Magnetic curvature

Field line curvature

The magnetic field line curvature is defined by

$\overrightarrow{\kappa }=\overrightarrow{b}\cdot \overrightarrow{\mathrm{\nabla }}\overrightarrow{b}$

where

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

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

$\overrightarrow{\kappa }\cdot \overrightarrow{\mathrm{\nabla }}p<0$

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

Normal curvature

The component of the curvature perpendicular to the flux surface is

${\kappa }_N=\overrightarrow{\kappa }\cdot \frac{\overrightarrow{\mathrm{\nabla }}\psi }{|\overrightarrow{\mathrm{\nabla }}\psi |}$

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

${\kappa }_G=\overrightarrow{\kappa }\cdot (\frac{\overrightarrow{\mathrm{\nabla }}\psi }{|\overrightarrow{\mathrm{\nabla }}\psi |}\times \frac{\overrightarrow{B}}{|\overrightarrow{B}|})$

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

${\overrightarrow{b}}_{\perp }=\frac{\overrightarrow{\mathrm{\nabla }}\psi }{|\overrightarrow{\mathrm{\nabla }}\psi |}\times \frac{\overrightarrow{B}}{|\overrightarrow{B}|}$

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

${\overrightarrow{\kappa }}_{\perp }={\overrightarrow{b}}_{\perp }\cdot \overrightarrow{\mathrm{\nabla }}{\overrightarrow{b}}_{\perp }$

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