Taylor state

In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.^[1] This was first proposed by John Bryan Taylor in 1974 and he backed up this claim using data from the ZETA machine^[2]. But since then relaxation of a plasma into a spheromak and Taylor State has been observed in the Compact Toroid Experiment (CTX) at Los Alamos in 1990^[3] and the S-1 machine at Princeton.

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface ${\displaystyle S}$ 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 ${\displaystyle \nabla \times {\vec {B}}=\lambda {\vec {B}}}$.

• John Bryan Taylor

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