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,

$\frac{\partial \overrightarrow{B}}{\partial t}=-\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{E},$

Ohm's Law,

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

where η is the resistivity (assumed homogeneous), 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:

$\frac{\partial \overrightarrow{B}}{\partial t}=-\frac{\eta }{{\mu }_0}\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\mathrm{\nabla }}\times \overrightarrow{B}=\frac{\eta }{{\mu }_0}{\mathrm{\nabla }}^2\overrightarrow{B},$

since

$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{B}=0.$

From this, 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.

See also

• Lundquist number