Resistive timescale

The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law,

$\frac{d\overrightarrow{B}}{dt}=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{E},$

Ohm's Law,

$\overrightarrow{E}=\eta \overrightarrow{j}$,

where η is the resistivity, and Ampère's Law,

$\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{B}={\mu }_0\overrightarrow{j}$,

one immediately derives a diffusion type equation for the magnetic field:

$\frac{d\overrightarrow{B}}{dt}=\frac{\eta }{{\mu }_0}\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\mathrm{\nabla }}\times \overrightarrow{B}$,

from which one can deduce the typical timescale

${\tau }_R\simeq \frac{{\mu }_0{L}^2}{\eta }$.

Here, L is the typical length scale of the problem, often taken equal to a, the minor radius of the toroidal plasma.