SIMULATING EXTREME ANISOTROPY

WITHOUT MESH ALIGNMENT

Carl Sovinec

Univ. of Wisconsin-Madison

Tom Gianakon

Los Alamos National Laboratory

MHD Working Group, Oct. 28 2001

APS 2001, Long Beach CA

· Nonlinear simulations of high-temperature plasmas must be able to resolve thermal transport anisotropy associated with the magnetic field direction.

· The saturation of pressure-driven modes is sensitive to changes in magnetic topology due to parallel thermal conduction.

· The ratio of parallel to perpendicular thermal conductivities leads to one threshold mechanism for neoclassical tearing modes.

· When the magnetic field is aligned with the grid or when the angle between and the grid is uniform, the anisotropy is accurately represented by standard techniques.

· A nonlinearly evolving magnetic topology requires more sophisticated approaches:

· There is curvature in the magnetic field.

· The topology of islands and stochastic regions is three-dimensional.

· Even 3D automated mesh refinement schemes would be severely challenged by these conditions; 3D refinement is possible but 3D alignment near a separatrix is not.

· The NIMROD code has addressed this challenge by using high-order finite-element basis functions, which represent curvature with or without alignment.

· The increase in spatial convergence rates with basis function order has been verified.

· Spatial convergence rates are retained with nonuniform meshing.
A quantitative measure of the numerical error can be determined by a simple test problem.

· The simplest thermal conduction problem is a 2D box with Dirichlet boundary conditions and a source. If the source drives the lowest eigenfunction only,

in the domain

,

it produces the temperature distribution

.

· A numerical test with B everywhere tangent to this temperature distribution and a uniform rectilinear grid has severely misaligned with the grid. With and , the inverse of the resulting T(0,0) is a good measure of the effective of the numerical algorithm.

· The magnetic field is created by inducing a perpendicular current density distribution that is proportional to the heat source.

Anisotropic Diffusion Demonstration

· Start from S=10⁴, P_m=0.1 saturation of Test Problem 1b (DIII-D-like equilibrium) that had k_iso=0.423 m²/s=0.1n.

· Freeze magnetic evolution and just run anisotropic thermal diffusion over the perpendicular time-scale (100 x resistive time-scale).

· k_┴=0.423, k_||=4.23x10⁸, Dt=1x10^-4 s.

· Island appears in pressure contours immediately.

Pressure contours (color) after 1.5 ms of anisotropic diffusion overlaid with Poincare surface of section.

a)

b)

a) "Probe" located at (R=1.312, Z=-0.248) and b) internal energy vs. time. Time-scale in a) is ms, and in b) it's 10 ms.

· Steady-state transport is regained after a perpendicular diffusion time.

a)

b)

a) "Probe" and b) internal energy. Time-scale is seconds.

· The n=0 pressure profile inside the island drops over the long time-scale, reflecting the loss of insulation over the magnetic island.