Flux tubes and field lines

Surfaces S₁ and S₂ are cross sections of a magnetic flux tube; the magnetic flux through S₁ is equal to the magnetic flux through S₂.

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.^[5]: 25

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:

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

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.^[6][7]

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

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

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.^[5]

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.

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

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

whereas, for flux to be conserved, the fluid must satisfy the stronger condition imposed by the ideal induction equation.^[8][9]

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:

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

However, the induction equation is linear, whereas there is a nonlinear relationship between ∇ × v and v in the vorticity equation.^[9]