Alfvén's Theorem

In magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, "states that in a fluid with infinite electric conductivity, the magnetic field is frozen into the fluid and has 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 later life, Alfvén changed his mind and advised against use of his own theorem. 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.

As an even stronger result, the magnetic flux through a co-moving surface is conserved in a perfectly conducting fluid. Alfvén's theorem also holds in "ideal-MHD" which applies even if the plasma is not perfectly conducting (i.e., it is resistive and the electrical conductivity is not infinite) as long as they are large in spatial scale which gives a large magnetic Reynolds number and means that the "convective term" dominates over the "diffusive term" in the magnetic induction equation.

Mathematical statement

Alfvén's theorem states that in a fluid with infinite electrical conductivity the magnetic flux $\Phi_B$ through an arbitrary open surface advected by a macroscopic, space- and time-dependent velocity field $\mathbf$ is constant, or
:$\frac = 0 ,$
where $D/Dt = \partial/\partial t + (\mathbf \cdot \mathbf)$ is the advective derivative.

Derivation

In a fluid with infinite electrical conductivity and a space- and time-dependent magnetic field $\mathbf$, 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 $\mathbf$. According to Gauss' theorem, the total magnetic flux through the closed surface enclosing a volume $V$ formed by $S_1$, $S_2$, and the surface $S_3$ that connects $S_1$ and $S_2$ at time $t + \delta t$ is zero:
:$\begin 0 &= \iiint_V \nabla \cdot \mathbf(t+\delta t)\ dV \\ &= -\iint_ \mathbf(t+\delta t)\cdot d\mathbf_1 + \iint_ \mathbf(t+\delta t)\cdot d\mathbf_2 + \iint_ \mathbf(t+\delta t)\cdot d\mathbf_3, \end$
where the divergence theorem was used and the sense of $S_1$ was reversed so that $d\mathbf_1$ points outwards from the enclosed volume. In the final term, $d\mathbf_3 = d\mathbf \times \mathbf\ \delta t$ where $d\mathbf$ is the line element around the boundary of $S_1$. Therefore,
:\begin
\iint_ \mathbf(t+\delta t) \cdot d\mathbf_3 &= \iint_ \mathbf(t+\delta t) \cdot \left(d\mathbf\times\mathbf\ \delta t\right) \\
&= \iint_ \left mathbf\ \delta t \times \mathbf(t+\delta t)\right\cdot d\mathbf.
\end
The change in flux through $S_1$ as it is advected to $S_2$ is
:$\delta\Phi_B = \iint_ \mathbf(t+\delta t) \cdot d\mathbf_2 - \iint_ \mathbf(t) \cdot d\mathbf_1.$
The surface integral over $S_2$ can be substituted using the expression derived from Gauss' theorem to give
:\begin
\delta\Phi_B &= \iint_ \mathbf(t+\delta t)\cdot d\mathbf_1 - \iint_ \mathbf(t) \cdot d\mathbf_1 - \iint_ \left mathbf\ \delta t \times \mathbf(t+\delta t)\right\cdot d\mathbf \\
&= \iint_ \left mathbf(t+\delta t) - \mathbf(t)\right\cdot d\mathbf_1 - \iint_ \left mathbf\ \delta t \times \mathbf(t+\delta t)\right\cdot d\mathbf.
\end
Dividing by $\delta t$ gives
:\begin
\frac &= \iint_ \frac \cdot d\mathbf_1 - \iint_ \left mathbf \times \mathbf(t)\right\cdot d\mathbf \\
&= \iint_ \frac \cdot d\mathbf_1 - \iint_ \nabla\times\left mathbf \times \mathbf(t)\right\cdot d\mathbf_1
\end
where the limit of a small $\delta t$ was used in the first term and Stoke's theorem was used in the second term. Writing $\delta\Phi_B / \delta t = D\Phi_B / Dt$ and combining the two integrals gives
:\frac = \iint_ \left( \frac - \nabla\times\left mathbf \times \mathbf(t)\right\right) \cdot d\mathbf_1.
Using the induction equation for an infinitely conductive fluid,
:$\frac = \nabla \times \left(\mathbf\times\mathbf\right)$,
the integrand vanishes and
:$\frac = 0.$

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.

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:
:$\nabla \times (\bf\times \bf)=\eta \nabla^2 \bf + \nabla \times (\bf \times \bf),$
in which, instead of fluid velocity, $\bf$, the flux velocity $\bf$ 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 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.

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.

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:

:$d=(,t)dt+\sqrt d(t)$

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

See also

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

References

{{DEFAULTSORT:Alfven's Theorem
Magnetohydrodynamics