Flux Freezing

is sometimes referred to as the

In order to verify the previous assertion, let us consider the magnetic flux, , through a loop, , that is co-moving with the plasma:

(7.14) |

Here, is some surface that spans . The time rate of change of is made up of two parts. First, there is the part due to the time variation of over the surface , which can be written

(7.15) |

Using the Faraday-Maxwell equation, this reduces to

(7.16) |

Second, there is the part due to the motion of . If is an element of then is the area swept out by per unit time. Hence, the flux crossing this area is . It follows that

(7.17) |

Using the curl theorem, we obtain

(7.18) |

Hence, the total time rate of change of is given by

(7.19) |

The condition

(7.20) |

clearly implies that remains constant in time for any arbitrary co-moving loop, . This, in turn, implies that magnetic field-lines must move with the plasma. In other words, the field-lines are frozen into the plasma.

A *flux-tube* is defined as a topologically cylindrical volume whose
sides are defined by magnetic field-lines. Suppose that, at some initial
time, a flux-tube is embedded in the plasma. According to the flux-freezing
constraint,

(7.21) |

the subsequent motion of the plasma and the magnetic field is always such that it maintains the integrity of the flux-tube. Because magnetic field-lines can be regarded as infinitely thin flux-tubes, we conclude that MHD plasma motion also maintains the integrity of field-lines. In other words, magnetic field-lines embedded in an MHD plasma can never break and reconnect: that is, MHD forbids any change in topology of the field-lines. It turns out that this is an extremely restrictive constraint. Later on, we shall discuss situations in which this constraint is relaxed. (See Section 7.14.)