Vlasov Equation - The Frozen-in Approximation

The Frozen-in Approximation

As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often say that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.

We introduce the scales, and for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in . By large we mean that

We then write

Vlasov equation can now be written

So far no approximations have been done. To be able to proceed we set, where is the gyro frequency and R is the gyroradius. By dividing by, we get

If and, the two first terms will be much less than one since, and due to the definitions of, and above. Since the last term is of the order of one, we can neglect the two first terms and write

This equation can be decomposed into a field aligned and a perpendicular part:

The next step is to write, where

It will soon be clear why this is done. With this substitution, we get

If the parallel electric field is small,

This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, is identical with the mean velocity, and we have

To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing with the thermal velocity or the Alfvén velocity. In the latter case is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.