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Magnetic curvature: Difference between revisions
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The magnetic curvature is defined by | == Field line curvature == | ||
The magnetic field line curvature is defined by | |||
:<math>\vec \kappa = \vec b \cdot \vec \nabla \vec b</math> | :<math>\vec \kappa = \vec b \cdot \vec \nabla \vec b</math> | ||
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is a unit vector along the magnetic field. | is a unit vector along the magnetic field. | ||
''κ'' points towards the local centre of curvature of ''B'', | |||
and its magnitude is equal to the inverse radius of curvature. | |||
A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when | A plasma is stable against curvature-driven [[Plasma instability|instabilities]] (e.g., ballooning modes) when | ||
:<math>\vec \kappa \cdot \vec \nabla p < 0</math> | :<math>\vec \kappa \cdot \vec \nabla p < 0</math> | ||
(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure. | (good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure. | ||
<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of | <ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref> | ||
=== Normal curvature === | |||
The component of the curvature perpendicular to the [[Flux surface|flux surface]] is | |||
:<math>\kappa_N = \vec \kappa \cdot \frac{\vec \nabla \psi}{|\vec \nabla \psi|}</math> | |||
Here, ψ is a [[Flux surface|flux surface]] label (such as the poloidal flux). | |||
=== Geodesic curvature === | |||
The component of the field line curvature parallel to the [[Flux surface|flux surface]] is | |||
:<math>\kappa_G = \vec \kappa \cdot \left (\frac{\vec \nabla \psi}{|\vec \nabla \psi|} \times \frac{\vec B}{|\vec B|}\right )</math> | |||
== Flux surface curvature == | |||
The tangent plane to any flux surface is spanned up by two tangent vectors: one is the normalized magnetic field vector (discussed above), and the other is | |||
:<math>\vec b_\perp = \frac{\vec \nabla \psi}{|\vec \nabla \psi|} \times \frac{\vec B}{|\vec B|}</math> | |||
The corresponding perpendicular curvature (the curvature of the flux surface in the direction perpendicular to the magnetic field) is | |||
:<math>\vec \kappa_\perp = \vec b_\perp \cdot \vec \nabla \vec b_\perp</math> | |||
and one can again define the corresponding normal and geodesic curvature components in analogy with the above. | |||
== References == | == References == | ||
<references /> | <references /> | ||