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.

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.