In magnetohydrodynamics, the Alfvén's theorem, also known as Alfvén's frozen in theorem, "states that in a fluid with infinite electric conductivity, the magnetic field is frozen into the fluid and has to move along with it". Hannes Alfvén put the idea forward for the first time in 1942. In his own words: "In view of the infinite conductivity, every motion of the liquid in relation to the lines of force is forbidden because it would give infinite eddy currents. Thus the matter of the liquid is “fastened” to the lines of force...". As an even stronger result, the magnetic flux through a co-moving surface is conserved in a perfectly conducting fluid.
Mathematical statement
In a fluid with infinite electric conductivity, the change of magnetic flux over time can be written as where and are the magnetic and velocity fields respectively. Here, is the surface enclosed by the curve with differential line element. Using the induction equation leads to These two integrals can be re-written using Stokes' theorem for the first one, and the vector identity for the second one. The result is This is the mathematical form of the Alfvén's theorem: the magnetic flux passing through a surface moving along with the fluid is conserved. This means that the plasma can move along with the local field lines. For the perpendicular motions of the fluid, the field lines will push the fluid or otherwise they will be dragged with the fluid.
Even for the non-ideal case, where the electric conductivity is not infinite, a similar result can be obtained by defining the magnetic flux transporting velocity by writing where instead of fluid velocity,, the flux velocity has been used. Although, in some cases, this velocity field can be found using magnetohydrodynamic equations, but the existence and uniqueness of this vector field depends on the underlying conditions.
Stochastic flux freezing
The flux freezing indicates that the magnetic field topology cannot change in a perfectly conducting fluid. However, this would lead to highly tangled magnetic fields with very complicated topologies that should impede the fluid motions. Nevertheless, astrophysical plasmas with high electrical conductivities do not generally show such complicated tangled fields. Also magnetic reconnection seems to occur in these plasmas unlike what is expected from the flux freezing conditions. This has important implications for magnetic dynamos. In fact, a very high electrical conductivity translates into high magnetic Reynolds numbers, which indicates that the plasma will be turbulent. In fact, the conventional views on flux freezing in highly conducting plasmas are inconsistent with the phenomenon of spontaneous stochasticity. It has become a standard argument even in textbooks, unfortunately, that magnetic flux freezing should hold better and better as magnetic diffusivity tends to zero. But the subtlety is that very large magnetic Reynolds numbers are usually associated with high kinetic Reynolds numbers. If kinematic viscosity tends to zero simultaneously with the resistivity, and if the plasma becomes turbulent, then Lagrangian trajectories will no longer be unique. The conventional "naive" flux freezing argument, discussed above, does not apply in general and stochastic flux freezing must be employed. The stochastic flux-freezing theorem for resistive magnetohydrodynamics generalizes ordinary flux-freezing discussed above. This generalized theorem states that magnetic field lines of the fine-grained magnetic field B are “frozen-in” to the stochastic trajectories solving the following stochastic differential equation, known as the Langevin equation; where is magnetic diffusivity and is a three-dimensional Gaussian white noise. The many “virtual” field-vectors that arrive to the same final point must be averaged to obtain the physical magnetic field at that point.
