Alfvén's theorem

In a fluid with infinite electric conductivity, the change in magnetic flux ${\displaystyle \Phi _{B}}$ over time ${\displaystyle t}$ through an open surface ${\displaystyle S}$ enclosed by a curve ${\displaystyle C}$ can be written as

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

where ${\displaystyle {\vec {B}}}$, ${\displaystyle {\vec {v}}}$, and ${\displaystyle d{\vec {l}}}$ are the magnetic field, velocity field, and differential line element, respectively. The velocity is the plasma velocity which is defined as the mean of the ion and electron velocities of the plasma weighted by their particle masses. Using the induction equation,

${\displaystyle {\partial {\vec {B}} \over \partial t}={\vec {\nabla }}\times ({\vec {v}}\times {\vec {B}}),}$

leads to

${\displaystyle {d\Phi _{B} \over dt}=\int _{S}{\vec {\nabla }}\times ({\vec {v}}\times {\vec {B}})\cdot d{\vec {S}}+\oint _{C}{\vec {B}}\cdot {\vec {v}}\times d{\vec {l}}.}$

The first integral can be rewritten using Stokes' theorem, and the second can be rewritten using the vector identity ${\displaystyle ({\vec {A}}\times {\vec {B}})\cdot {\vec {C}}=-{\vec {B}}\cdot ({\vec {A}}\times {\vec {C}})}$. This results in

${\displaystyle \int _{S}{\vec {B}}\cdot d{\vec {S}}=const.}$

This is the mathematical form of 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.

The curve ${\displaystyle C}$ sweeps out a cylindrical boundary along the local magnetic field lines in the fluid which forms a tube known as the flux tube. When the diameter of this tube goes to zero, it is called a magnetic field line.[4][5]

Even for the non-ideal case, in which the electric conductivity is not infinite, a similar result can be obtained by defining the magnetic flux transporting velocity by writing:

${\displaystyle {\vec {\nabla }}\times ({\vec {w}}\times {\vec {B}})=\eta {\vec {\nabla }}^{2}{\vec {B}}+{\vec {\nabla }}\times ({\vec {v}}\times {\vec {B}}),}$

in which, instead of fluid velocity, ${\displaystyle v}$, the flux velocity ${\displaystyle w}$ has been used. Although, in some cases, this velocity field can be found using magnetohydrodynamic equations, the existence and uniqueness of this vector field depends on the underlying conditions.[6]

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 would be 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.[7]

In fact, the conventional views on flux freezing in highly conducting plasmas are inconsistent with the phenomenon of spontaneous stochasticity. Unfortunately, it has become a standard argument, even in textbooks, that magnetic flux freezing should hold increasingly better as magnetic diffusivity tends to zero (non-dissipative regime). But the subtlety is that very large magnetic Reynolds numbers (i.e., small electric resistivity or high electrical conductivities) are usually associated with high kinetic Reynolds numbers (i.e., very small viscosities). If kinematic viscosity tends to zero simultaneously with the resistivity, and if the plasma becomes turbulent (associated with high Reynolds numbers), 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.[8]

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:

${\displaystyle d{\bf {x}}={\bf {u}}({\bf {x}},t)dt+{\sqrt {2\eta }}d{\bf {W}}(t)}$

in which ${\displaystyle \eta }$ is magnetic diffusivity and ${\displaystyle W}$ is the three-dimensional Gaussian white noise. (See also Wiener process.) The many “virtual” field-vectors ${\displaystyle {\tilde {\bf {B}}}}$ that arrive at the same final point must be averaged to obtain the physical magnetic field ${\displaystyle {\bf {B}}}$ at that point.[9]

• Alfvén wave
• Induction equation
• Kelvin's circulation theorem