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, :math:`\hat{x}`.


We keep the grid spacing and time step, :math:`\Delta x` and :math:`\Delta t`, explicitly written in the equations even though they are set equal to :math:`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 :math:`\omega_{p,s}`, for a particle species :math:`s` with number density :math:`n_s` and charge :math:`q_s` is

.. math::
    \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 :math:`\Gamma` or relativistically hot plasmas with mean Lorentz factor :math:`\langle \gamma \rangle \gtrsim 1` the plasma frequency becomes :math:`\omega_p \rightarrow \omega_p/\sqrt{\gamma}`.


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


Plasma skin depth
-----------------

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

.. math::
    d_s = \frac{c}{\omega_{p,s}}
    \quad\mathrm{and}\quad
    \

One skin depth in the code is :math:`\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, :math:`(B \cdot \nabla) B/4\pi \propto B^2/4\pi`, to the plasma rest mass (relativistic enthalpy density), :math:`\langle \gamma \rangle n m c^2`, is known as the plasma magnetization,

.. math::
    \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

.. math::
    \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, :math:`U_B = B^2/8\pi` to the plasma rest mass, so that the ratio is :math:`\sigma = B^2/8\pi n m c^2`, i.e., there is a difference of a factor :math:`4\pi` vs :math:`8\pi`.



Gyrofrequency
-------------

The angular frequency of a charged particle gyrating around a magnetic field :math:`B` is known as the gyrofrequency,

.. math::
    \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,

.. math::
    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,

.. math::
    \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, :math:`\omega_B \propto \sqrt{\sigma}` and Larmor radius decreases, :math:`r_L \propto \gamma/\sqrt{\sigma}`;
therefore, we need to be careful that the particle gyrations are still resolved, :math:`\Delta x < r_L = \sqrt{\sigma}/\hat{R} \hat{c}`.