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The magnetic well, along with the magnetic shear, is a fundamental concept for the stability of magnetically confined plasmas. | The magnetic well, along with the [[Magnetic shear|magnetic shear]], is a fundamental concept for the stability of magnetically confined plasmas. <ref>[https://fusion.gat.com/pubs-ext/ComPlasmaPhys/A22135.pdf J.M. Greene, ''A brief review of magnetic wells'', General Atomics Report GA-A22135 (1998)]</ref> | ||
A toroidally confined plasma with given pressure has a tendency to expand. | A toroidally confined plasma with given pressure has a tendency to expand. | ||
However, an ideal, collisionless plasma is bound to the magnetic field lines, and the flux in a magnetic flux tube is conserved. | However, an ideal, collisionless plasma is bound to the magnetic field lines, and the flux in a magnetic [[Flux tube|flux tube]] is conserved. | ||
The plasma will therefore prefer to move to a location where the volume to flux ratio (the ''specific volume'') is maximum. | The plasma will therefore prefer to move to a location where the volume to flux ratio (the ''specific volume'') is maximum. | ||
Assuming the existence of nested (toroidal) flux surfaces, labelled by | Assuming the existence of nested (toroidal) flux surfaces, labelled by <math>\psi</math>, and with volume <math>V(\psi)</math> and toroidal flux <math>\Phi(\psi)</math>, this specific volume is | ||
:<math>U = \frac{dV}{d\Phi}</math> | :<math>U = \frac{dV}{d\Phi}</math> | ||
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where <math>\bar L</math> is the length of the (toroidal) magnetic axis, one sees that the condition of maximum specific volume is equivalent to minimum B. | where <math>\bar L</math> is the length of the (toroidal) magnetic axis, one sees that the condition of maximum specific volume is equivalent to minimum B. | ||
The vacuum magnetic well is defined via an average of the magnetic pressure over a [[flux surface]]: | |||
:<math>W = 2 \frac{V}{ \left \langle B^2 \right \rangle } \frac{d}{dV} \left \langle \frac{B^2}{2} \right \rangle </math> | :<math>W = 2 \frac{V}{ \left \langle B^2 \right \rangle } \frac{d}{dV} \left \langle \frac{B^2}{2} \right \rangle </math> | ||
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:<math>\left \langle f \right \rangle = \int{\frac{f dl}{B}} \bigg / \int{\frac{dl}{B}}</math> | :<math>\left \langle f \right \rangle = \int{\frac{f dl}{B}} \bigg / \int{\frac{dl}{B}}</math> | ||
where ''dl'' is an arc segment along the field line. | where ''dl'' is an [[:Wikipedia:Arc_(geometry)|arc segment]] along the field line. | ||
The magnetic well is related to the average [[Magnetic curvature|magnetic field line curvature κ]]. | |||
<ref>M. Wakatani, ''Stellarator and Heliotron devices'', Oxford University Press, New York and Oxford (1998) {{ISBN|0-19-507831-4}}</ref> | |||
The magnetic well is | The relative magnetic well depth is defined as: | ||
:<math>w(\bar r) = \frac{(dV(0)/d\Phi - dV(\bar r)/d\Phi)}{dV(0)/d\Phi }</math> | |||
where <math>\bar r</math> is the [[Effective plasma radius|average radius of a flux surface]]. | |||
A positive gradient of ''w'' corresponds to a magnetic well (stable), and a negative gradient to a magnetic hill (unstable). | |||
== References == | == References == | ||
<references /> | <references /> |