MHD Simulations of Spheromaks and Low Aspect Ratio Tokamaks with Electrostatic Current Drive


Carl R. Sovinec, John M. Finn, and Diego del Castillo Negrete,

Los Alamos National Laboratory


A. Tarditi, D. D. Schnack

Science Applications International Corporation-San Diego


presented at the


41st Annual Meeting of the American Physical Society, Division of Plasma Physics


November 15-19, 1999





NIMROD code development team and advisors:

Ahmet Aydemir IFS

James Callen U-WI

Ming Chu GA

John Finn LANL

Tom Gianakon LANL

Charlson Kim CU-Boulder

Scott Kruger SAIC

Jean-Noel Leboeuf UCLA

Richard Nebel LANL

Scott Parker CU-Boulder

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






  2. Describe the numerical solutions of fluid models for spheromak configurations, showing the nonlinear evolution from an unstable pinch to a sustained spheromak.

  4. Examine the influence of including Hall and electron inertia terms in sustained configurations.

  6. Determine what conditions lead to closed flux surfaces.

  8. Examine the influence of geometry and plasma parameters on the flux conversion process.

  10. Investigate the transition from spheromak to electrostatically driven, low aspect ratio tokamak through the addition of a central post and external current.




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. Simulations with a central post are also based on this configuration.



Spheromaks driven with a plasma gun have concentric electrodes in the gun region and an open confinement region downstream.






In most of the simulations presented here, the physical behavior of the system is modeled with the resistive MHD equations, where uniform density and vanishing pressure are assumed:



We have begun to investigate the importance of electron fluid effects with the Hall and electron inertia terms. Here the generalized Ohm's law is:



Gyroviscous terms have not been included in these simulations.


The equations are solved with the NIMROD simulation code.




[See the discussion of poloidally symmetric reversal in Sect. II. A., "Single and multiple helicity Ohmic states in reversed-field pinches," Finn, Nebel, and Bathke, Phys. Fluids B 4, 1262 (1992), and interpret the poloidal RFP direction as the toroidal spheromak direction, and vice versa.]




When the compressible 2D MHD equations are solved for conditions that lead to a spheromak, the result is a "stabilized" or paramagnetic pinch. The net electric field is poloidal, and force balance leads to concentrated poloidal flux.


Contour plots of poloidal flux illustrate this concentration in 2D simulations.




All of these configurations are unstable in resistive 3D MHD.


The growth rate and eigenfunction of the linear (n=1) mode of the pinch are strongly influenced by the field line tying conditions at the electrodes.






The linearly unstable eigenfunction of the "concentrated configuration."


Saturation of the instability results from feedback to the average field and from coupling to larger n.


Evolution of fluctuation energy in the "concentrated configuration". Labels indicate n-number, and time is in diffusion times. The n=1 fluctuation dominates the saturated spectrum.


Saturation generates the spheromak configuration itself! MHD fluctuations convert toroidal flux to poloidal flux.

The "distributed" and "gun" configurations show qualitatively similar behavior when nonlinear 3D simulations with random perturbations are evolved from the 2D pinch states.








Saturation relaxes the parallel current profile from the initial pinch configuration, but the relaxation is incomplete, and O(1) variations persist in this driven/damped system.

The "parallel current," , is initially peaked at 12.5 in the pinch ("concentrated configuration"). In the final state, the toroidal average l profile has a plateau with .

Examining contour plots of local l shows large variations within the plateau region.


In MHD, fluctuations transfer power from and to the mean field through the correlation of perturbed velocity and magnetic field.

The quantity, , represents energy density transferred between the average current density and Fourier component n. [Ho and Craddock, Phys. Fluids B 3, 721 (1991).]

The n=1 contribution from the end of the "concentrated configuration" shows power absorbed from average current for r<0.22 and power delivered to average current for r>0.22.



The amount of generated poloidal flux increases with pinch l, or equivalently applied potential, for a large range of l in the electrode configuration.


The following plot shows computed flux amplification from a series of simulations with varied l(r=0) of the equilibrium pinch.



Flux amplification falls to zero at the marginal stability point of the pinch.




The "distributed" configuration has been simulated with Hall and electron inertia terms in Ohm's law.

and 2) the fluctuations rotate in the toroidal direction.

Flux amplification is still within 0.5% of the MHD result.




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



In weakly-driven cases that are just above marginal stability, flux surfaces form and are sustained in steady state.


Here the pinch column remains intact and is only helically distorted by the nonlinear saturation.

The helical flux surfaces that form around the distorted current column are analogous to flux surfaces in stellarators. The distorted plasma current column threading the surfaces plays the role of helical external coils in a stellarator.



Two views of the same surface are plotted with color indicating radial position to emphasize the helical distortion of the inboard side.



Plots of parallel current at different axial positions show the helical nature of the central current column.



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.



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.


Flux surfaces have also been observed during decay from strongly-driven configurations.



When the applied potential is decreased over 0.1 tr in a simulation with h=1.5 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 following shows the safety factor of a structure resulting from decay.



The independent variable in the safety factor plot is <r2>1/2, not <r2>.




For a fixed applied electric field, changing the height of the can in the electrode configurations has a large impact on flux amplification.



The curve labeled "0" represents a series of distributed flux simulations, and the curve labeled "1" has concentrated flux. All have an initial pinch l(r=0) of 15.




The gun configuration shows at least qualitatively similar behavior to the electrode configurations.


Of the distributed and gun simulations complete to date, input parameters for the two simulations in the following table are most similar.





applied poloidal flux







resulting flux amplification



S based on final peak |B| in spheromak



peak db/B at







As S is varied over values of O(103) to O(104), flux amplification increases.

The two curves show the concentrated configuration (label "0") at initial pinch l=10 and the gun configuration (label "1") with a value of 42 for the applied potential. In both simulation series, S is varied by changing mass density.

Fluctuation scaling is a critical issue, but the S values used so far are too low to make any predictions about high-S scaling.



As S is increased in the gun configuration, more average poloidal current is entrained with the spheromak.






In cases with S greater than ~3000, a limit cycle is observed in the temporal evolution as the spheromak is being sustained. This occurs in the electrode and gun configurations.


The evolution of the magnetic energies of the different Fourier components in the S=8500 electrode simulation demonstrates this behavior.



Small islands form at the low perturbed-b point of the cycle.




When the domain is modified to a toroidal configuration, vacuum toroidal field may be added. A study of the transition from spheromak-like configurations to low aspect ratio tokamaks is now underway.












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