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**SIMULATING
EXTREME ANISOTROPY**

**WITHOUT MESH
ALIGNMENT**

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**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

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

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**Anisotropic
Diffusion Demonstration**

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·
Start
from *S*=10^{4}, *P _{m}*=0.1 saturation of Test
Problem 1b (DIII-D-like equilibrium) that had

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

·
*k** _{┴}*=0.423,

·
Island
appears in pressure contours immediately.

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

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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.