Simulation of a sawtooth crash that occurred during DIII-D shot 86144. NIMROD solves the time-dependent resistive MHD equations. Both pressure contours and magnetic field lines are shown. The field lines are color coded with the pressure value. The nonlinear simulation was carried out with a Lundquist number of S = 107. Calculations of this type demonstrate significant progress toward performing numerical simulations with experimentally realistic values of important parameters.

Dalton Schnack and Scott Kruger, Science Applications International Corporation
Carl Sovinec, University of Wisconsin, Madison
Rick Nebel and Tom Gianakon, Los Alamos National Laboratory
Charlson Kim and Scott Parker, University of Colorado, Boulder
Eric Held, Utah State University

Research Objectives
The goal of this research is to develop a code which provides both flexibility in the physics, by using two-fluid or magnetohydrodynamic (MHD) models with analytic or gyrokinetic closures, and flexibility in the geometry, allowing for studies of any axisymmetric fusion concept, no matter how complicated the geometric configuration.

Computational Approach
The NIMROD code uses the extended MHD model to simulate the electromagnetic plasma behavior. The code has a time-split, semi-implicit advance and a combined finite element/Fourier series spatial representation. A major advance in the past year has been the generalization of the NIMROD code to use higher-order Lagrangian elements. This algorithm has been designed to run on massively parallel computers, while being able to handle the extreme stiffness of MHD problems in fusion plasmas. Normal modes of the system propagate across the domain in times that are orders of magnitude smaller than the time scales of the instabilities that we wish to study. Therefore, we have paid particular attention to avoiding numerical dissipation in the part of the algorithm associated with wave propagation. We have also paid considerable attention to ensure that truncation errors do not lead to unphysical coupling of compressional and shear waves.

Accomplishments
Extensive simulations of spheromak formation using NIMROD show that the spheromak does not have large regions of closed flux surfaces, but rather is chaotic over most of its domain. Results of the simulations agree well with many experimental observations. Simulations of the stabilization of tearing modes using NIMROD show that an important aspect of stabilization is the localization of the rf current source. Because realistic current sources cannot be perfectly localized, their effectiveness may not be as great as analytic theory predicts. Simulations with NIMROD show that a new heuristic model for the electron and ion stress tensors gives many of the effects analytic theory predicts, yet avoids many of the numerical problems that more rigorous closures give. The new closure allows for more realistic simulations of neoclassical tearing modes to be performed.

We are in the process of incorporating energetic particle effects into nonlinear MHD simulations. This has involved development of a finite element formulation of particle-in-cell simulation. We now have particles evolving in the time varying NIMROD fields and are calculating the energetic perturbed pressure. As an intermediate step, we are running a linear eigenmode in NIMROD and using the energy exchange between the particles and the MHD eigenmode to determine linear growth or damping. This is done by scaling the MHD field quantities by an appropriate factor, each time step representing the energy exchange between the particles and the MHD fields.

Significance
The NIMROD code is designed to do nonlinear, initial-value simulations of long-wavelength phenomena in fusion-reactor-relevent plasmas. These phenomena severely constrain the operating regime of fusion experiments, and improved understanding should lead to a better approach to providing fusion energy. Our development initiatives represent a consensus based on community feedback, especially feedback from the experimental community.

Publications
C. R. Sovinec, J. M. Finn, and D. del-Castillo-Negrete, "Formation and sustainment of electrostatically driven spheromaks in the resistive magnetohydrodynamic model," Phys. Plasmas 8, 475 (2001).

J. M. Finn, C. R. Sovinec, and D. del-Castillo-Negrete, "Chaotic scattering and self-organization in spheromak sustainment," Phys. Rev. Lett. 85, 4538 (2001).

T. A. Gianakon, "Limitations on the stabilization of resistive tearing modes," Phys. Plasmas 8, 4105 (2001).

http://www.nimrodteam.org