# Magnetic curvature

## 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.