Alfvén's theorem

Overview

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 (R_m)—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. Magnetic flux conservation implies that the magnetic flux through a surface moving with the bulk fluid velocity is constant, and magnetic field line conservation implies that, if two fluid elements are connected by a magnetic field line, they will always be.

Flux tubes and field lines

Alfvén's theorem is frequently expressed in terms of magnetic flux tubes and magnetic field lines.

A magnetic flux tube is a tube- or cylinder-like region of space containing a magnetic field such that its sides are everywhere parallel to the field. Consequently, the magnetic flux through these sides is zero, and the cross-sections along the tube's length have constant, equal magnetic flux. In the limit of a large magnetic Reynolds number, Alfvén's theorem requires that these surfaces of constant flux move with the fluid that they are embedded in. As such, magnetic flux tubes are frozen into the fluid.

The intersection of the sides of two magnetic flux tubes form a magnetic field line, a curve that is everywhere parallel to the magnetic field. In fluids where flux tubes are frozen-in, it then follows that magnetic field lines must also be frozen-in. However, the conditions for frozen-in field lines are weaker than the conditions for frozen-in flux tubes, or, equivalently, for conservation of flux.

Mathematical statement

In mathematical terms, Alfvén's theorem states that, in an electrically conducting fluid in the limit of a large magnetic Reynolds number, the magnetic flux Φ_B through an orientable, open material surface advected by a macroscopic, space- and time-dependent velocity field v is constant, or

${\displaystyle \frac{D{\mathrm{\Phi }}_B}{Dt}=0,}$

where D/Dt = ∂/∂t + (v ⋅ ∇) is the advective derivative.

Flux conservation

In ideal magnetohydrodynamics, magnetic induction dominates over magnetic diffusion at the velocity and length scales being studied. The diffusion term in the governing induction equation is then assumed to be small relative to the induction term and is neglected. The induction equation then reduces to its ideal form:

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

The conservation of magnetic flux through material surfaces embedded in the fluid follows directly from the ideal induction equation and the assumption of no magnetic monopoles through Gauss's law for magnetism.

In an electrically conducting fluid with a space- and time-dependent magnetic field B and velocity field v, an arbitrary, orientable, open surface S₁ at time t is advected by v in a small time δt to the surface S₂. The rate of change of the magnetic flux through the surface as it is advected from S₁ to S₂ is then

${\displaystyle \frac{D{\mathrm{\Phi }}_B}{Dt}=\underset{\delta t\to 0}{lim}\frac{{\iint }_{S_2}\mathbf{B}(t+\delta t)\cdot d{\mathbf{S}}_2-{\iint }_{S_1}\mathbf{B}(t)\cdot d{\mathbf{S}}_1}{\delta t}.}$

The surface integral over S₂ can be re expressed by applying Gauss's law for magnetism to assume that the magnetic flux through a closed surface formed by S₁, S₂, and the surface S₃ that connects the boundaries of S₁ and S₂ is zero. At time t + δt, this relationship can be expressed as

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

where the sense of S₁ was reversed so that dS₁ points outwards from the enclosed volume. In the surface integral over S₃, the differential surface element dS₃ = dl × v δt where dl is the line element around the boundary ∂S₁ of the surface S₁. Solving for the surface integral over S₂ then gives

${\displaystyle {\iint }_{S_2}\mathbf{B}(t+\delta t)\cdot d{\mathbf{S}}_2={\iint }_{S_1}\mathbf{B}(t+\delta t)\cdot d{\mathbf{S}}_1-{\oint }_{\mathrm{\partial }{S}_1}\left(\mathbf{v}\delta t\times \mathbf{B}(t)\right)\cdot d\mathbf{l},}$

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 DΦ_B/Dt and simplifying results in

${\displaystyle \begin{array}{c}\frac{D{\mathrm{\Phi }}_B}{Dt}=\underset{\delta t\to 0}{lim}{\iint }_{S_1}\frac{\mathbf{B}(t+\delta t)-\mathbf{B}(t)}{\delta t}\cdot d{\mathbf{S}}_1-{\oint }_{\mathrm{\partial }{S}_1}\left(\mathbf{v}\times \mathbf{B}(t)\right)\cdot d\mathbf{l}.\end{array}}$

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

${\displaystyle \frac{D{\mathrm{\Phi }}_B}{Dt}={\iint }_{S_1}\left(\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}-\mathrm{\nabla }\times \left(\mathbf{v}\times \mathbf{B}\right)\right)\cdot d{\mathbf{S}}_1.}$

Using the ideal induction equation, the integrand vanishes, and

${\displaystyle \frac{D{\mathrm{\Phi }}_B}{Dt}=0.}$

Field line conservation

Field line conservation can also be derived mathematically using the ideal induction equation, Gauss's law for magnetism, and the mass continuity equation.

The ideal induction equation can be rewritten using a vector identity and Gauss's law for magnetism as

${\displaystyle \frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}=(\mathbf{B}\cdot \mathrm{\nabla })\mathbf{v}-(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{B}-\mathbf{B}(\mathrm{\nabla }\cdot \mathbf{v}).}$

Using the mass continuity equation,

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+(\mathbf{v}\cdot \mathrm{\nabla })\rho =-\rho \mathrm{\nabla }\cdot \mathbf{v},}$

the ideal induction equation can be further rearranged to give

${\displaystyle \frac{D}{Dt}\left(\frac{\mathbf{B}}{\rho }\right)=\left(\frac{\mathbf{B}}{\rho }\cdot \mathrm{\nabla }\right)\mathbf{v}.}$

Similarly, for a line segment δl where v is the bulk plasma velocity at one end and v + δv is the velocity at the other end, the differential velocity between the two ends is δv = (δl ⋅ ∇)v and

${\displaystyle \frac{D\delta \mathbf{l}}{Dt}=(\delta \mathbf{l}\cdot \mathrm{\nabla })\mathbf{v}}$,

which has the same form as the equation obtained previously for B/ρ. Therefore, if δl and B are initially parallel, they will remain parallel.

While flux conservation implies field line conservation (see § Flux tubes and field lines), the conditions for the latter are weaker than the conditions for the former. Unlike the conditions for flux conservation, the conditions for field line conservation can be satisfied when an additional, source term parallel to the magnetic field is present in the ideal induction equation.

Mathematically, for field lines to be frozen-in, the fluid must satisfy

${\displaystyle \left(\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}-\mathrm{\nabla }\times \left(\mathbf{v}\times \mathbf{B}\right)\right)\times \mathbf{B}=0,}$

whereas, for flux to be conserved, the fluid must satisfy the stronger condition imposed by the ideal induction equation.

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 ω = ∇ × v in an ideal fluid where v is the velocity field:

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

However, the induction equation is linear, whereas there is a nonlinear relationship between ∇ × v and v in the vorticity equation.

Implications

Alfvén's theorem indicates that the magnetic field topology cannot change in a perfectly conducting fluid. However, in the case of complicated or turbulent flows, 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:

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

in which, instead of fluid velocity v, the flux velocity 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 Alfvén theorem

Research in the 21st century has claimed that the classical Alfvén theorem is inconsistent with the phenomenon of spontaneous stochasticity. Stochastic conservation laws developed to describe hydrodynamic behavior are shown to apply in the magnetohydrodynamic regime as well. Using the same tools produces results equivalent to that of classical Alfvén's theorem under ideal conditions, while also describing flux conservation and magnetic reconnection under non-ideal (real-world) conditions. Thus stochastic flux-freezing solutions can provide better descriptions of observed phenomena without relying on idealized conditions that are rare or even absent in the observed environment.

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:

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

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