SIMULATING EXTREME ANISOTROPY
WITHOUT MESH ALIGNMENT
Univ. of Wisconsin-Madison
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.
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, Pm=0.1 saturation of Test Problem 1b (DIII-D-like equilibrium) that had kiso=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) "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) "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.