Moment kinetic equations for electrons

Moment equations

Quasineutrality gives

\[\begin{align} n_e &= n_i \end{align}\]

while the continuity equation is

\[\begin{align} \frac{\partial n_e}{\partial t} + \frac{\partial}{\partial z}\left( n_e u_{e\parallel} \right) &= n_e n_n R_\mathrm{ioniz} + S_{e,n} \end{align}\]

Subtracting from ion continuity and using quasineutrality

\[\begin{align} \frac{\partial}{\partial z}\left(n_e (u_{i\parallel} - u_{e\parallel}) \right) &= S_{i,n} - S_{e,n} \end{align}\]

giving an equation for $u_{e\parallel}$. Assuming that the sources do not add any charge $S_{i,n} = S_{e,n}$ then implies that $u_{e\parallel} = u_{i\parallel} + u_0$ assuming in addition (at least for the moment) that there is no current through the sheath sets $u_0 = 0$ so that

\[\begin{align} u_{e\parallel} &= u_{i\parallel} \end{align}\]

In the parallel momentum equation, the inertial terms can be neglected as they are $O(m_e/m_i)$, leaving

\[\begin{align} 0 &= -\frac{\partial p_{e\parallel}}{\partial z} + e n_e \frac{\partial \phi}{\partial z} + F_{ei\parallel} + m_e \int v_\parallel C_{en} d^3 v + S_{e,\mathrm{mom}} \end{align}\]

where $C_{en}$ represents elastic electron-neutral collisions (for collisions see Electron collisions), which gives an equation for $E_\parallel = -\partial \phi/\partial z$

\[\begin{align} e E_\parallel &= -\frac{1}{n_e} \frac{\partial p_{e\parallel}}{\partial z} + \frac{F_{ei\parallel}}{n_e} + \frac{m_e}{n_e} \int v_\parallel C_{en} d^3 v + \frac{1}{n_e} S_{e,\mathrm{mom}} \\ e E_\parallel &= -\frac{1}{n_e} \frac{\partial p_{e\parallel}}{\partial z} + m_e \nu_{ei} \left( u_{i\parallel} - u_{e\parallel} \right) + \frac{1}{n_e} S_{e,\mathrm{mom}} \\ \end{align}\]

where the second line assumes the current Krook collision operator and neglects electron-neutral elastic collisions.

The energy equation is similar to the ion one

\[\begin{align} &\frac{3}{2} \frac{\partial p_e}{\partial t} + \frac{\partial q_{e\parallel}}{\partial z} + p_{e\parallel} \frac{\partial u_{e\parallel}}{\partial z} + \frac{3}{2} u_{e\parallel} \frac{\partial p_e}{\partial z} + \frac{3}{2} p_e \frac{\partial u_{e\parallel}}{\partial z} \nonumber \\ &\quad= -E_\mathrm{ioniz} n_e n_n R_\mathrm{ioniz} + \int \frac{1}{2} m_e |\boldsymbol{v} - u_{e\parallel} \hat{\boldsymbol{z}}|^2 C_{ei} d^3 v + \int \frac{1}{2} m_e |\boldsymbol{v} - u_{e\parallel} \hat{\boldsymbol{z}}|^2 C_{en} d^3 v \nonumber \\ &\qquad + \frac{3}{2} S_{e,p} \\ &\frac{3}{2} \frac{\partial p_e}{\partial t} + \frac{\partial q_{e\parallel}}{\partial z} + p_{e\parallel} \frac{\partial u_{e\parallel}}{\partial z} + \frac{3}{2} u_{e\parallel} \frac{\partial p_e}{\partial z} + \frac{3}{2} p_e \frac{\partial u_{e\parallel}}{\partial z} \nonumber \\ &\quad= -E_\mathrm{ioniz} n_e n_n R_\mathrm{ioniz} + 3 n_e \frac{m_e}{m_i} \nu_{ei} \left( T_i - T_e \right) + m_e n_e \nu_{ei} \left( u_{i\parallel} - u_{e_\parallel} \right)^2 \nonumber \\ &\qquad + \frac{3}{2} S_{e,p} \\ \end{align}\]

To get the second version we treat the collision operators as described in Electron collisions.

Electron collisions

Currently electron-neutral elastic collisions $C_{en}$ are not implemented.

Electron-electron collisions use a Krook operator

\[\begin{align} C_{K,ee} &= -\nu_{ee}(n_e,T_e) \left(f_e - \frac{n_e}{\pi^{3/2} v_{Te}^3} \exp\left(-\frac{|\boldsymbol{v} - u_{e\parallel}\hat{\boldsymbol{z}}|^2}{v_{Te}^2}\right) \right) \end{align}\]

with the electron-electron collision frequency

\[\begin{align} \nu_{ee}(n_e,T_e) = \frac{n_e e^4 \log\Lambda_{ee}}{4 \pi \epsilon_0^2 m_e^2 v_{Te}^3} \end{align}\]

Electron-ion collisions also use a Krook operator

\[\begin{align} C_{K,ei} &= -\nu_{ei}(n_e,T_e) \left(f_e - \frac{n_e}{\pi^{3/2} v_{Te}^3} \exp\left(-\frac{|\boldsymbol{v} - u_{i\parallel}\hat{\boldsymbol{z}}|^2}{v_{Te}^2}\right) \right) \end{align}\]

with the electron-ion collision frequency

\[\begin{align} \nu_{ei}(n_e,T_e) = \frac{n_e e^4 \log\Lambda_{ei}}{4 \pi \epsilon_0^2 m_e^2 v_{Te}^3} \end{align}\]

For the Krook operator, the friction is

\[\begin{align} F_{ei\parallel} &= -m_e \int v_\parallel C_{K,ei} d^3 v \nonumber \\ &= m_e n_e \nu_{ei} \left( u_{i\parallel} - u_{e\parallel} \right) \end{align}\]

Energy exchange with ions is kept assuming the distributions are Maxwellian to allow different temperatures (this form comes from the Fokker-Planck collision operator assuming Maxwellian distributions)

\[\begin{align} \int \frac{1}{2} m_i |\boldsymbol{v} - u_{i\parallel} \hat{\boldsymbol{z}}|^2 C_{ie}[f_i,f_e] d^3 v &\approx 3 \frac{n_e m_e \nu_{ei}}{m_i} (T_e - T_i) \end{align}\]

in the ion energy equation. In the electron energy equation, need the conversion

\[\begin{align} \int \frac{1}{2} m_e |\boldsymbol{v} - u_{e\parallel} \hat{\boldsymbol{z}}|^2 C_{ei}[f_e,f_i] d^3 v &= \int \left(\frac{1}{2} m_e v^2 - m_e u_{e\parallel} v_\parallel + \underbrace{\frac{1}{2} m_e u_{e\parallel}^2}_{=0\text{ particle conservation}} \right) C_ei d^3 v \nonumber \\ &= -\int \frac{1}{2} m_i v^2 C_{ie} - m_e u_{e\parallel} \underbrace{\int v_\parallel C_{ei} d^3 v}_{F_{ei\parallel}/m_e} \nonumber \\ &= -\int \frac{1}{2} m_i |\boldsymbol{v} - u_{i\parallel} \hat{\boldsymbol{z}}|^2 C_ie d^3 v - \underbrace{\int m_i u_{i\parallel} v_\parallel C_{ie} d^3 v}_{\text{momentum conservation } = u_{i\parallel} (-F_{ie\parallel}) = u_{i\parallel} F_{ei\parallel}} - F_{ei\parallel} u_{e\parallel} \nonumber \\ &= 3 n_e \frac{m_e}{m_i} \nu_{ei} \left(T_i - T_e \right) + F_{ei\parallel} \left(u_{i\parallel} - u_{e\parallel} \right) \\ \end{align}\]

1D2V kinetic equation

For the electrons $u_{e\parallel} \sim u_{i\parallel}$ by quasineutrality so $u_{e\parallel} \sim v_{Ti} \ll v_{Te}$, which means we can neglect $u_{e\parallel}$ in most of the terms for evolution of $f_e$.

Similar to the ion shape function equation, but $\partial F_e/\partial t$ is negligible, i.e. electrons move on timescales faster than the system evolution.

\[\begin{align} \dot{z}_e \frac{\partial F_e}{\partial z} + \dot{w}_{\parallel,e} \frac{\partial F_e}{\partial w_\parallel} + \dot{w}_{\perp,e} \frac{\partial F_e}{\partial w_\perp} &= \dot{F}_e + \mathcal{C}_{ee} + \mathcal{C}_{ei} + \mathcal{C}_{en} + \mathcal{S}_e \end{align}\]

where

\[\begin{align} \dot{z}_e &= v_{Te} w_\parallel \\ \dot{w}_{\parallel,e} &= \frac{1}{n_e m_e v_{Te}} \frac{\partial p_{e\parallel}}{\partial z} + \frac{w_\parallel}{3 p_e} \frac{\partial q_{e\parallel}}{\partial z} - w_\parallel^2 \frac{\partial v_{T_e}}{\partial z} \nonumber \\ &\quad - \frac{1}{m_e n_e v_{Te}} (S_{e,\mathrm{mom}} - m_e u_{e\parallel} S_{e,n}) - \frac{w_\parallel}{2 p_e} (S_{e,p} - T_e S_{e,n}) \\ \dot{w}_{\perp,e} &= \frac{w_\perp}{3 p_e} \frac{\partial q_{e\parallel}}{\partial z} - w_\perp w_\parallel \frac{\partial v_{Te}}{\partial z} \nonumber \\ &\quad - \frac{w_\perp}{2 p_e} (S_{e,p} - T_e S_{e,n}) \\ \frac{\dot{F}_e}{F_e} &= w_\parallel \left( 3 \frac{\partial v_{Te}}{\partial z} - \frac{v_{Te}}{n_e} \frac{\partial n_e}{\partial z} \right) - \frac{1}{p_e} \frac{\partial q_{e\parallel}}{\partial z} \nonumber \\ &\quad + \frac{3}{2 p_e} S_{e,p} - \frac{5}{2 n_e} S_{e,n} \\ \mathcal{C}_{ee} &= \frac{v_{Te}^3}{n_e} C_{K,ee} \\ &= -\frac{v_{Te}^3}{n_e} \nu_{ee} \left( f_e - \frac{n_e}{\pi^{3/2} v_{Te}^3} \exp\left( -\frac{|\boldsymbol{v} - u_{e\parallel}\hat{\boldsymbol{z}}|^2}{v_{Te}^2} \right) \right) \\ &= - \nu_{ee} \left( F_e - \frac{1}{\pi^{3/2}} \exp\left( -w^2 \right) \right) \\ \mathcal{C}_{ei} &= \frac{v_{Te}^3}{n_e} C_{K,ei} \\ &= -\frac{v_{Te}^3}{n_e} \nu_{ei} \left( f_e - \frac{n_e}{\pi^{3/2} v_{Te}^3} \exp\left( -\frac{|\boldsymbol{v} - u_{i\parallel}\hat{\boldsymbol{z}}|^2}{v_{Te}^2} \right) \right) \\ &= - \nu_{ei} \left( F_e - \frac{1}{\pi^{3/2}} \exp\left( -\left( w_\parallel - \frac{(u_{i\parallel} - u_{e\parallel})}{v_{Te}} \right)^2 - w_\perp^2 \right) \right) \\ \mathcal{C}_{en} &= \text{not implemented yet} \\ \mathcal{S}_{e} &= \frac{v_{Te}^3}{n_e} S_e \\ \end{align}\]

Although $S_e$ should mostly be small in $\sqrt{m_e/m_i}$, we keep it for the case when the source is a Maxwellian with a temperature significantly higher than the electron temperature because then the source can contribute significantly to the high energy tail of electrons. As we keep $\mathcal S_e$, to ensure that the first 3 moments of the shape function equation vanish, we must also keep the source contributions to $\dot w_{\parallel,e}$, $\dot w_{\perp,e}$, and $\dot F_e/F_e$ in the same form as the ion equations.

Reduction to 1D1V

We can take the $T_{e\perp}=0$ limit of the equations and marginalise over $v_\perp$/$w_\perp$ to reduce to 1D1V in a very similar way as for the ions.

Quasineutrality and force balance keep the same form, and the energy equation becomes

\[\begin{align} &\frac{3}{2} \frac{\partial p_e}{\partial t} + \frac{\partial q_{e\parallel}}{\partial z} + p_{e\parallel} \frac{\partial u_{e\parallel}}{\partial z} + \frac{3}{2} u_{e\parallel} \frac{\partial p_e}{\partial z} + \frac{3}{2} p_e \frac{\partial u_{e\parallel}}{\partial z} \nonumber \\ &\quad= -E_\mathrm{ioniz} n_e n_n R_\mathrm{ioniz} + \int \frac{1}{2} m_e (v_\parallel - u_{e\parallel})^2 \bar{C}_{ei} dv_\parallel + \int \frac{1}{2} m_e (v_\parallel - u_{e\parallel})^2 \bar{C}_{en} dv_\parallel \nonumber \\ &\qquad + \frac{3}{2} S_{e,p} \\ &\frac{3}{2} \frac{\partial p_e}{\partial t} + \frac{\partial q_{e\parallel}}{\partial z} + p_{e\parallel} \frac{\partial u_{e\parallel}}{\partial z} + \frac{3}{2} u_{e\parallel} \frac{\partial p_e}{\partial z} + \frac{3}{2} p_e \frac{\partial u_{e\parallel}}{\partial z} \nonumber \\ &\quad= -E_\mathrm{ioniz} n_e n_n R_\mathrm{ioniz} + 3 n_e \frac{m_e}{m_i} \nu_{ei} \left( T_i - T_e \right) + m_e n_e \nu_{ei} \left( u_{i\parallel} - u_{e_\parallel} \right)^2 \nonumber \\ &\qquad + \frac{3}{2} S_{e,p} \\ \end{align}\]

The kinetic equation for $\bar F_e = \int F_e d^2 w_\perp$ is

\[\begin{align} \dot{z}_e \frac{\partial \bar{F}_e}{\partial z} + \dot{w}_{\parallel,e} \frac{\partial \bar{F}_e}{\partial w_\parallel} &= \dot{\bar{F}}_e + \bar{\mathcal{C}}_{ee} + \bar{\mathcal{C}}_{ei} + \bar{\mathcal{C}}_{en} + \bar{\mathcal{S}}_e \end{align}\]

\[\begin{align} \dot{z}_e &= v_{Te} w_\parallel \\ \dot{w}_{\parallel,e} &= \frac{1}{n_e m_e v_{Te}} \frac{\partial p_{e\parallel}}{\partial z} + \frac{w_\parallel}{3 p_e} \frac{\partial q_{e\parallel}}{\partial z} - w_\parallel^2 \frac{\partial v_{T_e}}{\partial z} \nonumber \\ &\quad - \frac{1}{m_e n_e v_{Te}} (S_{e,\mathrm{mom}} - m_e u_{e\parallel} S_{e,n}) - \frac{w_\parallel}{2 p_e} (S_{i,p} - T_e S_{e,n}) \\ \frac{\dot{\bar{F}}_e}{\bar{F}_e} &= w_\parallel \left( \frac{\partial v_{Te}}{\partial z} - \frac{v_{Te}}{n_e} \frac{\partial n_e}{\partial z} \right) - \frac{1}{3 p_e} \frac{\partial q_{e\parallel}}{\partial z} \nonumber \\ &\quad + \frac{1}{2 p_e} S_{e,p} - \frac{3}{2 n_e} S_{e,n} \\ \bar{\mathcal{C}}_{ee} &= \frac{v_{Te}}{n_e} \bar{C}_{K,ee} \\ &= -\frac{v_{Te}}{n_e} \nu_{ee} \left( \bar{f}_e - \frac{n_e}{\sqrt{\pi} \sqrt{2 T_{e\parallel}/m_e}} \exp\left( -\frac{m_e (v_\parallel - u_{e\parallel})^2}{2 T_{e\parallel}} \right) \right) \\ &= - \nu_{ee} \left( \bar{F}_e - \frac{1}{\sqrt{3 \pi}} \exp\left( - \frac{w_\parallel^2}{3} \right) \right) \\ \bar{\mathcal{C}}_{ei} &= \frac{v_{Te}}{n_e} \bar{C}_{K,ei} \\ &= -\frac{v_{Te}}{n_e} \nu_{ei} \left( \bar{f}_e - \frac{n_e}{\sqrt{\pi} \sqrt{2 T_{e\parallel}/m_e}} \exp\left( -\frac{m_e (v_\parallel - u_{i\parallel})^2}{2 T_{e\parallel}} \right) \right) \\ &= - \nu_{ei} \left( \bar{F}_e - \frac{1}{\sqrt{3 \pi}} \exp\left( - \frac{\left( w_\parallel - \frac{(u_{i\parallel} - u_{e\parallel})}{v_{Te}} \right)^2}{3} \right) \right) \\ \bar{\mathcal{C}}_{en} &= \text{not implemented yet} \\ \bar{\mathcal{S}}_{e} &= \frac{v_{Te}}{n_e} \bar{S}_e = \frac{v_{Te}}{n_e} \int S_e d^2 v_\perp \\ \end{align}\]

recalling that $\int w_\perp \frac{\partial F_e}{\partial w_\perp} d^2 w_\perp = 2 \bar F_e$ as for the ions, and noting that in the $T_{e,\perp} = 0$ limit, $T_{e\parallel} = T_e/3$ so that $v_{Te}^2 = 2T_e/m_e = 6 T_{e\parallel}/m_e$.

Dimensionless equations

We make the equations dimensionless using the conversions defined here.

1D2V

The moment equations become

\[\begin{align} \hat{n}_e &= \hat{n}_i \\ \hat{u}_{e\parallel} &= \hat{u}_{i\parallel} \\ \hat{E}_\parallel &= -\frac{1}{\hat{n}_e} \frac{\partial \hat{p}_{e\parallel}}{\partial \hat{z}} + \frac{\hat{F}_{ei\parallel}}{\hat{n}_e} + \frac{\hat{m}_e}{\hat{n}_e} \int \hat{v}_\parallel \hat{C}_{en} d^3 \hat{v} + \frac{1}{\hat{n}_e} \hat{S}_{e,\mathrm{mom}} \\ \hat{E}_\parallel &= -\frac{1}{\hat{n}_e} \frac{\partial \hat{p}_{e\parallel}}{\partial \hat{z}} + \hat{m}_e \hat{\nu}_{ei} \left( \hat{u}_{i\parallel} - \hat{u}_{e\parallel} \right) + \frac{1}{\hat{n}_e} \hat{S}_{e,\mathrm{mom}} \\ \end{align}\]

\[\begin{align} &\frac{3}{2} \frac{\partial \hat{p}_e}{\partial \hat{t}} + \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} + \hat{p}_{e\parallel} \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{e\parallel} \frac{\partial \hat{p}_e}{\partial \hat{z}} + \frac{3}{2} \hat{p}_e \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{E}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{R}_\mathrm{ioniz} + \int \frac{1}{2} \hat{m}_e |\hat{\boldsymbol{v}} - \hat{u}_{e\parallel} \hat{\boldsymbol{z}}|^2 \hat{C}_{ei} d^3 v + \int \frac{1}{2} \hat{m}_e |\hat{\boldsymbol{v}} - \hat{u}_{e\parallel} \hat{\boldsymbol{z}}|^2 \hat{C}_{en} d^3 v \nonumber \\ &\qquad + \frac{3}{2} \hat{S}_{e,p} \\ &\frac{3}{2} \frac{\partial \hat{p}_e}{\partial \hat{t}} + \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} + \hat{p}_{e\parallel} \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{e\parallel} \frac{\partial \hat{p}_e}{\partial \hat{z}} + \frac{3}{2} \hat{p}_e \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{E}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{R}_\mathrm{ioniz} + 3 \hat{n}_e \frac{\hat{m}_e}{\hat{m}_i} \hat{\nu}_{ei} \left( \hat{T}_i - \hat{T}_e \right) + \hat{m}_e \hat{n}_e \hat{\nu}_{ei} \left( \hat{u}_{i\parallel} - \hat{u}_{e_\parallel} \right)^2 \nonumber \\ &\qquad + \frac{3}{2} \hat{S}_{e,p} \\ \end{align}\]

and the dimensionless kinetic equation is

\[\begin{align} \hat{\dot{z}}_e \frac{\partial F_e}{\partial \hat{z}} + \hat{\dot{w}}_{\parallel,e} \frac{\partial F_e}{\partial w_\parallel} + \hat{\dot{w}}_{\perp,e} \frac{\partial F_e}{\partial w_\perp} &= \hat{\dot{F}}_e + \hat{\mathcal{C}}_{ee} + \hat{\mathcal{C}}_{ei} + \hat{\mathcal{C}}_{en} + \hat{\mathcal{S}}_e \end{align}\]

where

\[\begin{align} \hat{\dot{z}}_e &= \hat{v}_{Te} w_\parallel \\ \hat{\dot{w}}_{\parallel,e} &= \frac{1}{\hat{n}_e \hat{m}_e \hat{v}_{Te}} \frac{\partial \hat{p}_{e\parallel}}{\partial \hat{z}} + \frac{w_\parallel}{3 \hat{p}_e} \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} - w_\parallel^2 \frac{\partial \hat{v}_{T_e}}{\partial \hat{z}} \nonumber \\ &\quad - \frac{1}{\hat{m}_e \hat{n}_e \hat{v}_{Te}} (S_{e,\mathrm{mom}} - m_e u_{e\parallel} S_{e,n}) - \frac{w_\parallel}{2 \hat{p}_e} (\hat{S}_{e,p} - T_e S_{e,n}) \\ \hat{\dot{w}}_{\perp,e} &= \frac{w_\perp}{3 \hat{p}_e} \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} - w_\perp w_\parallel \frac{\partial \hat{v}_{Te}}{\partial \hat{z}} \nonumber \\ &\quad - \frac{w_\perp}{2 \hat{p}_e} ()\hat{S}_{e,p} - T_e \hat{S}_{e,n}) \\ \frac{\hat{\dot{F}}_e}{F_e} &= w_\parallel \left( 3 \frac{\partial \hat{v}_{Te}}{\partial \hat{z}} - \frac{\hat{v}_{Te}}{\hat{n}_e} \frac{\partial \hat{n}_e}{\partial \hat{z}} \right) - \frac{1}{\hat{p}_e} \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad + \frac{3}{2 \hat{p}_e} \hat{S}_{e,p} - \frac{5}{2 \hat{n}_e} \hat{S}_{e,n} \\ \hat{\mathcal{C}}_{ee} &= \frac{\hat{v}_{Te}^3}{\hat{n}_e} \hat{C}_{K,ee} \\ &= -\frac{\hat{v}_{Te}^3}{\hat{n}_e} \hat{\nu}_{ee} \left( \hat{f}_e - \frac{\hat{n}_e}{\pi^{3/2} \hat{v}_{Te}^3} \exp\left( -\frac{|\hat{\boldsymbol{v}} - \hat{u}_{e\parallel}\hat{\boldsymbol{z}}|^2}{\hat{v}_{Te}^2} \right) \right) \\ &= - \hat{\nu}_{ee} \left( F_e - \frac{1}{\pi^{3/2}} \exp\left( -w^2 \right) \right) \\ \hat{\mathcal{C}}_{ei} &= \frac{\hat{v}_{Te}^3}{\hat{n}_e} \hat{C}_{K,ei} \\ &= -\frac{\hat{v}_{Te}^3}{\hat{n}_e} \hat{\nu}_{ei} \left( \hat{f}_e - \frac{\hat{n}_e}{\pi^{3/2} \hat{v}_{Te}^3} \exp\left( -\frac{|\hat{\boldsymbol{v}} - \hat{u}_{i\parallel}\hat{\boldsymbol{z}}|^2}{\hat{v}_{Te}^2} \right) \right) \\ &= - \hat{\nu}_{ei} \left( F_e - \frac{1}{\pi^{3/2}} \exp\left( -\left( w_\parallel - \frac{(\hat{u}_{i\parallel} - \hat{u}_{e\parallel})}{\hat{v}_{Te}} \right)^2 - w_\perp^2 \right) \right) \\ \hat{\mathcal{C}}_{en} &= \text{not implemented yet} \\ \hat{\mathcal{S}}_{e} &= \frac{\hat{v}_{Te}^3}{\hat{n}_e} \hat{S}_e \\ \end{align}\]

1D1V

In 1D1V the dimsionless energy equation is

\[\begin{align} &\frac{3}{2} \frac{\partial \hat{p}_e}{\partial \hat{t}} + \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} + \hat{p}_{e\parallel} \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{e\parallel} \frac{\partial \hat{p}_e}{\partial \hat{z}} + \frac{3}{2} \hat{p}_e \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{E}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{R}_\mathrm{ioniz} + \int \frac{1}{2} \hat{m}_e (\hat{v}_\parallel - \hat{u}_{e\parallel})^2 \hat{\bar{C}}_{ei} d\hat{v}_\parallel + \int \frac{1}{2} \hat{m}_e (\hat{v}_\parallel - \hat{u}_{e\parallel})^2 \hat{\bar{C}}_{en} d\hat{v}_\parallel \nonumber \\ &\qquad + \frac{3}{2} \hat{\bar{S}}_{e,p} \\ &\frac{3}{2} \frac{\partial \hat{p}_e}{\partial \hat{t}} + \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} + \hat{p}_{e\parallel} \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{e\parallel} \frac{\partial \hat{p}_e}{\partial \hat{z}} + \frac{3}{2} \hat{p}_e \frac{\partial \hat{u}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{E}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{R}_\mathrm{ioniz} + 3 \hat{n}_e \frac{\hat{m}_e}{\hat{m}_i} \hat{\nu}_{ei} \left( \hat{T}_i - \hat{T}_e \right) + \hat{m}_e \hat{n}_e \hat{\nu}_{ei} \left( \hat{u}_{i\parallel} - \hat{u}_{e_\parallel} \right)^2 \nonumber \\ &\qquad + \frac{3}{2} \hat{\bar{S}}_{e,p} \\ \end{align}\]

and the dimensionless kinetic equation is

\[\begin{align} \hat{\dot{z}}_e \frac{\partial \bar{F}_e}{\partial \hat{z}} + \hat{\dot{w}}_{\parallel,e} \frac{\partial \bar{F}_e}{\partial w_\parallel} &= \hat{\dot{\bar{F}}}_e + \hat{\bar{\mathcal{C}}}_{ee} + \hat{\bar{\mathcal{C}}}_{ei} + \hat{\bar{\mathcal{C}}}_{en} + \hat{\bar{\mathcal{S}}}_e \end{align}\]

\[\begin{align} \hat{\dot{z}}_e &= \hat{v}_{Te} w_\parallel \\ \hat{\dot{w}}_{\parallel,e} &= \frac{1}{\hat{n}_e \hat{m}_e \hat{v}_{Te}} \frac{\partial \hat{p}_{e\parallel}}{\partial \hat{z}} + \frac{w_\parallel}{3 \hat{p}_e} \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} - w_\parallel^2 \frac{\partial \hat{v}_{T_e}}{\partial \hat{z}} \nonumber \\ &\quad - \frac{1}{\hat{m}_e \hat{n}_e \hat{v}_{Te}} (\hat{S}_{e,\mathrm{mom}} - \hat{m}_e \hat{u}_{e\parallel} \hat{S}_{e,n}) - \frac{w_\parallel}{2 \hat{p}_e} (\hat{S}_{e,p} - \hat{T}_e \hat{S}_{e,n}) \\ \frac{\hat{\dot{\bar{F}}}_e}{\bar{F}_e} &= w_\parallel \left( \frac{\partial \hat{v}_{Te}}{\partial \hat{z}} - \frac{\hat{v}_{Te}}{\hat{n}_e} \frac{\partial \hat{n}_e}{\partial \hat{z}} \right) - \frac{1}{3 \hat{p}_e} \frac{\partial \hat{q}_{e\parallel}}{\partial \hat{z}} \nonumber \\ &\quad + \frac{1}{2 \hat{p}_e} \hat{S}_{e,p} - \frac{3}{2 \hat{n}_e} \hat{S}_{e,n} \\ \hat{\bar{\mathcal{C}}}_{ee} &= \frac{\hat{v}_{Te}}{\hat{n}_e} \hat{\bar{C}}_{K,ee} \\ &= - \hat{\nu}_{ee} \left( \bar{F}_e - \frac{1}{\sqrt{3 \pi}} \exp\left( - \frac{w_\parallel^2}{3} \right) \right) \\ \hat{\bar{\mathcal{C}}}_{ei} &= \frac{\hat{v}_{Te}}{\hat{n}_e} \hat{\bar{C}}_{K,ei} \\ &= - \hat{\nu}_{ei} \left( \bar{F}_e - \frac{1}{\sqrt{3 \pi}} \exp\left( - \frac{\left( w_\parallel - \frac{(\hat{u}_{i\parallel} - \hat{u}_{e\parallel})}{\hat{v}_{Te}} \right)^2}{3} \right) \right) \\ \hat{\bar{\mathcal{C}}}_{en} &= \text{not implemented yet} \\ \hat{\bar{\mathcal{S}}}_{e} &= \frac{\hat{v}_{Te}}{\hat{n}_e} \hat{\bar{S}}_e = \frac{\hat{v}_{Te}}{\hat{n}_e} \int \hat{S}_e d^2 \hat{v}_\perp \\ \end{align}\]

The conversion to the dimensionless equations in the 1D1V Excalibur reports, and the original version of the code, uses the conversions given here.

Old 1D1V kinetic electron equations

These were the form of equations implemented in the code for kinetic electrons before PR #322, April 2025.