flux freezing

The magnetic field lines of a plasma, under conditions of low resistivity, can be frozen to the plasma in which it is embedded. This means that as a volume of plasma moves, the density of magnetic field lines through the volume remains the same. This phenonomen, known as flux freezing, is a feature of the theory of ideal magnetohydrodynamics(or MHD) (i.e. when the plasma can be assumed to be perfectly conducting). If magnetic flux is frozen to the plasma it ensures that the magnetic topology of the plasma remains invariant. This means that various plasma and magnetic field cells will remain separate from each other. Flux freezing places restrictions on the allowable magnetic topology of a plasma.

Derivation

This flux freezing effect follows from one of the equations of magnetohydrodynamics, namely

E + v X B= η j

where E, B are the electric and magnetic fields, j is the current density and η is the resistivity of the plasma. In the case of ideal MHD the resistivity is assumed to be zero

E + v X B= 0    *

In other words the electric field is purely convective; there is no inductive component. Now consider any closed loop C enclosing an area S. The magnetic flux flux through this loop is simply given by impinging magnetic field integrated over the area.

Φ=∫B.dS

The question is - how much does the magnetic flux through the loop change if it moves with the plasma¹

. The time derivative of the magnetic flux will have two components. Firstly, the flux will change even if the loop didn't move due to time variations in the magnetic field itself.

dΦ/dt₁= ∫dB/dt.dS

Using Faraday's law this equation can be rewritten

dΦ/dt₁=-∫∇XE.dS

Secondly, as the loop moves into regions of different magnetic field the flux will change, even without a time variation in the field. In a time increment dt an element of the loop dl will move a distance vxdl and flux change over this distance is B.vxdl which can be written BXv.dl. And so the second contribution to the flux derivative can be written as the following line integral

dΦ/dt₂=∫BXvdl

which can be transformed to a surface integral using Stoke's Theorem.

dΦ/dt₂=-∫ ∇X(vXB)dS

Putting the two contributions together the total time derivative of the magnetic flux is

dΦ/dt=-∫ ∇X(E+vXB)dS

From by equation * above it is immediately evident that the R.H.S of the above equation is zero. It follows that the flux derivative is zero. As the plasma moves with fluid velocity v the magnetic flux Φ remains the same, meaning the magnetic flux lines are frozen to the plasma.

Applications

In real life plasmas the resistivity is never precisely zero, so there will always be flux diffusion and magnetic reconnection. In space plasmas the resistivity is extremely low and so flux freezing is an important effect (e.g. the Earth's magnetosphere and its interaction with the solar wind). The Lundquist number describes to what degree a plasma exhibits flux freezing or magnetic reconnection.

¹ Note, in MHD the plasma is assumed to be a single-species fluid and the separate identities of the ions and electrons are ignored. In this picture the plasma has a fluid velocity v

Sources:
'Plasma physics, an introduction', edited by R.Dendy
My own plasma physics knowledge.