Rotation of Spherical Plasma: Ferraro’s Theorem and the MRI
Robert Siller, C. B. Forest, V. Mirnov
Egedal-Forest Group, University of Wisconsin - Madison
Current Driven
Conclusion
The applicability of Ferraro’s Isorotation Theorem is seen in the
limit of strong magnetic fields/low velocity. At weak magnetic
fields/high velocity isorotation is complicated by the develop-
ment of the MRI.
Future work is to look at gaining better agreement with experi-
mental observations by understanding the boundary conditions,
and explore profiles that are never unstable to the MRI.
Comparison at
experimental
parameters of
simulation
output and
measured veloc-
ity, with neutral
drag added to
the system of
equations.
Velocity Driven
The boundary condition for these runs create faster rotation at the
poles, and no rotation at the equation.
Theory
Ferraro’s Isorotation Theorem is understood in the context of a cy-
lindrical, ideal, purely rotating plasma and states that angular fre-
quency is constant along magnetic field lines.
Method/Setup
Numerically solve the model equations with the following as-
sumptions: axisymmetric, time independent, no slip velocity, no
radial current, and no perpendicular magnetic field at the surface.
The solver for the listed conditions is a relaxation method with
specified magnetic (current), and velocity boundary conditions.
Two different types of boundary conditions are used: specified ve-
locity without magnetic field, and specified radial current (toroidal
magnetic field) without velocity.
The specified radial current method is used to more accurately
simulate the experimental setup in use, using a radial current be-
tween the equation and the poles in a global magnetic field to stir
the plasma. An example is shown below.
The Magnetorotational Instability is an important instability for ro-
tating plasma when the angular frequency decreases outwards.
For certain velocity profiles there exists flows inbetween where
Isorotation is almost true, and hydrodynamically dominated flows.
Locally the condition for the MRI is
Motivation
Understand the transition from hydrodynamic flow profiles to
magnetically dominated flow in a sphere.
Investigate the impact of spherical geometry, and the resultant
poloidal flow on Ferraro’s Isorotation Theorem and the develop-
ment of the Magnetorotational Instability.
Model Equations
Experiment Overview
• Multicusp confinement
using SmCo permanent
magnets.
• Large (1.5 m radius)
• Hot (T
e
~10 eV), isothermal
• Weakly magnetized bulk (~2 G)
• Fast flowing (~1 km/s)
Apply J x B torque at the edge of
plasma to spin in the toroidal direc
-
tion (for dynamo).
• Momentum viscously couples into
umagnetized region.
• Possible to search a wide range of
plasma parameters.
Apply J x B torque in the bulk of
plasma to spin in the toroidal direc
tion (for MRI).
S
S
S
N
N
N
–
+
V
B
J