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 $\nabla \times {\vec {B}}=\lambda {\vec {B}}$ .

References

↑ Paul M. Bellan (2000). Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization. pp. 71–79. ISBN 978-1-86094-141-2.

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[1] Paul M. Bellan (2000). Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization. pp. 71–79. ISBN 978-1-86094-141-2.

Taylor state

Derivation

See also

References