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 (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 = 1/ι is preferred (called the "safety factor").

The 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 related to the shearing apart of convective cells, produced by curvature-driven instabilities, as the field lines twist around the torus.