Plasma model
Equations for the plasma density \(n\), pressure \(p\) and momentum \(m_inV_{||i}\) are evolved:
\[\begin{split}\begin{aligned}
\frac{\partial n}{\partial t} &=& - \nabla\cdot\left( \mathbf{b}V_{||} n\right) + S_n - S\\
\frac{\partial}{\partial t}\left(\frac{3}{2}p\right) &=& -\nabla\cdot\mathbf{q} + V_{||}\partial_{||}p + S_p - E - R \\
\frac{\partial}{\partial t}\left(m_i nV_{||}\right) &=& -\nabla\cdot\left(m_inV_{||}\mathbf{b}V_{||}\right) - \partial_{||} p - F\\
j_{||} &=& 0 \\
T_i &=& T_e = \frac{1}{2}\frac{p}{en} \\
\mathbf{q} &=& \frac{5}{2}p\mathbf{b}V_{||} - \kappa_{||e}\partial_{||}T_e\end{aligned}\end{split}\]
Which has a conserved energy:
\[\int_V \left[ \frac{1}{2}m_inV_{||i}^2 + \frac{3}{2}p \right] dV\]
The heat conduction coefficient \(\kappa_{||e}\) is a nonlinear function of temperature \(T_e\):
\[\kappa_{||e} = \kappa_0 T_e^{5/2}\]
where \(\kappa_0\) is a constant. See section 8 for details.
Operators are:
\[\partial_{||}f = \mathbf{b}\cdot\nabla f \qquad \nabla_{||} f = \nabla\cdot\left(\mathbf{b} f\right)\]
Heat conduction
Spitzer heat conduction is used
\[\kappa_{||e} = 3.2\frac{ne^2T\tau_e}{m_e} \simeq 3.1\times 10^4 \frac{T^{5/2}}{\ln \Lambda}\]
which has units of W/m/eV so that in the formula \(q = -\kappa_{||e} \nabla T\), \(q\) has units of Watts per m\(^2\) and \(T\) has units of \(eV\). This uses the electron collision time:
\[\tau_e = \frac{6\sqrt{2}\pi^{3/2}\epsilon_0^2\sqrt{m_e}T_e^{3/2}}{\ln \Lambda e^{2.5} n} \simeq 3.44\times 10^{11} \frac{T_e^{3/2}}{\ln \Lambda n}\]
in seconds, where \(Te\) is in eV, and \(n\) is in m\(^{-3}\).
Normalising by the quantities in table 1 gives
\[\hat{\kappa}_{||e} = 3.2 \hat{n}\hat{T}_e\frac{m_i}{m_e}\tau_e\Omega_{ci}\]
where hats indicate normalised (dimensionless) variables.