Plasma parameters

Here we list a few typical plasma parameters that come up often when running simulations. For each quantity we give both the physical form and its numerical “code counterpart”. Code variables are marked with a hat, \(\hat{x}\).

We keep the grid spacing and time step, \(\Delta x\) and \(\Delta t\), explicitly written in the equations even though they are set equal to \(1\) in the actual numerical calculations; this allows converting code quantities to physical quantities when needed, by plugging in the spatial and temporal scales.

Plasma frequency

The plasma oscillation frequency, also known as the Langmuir frequency \(\omega_{p,s}\), for a particle species \(s\) with number density \(n_s\) and charge \(q_s\) is

\[\omega_{p,s} = \sqrt{\frac{4\pi n_s q_s^2}{m_s}} \quad\mathrm{and}\quad \hat{\omega}_p = \frac{\hat{c}}{\hat{R}} \frac{1}{\Delta t}\]

For relativistic bulk flows with Lorentz factor \(\Gamma\) or relativistically hot plasmas with mean Lorentz factor \(\langle \gamma \rangle \gtrsim 1\) the plasma frequency becomes \(\omega_p \rightarrow \omega_p/\sqrt{\gamma}\).

Simulation laps are typically in units of total plasma frequency so that one time step \(\Delta t = \hat{\omega}_p^{-1}\) in physical units.

Plasma skin depth

Plasma perturbations that move at the speed of light \(c\) and oscillate at the Langmuir frequency define a length scale known as the plasma skin depth,

\[d_s = \frac{c}{\omega_{p,s}} \quad\mathrm{and}\quad \\]

One skin depth in the code is \(\hat{R}\) grid cells.

Plasma magnetization

A finite magnetic field introduces another set of degrees of freedom to the plasma. The ratio of the magnetic field line tension, \((B \cdot \nabla) B/4\pi \propto B^2/4\pi\), to the plasma rest mass (relativistic enthalpy density), \(\langle \gamma \rangle n m c^2\), is known as the plasma magnetization,

\[\sigma_s = \frac{B^2}{4\pi n_s \langle \gamma \rangle m_s c^2}\]

The magnetization can be used to express the code magnetic field as

\[\hat{B} = \frac{ \hat{c}^2 \sqrt{\sigma} }{\hat{R}} \frac{\Delta x}{\Delta t^2}\]

Note

The magnetization can also be expressed as a ratio of the magnetic energy density, \(U_B = B^2/8\pi\) to the plasma rest mass, so that the ratio is \(\sigma = B^2/8\pi n m c^2\), i.e., there is a difference of a factor \(4\pi\) vs \(8\pi\).

Gyrofrequency

The angular frequency of a charged particle gyrating around a magnetic field \(B\) is known as the gyrofrequency,

\[\omega_B = \frac{|q_s| B}{\gamma m_s c} \quad\mathrm{and}\quad \hat{\omega_B} = \frac{\hat{c} \sqrt{\sigma} }{\hat{R}} \frac{1}{\Delta t}\]

Larmor radius

A charged particle gyrating in a magnetic field forms a “ring” around the magnetic field line with a radius known as the gyroradius, or Larmor radius,

\[r_L = \frac{c}{\omega_B} = \frac{\gamma \beta m_s c^2}{|q_s| B} \quad\mathrm{and}\quad \hat{r}_L = \frac{\hat{c}}{ \sqrt{\sigma}} \Delta x\]

Additional ratios of scales

These definitions also allow a slightly different way of expressing the magnetization, as a ratio of the gyrofrequency to the plasma frequency, or as a ratio of the Larmor radius to the skin depth,

\[\sigma = \left( \frac{\omega_B}{\omega_p} \right)^2 = \left( \frac{d_e}{r_L} \right)^2\]

Note that a high magnetization means that the gyrofrequency increases, \(\omega_B \propto \sqrt{\sigma}\) and Larmor radius decreases, \(r_L \propto \gamma/\sqrt{\sigma}\); therefore, we need to be careful that the particle gyrations are still resolved, \(\Delta x < r_L = \sqrt{\sigma}/\hat{R} \hat{c}\).