Flux Freezing

The MHD Ohm's law,

$${\bf E} + {\bf V}\times {\bf B} = {\bf0},$$ (8.13)

is sometimes referred to as the perfect conductivity equation (for obvious reasons), and sometimes as the flux freezing equation. The latter nomenclature comes about because Equation (8.13) implies that the magnetic flux through any loop in the plasma, each element of which moves with the local plasma velocity, is a conserved quantity.

In order to verify the previous assertion, let us consider the magnetic flux, ${\mit\Psi}$, through a loop, $C$, that is co-moving with the plasma:

$${\mit\Psi} = \int_{\bf S} {\bf B}\cdot d{\bf S}.$$ (8.14)

Here, ${\bf S}$ is some surface that spans $C$. The time rate of change of ${\mit\Psi}$ is made up of two parts. First, there is the part due to the time variation of ${\bf B}$ over the surface ${\bf S}$, which can be written

$$\left(\frac{\partial{\mit\Psi}}{\partial t}\right)_1 = \int_{\bf S} \frac{\partial{\bf B}}{\partial t}\cdot d{\bf S}.$$ (8.15)

Using the Faraday-Maxwell equation, this reduces to

$$\left(\frac{\partial{\mit\Psi}}{\partial t}\right)_1 = - \int_{\bf S} \nabla \times{\bf E} \cdot d{\bf S}.$$ (8.16)

Second, there is the part due to the motion of $C$. If $d{\bf r}$ is an element of $C$ then ${\bf V}\times d{\bf r}$ is the area swept out by $d{\bf r}$ per unit time. Hence, the flux crossing this area is ${\bf B} \cdot{\bf V}\times d{\bf r}$. It follows that

$$\left(\frac{\partial{\mit\Psi}}{\partial t}\right)_2 = \int_C {\bf B} \cdot{\bf V}\times d{\bf r} = \int_C {\bf B}\times{\bf V} \cdot d{\bf r}.$$ (8.17)

Using the curl theorem, we obtain

$$\left(\frac{\partial{\mit\Psi}}{\partial t}\right)_2= \int_{\bf S} \nabla\times ({\bf B}\times {\bf V})\cdot d{\bf S}.$$ (8.18)

Hence, the total time rate of change of ${\mit\Psi}$ is given by

$$\frac{d{\mit\Psi}}{dt} = - \int_{\bf S}\nabla\times\left({\bf E} + {\bf V}\times {\bf B}\right)\cdot d{\bf S}.$$ (8.19)

The condition

$${\bf E} + {\bf V}\times{\bf B} ={\bf0}$$ (8.20)

clearly implies that ${\mit\Psi}$ remains constant in time for any arbitrary co-moving loop, $C$. 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,

$$\frac{d{\mit\Psi}}{dt} = 0,$$ (8.21)

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. We shall discuss situations in which this constraint is relaxed in Chapter 9.