode is stabilized. The ideal stability of the m=1 mode for low‐shear inflection‐point profiles is broken at very low, but finite, beta by the appearance of a radially localized ideal interchange mode. As beta is increased, the m=1 mode becomes a global ideal internal kink. These ideal modes are stabilized by decreasing the aspect ratio and, to a lesser extent, by increasing shear at the q=1 surface. Nonlinear evolution of the m=1 mode is found to follow the K...