Taylor state

In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.^[1]

Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface $S$ surrounding a plasma with negligible thermal energy ($\beta \rightarrow 0$).

Since ${\vec {B}}\cdot {\vec {ds}}=0$ 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 $\delta {\vec {B}}\cdot {\vec {ds}}=0$ and $\delta {\vec {A}}_{{||}}=0$ on $S$.

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

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

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

See also

• John Bryan Taylor

References

1. Paul M. Bellan (2000). Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization. pp. 71–79. ISBN 1-86094-141-9.