Magnetic curvature

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Field line curvature

The magnetic field line curvature is defined by

${\displaystyle {\vec {\kappa }}={\vec {b}}\cdot {\vec {\nabla }}{\vec {b}}}$

where

${\displaystyle {\vec {b}}={\frac {\vec {B}}{|B|}}}$

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

${\displaystyle {\vec {\kappa }}\cdot {\vec {\nabla }}p<0}$

(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

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

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

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

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

${\displaystyle {\vec {b}}_{\perp }={\frac {{\vec {\nabla }}\psi }{|{\vec {\nabla }}\psi |}}\times {\frac {\vec {B}}{|{\vec {B}}|}}}$

The corresponding perpendicular curvature (the curvature of the flux surface in the direction perpendicular to the magnetic field) is

${\displaystyle {\vec {\kappa }}_{\perp }={\vec {b}}_{\perp }\cdot {\vec {\nabla }}{\vec {b}}_{\perp }}$

and one can again define the corresponding normal and geodesic curvature components in analogy with the above.