Alfvén's theorem

The concept of magnetic fields being frozen into fluids with infinite electrical conductivity was first proposed by Hannes Alfvén in a 1943 paper titled "On the Existence of Electromagnetic-Hydrodynamic Waves", published in the journal Arkiv för matematik, astronomi och fysik.

[3] Informally, Alfvén's theorem refers to the fundamental result in ideal magnetohydrodynamic theory that electrically conducting fluids and the magnetic fields within are constrained to move together in the limit of large magnetic Reynolds numbers (Rm)—such as when the fluid is a perfect conductor or when velocity and length scales are infinitely large.

In the limit of a large magnetic Reynolds number, Alfvén's theorem requires that these surfaces of constant flux move with the fluid that they are embedded in.

[5]: 25 In mathematical terms, Alfvén's theorem states that, in an electrically conducting fluid in the limit of a large magnetic Reynolds number, the magnetic flux ΦB through an orientable, open material surface advected by a macroscopic, space- and time-dependent velocity field[note 1] v is constant, or where D/Dt = ∂/∂t + (v ⋅ ∇) is the advective derivative.

At time t + δt, this relationship can be expressed as where the sense of S1 was reversed so that dS1 points outwards from the enclosed volume.

Solving for the surface integral over S2 then gives where the final term was rewritten using the properties of scalar triple products and a first-order approximation was taken.

Substituting this into the expression for DΦB/Dt and simplifying results in Applying the definition of a partial derivative to the integrand of the first term, applying Stokes' theorem to the second term, and combining the resultant surface integrals gives Using the ideal induction equation, the integrand vanishes, and Field line conservation can also be derived mathematically using the ideal induction equation, Gauss's law for magnetism, and the mass continuity equation.

Astrophysical plasmas with high electrical conductivities do not generally show such complicated tangled fields.

[11] Research in the 21st century has claimed that the classical Alfvén theorem is inconsistent with the phenomenon of spontaneous stochasticity.

Stochastic conservation laws developed to describe hydrodynamic behavior are shown to apply in the magnetohydrodynamic regime as well.