Taylor state

A Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity. [cite book
author = Paul M. Bellan
year = 2000
title = Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization
pages = 71-79
id = ISBN 1-86094-141-9]

Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface $S$ surrounding a plasma with negligible thermal energy ().

Since $vec\{B\}.vec\{ds\}=0$ on $S$. This implies that $vec\{A\}\_=0$.

As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies $delta\; vec\{B\}.vec\{ds\}=0$ and $deltavec\{A\}\_=0$ on $S$.

We formulate a variational problem of minimizing the plasma energy $W=int\; d^3rB^2/2mu\_circ$ while conserving magnetic helicity $K=int\; d^3rvec\{A\}.vec\{B\}$.

The variational problem is$delta\; W\; -lambda\; delta\; K\; =\; 0$.

After some algebra this leads to the following constraint for the minimum energy state $abla\; imes\; vec\{B\}\; =\; lambda\; vec\{B\}$.

References