# Magnetic curvature

## Contents

## 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.
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### 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.