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