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,

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

Ohm's Law,

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

where η is the resistivity (assumed homogeneous), and Ampère's Law,

${\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {j}},$

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

${\frac {d{\vec {B}}}{dt}}={\frac {\eta }{\mu _{0}}}{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {B}}=-{\frac {\eta }{\mu _{0}}}\nabla ^{2}{\vec {B}},$

since

${\vec {\nabla }}\cdot {\vec {B}}=0.$

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