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Concept
Alfvén's theorem
Summary
In ideal magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, states that electrically conducting fluids and embedded magnetic fields are constrained to move together in the limit of large magnetic Reynolds numbers. It is named after Hannes Alfvén, who put the idea forward in 1943.
Alfvén's theorem implies that the magnetic topology of a fluid in the limit of a large magnetic Reynolds number cannot change. This approximation breaks down in current sheets, where magnetic reconnection can occur.
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. He wrote:
In view of the infinite conductivity, every motion (perpendicular to the field) 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...
"On the Existence of Electromagnetic-Hydrodynamic Waves" interpreted the results of Alfvén's earlier paper "Existence of Electromagnetic-Hydrodynamic Waves" published in the journal Nature in 1942.
Later in life, Alfvén advised against the use of his own theorem.
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. Motions of the two are constrained in that all bulk fluid motions perpendicular to the magnetic field result in matching perpendicular motion of the field at the same velocity and vice versa.
Formally, the connection between the movement of the fluid and the movement of the magnetic field is detailed in two primary results often referred to as magnetic flux conservation and magnetic field line conservation.
Official source
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