Taylor state

One of the ways that a taylor state can be applied is in the formation of plasma structures called Spheromaks. They are predicted to form naturally, through a magnetic relaxation process inside a plasma that is conserving magnetic flux. This behavior has been observed in the Compact Toroid Experiment (CTX) at Los Alamos in 1990[3] at the S-1 machine at Princeton[4] and on the HIT-SI machine at the University of Washington. This is the condition that underpins the Dynomak.

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