Taylor state

In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity. In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity. Consider a closed, simply-connected, flux-conserving, perfectly conducting surface S {displaystyle S} surrounding a plasma with negligible thermal energy ( β → 0 {displaystyle eta ightarrow 0} ). Since B → ⋅ d s → = 0 {displaystyle {vec {B}}cdot {vec {ds}}=0} on S {displaystyle S} . This implies that A → | | = 0 {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 δ B → ⋅ d s → = 0 {displaystyle delta {vec {B}}cdot {vec {ds}}=0} and δ A → | | = 0 {displaystyle delta {vec {A}}_{||}=0} on S {displaystyle S} . We formulate a variational problem of minimizing the plasma energy W = ∫ d 3 r B 2 / 2 μ ∘ {displaystyle W=int d^{3}rB^{2}/2mu _{circ }} while conserving magnetic helicity K = ∫ d 3 r A → ⋅ B → {displaystyle K=int d^{3}r{vec {A}}cdot {vec {B}}} . The variational problem is δ W − λ δ K = 0 {displaystyle delta W-lambda delta K=0} . After some algebra this leads to the following constraint for the minimum energy state ∇ × B → = λ B → {displaystyle abla imes {vec {B}}=lambda {vec {B}}} .