Dpp siller 1

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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