MHD Simulations of Spheromaks and Low Aspect Ratio Tokamaks with Electrostatic Current Drive
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
ACKNOWLEDGEMENTS
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
OBJECTIVES
GEOMETRY
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.
FLUID MODELS
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.
SPHEROMAK BASICS
[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.]
RELATING PINCHES AND SPHEROMAKS
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.
The "distributed" and "gun" configurations show qualitatively similar behavior when nonlinear 3D simulations with random perturbations are evolved from the 2D pinch states.
DISTRIBUTED CONFIGURATION
GUN CONFIGURATION
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.
HALL AND ELECTRON INERTIA
The "distributed" configuration has been simulated with Hall and electron inertia terms in Ohm's law.
CHOATIC SCATTERING AND FLUX SURFACES
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 <r^{2}>^{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 t_{r} 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 <r^{2}>^{1/2}, not <r^{2}>.
VARYING CAN HEIGHT
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.
GUN CONFIGURATION
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.
configuration |
distributed |
gun |
applied poloidal flux |
0.089 |
0.10 |
appliedpotential |
30 |
42 |
resulting flux amplification |
229% |
113% |
S based on final peak |B| in spheromak |
1100 |
900 |
peak db/B at r=0 |
36% |
90% |
LUNDQUIST NUMBER
As S is varied over values of O(10^{3}) to O(10^{4}), 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.
As S is increased in the gun configuration, more average poloidal current is entrained with the spheromak.
LIMIT CYCLE BEHAVIOR
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.
TRANSITION TO HELICITY INJECTED TOKAMAKS
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.
DISCUSSION AND CONCLUSIONS
This poster will be available on our web site, http://nimrodteam.org or http://nimrod.saic.com .