Braginskii Closure#

Collision Frequencies#

The first step in calculating the Braginskii heat fluxes and frictions is to calculate collsiion frequencies between all pairs of species.

The equation for collision frequency of species a with species b is:

\[\nu_{ab} = \frac{Z_a^2 Z_b^2 e^4 \ln{\Lambda_{ab}}}{3\pi^{3/2}\epsilon_0^2} \frac{n_b}{\left(v_a^2+v_b^2\right)^{3/2}} \frac{1}{\mu_{ab}m_a}\]

Note that the collision rate is for Coulomb scattering events, thus \(\nu_{ab}\neq\nu_{ba}\). \(nu_{ab}\) is the frequency at which species a undergoes Coulomb scattering due to collisions with species b.

In fact, due to conservation of momentum:

\[m_a n_a \nu_{ab} = m_b n_b \nu_{ba}\]

Coulomb Logarithm#

The Coulomb logarithm \(\ln\Lambda\) appears in the collision frequency equation in order to account for the cumulative effect of multiple small-angle collisions. Its value depends on the species involved in the collisions, and has qualitatively different forms for electron and ion collisions.

\[\ln{\Lambda_{ee}} = 30.4 - \frac{1}{2}\ln{n_e} + \frac{5}{4}\ln{T_e} - \sqrt{10^{-5}+\left(\ln{T_e}-2\right)^2/16}\]
\[\begin{split}\ln{\Lambda_{ei}} = \begin{cases} 10 & \text{if}\ T_e \lt0.1eV\ \text{or}\ n_e \lt 10^{10}m^{-3} \\ 30 - \frac{1}{2}\ln{n_e} - \ln{Z} + \frac{3}{2}\ln{T_e} & \text{if}\ T_i m_e/m_i \lt T_e \lt 10Z^2 \\ 31 - \frac{1}{2}\ln{n_e} + \ln{T_e} & \text{if}\ T_i m_e/m_i \lt 10Z^2 \lt T_e \\ 23 - \frac{1}{2}\ln{n_e} + \frac{3}{2}\ln{T_e} - \ln{\left(Z^2A\right)} & \text{if}\ T_e \lt T_i m_e/m_i \end{cases}\end{split}\]
\[\ln{\Lambda_{ii}} = 29.91 - \ln{\left[\frac{Z_aZ_b\left(m_a+m_b\right)}{m_aT_b+m_bT_a} \sqrt{\frac{n_aZ_a^2}{T_a} + \frac{n_bZ_b^2}{T_b}}\right]}\]

Heat flux#

The collisional parallel heat flux for species i is given by:

\[\mathbf{q}_i = C_i\frac{n T_i}{m_i \nu_i} \left(\mathbf{b}\cdot\nabla T_i\right)\mathbf{b}\]

where

\[\nu_i = \sum_j \nu_{ij}\]

\(C_{ion} = 3.9\) and \(C_{electron} = 3.19\)

Friction#

Momentum is exchanged between two species by collisions according to:

\[F_{ab} = \nu_{ab} m_a n_a \left(v_b - v_a\right)\]

Associated with this frictional momentum exchange is a frictional heating (Joule heating):

\[Q_{ab} = \frac{m_b}{m_a + m_b} \left(v_b - v_a\right) F_{ab}\]

Both species’ internal energy increases as a result of collisions.

Heat Exchange#

Two collisional species exchange heat if they are at different temperatures

The heat exchanged from species b to species a by this mechanism is:

\[W_{ab} = 3 \nu_{ab} n_a \frac{m_a}{m_a+m_b} \left(T_b - T_a\right)\]