Three-dimensional Resistive MHD Investigation of Spheromak Sustainment


Carl R. Sovinec, John M. Finn,

and Diego del Castillo-Negrete,

Los Alamos National Laboratory

presented at the

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











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.


Simulation parameters:

  1. Cylinder radius is 1 m.
  2. Computations have been performed with cylinder height=0.75-1.5 m.
  3. Most cases have S=2000 on-axis; some have S=10,000.
  4. Most cases have =1 m/s2.
  5. Computations use either a "stabilized" pinch equilibrium or a simulated applied axial electric field.












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.




Auxiliary slide




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.




Auxiliary slide



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.





The robust generation of poloidal flux is usually accompanied by chaotic scattering of the magnetic field.







Auxiliary slide

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.




Auxiliary slide


Contours of l at axial positions spaced from the bottom to the top of the can show the helically deformed pinch current.




Auxiliary slide



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.

Note that the independent variable is <r2>1/2, mean square distance from the magnetic axis.




In steady state, flux surface averages of the electrostatic field vanish. This is satisfied in the computations.




Due to the low current, we would not expect much Ohmic heating within these stellarator-like flux surfaces.




Auxiliary slide


We are attempting to optimize the volume of flux surfaces and reduce q. One idea is to use a nonsymmetric perturbation of the magnetic field.


without nonsymmetric perturbation



Note that the independent variable is <r2>1/2, mean square distance from the magnetic axis, not <r2>.




Auxiliary slide


with 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.


Note that the independent variable is <r2>1/2, mean square distance from the magnetic axis, not <r2>.




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.






Auxiliary slide



This plot shows the safety factor of a structure resulting from decay.


Note that the independent variable is <r2>1/2, mean square distance from the magnetic axis.






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.










  1. More analysis of:
    1. flux conversion process
    2. limit cycle behavior
    3. S dependence
  2. Optimize the stellarator transform effect (nonsymmetric perturbations).
  3. Investigate gun-driven configurations.
  4. Investigate realistic geometries (SAIC-LLNL).