A. Bokshi

arka.bokshi@gmail.com

Collision frequency

The frequency with which a species \(a\), characterized by its mass \(m_a\) and temperature \(T_a\), undergoes collisions with a background species \(b\) with number density \(n_b\), is given by (in SI units)

(1)\[ \begin{equation} \nu_{ab} = \frac{ \sqrt{2}\pi n_b Z_a^2 Z_b^2 e^4 \ln \Lambda^{a/b} }{ (4\pi \epsilon_0)^2 \sqrt{m_a} T_a^{3/2} } \end{equation}\]

The collision frequency as defined in various codes is written in the Pyrokinetics normalization convention.

CGYRO

CGYRO defines the normalized electron-electron collision frequency \(\nu_{ee}\) in Gaussian units – without the factor \((4\pi\epsilon_0)^2\), i.e.

\[\begin{split} \nu_{ee} & = \frac{ 4\pi n_e e^4 \ln \Lambda }{ \sqrt{m_e} (2T_e)^{3/2} } \end{split} \label{eq:nu_cgyro}\]

All other collision rates are self-consistently determined from this \(\nu_{ee}a/c_s\), which is equal to NU_EE in the input deck.

GENE

In GENE, the parameter coll is defined as

\[\begin{split}\begin{split} \nu_{c} &= \pi \ln \Lambda e^4 n_\mathrm{ref} L_\mathrm{ref} / (2^{3/2}T^2_\mathrm{ref}) \\ &= 2.3031\cdot 10^{-5} \frac{ L_\mathrm{ref} n^{19}_\mathrm{ref} } {(T^k_\mathrm{ref})^2} \ln \Lambda \end{split} \label{eq:nu_c}\end{split}\]

(note the division by \((4\pi \epsilon_0)^2\)) and where

\[\begin{split} \ln \Lambda = 24 - \ln \left( \frac{\sqrt{n_e 10^{-6}}}{T_e} \right) \end{split}\]

Working with the assumptions that in the calculation of coll we have used

\[\begin{split}\begin{split} n_\mathrm{ref}&=n_e\\ T_\mathrm{ref}&=T_e\\ m_\mathrm{ref}&=m_D \end{split}\end{split}\]

i.e. \(c_s=\sqrt{T_e/m_D}\), we arrive at

\[\begin{split} \hat{\nu}_{ee} &= \nu_{ee} \frac{a}{c_s} = 4 \sqrt{\frac{m_D}{m_e}} \nu_c \frac{a}{L_\mathrm{ref}} \end{split}\]

GS2

GS2 defines the normalized collision frequency vnewk = nu_ss Lref/vref, where from equation (1) it naturally follows that

\[\begin{split} \nu_{ss} & = \frac{ \sqrt{2}\pi n_s Z_s^4 e^4 \ln \Lambda }{ (4\pi \epsilon_0)^2 \sqrt{m_s} T_s^{3/2} } \end{split} \label{eq:nu_gs2}\]

Expressed in terms of the electron-electron collisions:

\[\begin{split} \nu_{ss} & = \frac{ \hat{n}_s Z_s^4 }{ \sqrt{\hat{m}_s} \hat{T}_s^{3/2} } \nu_{ee} \end{split} \label{eq:nu_ee}\]

where we have assumed that the Coulomb logarithm is roughly invariant and hat refers to normalization by the corresponding electron parameters.

GKW

GKW defines the normalized collision frequency \(\Gamma_N^{a/b} = \nu_{ab} R_\mathrm{ref}/v_\mathrm{tha}\), which yields

\[\begin{split} \Gamma_{N}^{a/b} & = 6.5141.10^{-5} \frac{R_\mathrm{ref} n^{19}_b}{(T^k_a)^2} Z_a^2 Z_b^2 \ln \Lambda^{a/b} \end{split}\]

GKW can define all other collision frequencies in terms of the singly charged ion-ion collision frequency taken at some reference values, \(\Gamma_N^{i/i}\) or coll_freq, by setting the switch freq_override=.true.

\[\begin{split} \Gamma_{N}^{i/i} = 6.5141 \cdot 10^{-5} \frac{R_\mathrm{ref} n^{19}_\mathrm{ref}}{(T^k_\mathrm{ref})^2} \ln \Lambda^{i/i} \end{split}\]

Here \(R_\mathrm{ref},n^{19}_\mathrm{ref},T^k_\mathrm{ref}\) are in \(\mathrm{m},10^{19}/\mathrm{m}^3,\mathrm{keV}\) units respectively. In the Pyrokinetics convention, beginning with (1), and noting that \(\Gamma_N^{i/i} = \nu_{ii} R_\mathrm{ref}/v_\mathrm{thi}\), where \(v_\mathrm{thi}=\sqrt{2T_i/m_i}\), we derive

\[\begin{split} \hat{\nu}_{ee} = \nu_{ee} \frac{a}{c_s} = \sqrt{2} \frac{a}{R_\mathrm{ref}} \left( \frac{T_i}{T_e} \right)^2 \frac{n_e}{n_i} \sqrt{\frac{m_D}{m_e}} \Gamma^{i/i}_N \end{split}\]

TGLF

In TGLF, we define the electron-ion collision frequency, XNUE, which equals \(\nu_{ei} a/c_s\)

\[\begin{split} \nu_{ei} = \frac{n_i}{n_e}\nu_{ee} \end{split}\]

Attention

It would appear that in the source code, TGLF uses the same definition of the collision frequency as CGYRO.