Alfvén's theorem

The closed surface formed by S₁, S₂, and S₃

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

${\displaystyle {\frac {D\Phi _{B}}{Dt}}=\lim _{\delta t\to 0}{\frac {\iint _{S_{2}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{2}-\iint _{S_{1}}\mathbf {B} (t)\cdot d\mathbf {S} _{1}}{\delta t}}.}$

The surface integral over 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

${\displaystyle \iint _{S_{2}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{2}=\iint _{S_{1}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{1}-\oint _{\partial S_{1}}\left(\mathbf {v} \ \delta t\times \mathbf {B} (t)\right)\cdot d\mathbf {l} ,}$

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

${\displaystyle {\begin{aligned}{\frac {D\Phi _{B}}{Dt}}=\lim _{\delta t\to 0}\iint _{S_{1}}{\frac {\mathbf {B} (t+\delta t)-\mathbf {B} (t)}{\delta t}}\cdot d\mathbf {S} _{1}-\oint _{\partial S_{1}}\left(\mathbf {v} \times \mathbf {B} (t)\right)\cdot d\mathbf {l} .\end{aligned}}}$

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

${\displaystyle {\frac {D\Phi _{B}}{Dt}}=\iint _{S_{1}}\left({\frac {\partial \mathbf {B} }{\partial t}}-\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)\right)\cdot d\mathbf {S} _{1}.}$

Using the ideal induction equation, the integrand vanishes, and

${\displaystyle {\frac {D\Phi _{B}}{Dt}}=0.}$

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

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

Using the mass continuity equation,

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

the ideal induction equation can be further rearranged to give

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

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

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

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.