Alfvén's Theorem

In magnetohydrodynamics, the Alfvén's theorem, also known as Alfvén's frozen in theorem, states that in a fluid with infinite electric conductivity, magnetic field lines are frozen into the fluid and have 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 (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...".

In most astrophysical environments, as well as laboratory plasmas, the electric conductivity is not infinite, so the magnetic field lines are not ideally frozen into the fluid. However, with a high electric conductivity, or equivalently a small resistivity, the frozen in theorem can be approximately applied. This is called the frozen flux approximation which is widely used in dynamo theory.

In a fluid with infinite electric conductivity, the change of magnetic flux over time can be written as

${\displaystyle {d\Phi _{B} \over dt}=\int _{S}{\partial {\vec {B}} \over \partial t}\cdot d{\vec {S}}+\oint _{C}{\vec {B}}\cdot {\vec {v}}\times d{\vec {l}},}$

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