Taylor state

Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface S surrounding a plasma with negligible thermal energy ( β → 0 ).

Since B → ⋅ d s → = 0 on S . This implies that 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 δ B → ⋅ d s → = 0 and δ A → | | = 0 on S .

We formulate a variational problem of minimizing the plasma energy W = ∫ d 3 r B 2 / 2 μ ∘ while conserving magnetic helicity K = ∫ d 3 r A → ⋅ B → .

The variational problem is δ W − λ δ K = 0 .

After some algebra this leads to the following constraint for the minimum energy state ∇ × B → = λ B → .