Units

We typically use four fundamental quantities to characterize the simulations:

numerical speed of light, \(\hat{c}\) (cfl or c)
particle number density, \(\hat{n_{s}}\) (ppc),
numerical skin-depth resolution, \(\hat{R}\) (c_omp), and
plasma magnetization, \(\sigma\) (sigma).

Note

Technically we are using the cgs-Gaussian unit system in the following discussions. This is an unrationalized unit system meaning that factors of \(4\pi\) appear in the Maxwell’s equations.

Electromagnetic field

Faraday’s and Ampere’s laws are

\[ \begin{align}\begin{aligned}\frac{\partial E}{\partial t} &=+c \nabla \times B - 4\pi J\\\frac{\partial B}{\partial t} &=-c \nabla \times E\end{aligned}\end{align} \]

that in code units corresponds to solving

\[ \begin{align}\begin{aligned}\Delta_t \hat{E} &=+\hat{c} \nabla \times \hat{B} - \hat{c} \hat{J}\\\Delta_t \hat{B} &=-\hat{c} \nabla \times \hat{E}\end{aligned}\end{align} \]

The benefits are clear: the equations appear symmetric and only have the numerical speed of light, \(\hat{c}\) as a natural constant in them.

Lorentz force

The Lorentz force experienced by charged particles is

\[\frac{d (\gamma \beta) }{d t} = \frac{q_s}{m_s} (E + \beta \times B)\]

that in code units corresponds to solving

\[\Delta_t (\gamma \beta) = \hat{c} \frac{\hat{q}}{\hat{m}} (\hat{E} + \beta \times \hat{B})\]

Note

Only the charge-to-mass ratio enters the Lorentz force.

Current density

Charge current density and its numerical counterparts are

\[4\pi J = 4\pi q \beta c \quad\mathrm{and}\quad \hat{J}\Delta t = \hat{q} \beta \hat{c}^2\]

Current density is technically a flux of charge through an area per time; therefore, the current has numerical units of \(\hat{J} = \frac{v \hat{q}}{\Delta x^2 \Delta t}\).

Note

Technically we store the current density per time step in the grid; hence the extra \(\hat{c} = \Delta t\) factor in the expression of the numerical charge current density.