Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 67

so

d φ/ dt = [ ∂ B /∂ t −∇× ( V × B ) ] · d a

S

σ

But for infinite we have shown that the bracket vanishes so

d φ/ dt = 0

Thus, the flux is constant through the surface. We say that the magnetic fluid
is “frozen in” and can be considered to move with the fluid. In this sense, if the
magnetic field line is labeled by particles on the line at time t , they still label
the line at any other time. The concept of frozen-in field lines is very useful and
allows visualization of complex flow if we know the magnetic field geometry.
Conversely, if the motion is known, the field geometry can be deduced.
If

σ =∞ , the field can slip through the fluid, and we have

2 B

∂ /∂ −∇× ( × ) = ( /μ 0 σ) ∇

B t V B 1

For a stationary case ( V = 0),

2 B

∂ B /∂ t = ( 1 /μ 0 σ) ∇

which is a diffusion equation. If the diffusion scale length is L , the diffusion time
constant is given by

L 2 τ = μ 0 σ

For typical laboratory dimensions, L is small and so, even for good conductors,

τ is short. In cosmic plasmas or conducting fluids, however, τ is large, and the

concept of frozen-in fields is correspondingly important.
Before leaving this section, it is useful to derive an energy relationship based
on MHD principles. The particle pressure p

nk B T is equivalent to the particle
energy density, and this is also true for magnetic pressure and magnetic energy
density. That is, B 2 /2

=

μ 0 yields the energy stored in a magnetic field per unit
volume. The total stored magnetic energy in a system is then

B 2 dV

W B = ( 1 / 2 μ 0 )

where dV is the volume element and we have used a single integral sign to
designate a triple integral. Changes of this quantity with time can be written

∂ W B /∂ t = ( 1 /μ 0 ) ( B · ∂ B /∂ t ) dV