Magnetic curvature

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Field line curvature

The magnetic field line curvature is defined by

$ \vec \kappa = \vec b \cdot \vec \nabla \vec b $

where

$ \vec b = \frac{\vec B}{|B|} $

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

$ \vec \kappa \cdot \vec \nabla p < 0 $

(good curvature) and unstable otherwise (bad curvature). Here, p is the pressure. [1]

Normal curvature

The component of the curvature perpendicular to the flux surface is

$ \kappa_N = \vec \kappa \cdot \frac{\vec \nabla \psi}{|\vec \nabla \psi|} $

Here, ψ is a flux surface label (such as the poloidal flux).

Geodesic curvature

The component of the field line curvature parallel to the flux surface is

$ \kappa_G = \vec \kappa \cdot \left (\frac{\vec \nabla \psi}{|\vec \nabla \psi|} \times \frac{\vec B}{|\vec B|}\right ) $

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

$ \vec b_\perp = \frac{\vec \nabla \psi}{|\vec \nabla \psi|} \times \frac{\vec B}{|\vec B|} $

The corresponding perpendicular curvature (the curvature of the flux surface in the direction perpendicular to the magnetic field) is

$ \vec \kappa_\perp = \vec b_\perp \cdot \vec \nabla \vec b_\perp $

and one can again define the corresponding normal and geodesic curvature components in analogy with the above.

References

  1. A.H. Boozer, Physics of magnetically confined plasmas, Rev. Mod. Phys. 76 (2004) 1071