Three-dimensional Resistive MHD Investigation of Spheromak Sustainment
and Diego del Castillo-Negrete,
Los Alamos National Laboratory
American Physical Society Centennial Meeting/
1999 International Sherwood Fusion Theory Conference
March 23, 1999
NIMROD code development team and advisors:
Ahmet Aydemir IFS
Curtis Bolton OFES
James Callen U-WI
Ming Chu GA
John Finn LANL
Tom Gianakon LANL
Alan Glasser LANL
Scott Kruger SAIC
Jean-Noel Leboeuf UCLA
Richard Nebel LANL
Steve Plimpton SNL
Nina Popova MSU
Dalton Schnack SAIC
Carl Sovinec LANL
Alfonso Tarditi SAIC
Computations have been performed at
National Energy Research Scientific Computing Center, LBNL
Advanced Computing Laboratory, LANL
GEOMETRY AND EQUATIONS
The geometry for simulating a "flux core" or "electrode" spheromak is a simple can with magnetic flux frozen into the top and bottom ends, which represent electrodes.
The physical behavior of the system is modeled with the resistive MHD equations, where uniform density and vanishing pressure are assumed:
The equations are solved with the NIMROD simulation code.
MHD INSTABILITY AND NONLINEAR SATURATION
Auxiliary slide: plots of the n=1 linear eigenfunction.
Saturation of the instability results from feedback to the n=0 field and from coupling to larger n.
The following shows the nonlinear result of flux amplification vs. the on-axis parallel current (l) of the equilibrium pinch for S=2000, h=1 m cases.
The final saturated state is NOT a Taylor state.
For the f=0 plot, the minimum value (green) is -8.8 and the maximum (red) is 13.7. For the f=p plot, the minimum value is -12.4 and the maximum is 16.5.
The maximum current density in the plot of the pinch is 31% larger than the largest vector in the f=0 plot, but the nonlinear state sustains 23% more current.
CHOATIC SCATTERING AND FLUX SURFACES
The robust generation of poloidal flux is usually accompanied by chaotic scattering of the magnetic field.
Here we take the steady-state results from a case with lpinch(0)=10, multiply the nonsymmetric part of the magnetic field by factors < 1, and redraw the surfaces of section.
In weakly-driven cases, just above marginal stability, flux surfaces form and are sustained in steady state.
A field line trace over a flux surface shows that most of the transform occurs on the inner part of the surface.
Color indicates axial position, z, on the surface.
A helical prominence exists on the inner part of the flux surface. It coincides with helical pinch current at smaller r.
Color indicates toroidal position, f, on the surface.
Contours of l at axial positions spaced from the bottom to the top of the can show the helically deformed pinch current.
The safety factor of the flux surfaces in a similar case is ~10 near the magnetic axis of the structure. q computed with just the symmetric (n=0) part of the field is in error by more than a factor of 2.
Due to the low current, we would not expect much Ohmic heating within these stellarator-like flux surfaces.
without nonsymmetric perturbation
So far, we have been able to reduce q with the perturbation, but we have not yet increased the volume of flux surfaces.
Flux surfaces have also been observed during decay.
When the applied potential is decreased over 0.1 tr in a simulation with h=1.5 m and uniform flux through the electrodes, flux surfaces form as the poloidal field decays.
There is net toroidal electric field during the decay, which produces relatively uniform magnetic transform over the flux surfaces. It would also lead to Ohmic heating.
The transform on the flux surfaces formed by decay is much more uniform than in the weakly driven cases, and q~1.
This plot shows the safety factor of a structure resulting from decay.
LIMIT CYCLE BEHAVIOR
In cases with larger S (10,000) or large current, a limit cycle is observed in the temporal evolution as the configuration is sustained.
The evolution of the magnetic energies of the different Fourier components in an S=10,000 demonstrates this behavior.
Small islands form at the low perturbed-B point of the cycle.