Regular conditional probability: Difference between revisions

Latest revision as of 20:00, 21 January 2014

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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 $\nabla \times {\vec {B}}=\lambda {\vec {B}}$ .

References

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+In [[plasma physics]], a '''Taylor state''' is the [[minimum energy state]] of a [[plasma (physics)|plasma]] satisfying the constraint of conserving [[magnetic helicity]].<ref>{{cite book
+| author = Paul M. Bellan
+| year = 2000
+| title = Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization
+| pages = 71–79
+| id = ISBN 1-86094-141-9
+}}</ref>
+== Derivation ==
+Consider a closed, simply-connected, flux-conserving, perfectly conducting surface <math>S</math> surrounding a plasma with negligible thermal energy (<math>\beta \rightarrow 0</math>).
+Since <math>\vec{B}\cdot\vec{ds}=0</math> on <math>S</math>. This implies that <math>\vec{A}_{||}=0</math>.
+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 <math>\delta \vec{B}\cdot\vec{ds}=0</math> and <math>\delta\vec{A}_{||}=0</math> on <math>S</math>.
+We formulate a variational problem of minimizing the plasma energy <math>W=\int d^3rB^2/2\mu_\circ</math> while conserving magnetic helicity <math>K=\int d^3r\vec{A}\cdot\vec{B}</math>.
+The variational problem is
+<math> \delta W -\lambda \delta K = 0</math>.
+After some algebra this leads to the following constraint for the minimum energy state
+<!-- Need to expand on the derivation, may be explain the terms - vector potential etc. --~~~~ -->
+<math> \nabla \times \vec{B} = \lambda \vec{B}</math>.
+==See also==
+*[[John Bryan Taylor]]
+== References ==
+{{Reflist}}
+[[Category:Plasma physics]]
+{{physics-stub}}

Regular conditional probability: Difference between revisions

Latest revision as of 20:00, 21 January 2014

Derivation

See also

References

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