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, 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.