Diffusion magnétique

Magnetic diffusion refers to the motion of magnetic fields, typically in the presence of a conducting solid or fluid such as a plasma. The motion of magnetic fields is described by the magnetic diffusion equation and is due primarily to induction and diffusion of magnetic fields through the material. The magnetic diffusion equation is a partial differential equation commonly used in physics. Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering. The magnetic diffusion equation is where is the permeability of free space and is the electrical conductivity of the material, which is assumed to be constant. denotes the (non-relativistic) velocity of the plasma. The first term on the right hand side accounts for effects from induction of the plasma, while the second accounts for diffusion. The latter acts as a dissipation term, resulting in a loss of magnetic field energy to heat. The relative importance of the two terms is characterized by the magnetic Reynolds number, . In the case of a non-uniform conductivity the magnetic diffusion equation is however, it becomes significantly harder to solve. Starting from the generalized Ohm's law: and the curl equations for small displacement currents (i.e. low frequencies) substitute into the Ampere-Maxwell law to get Taking the curl of the above equation and substituting into Faraday's law, This expression can be simplified further by writing it in terms of the i-th component of and the Levi-Cevita tensor : Using the identity and recalling , the cross products can be eliminated: Written in vector form, the final expression is where is the material derivative. This can be rearranged into a more useful form using vector calculus identities and : In the case , this becomes a diffusion equation for the magnetic field, where is the magnetic diffusivity. In some cases it is possible to neglect one of the terms in the magnetic diffusion equation.