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Resistive timescale: Difference between revisions
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(Created page with 'The resistive timescale is the typical time for the diffusion of a magnetic field into a resistive plasma. Based on Faraday's Law, :<math>\frac{d\vec B}{dt} = \vec \nabla \times …') |
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:<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math> | :<math>\frac{d\vec B}{dt} = \vec \nabla \times \vec E,</math> | ||
Ohm's Law, | Ohm's Law, | ||
:<math>\vec E = \eta \vec j</math> | :<math>\vec E = \eta \vec j,</math> | ||
where ''η'' is the resistivity, and Ampère's Law, | where ''η'' is the resistivity (assumed homogeneous), and Ampère's Law, | ||
:<math> \vec \nabla \times \vec B = \mu_0 \vec j</math> | :<math> \vec \nabla \times \vec B = \mu_0 \vec j,</math> | ||
one immediately derives a diffusion type equation for the magnetic field: | one immediately derives a diffusion type equation for the magnetic field: | ||
:<math>\frac{d\vec B}{dt} = \frac{\eta}{\mu_0} \vec \nabla \times \vec \nabla \times \vec B</math>, | :<math>\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,</math> | ||
since | |||
:<math> \tau_R \simeq \frac{\mu_0 L^2}{\eta}</math> | : <math>\vec \nabla \cdot \vec B = 0.</math> | ||
From this, one can deduce the typical timescale | |||
:<math> \tau_R \simeq \frac{\mu_0 L^2}{\eta}.</math> | |||
Here, ''L'' is the typical length scale of the problem, often taken equal to ''a'', the [[Toroidal coordinates|minor radius]] of the toroidal plasma. | Here, ''L'' is the typical length scale of the problem, often taken equal to ''a'', the [[Toroidal coordinates|minor radius]] of the toroidal plasma. | ||
== See also == | |||
* [[:Wikipedia:Lundquist number|Lundquist number]] | |||