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.