Magnetized Plasmas

A magnetized plasma is one in which the ambient magnetic field, ${\bf B}$, is strong enough to significantly alter particle trajectories. In particular, magnetized plasmas are highly anisotropic, responding differently to forces that are parallel and perpendicular to the direction of ${\bf B}$. Incidentally, a magnetized plasma moving with mean velocity ${\bf V}$ contains an electric field ${\bf E} = - {\bf V}\times {\bf B}$ that is not affected by Debye shielding. Of course, the electric field is essentially zero in the rest frame of the plasma.

As is well known, charged particles respond to the Lorentz force,

$${\bf F} = q\,{\bf v}\times {\bf B},$$ (1.28)

by freely streaming in the direction of ${\bf B}$, while executing circular Larmor orbits, or gyro-orbits, in the plane perpendicular to ${\bf B}$ (Fitzpatrick 2008). As the field-strength increases, the resulting helical orbits become more tightly wound, effectively tying particles to magnetic field-lines.

The typical Larmor radius, or gyroradius, of a charged particle gyrating in a magnetic field is given by

$$\rho\equiv \frac{v_t}{\mit\Omega},$$ (1.29)

where

$${\mit\Omega} = \frac{e B}{m}$$ (1.30)

is the cyclotron frequency, or gyrofrequency, associated with the gyration. As usual, there is a distinct gyroradius for each species. When species temperatures are comparable, the electron gyroradius is distinctly smaller than the ion gyroradius:

$$\rho_e \sim \left(\frac{m_e}{m_i}\right)^{1/2}\!\rho_i.$$ (1.31)

A plasma system, or process, is said to be magnetized if its characteristic lengthscale, $L$, is large compared to the gyroradius. In the opposite limit, $\rho\gg L$, charged particles have essentially straight-line trajectories. Thus, the ability of the magnetic field to significantly affect particle trajectories is measured by the magnetization parameter,

$$\delta \equiv \frac{\rho}{L}.$$ (1.32)

There are some cases of interest in which the electrons are magnetized, but the ions are not. However, a “magnetized” plasma conventionally refers to one in which both species are magnetized. This state is generally achieved when

$$\delta_i \equiv \frac{\rho_i}{L} \ll 1.$$ (1.33)