.
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 {\displaystyle S} surrounding a plasma with negligible thermal energy ( \( {\displaystyle \beta \rightarrow 0} \) ).
Since \( {\displaystyle {\vec {B}}\cdot {\vec {ds}}=0} \) on \( {\displaystyle S} \) . This implies that \({\displaystyle {\vec {A}}_{||}=0} \) .
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 \( {\displaystyle \delta {\vec {B}}\cdot {\vec {ds}}=0} \) and \( {\displaystyle \delta {\vec {A}}_{||}=0} \) on \( {\displaystyle S} \) .
We formulate a variational problem of minimizing the plasma energy \( {\displaystyle W=\int d^{3}rB^{2}/2\mu _{\circ }} \) while conserving magnetic helicity \( {\displaystyle K=\int d^{3}r{\vec {A}}\cdot {\vec {B}}} \).
The variational problem is \({\displaystyle \delta W-\lambda \delta K=0} \).
After some algebra this leads to the following constraint for the minimum energy state \({\displaystyle \nabla \times {\vec {B}}=\lambda {\vec {B}}} \) .
See also
John Bryan Taylor
References
Paul M. Bellan (2000). Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization. pp. 71–79. ISBN 1-86094-141-9.
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
