Magnetic shear

The shear of a vector field F is

${\vec {\nabla }}{\vec {F}}$

Thus, in 3 dimensions, the shear is a 3 x 3 tensor.

In the context of magnetic confinement, and assuming the existence of toroidally nested magnetic flux surfaces, the only relevant variation of the direction of the magnetic field is the radial gradient of the rotational transform (field line pitch). The latter is defined as

${\frac {\iota }{2\pi }}={\frac {d\psi }{d\phi }}$

where ψ is the poloidal magnetic flux, and φ the toroidal magnetic flux. Thus, ι/2π is the mean number of toroidal transits (n) divided by the mean number of poloidal transits (m) of a field line on a flux surface. In tokamak research, the quantity q = 2π/ι is preferred (called the "safety factor").

The (global) magnetic shear is defined as

$s={\frac {r}{q}}{\frac {dq}{dr}}=-{\frac {r}{\iota }}{\frac {d\iota }{dr}}$

High values of magnetic shear provide stability, since the radial extension of helically resonant modes is reduced. Negative shear also provides stability, possibly because convective cells, generated by curvature-driven instabilities, are sheared apart as the field lines twist around the torus.

The (local) magnetic shear is defined as

$s_{\rm {local}}=2\pi {\vec {h}}\cdot {\vec {\nabla }}\times {\vec {h}}$

where

${\vec {h}}={\frac {{\vec {\nabla }}\psi }{|{\vec {\nabla }}\psi |}}\times {\vec {b}}$

and

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