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 anisotropic, responding differently to forces which are parallel and perpendicular to the direction of ${\bf B}$. Note that a magnetized plasma moving with mean velocity ${\bf V}$ contains an electric field ${\bf E} = - {\bf V}\times {\bf B}$ which is not affected by Debye shielding. Of course, in the rest frame of the plasma the electric field is essentially zero.

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

$${\bf F} = q\,{\bf v}\times {\bf B},$$ (27)

by freely streaming in the direction of ${\bf B}$, whilst executing circular Larmor orbits, or gyro-orbits, in the plane perpendicular to ${\bf B}$. 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

$$\rho\equiv \frac{v_t}{\Omega},$$ (28)

where

$${\Omega} = e B/m$$ (29)

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:

$$\rho_e \sim \left(\frac{m_e}{m_i}\right)^{1/2}\!\rho_i.$$ (30)

A plasma system, or process, is said to be magnetized if its characteristic length-scale $L$ 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

$$\delta \equiv \frac{\rho}{L}.$$ (31)

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

$$\delta_i \equiv \frac{\rho_i}{L} \ll 1.$$ (32)