J. A. Holmes

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.

Nonlinear dynamics of tearing modes in the reversed field pinch

The results of investigations of nonlinear tearing‐mode dynamics in reversed field pinch plasmas are described. The linear instabilities have poloidal mode number m=1 and toroidal mode numbers 10≲n≲20, and the resonant surfaces are therefore in the plasma core. The nonlinear dynamics result in dual cascade processes. The first process is a rapid m=1 spectral broadening toward high n, with a simultaneous spreading of magnetic turbulence radially outward toward the field‐reversal surface. Global m=0 perturbations, which are driven to large amplitudes by the m=1 instabilities, in turn trigger the m=1 spectral broadening by back coupling to the higher n. The second process is a cascade toward large m and is mediated by m=2 modes. The m=2 perturbations have the structure of localized, driven current sheets and nonlinearly stabilize the m=1 modes by transferring m=1 energy to small‐scale dissipation. The calculated spectrum has many of the qualitative features observed in experiments.

3D nonlinear calculations of resistive tearing modes

Abstract Recent numerical calculations of the evolution of resistive tearing modes have been central to the understanding of MHD activity and disruptions in tokamaks. The nonlinear three-dimensional initial value computer code RSF has provided many of these results. RSF assumes cylindrical geometry with a Fourier series representation in the two periodic coordinates and a finite difference representation in the radial direction. This choice makes RSF considerably more accurate and efficient than previous codes.

Equilibrium and stability properties of high-beta torsatrons

Equilibrium and stability properties of high‐beta torsatrons are investigated using numerical and semianalytical techniques based on the method of toroidal averaging. The averaged equilibria are compared with those obtained using full three‐dimensional codes. Good agreement is obtained, thus validating the averaged method approach. The stability of plasmas for configurations with different aspect ratios and numbers of field periods is studied. The role of the vertical field is also studied in detail. The main conclusion is that for moderate‐aspect‐ratio torsatrons (Ap≲8), the self‐stabilizing effect of the magnetic axis shift is large enough to open a direct path to the second stability region.

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.

Resistive MHD studies of high β tokamak plasmas

Abstract Numerical calculations have been performed to study the MHD activity in high-β tokamaks such as ISX-B. These initial value calculations build on earlier low β techniques, but the β effects create several new numerical issues. These issues are discussed and resolved. In addition to time-stepping modules, our system of computer codes includes equilibrium solvers (used to provide an initial condition) and output modules, such as a magnetic field line follower and an X-ray diagnostic code. The transition from current driven modes at low β to predominantly pressure driven modes at high β is described. The nonlinear studies yield X-ray emissivity plots which are compared with experiment.

Zero-current, high beta stellarator equilibria with rotational transform profile control

High beta, zero-current equilibria for a stellarator device are calculated using the averaging method. It is found that, by shaping the vertical field, the rotational transform can be controlled in an approximate way as beta is increased. At the same time, the Pfirsch-Schlueter currents are reduced - with no modification of the magnetic well. This permits access to the high beta regime with more favorable rotational transform profiles. Results are presented for the Advanced Toroidal Facility (ATF) device.

Nonlinear interaction of tearing modes: A comparison between the tokamak and the reversed field pinch configurations

The multiple helicity nonlinear interaction of resistive tearing modes is compared for the tokamak and reversed field pinch configurations using the magnetohydrodynamic equations. Unlike the case of the tokamak disruption, for which this interaction is destabilizing when islands overlap, the nonlinear coupling of the dominant helicities is shown to be a stabilizing influence in the reversed field pinch. The behavior of the coupled instabilities in the two configurations can be understood as a consequence of the stability properties of the nonlinearly driven modes. In the case of the tokamak disruption, quasilinear effects linearly destabilize the dominant driven mode, which then feeds energy to the driving mode. For the reversed field pinch the driven modes remain stable, acting as a brake on the growth of the dominant instabilities. Furthermore, for the reversed field pinch configuration, numerical results indicate that nonlinear coupling of different helicities results in noticeably more rapid saturation of the dominant instabilities than was observed in single helicity studies.

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...