Magnetic curvature

The magnetic 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.

Normal curvature

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

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

Geodesic curvature

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