Alfvén's theorem

In ideal magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, states that magnetic field lines and flux tubes in a fluid with a large magnetic Reynolds number are frozen into the fluid and have to move along with it. It is named after Hannes Alfvén, who put the idea forward in 1943. However, Alfvén's theorem is much used today because of a second mechanism, magnetic reconnection. This is a breakdown of Alfvén's theorem in thin current sheets and is important as it can untangle field lines that would become increasingly tangled by plasma velocity shears and vortices in regions of low plasma beta if Alfvén's theorem applied everywhere.

History

Alfvén's theorem was first proposed by Hannes Alfvén in his 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 use of his own theorem.

Mathematical statement

Alfvén's theorem states that, in an electrically conducting fluid with a very large magnetic Reynolds number, the magnetic flux $\Phi_B$ through an orientable, open material surface advected by a macroscopic, space- and time-dependent velocity field $\mathbf{v}$ is constant, or

$\frac{D\Phi_B}{Dt} = 0 ,$

where $D/Dt = \partial/\partial t + (\mathbf{v} \cdot \mathbf{\nabla})$ is the advective derivative.

In fluids where the magnetic Reynolds number is very large, magnetic induction dominates over magnetic diffusion, and the diffusion term in the induction equation becomes negligible. The induction equation then reduces to its ideal form:

$\frac{\partial\mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{v}\times\mathbf{B}\right).$

Derivation

File:Alfvén's frozen-in flux theorem proof.svg

When applying Gauss's law for magnetism, the differential surface element $d\mathbf{S}_1$ of surface $S_1$ must be reversed so that it points outwards from the volume enclosed by the three surfaces.

In a fluid with a large magnetic Reynolds number and a space- and time-dependent magnetic field $\mathbf{B}$, an arbitrary open surface $S_1$ at time $t$ is advected in a small time $\delta t$ to the surface $S_2$ by a macroscopic, space- and time-dependent velocity field $\mathbf{v}$. The rate of change of the magnetic flux through the surface as it is advected from $S_1$ to $S_2$ is then

$\frac{D\Phi_B}{Dt} = \lim_{\delta t \to 0} \frac{\iint_{S_2} \mathbf{B}(t+\delta t) \cdot d\mathbf{S}_2 - \iint_{S_1} \mathbf{B}(t) \cdot d\mathbf{S}_1}{\delta t}.$

The surface integral over $S_2$ can be re expressed by applying Gauss's law for magnetism to assume that the magnetic flux through a closed surface formed by $S_1$, $S_2$, and the surface $S_3$ that connects the boundaries of $S_1$ and $S_2$ is zero. At time $t + \delta t$, this relationship can be expressed as

$0 = -\iint_{S_1} \mathbf{B}(t+\delta t)\cdot d\mathbf{S}_1 + \iint_{S_2} \mathbf{B}(t+\delta t)\cdot d\mathbf{S}_2 + \iint_{S_3} \mathbf{B}(t+\delta t)\cdot d\mathbf{S}_3 ,$

where the sense of $S_1$ was reversed so that $d\mathbf{S}_1$ points outwards from the enclosed volume. In the surface integral over $S_3$, the differential surface element $d\mathbf{S}_3 = d\mathbf{l} \times \mathbf{v}\ \delta t$ where $d\mathbf{l}$ is the line element around the boundary $\partial S_1$ of the surface $S_1$. Solving for the surface integral over $S_2$ then gives

$\iint_{S_2} \mathbf{B}(t+\delta t)\cdot d\mathbf{S}_2 = \iint_{S_1} \mathbf{B}(t+\delta t)\cdot d\mathbf{S}_1 - \oint_{\partial S_1} \left(\mathbf{v}\ \delta t \times \mathbf{B}(t)\right) \cdot d\mathbf{l},$

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 $D\Phi_B/Dt$ and simplifying results in

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \frac{D\Phi_B}{Dt} = \lim_{\delta t \to 0} \iint_{S_1} \frac{\mathbf{B}(t+\delta t) - \mathbf{B}(t)}{\delta t} \cdot d\mathbf{S}_1 - \oint_{\partial S_1} \left(\mathbf{v} \times \mathbf{B}(t)\right) \cdot d\mathbf{l}. \end{align}

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

$\frac{D\Phi_B}{Dt} = \iint_{S_1} \left( \frac{\partial\mathbf{B}}{\partial t} - \nabla\times\left(\mathbf{v} \times \mathbf{B}\right) \right) \cdot d\mathbf{S}_1.$

Using the ideal induction equation, the integrand vanishes, and

$\frac{D\Phi_B}{Dt} = 0.$

Flux tubes and field lines

The curve $C$ 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.

The frozen-in property of magnetic field lines is a consequence of magnetic flux-conservation through material surfaces. Since flux-conservation necessitates that the field lines intersecting a material surface remain with the material surface as it is advected, the field lines can be considered "frozen-in" to the fluid.

Kelvin's circulation theorem

Further information: Kelvin's circulation theorem

Kelvin's circulation theorem states that vortex tubes moving with an ideal fluid are frozen to the fluid, analogous to how magnetic flux tubes moving with a perfectly conducting ideal-MHD fluid are frozen to the fluid. The ideal induction equation takes the same form as the equation for vorticity $\boldsymbol{\omega} = \nabla\times\mathbf{v}$ in an ideal fluid where $\mathbf{v}$ is the velocity field:

$\frac{\partial \boldsymbol{\omega}}{\partial t} = \nabla \times (\mathbf{v}\times\boldsymbol{\omega}).$

However, the induction equation is linear, whereas there is a nonlinear relationship between $\nabla\times\mathbf{v}$ and $\mathbf{v}$ in the vorticity equation.

Implications

Alfvén's theorem 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. Astrophysical plasmas with high electrical conductivities do not generally show such complicated tangled fields. 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.

Resistive fluids

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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \nabla \times (\bf{w}\times \bf{B})=\eta \nabla^2 \bf{B} + \nabla \times (\bf{v} \times \bf{B}),

in which, instead of fluid velocity, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \bf{v} , the flux velocity Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \bf{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.

Stochastic flux freezing

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.

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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): d{\bf{x}}={\bf{u}}({\bf{x}},t)dt+\sqrt{2\eta} d{\bf{W}}(t)

in which $\eta$ is magnetic diffusivity and $W$ is the three-dimensional Gaussian white noise. (See also Wiener process.) The many “virtual” field-vectors Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \tilde {\bf{B}}

that arrive at the same final point must be averaged to obtain the physical magnetic field Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\bf{B}}
at that point.

See also

• Alfvén wave
• Magnetic pressure
• Magnetic tension