Magnetic curvature

The magnetic 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

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

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

Geodesic curvature

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