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Concept
Induction equation
Résumé
In magnetohydrodynamics, the induction equation is a partial differential equation that relates the magnetic field and velocity of an electrically conductive fluid such as a plasma. It can be derived from Maxwell's equations and Ohm's law, and plays a major role in plasma physics and astrophysics, especially in dynamo theory.
Maxwell's equations describing the Faraday's and Ampere's laws read:
and
where:
is the electric field.
is the magnetic field.
is the electric current density.
The displacement current can be neglected in a plasma as it is negligible compared to the current carried by the free charges. The only exception to this is for exceptionally high frequency phenomena: for example, for a plasma with a typical electrical conductivity of , the displacement current is smaller than the free current by a factor of for frequencies below .
The electric field can be related to the current density using the Ohm's law:
where
is the velocity field.
is the electric conductivity of the fluid.
Combining these three equations, eliminating and , yields the induction equation for an electrically resistive fluid:
Here is the magnetic diffusivity (in the literature, the electrical resistivity, defined as , is often identified with the magnetic diffusivity).
If the fluid moves with a typical speed and a typical length scale , then
The ratio of these quantities, which is a dimensionless parameter, is called the magnetic Reynolds number:
For a fluid with infinite electric conductivity, , the first term in the induction equation vanishes. This is equivalent to a very large magnetic Reynolds number. For example, it can be of order in a typical star. In this case, the fluid can be called a perfect or ideal fluid. So, the induction equation for an ideal conductive fluid such as most astrophysical plasmas is
This is taken to be a good approximation in dynamo theory, used to explain the magnetic field evolution in the astrophysical environments such as stars, galaxies and accretion discs.
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