Dalton Schnack, Ming Chu, Scott Parker, Jean-Noel Lebouf, Eric Held, John Wright, Jim Callen, Rick Nebel, Charlson Kim, Tom Gianakon, Carl Sovinec, Alfonso Tarditi

INTRODUCTION

DOE is aware of the mighty challenges and will do its best to alleviate problems that the team may be facing in the near future.

CODE DEVELOPMENT

Allowing for solutions that contain coupling along the toroidal direction has become a necessity. Examples are anisotropic thermal conduction problems or a 3D continuity equation, where the density r(r,z,j), for each toroidal mode, has still the full 3D dependence.

For this purpose a 3D CG solver has been added to NIMROD. 3D domain decomposition in layers is still allowed. Direct solve for quadrature points internal to the cell is not possible anymore, but impact on CPU time performance seems not significant (at least based on preliminary tests).

Other significant consequences are that now maintaining the efficiency of the FEM computations has become more important..

Data storage at quadrature points has changed and likely will need to be considered while making changes to the physics kernel of the code.

Tolerance input parameter needs to be reduced for problems with high thermal anisotropy.

A demonstration run for anisotropic thermal conduction for a D3D problem was presented.

Original "step function" choice for the shape variable caused problems: new formulation uses a smooth initial condition for the shape variable that dramatically reduces the high-k noise. The new coding is being ported to the latest release of NIMROD.

A simple test with "tracing" particles trapped in banana orbits (on a r-z grid) shows that the energy is fairly well conserved. A version that tracks the particles (w/o self consistent effects) could be made deliverable to the team quickly.

Experience was developed with Open-DX at U. Colorado. Help for setting up Open DX based visualization will be provided to team members by U. Colorado upon request.

Theoretical work on the kinetic closure or parallel Ohm's law was reported. This development interests electron transit time scale (for passing particles, the time for an electron to go one poloidal loop around). Electron inertia can be included. A slab tearing mode test case was suggested.

In Chapman-Enskog like approach the electron distribution is written as sum of a Maxwellian and a distortion term. This term is then inserted and solved for in a drift kinetic equation. This model actually solves the gyro averaged drift kinetic equation for electron parallel dynamics.

In the calculation of parallel stress tensor and heat flow moments time dependent closures retain free streaming and collisional effects.

APPLICATIONS

Neoclassical effects through viscous stress tensor: a successful test case was completed.

Resistive MHD shows fast growing ballooning instabilities: neo-classical enhancements of polarization current stabilize medium-n ballooning modes. With extended MHD neo-classical tearing modes (TM) are generated. A TM stabilization case was demonstrated with a customized version of the code: slab geometry is considered and a source term for an auxiliary current drive is added in the Ohm's law. A TM is loaded as initial condition, the evolution of the island size is then observed for different values of the source current width. It was shown that there is lower limit that island size can reach.

The DIII-D shot #87009 scenario is considered. It was shown experimentally and theoretically that increasing beta through the marginal stability threshold produces a mode growing faster than exponentially (growth proportional to t^3/2).

For the NIMROD runs a normalized beta (b_N ) higher than in the experiment needs to be considered because the lack of vacuum region. A run with high order finite elements (3^rd order) was proven to be numerically stable with 65 poloidal mesh points (previous efforts with linear elements required much higher poloidal resolution). Linear growth of global modes for b_N 5 and 6 was observed with fair agreement with GATO linear results, as found with earlier simulations with linear elements. For b_N=4, below the stability threshold, a resistive interchange mode was also detected. Results from nonlinear test runs were also shown.

Several applications of the NIMROD code are being pursued at GA. Simulation of FRC formation through merging Spheromaks will be attempted to investigate the a suggested tilt stabilization effect due to the presence of a residual toroidal field.

The problem of heat conduction in tokamaks will be investigated in connection with experimental results in ASDEX suggesting that neoclassical tearing modes appear to be stabilized by externally applied helical field in presence of magnetic island and other non-resonant harmonic components.

Another problem of interest is the Electron Cyclotron Current Drive (ECCD) suppression of neoclassical tearing modes (in connection with experimental result in DIII-D).

Simulation of R/a=1.75 RFP for single helicity modes were reported. Poincare plots of runs in periodic linear geometry shows single helicity state while runs in toroidal geometry does not. An example of a new diagnostic able to resolve modes in the poloidal direction was shown.

A model of an accretion disk considers differential rotation that in turn causes radial electric field. A NIMROD test case was set where a dipole flux distribution expands as the differential rotation is applied at the disk.

PSACI PROPOSAL EFFORT

NIMROD benchmark with M3D will include Wonchull Park proposed ideal MHD case. Due to lacking efforts of the M3D team in this area, no M3D run with same parameters as any NIMROD run was made available yet.

A test case for a M3D vs. analytic theory comparison has been made available. The test refers to a tokamak in a periodic cylinder, with m=1 ideal mode.

NIMROD runs two-fluid runs (with Hall and diamagnetic terms only) were performed and linear eigenfunction data were collected for the purpose of comparing with M3D recent results by Linda Sujiama. The NIMROD two-fluid semi-implicit operator seems to work satisfactory. Gyro-viscous terms and polarization drift need to be included.

PLANNING

A new version of the code with 3D linear system capabilities will be made available to the team members before Carl Sovinec's transfer to U. of Wisconsin - Madison. The new version of the code is expected to include also the vacuum region formulation and the full 3D continuity equation.

- NIMROD workshop (Eric Held and Carl Sovinec)

An informal NIMROD "code developer" workshop is tentatively planned for the late summer with the purpose of transferring some of Carl Sovinec's code development and maintenance expertise to other team members.