Moment kinetic equations

In 1D the ion kinetic equation for the distribution function $f_i(t, \boldsymbol{r}, \boldsymbol{v})=f_i(t, z, v_\parallel, v_\perp)$ is

\[\begin{align} \frac{\partial f_i}{\partial t} + v_\parallel \frac{\partial f_i}{\partial z} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \frac{\partial f_i}{\partial v_\parallel} = C_{ii}[f_i,f_i] + C_{in}[f_i,f_n] + S_i \end{align}\]

$C_{ii}$ are ion-ion collisions, $C_{in}$ ion-neutral collisions/reactions, and $S_i$ a source term.

Simplified neutral interactions $C_{in}$ including charge exchange and ionization with constant $R_\mathrm{CX}$ and $R_\mathrm{ioniz}$

\[\begin{align} C_{in} = -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \end{align}\]

Want to normalise the distribution for any species $s$ to extract low-order moments.

\[\begin{align} F_s(t,z,w_\parallel,w_\perp) = \frac{v_{Ts}^3}{n_s} f_s(t, z, u_{s\parallel}(t,z) + v_{Ts}(t,z)w_\parallel, v_{Ts}(t,z)w_\perp) \end{align}\]

with normalised velocities

\[\begin{align} w_\parallel(t,z,v_\parallel) &= \frac{v_\parallel - u_{s\parallel}(t,z)}{v_{Ts}(t,z)} \\ w_\perp(t,z,v_\perp) &= \frac{v_\perp}{v_{Ts}(t,z)} \end{align}\]

the density

\[\begin{align} n_s(t,z) = 2\pi\int_{-\infty}^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp f_s(t,z,v_\parallel,v_\perp) \end{align}\]

the average parallel velocity

\[\begin{align} u_{s\parallel}(t,z) = \frac{2\pi}{n_s}\int_{-\infty}^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp v_\parallel f_s(t,z,v_\parallel,v_\perp) \end{align}\]

and the thermal speed

\[\begin{align} v_{Ts}^2(t,z) = \frac{4\pi}{3n_s}\int_{-\infty}^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp \left[ (v_\parallel - u_{s\parallel}(t,z))^2 + v_\perp^2 \right] f_s(t,z,v_\parallel,v_\perp) \end{align}\]

For use later, we also give the definitions of the parallel and perpendicular pressure

\[\begin{align} p_{s\parallel}(t,z) &= 2\pi\int_{-\infty}^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp m_s \left( v_\parallel - u_{s\parallel}(t,z) \right)^2 f_s(t,z,v_\parallel,v_\perp) \nonumber \\ &= m_s n_s v_{Ts}^2 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp w_\parallel^2 F_s(t,z,w_\parallel,w_\perp) \\ p_{s\perp}(t,z) &= 2\pi\int_{-\infty}^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp m_s \frac{v_\perp^2}{2} f_s(t,z,v_\parallel,v_\perp) \\ &= \frac{1}{2} m_s n_s v_{Ts}^2 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp w_\perp^2 F_s(t,z,w_\parallel,w_\perp) \\ \end{align}\]

and $T_{s\parallel} = p_{s\parallel}/n_s$, $T_{s\perp} = p_{s\perp}/n_s$, which we can note means that

\[\begin{align} p_s &= n_s T_s = \frac{m_s}{2} n_s v_{Ts}^2 = \frac{1}{3}(p_{s\parallel} + 2p_{s\perp}) \\ T_s &= \frac{1}{3}(T_{s\parallel} + 2T_{s\perp}) \\ v_{Ts} &= \sqrt{\frac{2T_s}{m_s}}= \sqrt{\frac{2(T_{s\parallel} + 2T_{s\perp})}{3m_s}} \\ \end{align}\]

The parallel heat flux is

\[\begin{align} q_{s\parallel} &= \int \frac{m_s}{2} \left( (v_\parallel - u_{s\parallel})^2 + v_\perp^2 \right) (v_\parallel - u_{s\parallel}) f_s d^3 v \\ &= \frac{n_s}{v_{Ts}^3} \int \frac{m_s}{2} \left( (v_\parallel - u_{s\parallel})^2 + v_\perp^2 \right) (v_\parallel - u_{s\parallel}) F_s d^3 v \nonumber \\ &= n_s \int \frac{m_s}{2} v_{Ts}^2 \left( w_\parallel^2 + w_\perp^2 \right) v_{Ts} w_\parallel F_s d^3 w \nonumber \\ &= n_s v_{Ts}^3 \int \frac{m_s}{2} \left( w_\parallel^2 + w_\perp^2 \right) w_\parallel F_s d^3 w \\ \end{align}\]

$F_s$ must therefore satisfy the conditions

\[\begin{align} 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp F_s(t,z,w_\parallel,w_\perp) &= 1 \\ 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp w_\parallel F_s(t,z,w_\parallel,w_\perp) &= 0 \\ 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp (w_\parallel^2 + w_\perp^2) F_s(t,z,w_\parallel,w_\perp) &= \frac{3}{2} \\ \end{align}\]

Ion moment equations

Can integrate the drift kinetic equation to give the moment equations:

• continuity

  [ intermediate steps ]

  \[\begin{align} & \int \frac{\partial f_i}{\partial t} d^3 v + \underbrace{\int v_\parallel \frac{\partial f_i}{\partial z} d^3 v}_\text{take derivative out of velocity integral} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \underbrace{\int \frac{\partial f_i}{\partial v_\parallel} d^3 v}_\text{total derivative integrates to 0} \nonumber \\ &= \underbrace{\int C_{ii}[f_i,f_i] d^3 v}_{=0\text{, collisions conserve particles}} + \int \left[ \underbrace{-R_\mathrm{CX}(n_n f_i - n_i f_n)}_{=0\text{, no particle source from CX}} + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \underbrace{\int S_i d^3 v}_{S_{i,n}} \\ \end{align}\]

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

  where the density source is $S_{s,n} = \int S_s d^3 v$

• momentum

  [ intermediate steps ]

  \[\begin{align} & \int v_\parallel \frac{\partial f_i}{\partial t} d^3 v + \int v_\parallel^2 \frac{\partial f_i}{\partial z} d^3 v - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int v_\parallel \frac{\partial f_i}{\partial v_\parallel} d^3 v \nonumber \\ &= \underbrace{\int v_\parallel C_{ii}[f_i,f_i] d^3 v}_{=0\text{, collisions conserve momentum}} + \int v_\parallel \left[ -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \underbrace{\int v_\parallel S_i d^3 v}_{S_{i,\mathrm{mom}} / m_i} \\ & \frac{\partial}{\partial t} \int v_\parallel f_i d^3 v + \frac{\partial}{\partial z} \int v_\parallel^2 f_i d^3 v + \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int f_i d^3 v \nonumber \\ &= \int v_\parallel \left[ -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \frac{1}{m_i} S_{i,\mathrm{mom}} \\ & \frac{\partial}{\partial t} \int v_\parallel f_i d^3 v + \frac{\partial}{\partial z}(u_{i\parallel}^2 n_i) + \frac{\partial}{\partial z} \int v_{Ti}^2 \left( v_\parallel - u_{i\parallel} \right)^2 f_i d^3 v + \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int f_i d^3 v \nonumber \\ &= \int v_\parallel \left[ -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \frac{1}{m_i} S_{i,\mathrm{mom}} \\ \end{align}\]

  \[\begin{align} & m_i \frac{\partial}{\partial t}(n_i u_{i\parallel}) + m_i \frac{\partial}{\partial z}(n_i u_{i\parallel}^2) + \frac{\partial p_{i\parallel}}{\partial z} + e n_i \frac{\partial \phi}{\partial z} \nonumber \\ &\quad= R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} m_i n_e n_n u_{n\parallel} + S_{i,\mathrm{mom}} \\ \end{align}\]

  where the momentum source is $S_{s,\mathrm{mom}} = m_s \int v_\parallel S_s d^3 v$, which can also be manipulated into an equation for $\partial u_{i\parallel}/\partial t$ using the continuity equation

  [ intermediate steps ]

  take $u_{i\parallel}\times$continuity $\rightarrow$

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

  \[\begin{align} m_i u_{i\parallel} \frac{\partial n_i}{\partial t} &= - m_i u_{i\parallel} \frac{\partial}{\partial z}(n_i u_{i\parallel}) + R_\mathrm{ioniz} n_e n_n u_{i\parallel} + u_{i\parallel} S_{i,n} \end{align}\]

  Momentum $\rightarrow$

  \[\begin{align} &m_i u_{i\parallel} \frac{\partial n_i}{\partial t} + m_i n_i \frac{\partial u_{i\parallel}}{\partial t} + m_i u_{i\parallel}^2 \frac{\partial n_i}{\partial z} + m_i n_i 2u_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{\partial p_{i\parallel}}{\partial z} + e n_i \frac{\partial\phi}{\partial z} \nonumber \\ &= R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} m_i n_e n_n u_{n\parallel} + S_{i,\mathrm{mom}} \\ \end{align}\]

  Sub from continuity $\rightarrow$ cancellation

  \[\begin{align} &m_i n_i \frac{\partial u_{i\parallel}}{\partial t} + m_i n_i u_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &= - \frac{\partial p_{i\parallel}}{\partial z} - e n_i \frac{\partial\phi}{\partial z} + R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} m_i n_e n_n (u_{n\parallel} - u_{i\parallel}) + S_{i,\mathrm{mom}} - m_i u_{i\parallel} S_{i,n} \\ \end{align}\]

• Energy

  [ intermediate steps ]

  \[\begin{align} & \frac{3}{2} \int v^2 \frac{\partial f_i}{\partial t} d^3 v + \frac{3}{2} \int v^2 v_\parallel \frac{\partial f_i}{\partial z} d^3 v - \frac{3}{2} \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int v^2 \frac{\partial f_i}{\partial v_\parallel} d^3 v \nonumber \\ &\quad= \frac{3}{2} \underbrace{\int v^2 C_{ii}[f_i,f_i] d^3 v}_{=0\text{, collisions conserve energy}} + \frac{3}{2} \int v^2 \left[ -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \underbrace{\frac{3}{2} \int v^2 S_i d^3 v}_{3 S_{i,E} / m_i} \\ & \frac{3}{2} \frac{\partial}{\partial t} \int v^2 f_i d^3 v + \frac{3}{2} \frac{\partial}{\partial z} \int v^2 v_\parallel f_i d^3 v + \frac{3}{2} \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int 2 v_\parallel f_i d^3 v \nonumber \\ &\quad= \frac{3}{2} \int v^2 \left[ -R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n \right] d^3 v + \frac{3}{m_i} S_{i,E} \\ & \frac{3}{2} \frac{\partial}{\partial t} \int \left((v_\parallel - u_{i\parallel})^2 + \cancel{2(v_\parallel - u_{i\parallel}) u_{i\parallel}} + u_{i\parallel}^2 + v_\perp^2 \right) f_i d^3 v \nonumber \\ &+ \frac{3}{2} \frac{\partial}{\partial z} \int \underbrace{\left((v_\parallel - u_{i\parallel})^2 + 2(v_\parallel - u_{i\parallel}) u_{i\parallel} + u_{i\parallel}^2 + v_\perp^2 \right) \left( (v_\parallel - u_{i\parallel}) + u_{i\parallel} \right)}_{(v_\parallel - u_{i\parallel})^3 + 2(v_\parallel - u_{i\parallel})^2 u_{i\parallel} + \cancel{u_{i\parallel}^2(v_\parallel - u_{i\parallel})} + v_\perp^2(v_\parallel - u_{i\parallel}) + (v_\parallel - u_{i\parallel})^2 u_{i\parallel} + \cancel{2(v_\parallel - u_{i\parallel})u_{i\parallel}^2} + u_{i\parallel}^3 + v_\perp^2 u_{i\parallel}} f_i d^3 v \nonumber \\ &+ \frac{3}{2} \frac{e}{m_i} \frac{\partial\phi}{\partial z} \int 2 \left( \cancel{(v_\parallel - u_{i\parallel})} + u_{i\parallel} \right) f_i d^3 v \nonumber \\ &\quad= - \frac{3}{2} \int \left((v_\parallel - u_{i\parallel})^2 + \cancel{2(v_\parallel - u_{i\parallel}) u_{i\parallel}} + u_{i\parallel}^2 + v_\perp^2 \right) R_\mathrm{CX} n_n f_i d^3 v \nonumber \\ &\qquad+ \frac{3}{2} \int \left((v_\parallel - u_{n\parallel})^2 + \cancel{2(v_\parallel - u_{n\parallel}) u_{n\parallel}} + u_{n\parallel}^2 + v_\perp^2 \right) R_\mathrm{CX} n_i f_n d^3 v \nonumber \\ &\qquad+ \frac{3}{2} \int \left((v_\parallel - u_{n\parallel})^2 + \cancel{2(v_\parallel - u_{n\parallel}) u_{n\parallel}} + u_{n\parallel}^2 + v_\perp^2 \right) R_\mathrm{ioniz} n_e f_n d^3 v \nonumber \\ &\qquad+ \frac{3}{m_i} S_{i,E} \\ & \frac{3}{2} \frac{\partial}{\partial t} \left( \frac{3 p_i}{m_i} + n_i u_{i\parallel}^2 \right) \nonumber \\ &+ \frac{3}{2} \frac{\partial}{\partial z} \left[ \int \left( (v_\parallel - u_{i\parallel})^3 + v_\perp^2(v_\parallel - u_{i\parallel}) \right) f_i d^3 v + 2\frac{p_{i\parallel}}{m_i} u_{i\parallel} + \frac{3 p_i}{m_i} u_{i\parallel} + n_i u_{i\parallel}^3 \right] \nonumber \\ &+ 3 \frac{e}{m_i} \frac{\partial\phi}{\partial z} n_i u_{i\parallel} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(\frac{3 p_i}{m_i} n_n + n_i n_n u_{i\parallel}^2 - \frac{3 p_n}{m_i} n_i - n_i n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} R_\mathrm{ioniz} n_e \left(\frac{3 p_n}{m_i} + n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ \frac{3}{m_i} S_{i,E} \\ & \frac{3}{2} \frac{\partial}{\partial t} \left( \frac{3 p_i}{m_i} + n_i u_{i\parallel}^2 \right) + \frac{3}{2} \frac{\partial}{\partial z} \left[ \frac{2 q_{i\parallel}}{m_i} + 2\frac{p_{i\parallel}}{m_i} u_{i\parallel} + \frac{3 p_i}{m_i} u_{i\parallel} + n_i u_{i\parallel}^3 \right] + 3 \frac{e}{m_i} \frac{\partial\phi}{\partial z} n_i u_{i\parallel} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(\frac{3 p_i}{m_i} n_n + n_i n_n u_{i\parallel}^2 - \frac{3 p_n}{m_i} n_i - n_i n_n u_{n\parallel}^2 \right) + \frac{3}{2} R_\mathrm{ioniz} n_e \left(\frac{3 p_n}{m_i} + n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ \frac{3}{m_i} S_{i,E} \\ & \frac{9}{2} \frac{\partial p_i}{\partial t} + 3 m_i u_{i\parallel} \frac{\partial(n_i u_{i\parallel})}{\partial t} - \frac{3}{2} m_i u_{i\parallel}^2 \frac{\partial n_i}{\partial t} + 3 \frac{\partial q_{i\parallel}}{\partial z} + 3 u_{i\parallel} \frac{\partial p_{i\parallel}}{\partial z} + 3 p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad+ \frac{9}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{9}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} + \frac{9}{2} m_i n_i u_{i\parallel}^2 \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} m_i u_{i\parallel}^3 \frac{\partial n_i}{\partial z} + 3 e \frac{\partial\phi}{\partial z} n_i u_{i\parallel} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(3 p_i n_n + m_i n_i n_n u_{i\parallel}^2 - 3 p_n n_i - m_i n_i n_n u_{n\parallel}^2 \right) + \frac{3}{2} R_\mathrm{ioniz} n_e \left(3 p_n + m_i n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ \frac{3}{m_i} S_{i,E} \\ & \frac{9}{2} \frac{\partial p_i}{\partial t} \nonumber \\ &\quad- 3 m_i u_{i\parallel} \frac{\partial}{\partial z}(n_i u_{i\parallel}^2) - 3 u_{i\parallel} \frac{\partial p_{i\parallel}}{\partial z} - 3 e n_i u_{i\parallel} \frac{\partial \phi}{\partial z} + 3 u_{i\parallel} R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + 3 u_{i\parallel} R_\mathrm{ioniz} m_i n_e n_n u_{n\parallel} + 3 u_{i\parallel} S_{i,\mathrm{mom}} \nonumber \\ &\quad+ \frac{3}{2} m_i u_{i\parallel}^2 \frac{\partial}{\partial z}(n_i u_{i\parallel}) - \frac{3}{2} m_i u_{i\parallel}^2 R_\mathrm{ioniz} n_e n_n - \frac{3}{2} m_i u_{i\parallel}^2 S_{i,n} \nonumber \\ &\quad+ 3 \frac{\partial q_{i\parallel}}{\partial z} + 3 u_{i\parallel} \frac{\partial p_{i\parallel}}{\partial z} + 3 p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad+ \frac{9}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{9}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} + \frac{9}{2} m_i n_i u_{i\parallel}^2 \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} m_i u_{i\parallel}^3 \frac{\partial n_i}{\partial z} + 3 e \frac{\partial\phi}{\partial z} n_i u_{i\parallel} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(3 p_i n_n + m_i n_i n_n u_{i\parallel}^2 - 3 p_n n_i - m_i n_i n_n u_{n\parallel}^2 \right) + \frac{3}{2} R_\mathrm{ioniz} n_e \left(3 p_n + m_i n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ 3 S_{i,E} \\ & \frac{9}{2} \frac{\partial p_i}{\partial t} \nonumber \\ &\quad- \cancel{6 m_i n_i u_{i\parallel}^2 \frac{\partial u_{i\parallel}}{\partial z}} - \cancel{3 m_i u_{i\parallel}^3 \frac{\partial n_i}{\partial z}} - \cancel{3 u_{i\parallel} \frac{\partial p_{i\parallel}}{\partial z}} - \cancel{3 e n_i u_{i\parallel} \frac{\partial \phi}{\partial z}} + 3 u_{i\parallel} R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + 3 u_{i\parallel} R_\mathrm{ioniz} m_i n_e n_n u_{n\parallel} + 3 u_{i\parallel} S_{i,\mathrm{mom}} \nonumber \\ &\quad+ \cancel{\frac{3}{2} m_i n_i u_{i\parallel}^2 \frac{\partial u_{i\parallel}}{\partial z}} + \cancel{\frac{3}{2} m_i u_{i\parallel}^3 \frac{\partial n_i}{\partial z}} - \frac{3}{2} m_i u_{i\parallel}^2 R_\mathrm{ioniz} n_e n_n - \frac{3}{2} m_i u_{i\parallel}^2 S_{i,n} \nonumber \\ &\quad+ 3 \frac{\partial q_{i\parallel}}{\partial z} + \cancel{3 u_{i\parallel} \frac{\partial p_{i\parallel}}{\partial z}} + 3 p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad+ \frac{9}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{9}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} + \cancel{\frac{9}{2} m_i n_i u_{i\parallel}^2 \frac{\partial u_{i\parallel}}{\partial z}} + \cancel{\frac{3}{2} m_i u_{i\parallel}^3 \frac{\partial n_i}{\partial z}} + \cancel{3 e \frac{\partial\phi}{\partial z} n_i u_{i\parallel}} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(3 p_i n_n + m_i n_i n_n u_{i\parallel}^2 - 3 p_n n_i - m_i n_i n_n u_{n\parallel}^2 \right) + \frac{3}{2} R_\mathrm{ioniz} n_e \left(3 p_n + m_i n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ 3 S_{i,E} \\ & \frac{9}{2} \frac{\partial p_i}{\partial t} + 3 \frac{\partial q_{i\parallel}}{\partial z} + 3 p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{9}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{9}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{3}{2} R_\mathrm{CX} \left(3 p_i n_n - m_i n_i n_n u_{i\parallel}^2 + 2 m_i n_i n_n u_{i\parallel}u_{n\parallel} - 3 p_n n_i - m_i n_i n_n u_{n\parallel}^2 \right) + \frac{3}{2} R_\mathrm{ioniz} n_e \left(3 p_n + m_i n_n u_{i\parallel}^2 - 2 m_i n_n u_{i\parallel} u_{n\parallel} + m_i n_n u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ 3 S_{i,E} - 3 u_{i\parallel} S_{i,\mathrm{mom}} + \frac{3}{2} m_i u_{i\parallel}^2 S_{i,n} \\ \end{align}\]

  \[\begin{align} & \frac{3}{2} \frac{\partial p_i}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{3}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} \\ \end{align}\]

  where the energy source is $S_{s,E} = \frac{1}{2} m_s \int v^2 S_s d^3 v$, and the pressure source is $S_{s,p} = \frac{1}{3} m_s \int |\boldsymbol{v} - u_{s\parallel}\hat{\boldsymbol{z}}|^2 S_s d^3 v = \frac{2}{3} S_{s,E} - \frac{2}{3} u_{s\parallel} S_{s,\mathrm{mom}} + \frac{1}{3} m_s u_{s\parallel}^2 S_{s,n}$. We use the pressure as an evolving variable in the code, so this is the energy equation used. It is also useful to subsitute in the continuity equation to convert this to a temperature equation and then a $v_{Ti}$ equation, as the latter will be used to form the kinetic equation for $F_i$.

  [ intermediate steps ]

  \[\begin{align} & \frac{3}{2} \frac{\partial n_i T_i}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} u_{i\parallel} \frac{\partial n_i T_i}{\partial z} + \frac{3}{2} n_i T_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} \\ & \frac{3}{2} n_i \frac{\partial T_i}{\partial t} + \frac{3}{2} T_i \frac{\partial n_i}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} n_i u_{i\parallel} \frac{\partial T_i}{\partial z} + \frac{3}{2} u_{i\parallel} T_i \frac{\partial n_i}{\partial z} + \frac{3}{2} n_i T_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} \\ & \frac{3}{2} n_i \frac{\partial T_i}{\partial t} - \frac{3}{2} n_i T_i \frac{\partial u_{i\parallel}}{\partial z} - \frac{3}{2} u_{i\parallel} T_i \frac{\partial n_i}{\partial z} + \frac{3}{2} T_i R_\mathrm{ioniz} n_e n_n + \frac{3}{2} T_i S_{i,n} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} n_i u_{i\parallel} \frac{\partial T_i}{\partial z} + \frac{3}{2} u_{i\parallel} T_i \frac{\partial n_i}{\partial z} + \frac{3}{2} n_i T_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} \\ \end{align}\]

  \[\begin{align} & \frac{3}{2} n_i \frac{\partial T_i}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} n_i u_{i\parallel} \frac{\partial T_i}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} - \frac{3}{2} T_i S_{i,n} \\ \end{align}\]

  [ intermediate steps ]

  \[\begin{align} \frac{\partial T_i}{\partial t} &= \frac{1}{2} m_i \frac{\partial v_{Ti}^2}{\partial t} \nonumber \\ &= m_i v_{Ti} \frac{\partial v_{Ti}}{\partial t} \\ \frac{\partial T_i}{\partial z} &= \frac{1}{2} m_i \frac{\partial v_{Ti}^2}{\partial z} \nonumber \\ &= m_i v_{Ti} \frac{\partial v_{Ti}}{\partial z} \\ \end{align}\]

  \[\begin{align} & \frac{3}{2} m_i n_i v_{Ti} \frac{\partial v_{Ti}}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} m_i n_i u_{i\parallel} v_{Ti} \frac{\partial v_{Ti}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} - \frac{3}{2} T_i S_{i,n} \\ \end{align}\]

  \[\begin{align} & \frac{3}{2} m_i n_i v_{Ti} \left( \frac{\partial v_{Ti}}{\partial t} + u_{i\parallel} \frac{\partial v_{Ti}}{\partial z} \right) \nonumber \\ &\quad= - \frac{\partial q_{i\parallel}}{\partial z} - p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\qquad- \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} - \frac{3}{2} T_i S_{i,n} \\ \end{align}\]

Ion kinetic equation

Before giving the full 'moment kinetic' equation, we consider two reduced versions which evolve separately only $n_i$, or only $n_i$ and $u_{i\parallel}$.

Separate $n_i$

Normalise $n_i$ out of the distribution function. Velocity coordinates do not need to be modified.

\[\begin{align} F_s(t,z,v_\parallel,v_\perp) = \frac{f_s(t,z,v_\parallel,v_\perp)}{n_s} \end{align}\]

\[\begin{align} & \frac{\partial F_i}{\partial t} + v_\parallel \frac{\partial F_i}{\partial z} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \frac{\partial F_i}{\partial v_\parallel} + \left( \frac{(v_\parallel - u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} - \frac{\partial u_{i\parallel}}{\partial z} + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} + \frac{1}{n_i} S_{i,n} \right) F_i \nonumber \\ &\quad= \frac{1}{n_i} C_{ii}[n_i F_i, n_i F_i] - R_\mathrm{CX} n_n (F_i - F_n) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} F_n + \frac{1}{n_i} S_i \\ \end{align}\]

Separate $n_i$ and $u_{i\parallel}$

Normalise $n_i$ out of the distribution function. Shift to peculiar velocity.

\[\begin{align} F_s(t,z,\hat{w}_\parallel,v_\perp) &= \frac{f_s(t,z,\hat{w}_\parallel,v_\perp)}{n_s} \\ \hat{w}_\parallel &= v_\parallel - u_{s\parallel} \end{align}\]

\[\begin{align} \left. \frac{\partial}{\partial t} \right|_{z,v_\parallel,v_\perp} &= \left. \frac{\partial t}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial t} \right|_{z,\hat{w}_\parallel,v_\perp} + \left. \frac{\partial z}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,\hat{w}_\parallel,v_\perp} + \left. \frac{\partial \hat{w}_\parallel}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} + \left. \frac{\partial v_\perp}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial v_\perp} \right|_{t,z,\hat{w}_\parallel} \\ &= \left. \frac{\partial}{\partial t} \right|_{z,\hat{w}_\parallel,v_\perp} - \left. \frac{\partial u_{s\parallel}}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} \\ &\equiv \left. \frac{\partial}{\partial t} \right|_{z,\hat{w}_\parallel,v_\perp} - \frac{\partial u_{s\parallel}}{\partial t} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} \\ \left. \frac{\partial}{\partial z} \right|_{t,v_\parallel,v_\perp} &= \left. \frac{\partial t}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,\hat{w}_\parallel,v_\perp} + \left. \frac{\partial z}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,\hat{w}_\parallel,v_\perp} + \left. \frac{\partial \hat{w}_\parallel}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} + \left. \frac{\partial v_\perp}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial v_\perp} \right|_{t,z,\hat{w}_\parallel} \\ &= \left. \frac{\partial}{\partial z} \right|_{t,\hat{w}_\parallel,v_\perp} - \left. \frac{\partial u_{s\parallel}}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} \\ &\equiv \left. \frac{\partial}{\partial z} \right|_{z,\hat{w}_\parallel,v_\perp} - \frac{\partial u_{s\parallel}}{\partial z} \left. \frac{\partial}{\partial \hat{w}_\parallel} \right|_{t,z,v_\perp} \\ \end{align}\]

$\partial / \partial v_\parallel |_{t,z,v_\perp} = \partial / \partial \hat{w}_\parallel |_{t,z,v_\perp}$ and $\partial / \partial v_\perp |_{t,z,v_\parallel} = \partial / \partial v_\perp |_{t,z,\hat{w}_\parallel}$ as $u_{s\parallel}$ does not depend on $v_\parallel$ or $v_\perp$.

\[\begin{align} & \frac{\partial F_i}{\partial t} + (\hat{w}_\parallel + u_{i\parallel}) \frac{\partial F_i}{\partial z} \nonumber \\ &\quad- \left( \hat{w}_\parallel \frac{\partial u_{i\parallel}}{\partial z} - \frac{1}{m_i n_i} \frac{\partial p_{i\parallel}}{\partial z} + R_\mathrm{CX} n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} u_{n\parallel} + \frac{1}{m_i n_i} S_{i,\mathrm{mom}} \right) \frac{\partial F_i}{\partial \hat{w}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{w}_\parallel}{n_i} \frac{\partial n_i}{\partial z} - \frac{\partial u_{i\parallel}}{\partial z} + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} + \frac{1}{n_i} S_{i,n} \right) F_i \nonumber \\ &\quad= \frac{1}{n_i} C_{ii}[n_i F_i, n_i F_i] - R_\mathrm{CX} n_n (F_i - F_n) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} F_n + \frac{1}{n_i} S_i \\ \end{align}\]

Full moment-kinetics (separate $n_i$, $u_{i\parallel}$ and $p_i$)

Form evolution equation for $F_i(t,z,w_\parallel,w_\perp)$, starting from kinetic equation for $f_i(t,z,v_\parallel,v_\perp)$

\[\begin{align} \left. \frac{\partial f_i}{\partial t} \right|_{z,v_\parallel,v_\perp} + v_\parallel \left. \frac{\partial f_i}{\partial z} \right|_{t,v_\parallel,v_\perp} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \left. \frac{\partial f_i}{\partial v_\parallel} \right|_{t,z,v_\perp} = C_{ii}[f_i, f_i] - R_\mathrm{CX}(n_n f_i - n_i f_n) + R_\mathrm{ioniz} n_e f_n + S_i \end{align}\]

and using the definitions of the normalised distribution function and coordinates (repeated here for convenience)

\[\begin{align} F_s(t,z,w_\parallel,w_\perp) &= \frac{v_{Ts}^3}{n_s} f_s(t, z, u_{s\parallel}(t,z) + v_{Ts}(t,z)w_\parallel, v_{Ts}(t,z)w_\perp) \nonumber \\ w_\parallel(t,z,v_\parallel) &= \frac{v_\parallel - u_{s\parallel}(t,z)}{v_{Ts}(t,z)} \nonumber \\ w_\perp(t,z,v_\perp) &= \frac{v_\perp}{v_{Ts}(t,z)} \nonumber \\ \end{align}\]

Substituting the definition of $F_i$ gives

\[\begin{align} \left. \frac{\partial f_i}{\partial t} \right|_{z,v_\parallel,v_\perp} &= \left. \frac{\partial}{\partial t} \right|_{z,v_\parallel,v_\perp} \left(\frac{n_i F_i}{v_{Ti}^3}\right) \nonumber \\ &= \left. \frac{\partial n_i}{\partial t} \right|_{z,v_\parallel,v_\perp} \frac{F_i}{v_{Ti}^3} - 3 \left. \frac{\partial v_{Ti}}{\partial t} \right|_{z,v_\parallel,v_\perp} \frac{n_i F_i}{v_{Ti}^4} + \frac{n_i}{v_{Ti}^3} \left. \frac{\partial F_i}{\partial t} \right|_{z,v_\parallel,v_\perp} \\ v_\parallel \left. \frac{\partial f_i}{\partial z} \right|_{t,v_\parallel,v_\perp} &= v_\parallel \left. \frac{\partial}{\partial z} \right|_{t,v_\parallel,v_\perp} \left(\frac{n_i F_i}{v_{Ti}^3}\right) \nonumber \\ &= v_\parallel \left. \frac{\partial n_i}{\partial z} \right|_{t,v_\parallel,v_\perp} \frac{F_i}{v_{Ti}^3} - 3 v_\parallel \left. \frac{\partial v_{Ti}}{\partial z} \right|_{t,v_\parallel,v_\perp} \frac{n_i F_i}{v_{Ti}^4} + v_\parallel \frac{n_i}{v_{Ti}^3} \left. \frac{\partial F_i}{\partial z} \right|_{t,v_\parallel,v_\perp} \\ \frac{e}{m_i} \frac{\partial \phi}{\partial z} \left. \frac{\partial f_i}{\partial v_\parallel} \right|_{t,z,v_\perp} &= \frac{e}{m_i} \frac{\partial \phi}{\partial z} \left. \frac{\partial}{\partial v_\parallel} \right|_{t,z,v_\perp} \left(\frac{n_i F_i}{v_{Ti}^3}\right) \nonumber \\ &= \frac{e n_i}{m_i v_{Ti}^3} \frac{\partial \phi}{\partial z} \left. \frac{\partial F_i}{\partial v_\parallel} \right|_{t,z,v_\perp} \\ \end{align}\]

making the kinetic equation

\[\begin{align} &\frac{n_i}{v_{Ti}^3} \left. \frac{\partial F_i}{\partial t} \right|_{z,v_\parallel,v_\perp} + v_\parallel \frac{n_i}{v_{Ti}^3} \left. \frac{\partial F_i}{\partial z} \right|_{t,v_\parallel,v_\perp} - \frac{e n_i}{m_i v_{Ti}^3} \frac{\partial\phi}{\partial z} \left. \frac{\partial F_i}{\partial v_\parallel} \right|_{t,z,v_\perp} \nonumber \\ &\quad + \left( \frac{1}{v_{Ti}^3} \frac{\partial n_i}{\partial t} - \frac{3 n_i}{v_{Ti}^4} \frac{\partial v_{Ti}}{\partial t} + \frac{v_\parallel}{v_{Ti}^3} \frac{\partial n_i}{\partial z} - \frac{3 v_\parallel n_i}{v_{Ti}^4} \frac{\partial v_{Ti}}{\partial z} \right) F_i \nonumber \\ &\quad= C_{ii}[\frac{n_i F_i}{v_{Ti}^3}, \frac{n_i F_i}{v_{Ti}^3}] - R_\mathrm{CX} \left( n_n \frac{n_i F_i}{v_{Ti}^3} - n_i \frac{n_n F_n}{v_{Tn}^3} \right) + R_\mathrm{ioniz} n_e \frac{n_n F_n}{v_{Tn}^3} + S_i \\ &\left. \frac{\partial F_i}{\partial t} \right|_{z,v_\parallel,v_\perp} + v_\parallel \left. \frac{\partial F_i}{\partial z} \right|_{t,v_\parallel,v_\perp} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \left. \frac{\partial F_i}{\partial v_\parallel} \right|_{t,z,v_\perp} \nonumber \\ &\quad + \left( \frac{1}{n_i} \frac{\partial n_i}{\partial t} - \frac{3}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \frac{v_\parallel}{n_i} \frac{\partial n_i}{\partial z} - \frac{3 v_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} \right) F_i \nonumber \\ &\quad= \frac{v_{Ti}^3}{n_i} C_{ii}[\frac{n_i F_i}{v_{Ti}^3}, \frac{n_i F_i}{v_{Ti}^3}] - R_\mathrm{CX} n_n \left( F_i - \frac{v_{Ti}^3}{v_{Tn}^3} F_n \right) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \frac{v_{Ti}^3}{v_{Tn}^3} F_n + \frac{v_{Ti}^3}{n_i} S_i \\ \end{align}\]

The change of coordinates transforms the derivatives as

\[\begin{align} \left. \frac{\partial}{\partial t} \right|_{z,v_\parallel,v_\perp} &= \left. \frac{\partial t}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial t} \right|_{z,w_\parallel,w_\perp} + \left. \frac{\partial z}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,w_\parallel,w_\perp} + \left. \frac{\partial w_\parallel}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial w_\parallel} \right|_{t,z,w_\perp} + \left. \frac{\partial w_\perp}{\partial t} \right|_{z,v_\parallel,v_\perp} \left. \frac{\partial}{\partial w_\perp} \right|_{t,z,w_\parallel} \\ &= \frac{\partial}{\partial t} + \left( -\frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial t} - \frac{(v_\parallel - u_{i\parallel})}{v_{Ti}^2} \frac{\partial v_{Ti}}{\partial t} \right) \frac{\partial}{\partial w_\parallel} - \frac{v_\perp}{v_{Ti}^2} \frac{\partial v_{Ti}}{\partial t} \frac{\partial}{\partial w_\perp} \\ &= \frac{\partial}{\partial t} + \left( -\frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial t} - \frac{w_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} \right) \frac{\partial}{\partial w_\parallel} - \frac{w_\perp}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} \frac{\partial}{\partial w_\perp} \\ \left. \frac{\partial}{\partial z} \right|_{t,v_\parallel,v_\perp} &= \left. \frac{\partial t}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial t} \right|_{z,w_\parallel,w_\perp} + \left. \frac{\partial z}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,w_\parallel,w_\perp} + \left. \frac{\partial w_\parallel}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial w_\parallel} \right|_{t,z,w_\perp} + \left. \frac{\partial w_\perp}{\partial z} \right|_{t,v_\parallel,v_\perp} \left. \frac{\partial}{\partial w_\perp} \right|_{t,z,w_\parallel} \\ &= \frac{\partial}{\partial z} + \left( -\frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial z} - \frac{(v_\parallel - u_{i\parallel})}{v_{Ti}^2} \frac{\partial v_{Ti}}{\partial z} \right) \frac{\partial}{\partial w_\parallel} - \frac{v_\perp}{v_{Ti}^2} \frac{\partial v_{Ti}}{\partial z} \frac{\partial}{\partial w_\perp} \\ &= \frac{\partial}{\partial z} + \left( -\frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial z} - \frac{w_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} \right) \frac{\partial}{\partial w_\parallel} - \frac{w_\perp}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} \frac{\partial}{\partial w_\perp} \\ \left. \frac{\partial}{\partial v_\parallel} \right|_{t,z,v_\perp} &= \left. \frac{\partial t}{\partial v_\parallel} \right|_{t,z,v_\perp} \left. \frac{\partial}{\partial t} \right|_{z,w_\parallel,w_\perp} + \left. \frac{\partial z}{\partial v_\parallel} \right|_{t,z,v_\perp} \left. \frac{\partial}{\partial z} \right|_{t,w_\parallel,w_\perp} + \left. \frac{\partial w_\parallel}{\partial v_\parallel} \right|_{t,z,v_\perp} \left. \frac{\partial}{\partial w_\parallel} \right|_{t,z,w_\perp} + \left. \frac{\partial w_\perp}{\partial v_\parallel} \right|_{t,z,v_\perp} \left. \frac{\partial}{\partial w_\perp} \right|_{t,z,w_\parallel} \\ &= \frac{1}{v_{Ti}} \frac{\partial}{\partial w_\parallel} \\ \left. \frac{\partial}{\partial v_\perp} \right|_{t,z,v_\parallel} &= \left. \frac{\partial t}{\partial v_\perp} \right|_{t,z,v_\parallel} \left. \frac{\partial}{\partial t} \right|_{z,w_\parallel,w_\perp} + \left. \frac{\partial z}{\partial v_\perp} \right|_{t,z,v_\parallel} \left. \frac{\partial}{\partial z} \right|_{t,w_\parallel,w_\perp} + \left. \frac{\partial w_\parallel}{\partial v_\perp} \right|_{t,z,v_\parallel} \left. \frac{\partial}{\partial w_\parallel} \right|_{t,z,w_\perp} + \left. \frac{\partial w_\perp}{\partial v_\perp} \right|_{t,z,v_\parallel} \left. \frac{\partial}{\partial w_\perp} \right|_{t,z,w_\parallel} \\ &= \frac{1}{v_{Ti}} \frac{\partial}{\partial w_\perp} \\ \end{align}\]

and so the kinetic equation becomes

\[\begin{align} &\frac{\partial F_i}{\partial t} + (v_{Ti} w_\parallel + u_{i\parallel}) \frac{\partial F_i}{\partial z} \nonumber \\ &\quad - \left( \frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial t} + \left( w_\parallel + \frac{u_{i_\parallel}}{v_{Ti}} \right) \frac{\partial u_{i\parallel}}{\partial z} + \frac{e}{m_i v_{Ti}} \frac{\partial\phi}{\partial z} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + w_\parallel \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) \frac{\partial v_{Ti}}{\partial z} \right) \frac{\partial F_i}{\partial w_\parallel} \nonumber \\ &\quad - \left( \frac{w_\perp}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) w_\perp \frac{\partial v_{Ti}}{\partial z} \right) \frac{\partial F_i}{\partial w_\perp} \nonumber \\ &\quad + \left( \frac{1}{n_i} \frac{\partial n_i}{\partial t} + \frac{(v_{Ti} w_\parallel + u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} - \frac{3}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} - \frac{3 (v_{Ti} w_\parallel + u_{i\parallel})}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} \right) F_i \nonumber \\ &\quad= \frac{v_{Ti}^3}{n_i} C_{ii}[\frac{n_i F_i}{v_{Ti}^3}, \frac{n_i F_i}{v_{Ti}^3}] - R_\mathrm{CX} n_n \left( F_i - \frac{v_{Ti}^3}{v_{Tn}^3} F_n \right) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \frac{v_{Ti}^3}{v_{Tn}^3} F_n + \frac{v_{Ti}^3}{n_i} S_i \\ &\frac{\partial F_i}{\partial t} + \dot{z} \frac{\partial F_i}{\partial z} + \dot{w}_\parallel \frac{\partial F_i}{\partial w_\parallel} + \dot{w}_\perp \frac{\partial F_i}{\partial w_\perp} = \dot{F}_i + \mathcal{C}_i + \frac{v_{Ti}^3}{n_i} S_i \\ \end{align}\]

where

\[\begin{align} \dot{z} &= v_{Ti} w_\parallel + u_{i\parallel} \\ \dot{w}_\parallel &= - \left( \frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial t} + \left( w_\parallel + \frac{u_{i_\parallel}}{v_{Ti}} \right) \frac{\partial u_{i\parallel}}{\partial z} + \frac{e}{m_i v_{Ti}} \frac{\partial\phi}{\partial z} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + w_\parallel \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) \frac{\partial v_{Ti}}{\partial z} \right) \\ \dot{w}_\perp &= -\left( \frac{w_\perp}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) w_\perp \frac{\partial v_{Ti}}{\partial z} \right) \\ \frac{\dot{F}_i}{F_i} &= \frac{3}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \frac{3 (v_{Ti} w_\parallel + u_{i\parallel})}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} - \frac{1}{n_i} \frac{\partial n_i}{\partial t} - \frac{(v_{Ti} w_\parallel + u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} \\ \mathcal{C}_i &= \frac{v_{Ti}^3}{n_i} C_{ii}[\frac{n_i F_i}{v_{Ti}^3}, \frac{n_i F_i}{v_{Ti}^3}] - R_\mathrm{CX} n_n \left( F_i - \frac{v_{Ti}^3}{v_{Tn}^3} F_n \right) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \frac{v_{Ti}^3}{v_{Tn}^3} F_n \\ \end{align}\]

We could substitute in the moment equations to eliminate the time-derivative-of-moment terms

\[\begin{align} \dot{w}_\parallel &= w_\parallel \frac{\partial u_{i\parallel}}{\partial z} + \frac{1}{m_i n_i v_{Ti}} \frac{\partial p_{i\parallel}}{\partial z} - R_\mathrm{CX} \frac{n_n}{v_{Ti}} (u_{n\parallel} - u_{i\parallel}) - R_\mathrm{ioniz} \frac{n_e n_n (u_{n\parallel} - u_{i\parallel})}{n_i v_{Ti}} - \frac{1}{m_i n_i v_{Ti}} (S_{i,\mathrm{mom}} - m_i u_{i\parallel} S_{i,n}) \nonumber \\ &\qquad\quad - w_\parallel^2 \frac{\partial v_{Ti}}{\partial z} + \frac{w_\parallel}{3 p_i} \frac{\partial q_{i\parallel}}{\partial z} + \frac{w_\parallel p_{i\parallel}}{3 p_i} \frac{\partial u_{i\parallel}}{\partial z} + \frac{w_\parallel}{6 p_i} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\parallel}{6 p_i} R_\mathrm{ioniz} n_e n_n \left( 3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\parallel}{2 p_i} S_{i,p} + \frac{w_\parallel}{2 p_i} T_i S_{i,n} \\ \dot{w}_\perp &= -w_\parallel w_\perp \frac{\partial v_{Ti}}{\partial z} + \frac{w_\perp}{3 p_i} \frac{\partial q_{i\parallel}}{\partial z} + \frac{w_\perp p_{i\parallel}}{3 p_i} \frac{\partial u_{i\parallel}}{\partial z} + \frac{w_\perp}{6 p_i} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\perp}{6 p_i} R_\mathrm{ioniz} n_e n_n \left( 3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\perp}{2 p_i} S_{i,p} + \frac{w_\perp}{2 p_i} T_i S_{i,n} \\ \frac{\dot{F}_i}{F_i} &= 3 w_\parallel \frac{\partial v_{Ti}}{\partial z} - \frac{1}{p_i} \frac{\partial q_{i\parallel}}{\partial z} - \frac{p_{i\parallel}}{p_i} \frac{\partial u_{i\parallel}}{\partial z} - \frac{1}{2 p_i} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2 p_i} R_\mathrm{ioniz} n_e n_n \left( 3 T_n - 3 T_i + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{3}{2 p_i} S_{i,p} - \frac{5}{2 n_i} S_{i,n} \nonumber \\ &\qquad\quad - \frac{v_{Ti} w_\parallel}{n_i} \frac{\partial n_i}{\partial z} + \frac{\partial u_{i\parallel}}{\partial z} - R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \nonumber \\ \end{align}\]

However, the expressions for $\dot{w}_\parallel$, $\dot{w}_\perp$, and $\dot F_i$ become much longer. In the code, we have to calculate $\partial n_i/\partial t$, etc. anyway to evolve the moment equations, so it will be simpler to save these values, and implement the kinetic equation coefficients in terms of $\partial n_i/\partial t$, etc. Especially if/when new terms (e.g. inter-species friction) are added to the model, this approach will minimise the number of places in the code that need to be updated.

Neutral equations

Neutrals do not see $\phi$ in their kinetic equation, and signs of ion-neutral reaction terms are flipped

\[\begin{align} & \frac{\partial f_n}{\partial t} + v_\parallel \frac{\partial f_n}{\partial z} = \mathcal{C}_{ni}[f_n,f_i] + S_n \\ & \mathcal{C}_{ni}[f_n,f_i] = -\mathcal{C}_{in}[f_i,f_n] = R_\mathrm{CX} (n_n f_i - n_i f_n) - R_\mathrm{ioniz} n_e f_n \end{align}\]

The moment equations are therefore very similar to those of the ions

\[\begin{align} & \frac{\partial n_n}{\partial t} + \frac{\partial}{\partial z}\left( n_n u_{n\parallel} \right) = - R_\mathrm{ioniz} n_e n_n + S_{n,n} \\ & m_n \frac{\partial}{\partial t}(n_n u_{n\parallel}) + m_n \frac{\partial}{\partial z}(n_n u_{n\parallel}^2) + \frac{\partial p_{n\parallel}}{\partial z} \nonumber \\ &\quad= -R_\mathrm{CX} m_n n_i n_n (u_{n\parallel} - u_{i\parallel}) - R_\mathrm{ioniz} m_n n_e n_n u_{n\parallel} + S_{n,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial}{\partial t} \left( \frac{3 p_n}{m_n} + n_n u_{n\parallel}^2 \right) + \frac{3}{2} \frac{\partial}{\partial z} \left[ \frac{2 q_{n\parallel}}{m_n} + 2\frac{p_{n\parallel}}{m_n} u_{n\parallel} + \frac{3 p_n}{m_n} u_{n\parallel} + n_n u_{n\parallel}^3 \right] \nonumber \\ &\quad= \frac{3}{2} R_\mathrm{CX} \frac{n_i n_n}{m_i} \left(3 T_i + m_i u_{i\parallel}^2 - 3 T_n - m_i u_{n\parallel}^2 \right) - \frac{3}{2} R_\mathrm{ioniz} \frac{n_e n_n}{m_i} \left(3 T_n + m_i u_{n\parallel}^2 \right) \nonumber \\ &\qquad+ \frac{3}{m_i} S_{n,E} \\ & \frac{3}{2} \frac{\partial p_n}{\partial t} + \frac{\partial q_{n\parallel}}{\partial z} + p_{n\parallel} \frac{\partial u_{n\parallel}}{\partial z} + \frac{3}{2} u_{n\parallel} \frac{\partial p_n}{\partial z} + \frac{3}{2} p_n \frac{\partial u_{n\parallel}}{\partial z} \nonumber \\ &\quad= \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{3}{2} R_\mathrm{ioniz} n_e n_n T_n \nonumber \\ &\qquad+ \frac{3}{2} S_{n,p} \\ \end{align}\]

The alternative forms of equation for $\partial u_{n\parallel}\partial t$ and $\partial v_{Tn} / \partial t$ may also be useful

\[\begin{align} &m_n n_n \frac{\partial u_{n\parallel}}{\partial t} + m_n n_n u_{n\parallel} \frac{\partial u_{n\parallel}}{\partial z} \nonumber \\ &= - \frac{\partial p_{n\parallel}}{\partial z} - R_\mathrm{CX} m_n n_i n_n (u_{n\parallel} - u_{i\parallel}) + S_{n,\mathrm{mom}} - m_n u_{n\parallel} S_{n,n} \\ & \frac{3}{2} m_n n_n v_{Tn} \left( \frac{\partial v_{Tn}}{\partial t} + u_{n\parallel} \frac{\partial v_{Tn}}{\partial z} \right) \nonumber \\ &\quad= - \frac{\partial q_{n\parallel}}{\partial z} - p_{n\parallel} \frac{\partial u_{n\parallel}}{\partial z} \nonumber \\ &\qquad+ \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{n,p} - \frac{3}{2} T_n S_{n,n} \\ \end{align}\]

The 3 variants on the moment-kinetic equation, given in the following subsections, are also very similar to the ion ones.

Separate $n_n$

\[\begin{align} & \frac{\partial F_n}{\partial t} + v_\parallel \frac{\partial F_n}{\partial z} + \left( \frac{(v_\parallel - u_{n\parallel})}{n_n} \frac{\partial n_n}{\partial z} - \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{n_n} S_{n,n} \right) F_n \nonumber \\ &\quad= R_\mathrm{CX} n_i (F_i - F_n) + \frac{1}{n_n} S_n \\ \end{align}\]

Separate $n_n$ and $u_{n\parallel}$

\[\begin{align} & \frac{\partial F_n}{\partial t} + (\hat{w}_\parallel + u_{n\parallel}) \frac{\partial F_n}{\partial z} \nonumber \\ &\quad- \left( \hat{w}_\parallel \frac{\partial u_{n\parallel}}{\partial z} - \frac{1}{m_n n_n} \frac{\partial p_{n\parallel}}{\partial z} - R_\mathrm{CX} n_i (u_{n\parallel} - u_{i\parallel}) + \frac{1}{m_n n_n} S_{n,\mathrm{mom}} \right) \frac{\partial F_n}{\partial \hat{w}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{w}_\parallel}{n_n} \frac{\partial n_n}{\partial z} - \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{n_n} S_{n,n} \right) F_n \nonumber \\ &\quad= R_\mathrm{CX} n_i (F_i - F_n) + \frac{1}{n_n} S_n \\ \end{align}\]

Full moment-kinetics (separate $n_n$, $u_{n\parallel}$ and $p_{n}$)

\[\begin{align} &\frac{\partial F_n}{\partial t} + \dot{z} \frac{\partial F_n}{\partial z} + \dot{w}_\parallel \frac{\partial F_n}{\partial w_\parallel} + \dot{w}_\perp \frac{\partial F_n}{\partial w_\perp} = \dot{F}_n + \mathcal{C}_n + \frac{v_{Tn}^3}{n_n} S_n \\ &\dot{z} = v_{Tn} w_\parallel + u_{n\parallel} \\ &\dot{w}_\parallel = - \left( \frac{1}{v_{Tn}} \frac{\partial u_{n\parallel}}{\partial t} + \left( w_\parallel + \frac{u_{n_\parallel}}{v_{Tn}} \right) \frac{\partial u_{n\parallel}}{\partial z} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{v_{Tn}} \frac{\partial v_{Tn}}{\partial t} + w_\parallel \left( w_\parallel + \frac{u_{n\parallel}}{v_{Tn}} \right) \frac{\partial v_{Tn}}{\partial z} \right) \\ &\dot{w}_\perp = -\left( \frac{w_\perp}{v_{Tn}} \frac{\partial v_{Tn}}{\partial t} + \left( w_\parallel + \frac{u_{n\parallel}}{v_{Tn}} \right) w_\perp \frac{\partial v_{Tn}}{\partial z} \right) \\ &\frac{\dot{F}_n}{F_n} = \frac{3}{v_{Tn}} \frac{\partial v_{Tn}}{\partial t} + \frac{3 (v_{Tn} w_\parallel + u_{n\parallel})}{v_{Tn}} \frac{\partial v_{Tn}}{\partial z} - \frac{1}{n_n} \frac{\partial n_n}{\partial t} - \frac{(v_{Tn} w_\parallel + u_{n\parallel})}{n_n} \frac{\partial n_n}{\partial z} \\ \mathcal{C}_n &= R_\mathrm{CX} n_i \left( \frac{v_{Tn}^3}{v_{Ti}^3} F_i - F_n \right) - R_\mathrm{ioniz} n_e F_n \\ \end{align}\]

Again, we could substitute in the moment equations

\[\begin{align} \dot{w}_\parallel &= w_\parallel \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{m_n n_n v_{Tn}} \frac{\partial p_{n\parallel}}{\partial z} + R_\mathrm{CX} \frac{n_i}{v_{Tn}} (u_{n\parallel} - u_{i\parallel}) - \frac{1}{m_n n_n v_{Tn}} (S_{n,\mathrm{mom}} - u_{n\parallel} S_{n,n}) \nonumber \\ &\qquad\quad - w_\parallel^2 \frac{\partial v_{Tn}}{\partial z} + \frac{w_\parallel}{3 p_n} \frac{\partial q_{n\parallel}}{\partial z} + \frac{w_\parallel p_{n\parallel}}{3 p_n} \frac{\partial u_{n\parallel}}{\partial z} - \frac{w_\parallel}{6 p_n} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\parallel}{2 p_n} S_{n,p} + \frac{w_\parallel}{2 p_n} T_n S_{n,n} \\ \dot{w}_\perp &= -w_\parallel w_\perp \frac{\partial v_{Tn}}{\partial z} + \frac{w_\perp}{3 p_n} \frac{\partial q_{n\parallel}}{\partial z} + \frac{w_\perp p_{n\parallel}}{3 p_n} \frac{\partial u_{n\parallel}}{\partial z} - \frac{w_\perp}{6 p_n} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{w_\perp}{2 p_n} S_{n,p} + \frac{w_\perp}{2 p_n} T_n S_{n,n} \\ \frac{\dot{F}_n}{F_n} &= 3 w_\parallel \frac{\partial v_{Tn}}{\partial z} - \frac{1}{p_n} \frac{\partial q_{n\parallel}}{\partial z} - \frac{p_{n\parallel}}{p_n} \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{2 p_n} R_\mathrm{CX} n_i n_n \left( 3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{3}{2 p_n} S_{n,p} - \frac{5}{2 n_n} S_{n,n} \nonumber \\ &\qquad\quad - \frac{v_{Tn} w_\parallel}{n_n} \frac{\partial n_n}{\partial z} + \frac{\partial u_{n\parallel}}{\partial z} - R_\mathrm{ioniz} n_e \nonumber \\ \end{align}\]

noting that ionization does not appear in the equations for $\partial u_{n\parallel} / \partial t$ or $\partial v_{Tn} / \partial t$ so that the only contributions are from $\partial n_n / \partial t$ that contributes to $\dot{F}$ and the explicit term in $\mathcal C_n$, and these will cancel.

Wall boundary condition (Knudsen cosine distribution)

Neutrals returning from the wall belong to a Knudsen cosine distribution, defined with a wall temperature $T_\mathrm{wall}$, which is given by [Excalibur report TN-05]

\[\begin{align} f_n(0,v_\parallel>0,v_\perp,t) &= \Gamma_0 f_\mathrm{Kw}(v_\parallel,v_\perp) \\ f_n(L,v_\parallel<0,v_\perp,t) &= \Gamma_L f_\mathrm{Kw}(v_\parallel,v_\perp) \\ \Gamma_0 &= \sum_{s=i,n} 2 \pi \int_{-\infty}^0 dv_\parallel \int_0^\infty dv_\perp v_\perp |v_\parallel| f_s(0,v_\parallel,v_\perp,t) \\ \Gamma_L &= \sum_{s=i,n} 2 \pi \int_0^\infty dv_\parallel \int_0^\infty dv_\perp v_\perp |v_\parallel| f_s(L,v_\parallel,v_\perp,t) \\ f_\mathrm{Kw}(v_\parallel,v_\perp) &= \frac{3}{4\pi} \left( \frac{m_n}{T_w} \right)^2 \frac{|v_\parallel|}{\sqrt{v_\parallel^2 + v_\perp^2}} \exp\left( -\frac{m_n(v_\parallel^2 + v_\perp^2)}{2T_w} \right) \\ \end{align}\]

Reduction to 1D1V

To reduce the model to 1D1V, we take the limit $T_{s\perp} \rightarrow 0$, and marginalise over $v_\perp$ to remove one velocity space dimension.

One way to do this formally is to assume that

\[\begin{align} f_s &= \bar{f}_s(t,z,v_\parallel) f_{s\perp}(v_\perp) \\ \text{with } f_{s\perp}(v_\perp) &= \frac{\exp(-v_\perp^2/v_{Ts\perp}^2)}{\pi v_{Ts\perp}^2} \\ \text{where } \frac{1}{2} m_s v_{Ts\perp}^2 &= T_\perp \\ \text{and similarly } S_s &= \bar{S}_s(t,z,v_\parallel) f_{s\perp}(v_\perp) \end{align}\]

$f_{s\perp}$ is defined so that

\[\begin{align} \int f_{s\perp} d^2 v_\perp &= 2\pi \int_0^\infty f_{s\perp} v_\perp dv_\perp \nonumber \\ &= 2\pi \int_0^\infty \frac{\exp(-v_\perp^2/v_{Ts\perp}^2)}{\pi v_{Ts\perp}^2} v_\perp dv_\perp \nonumber \\ &= 2 \int_0^\infty \exp(-x^2) x dx \nonumber \\ &= 2 \left[ -\frac{1}{2} \exp(-x^2) \right]_0^\infty \nonumber \\ &= 1 \\ \int v_\perp f_{s\perp} d^2 v_\perp &= 2\pi \int_0^\infty v_\perp f_{s\perp} v_\perp dv_\perp \nonumber \\ &= \lim_{v_{Ts\perp}\rightarrow 0} 2\pi \int_0^\infty v_\perp^2 \frac{\exp(-v_\perp^2/v_{Ts\perp}^2)}{\pi v_{Ts\perp}^2} dv_\perp \nonumber \\ &= \lim_{v_{Ts\perp}\rightarrow 0} 2 v_{Ts\perp} \int_0^\infty x^2 \exp(-x^2) dx \nonumber \\ &= \lim_{v_{Ts\perp}\rightarrow 0} 2 v_{Ts\perp} \left( \left[ -\frac{1}{2} x \exp(-x^2) \right]_0^\infty + \frac{1}{2} \int_0^\infty \exp(-x^2) \right) \nonumber \\ &= \lim_{v_{Ts\perp}\rightarrow 0} 2 v_{Ts\perp} \left( 0 + \frac{1}{2} \frac{\sqrt{\pi}}{2} \right) \nonumber \\ &= 0 \end{align}\]

and so

\[\begin{align} \int f_s(t,z,v_\parallel,v_\perp) d^2 v_\perp = \bar{f}_s(t,z,v_\parallel) \\ \int S_s(t,z,v_\parallel,v_\perp) d^2 v_\perp = \bar{S}_s(t,z,v_\parallel) \\ \end{align}\]

Integrals with any higher powers of $v_\perp$ also vanish.

The source integrals are the same, but can be written in terms of $\bar S_s$, $S_{s,n} = \int \bar S_{s} dv_\parallel$, $S_{s,\mathrm{mom}} = \int m_s v_\parallel \bar S_{s} dv_\parallel$, $S_{s,E} = \int \frac{1}{2} m_s v_\parallel^2 \bar S_{s} dv_\parallel$, and $S_{s,p} = \frac{2}{3} S_{i,E} - \frac{2}{3} u_{i\parallel} S_{i,\mathrm{mom}} + \frac{1}{3} m_s u_{s\parallel}^2 S_{s,n}$.

The marginalised shape function must be marginalised over $w_\perp$, not $v_\perp$ because $F_s$ is dimensionless, and $\bar F_s$ should be dimensionless too.

\[\begin{align} \bar{F}_s(t,z,w_\parallel) &= \int F_s(t,z,w_\parallel,w_\perp) d^2 w_\perp \nonumber \\ &= \int F_s(t,z,w_\parallel,w_\perp) \frac{1}{v_{Ts}^2} d^2 v_\perp \nonumber \\ &= \int \frac{v_{Ts}^3}{n_s} f_s(t,z,v_\parallel,v_\perp) \frac{1}{v_{Ts}^2} d^2 v_\perp \nonumber \\ &= \frac{v_{Ts}}{n_s} \int f_s(t,z,v_\parallel,v_\perp) d^2 v_\perp \nonumber \\ &= \frac{v_{Ts}}{n_s} \bar{f}_s(t,z,v_\parallel) \\ \end{align}\]

and like the integrals above, $\int w_\perp F_s(t,z,w_\parallel,w_\perp) d^2 w_\perp = 0$.

Setting $T_\perp = 0$ in the expressions in section Moment kinetic equations,

\[\begin{align} p_s &= \frac{1}{3}(p_{s\parallel} + 2p_{s\perp}) = \frac{p_{s\parallel}}{3} \\ T_s &= \frac{1}{3}(T_{s\parallel} + 2T_{s\perp}) = \frac{T_{s\parallel}}{3} \\ v_{Ts} &= \sqrt{\frac{2(T_{s\parallel} + 2 T_{s\perp})}{3 m_s}} = \sqrt{\frac{2T_{s\parallel}}{3 m_s}} \\ \end{align}\]

The parallel heat flux reduces to

\[\begin{align} q_{s\parallel} &= \int \frac{m_s}{2} \left( (v_\parallel - u_{s\parallel})^2 + v_\perp^2 \right) (v_\parallel - u_{s\parallel}) f_s d^3 v \nonumber \\ &= \int \frac{m_s}{2} \left( (v_\parallel - u_{s\parallel})^2 \right) (v_\parallel - u_{s\parallel}) \left(\int f_s d^2 v_\perp \right) dv_\parallel \nonumber \\ &\quad + \cancel{\int \frac{m_s}{2} (v_\parallel - u_{s\parallel}) \left(\int v_\perp^2 f_s d^2 v_\perp \right) d^3 v} \nonumber \\ &= \int \frac{m_s}{2} (v_\parallel - u_{s\parallel})^3 \bar{f}_s dv_\parallel \nonumber \\ \end{align}\]

or in terms of $\bar F_s$

\[\begin{align} q_{s\parallel} &= n_s v_{Ts}^3 \int \frac{m_s}{2} \left( w_\parallel^2 + \cancel{w_\perp^2} \right) w_\parallel F_s d^3 w \nonumber \\ q_{s\parallel} &= n_s v_{Ts}^3 \int \frac{m_s}{2} w_\parallel^3 \left(\int F_s d^2 w_\perp \right) dw_\parallel \\ q_{s\parallel} &= n_s v_{Ts}^3 \int \frac{m_s}{2} w_\parallel^3 \bar{F}_s dw_\parallel \\ \end{align}\]

The moment constraints reduce to

\[\begin{align} 1 &= \int dw_\parallel \int d^2 w_\perp F_s(t,z,w_\parallel,w_\perp) = \int dw_\parallel \bar{F}_s(t,z,w_\parallel) \\ 0 &= \int dw_\parallel \int d^2 w_\perp w_\parallel F_s(t,z,w_\parallel,w_\perp) = \int dw_\parallel w_\parallel \bar{F}_s(t,z,w_\parallel) \\ \frac{3}{2} &= \int dw_\parallel \int d^2 w_\perp (w_\parallel^2 + \cancel{w_\perp^2}) F_s(t,z,w_\parallel,w_\perp) = \int dw_\parallel w_\parallel^2 \bar{F}_s(t,z,w_\parallel,w_\perp) \\ \end{align}\]

1D1V ion moment equations

By setting $T_{i\perp} = 0$ in the moment equations for ions

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

\[\begin{align} & m_i \frac{\partial}{\partial t}(n_i u_{i\parallel}) + m_i \frac{\partial}{\partial z}(n_i u_{i_\parallel}^2) + \frac{\partial p_{i\parallel}}{\partial z} + e n_i \frac{\partial \phi}{\partial z} \nonumber \\ &\quad = R_\mathrm{CX} m_i n_i n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} m_i n_e n_n u_{n\parallel} + S_{i,\mathrm{mom}} \\ \end{align}\]

\[\begin{align} & \frac{3}{2} \frac{\partial p_i}{\partial t} + \frac{\partial q_{i\parallel}}{\partial z} + p_{i\parallel} \frac{\partial u_{i\parallel}}{\partial z} + \frac{3}{2} u_{i\parallel} \frac{\partial p_i}{\partial z} + \frac{3}{2} p_i \frac{\partial u_{i\parallel}}{\partial z} \nonumber \\ &\quad= - \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n - m_i (u_{i\parallel} - u_{n\parallel})^2 \right) + \frac{1}{2} R_\mathrm{ioniz} n_e n_n \left(3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} S_{i,p} \\ \end{align}\]

1D1V ion kinetic equation

When we marginalise the ion kinetic equation to reduce it to 1D1V form, we note that $\dot w_\perp \propto w_\perp$, so the term

\[\begin{align} \dot{w}_\perp \frac{\partial F_i}{\partial w_\perp} = -w_\perp \left( \frac{1}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) \right) \frac{\partial F_i}{\partial w_\perp} \end{align}\]

and marginalising

\[\begin{align} \int w_\perp \frac{\partial F_i}{\partial w_\perp} d^2 w_\perp &= -2 \bar{F}_i \\ \end{align}\]

This term cancels with the contribution to $\dot F_i$ that was associated with having three powers of $v_{Ti}$ in $F_s(t,z,w_\parallel,w_\perp) = \frac{v_{Ti}^3}{n_i} f_s(t,z,v_\parallel,v_\perp)$ rather than the one power of $v_{Ti}$ in $\bar{F}_s(t,z,w_\parallel,w_\perp) = \frac{v_{Ti}}{n_i} \bar{f}_s(t,z,v_\parallel,v_\perp)$. Finally

\[\begin{align} &\frac{\partial \bar{F}_i}{\partial t} + \dot{z} \frac{\partial \bar{F}_i}{\partial z} + \dot{w}_\parallel \frac{\partial \bar{F}_i}{\partial w_\parallel} + \dot{w}_\perp \frac{\partial \bar{F}_i}{\partial w_\perp} = \dot{\bar{F}}_i + \bar{\mathcal{C}}_i + \frac{v_{Ti}}{n_i} \bar{S}_i \\ \end{align}\]

(noting that $\int S_i d^2 w_\perp = v_{Ti}^{-2} \int S_i d^2 v_\perp = v_{Ti}^{-2} \bar S_i$), where

\[\begin{align} \dot{z} &= v_{Ti} w_\parallel + u_{i\parallel} \\ \dot{w}_\parallel &= - \left( \frac{1}{v_{Ti}} \frac{\partial u_{i\parallel}}{\partial t} + \left( w_\parallel + \frac{u_{i_\parallel}}{v_{Ti}} \right) \frac{\partial u_{i\parallel}}{\partial z} + \frac{e}{m_i v_{Ti}} \frac{\partial\phi}{\partial z} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + w_\parallel \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) \frac{\partial v_{Ti}}{\partial z} \right) \\ \frac{\dot{\bar{F}}_i}{\bar{F}_i} &= \frac{3}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \frac{3 (v_{Ti} w_\parallel + u_{i\parallel})}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} - \frac{1}{n_i} \frac{\partial n_i}{\partial t} - \frac{(v_{Ti} w_\parallel + u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} - \underbrace{2 \left( \frac{1}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \left( w_\parallel + \frac{u_{i\parallel}}{v_{Ti}} \right) \right)}_\text{term coming from $\dot{w}_\perp \partial F_i / \partial w_\perp$} \nonumber \\ &= \frac{1}{v_{Ti}} \frac{\partial v_{Ti}}{\partial t} + \frac{(v_{Ti} w_\parallel + u_{i\parallel})}{v_{Ti}} \frac{\partial v_{Ti}}{\partial z} - \frac{1}{n_i} \frac{\partial n_i}{\partial t} - \frac{(v_{Ti} w_\parallel + u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} \\ \bar{\mathcal{C}}_i &= \frac{v_{Ti}}{n_i} \int C_{ii}[f_i, f_i] d^2 v_\perp - R_\mathrm{CX} n_n \left( \bar{F}_i - \frac{v_{Ti}}{v_{Tn}} \bar{F}_n \right) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \frac{v_{Ti}}{v_{Tn}} \bar{F}_n \\ \end{align}\]

The separate-$n_i$ and separate-$n_i$,$u_{i\parallel}$ equations are simpler to marginalise, as they contained no $\partial F_i / \partial w_{i\perp}$ term.

• Separate $n_i$

  \[\begin{align} & \frac{\partial \bar{F}_i}{\partial t} + v_\parallel \frac{\partial \bar{F}_i}{\partial z} - \frac{e}{m_i} \frac{\partial\phi}{\partial z} \frac{\partial \bar{F}_i}{\partial v_\parallel} + \left( \frac{(v_\parallel - u_{i\parallel})}{n_i} \frac{\partial n_i}{\partial z} - \frac{\partial u_{i\parallel}}{\partial z} + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} + \frac{1}{n_i} S_{i,n} \right) \bar{F}_i \nonumber \\ &\quad= \frac{1}{n_i} \int C_{ii}[n_i F_i, n_i F_i] d^2 v_\perp - R_\mathrm{CX} n_n (\bar{F}_i - \bar{F}_n) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \bar{F}_n + \frac{1}{n_i} \bar{S}_i \\ \end{align}\]

• Separate $n_i$ and $u_{i\parallel}$, with again $\hat w_\parallel = v_\parallel - u_{s\parallel}$

  \[\begin{align} & \frac{\partial \bar{F}_i}{\partial t} + (\hat{w}_\parallel + u_{i\parallel}) \frac{\partial \bar{F}_i}{\partial z} \nonumber \\ &\quad- \left( \hat{w}_\parallel \frac{\partial u_{i\parallel}}{\partial z} - \frac{1}{m_i n_i} \frac{\partial p_{i\parallel}}{\partial z} + R_\mathrm{CX} n_n (u_{n\parallel} - u_{i\parallel}) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} u_{n\parallel} + \frac{1}{m_i n_i} S_{i,\mathrm{mom}} \right) \frac{\partial \bar{F}_i}{\partial \hat{w}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{w}_\parallel}{n_i} \frac{\partial n_i}{\partial z} - \frac{\partial u_{i\parallel}}{\partial z} + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} + \frac{1}{n_i} S_{i,n} \right) \bar{F}_i \nonumber \\ &\quad= \frac{1}{n_i} \int C_{ii}[n_i F_i, n_i F_i] d^2 v_\perp - R_\mathrm{CX} n_n (\bar{F}_i - \bar{F}_n) + R_\mathrm{ioniz} \frac{n_e n_n}{n_i} \bar{F}_n + \frac{1}{n_i} \bar{S}_i \\ \end{align}\]

1D1V neutral equations

Similarly setting $T_{i\perp} = 0$ in the moment equations for neutrals

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

\[\begin{align} & m_n \frac{\partial}{\partial t}(n_n u_{n\parallel}) + m_n \frac{\partial}{\partial z}(n_n u_{n\parallel}^2) + \frac{\partial p_{n\parallel}}{\partial z} \nonumber \\ &\quad= -R_\mathrm{CX} m_n n_i n_n (u_{n\parallel} - u_{i\parallel}) - R_\mathrm{ioniz} m_n n_e n_n u_{n\parallel} + S_{n,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial p_n}{\partial t} + \frac{\partial q_{n\parallel}}{\partial z} + p_{n\parallel} \frac{\partial u_{n\parallel}}{\partial z} + \frac{3}{2} u_{n\parallel} \frac{\partial p_n}{\partial z} + \frac{3}{2} p_n \frac{\partial u_{n\parallel}}{\partial z} \nonumber \\ &\quad= \frac{1}{2} R_\mathrm{CX} n_i n_n \left(3 T_i - 3 T_n + m_i (u_{i\parallel} - u_{n\parallel})^2 \right) - \frac{3}{2} R_\mathrm{ioniz} n_e n_n T_n \nonumber \\ &\qquad+ \frac{3}{2} S_{n,p} \\ \end{align}\]

Marginalising gives the 1D1V moment-kinetic neutral equation in a very similar form as for the ions

\[\begin{align} &\frac{\partial \bar{F}_n}{\partial t} + \dot{z} \frac{\partial \bar{F}_n}{\partial z} + \dot{w}_\parallel \frac{\partial \bar{F}_n}{\partial w_\parallel} = \dot{\bar{F}}_n + \bar{\mathcal{C}}_n + \frac{v_{Tn}}{n_n} \bar{S}_n \\ \end{align}\]

where

\[\begin{align} \dot{z} &= v_{Tn} w_\parallel + u_{n\parallel} \\ \dot{w}_\parallel &= - \left( \frac{1}{v_{Tn}} \frac{\partial u_{n\parallel}}{\partial t} + \left( w_\parallel + \frac{u_{n_\parallel}}{v_{Tn}} \right) \frac{\partial u_{n\parallel}}{\partial z} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{v_{Tn}} \frac{\partial v_{Tn}}{\partial t} + w_\parallel \left( w_\parallel + \frac{u_{n\parallel}}{v_{Tn}} \right) \frac{\partial v_{Tn}}{\partial z} \right) \\ \frac{\dot{\bar{F}}_n}{\bar{F}_n} &= \frac{1}{v_{Tn}} \frac{\partial v_{Tn}}{\partial t} + \frac{(v_{Tn} w_\parallel + u_{n\parallel})}{v_{Tn}} \frac{\partial v_{Tn}}{\partial z} - \frac{1}{n_n} \frac{\partial n_n}{\partial t} - \frac{(v_{Tn} w_\parallel + u_{n\parallel})}{n_n} \frac{\partial n_n}{\partial z} \\ \bar{\mathcal{C}}_n &= R_\mathrm{CX} n_i \left( \frac{v_{Tn}}{v_{Ti}} \bar{F}_i - \bar{F}_n \right) - R_\mathrm{ioniz} n_e \bar{F}_n \\ \end{align}\]

• Separate $n_n$

  \[\begin{align} & \frac{\partial \bar{F}_n}{\partial t} + v_\parallel \frac{\partial \bar{F}_n}{\partial z} + \left( \frac{(v_\parallel - u_{n\parallel})}{n_n} \frac{\partial n_n}{\partial z} - \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{n_n} S_{n,n} \right) \bar{F}_n \nonumber \\ &\quad= R_\mathrm{CX} n_i (\bar{F}_i - \bar{F}_n) + \frac{1}{n_n} \bar{S}_n \\ \end{align}\]

• Separate $n_n$ and $u_{n\parallel}$

  \[\begin{align} & \frac{\partial \bar{F}_n}{\partial t} + (\hat{w}_\parallel + u_{n\parallel}) \frac{\partial \bar{F}_n}{\partial z} \nonumber \\ &\quad- \left( \hat{w}_\parallel \frac{\partial u_{n\parallel}}{\partial z} - \frac{1}{m_n n_n} \frac{\partial p_{n\parallel}}{\partial z} - R_\mathrm{CX} n_i (u_{n\parallel} - u_{i\parallel}) + \frac{1}{m_n n_n} S_{n,\mathrm{mom}} \right) \frac{\partial \bar{F}_n}{\partial \hat{w}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{w}_\parallel}{n_n} \frac{\partial n_n}{\partial z} - \frac{\partial u_{n\parallel}}{\partial z} + \frac{1}{n_n} S_{n,n} \right) \bar{F}_n \nonumber \\ &\quad= R_\mathrm{CX} n_i (\bar{F}_i - \bar{F}_n) + \frac{1}{n_n} \bar{S}_n \\ \end{align}\]

Wall boundary condition (Knudsen cosine distribution)

For the 1D1V model, the Knudsen cosine distribution is marginalised over $v_\perp$ [Excalibur report TN-08]

\[\begin{align} \bar{f}_\mathrm{Kw}(v_\parallel) &= 2 \pi \int_0^\infty dv_\perp v_\perp f_\mathrm{Kw}(v_\parallel,v_\perp) \nonumber \\ &= 3 \sqrt{\pi} \left( \frac{m_n}{2T_w} \right)^{3/2} |v_\parallel| \,\mathrm{erfc}\!\left( \sqrt{\frac{m_n}{2 T_w}} |v_\parallel| \right) \\ \end{align}\]

Conversion to conventions of 1D1V Excalibur reports

The notes above follow the conventions of the nD2V systems of equations in the Excalibur reports. The original version of the 1D1V system as defined in the Excalibur reports had some conventions chosen differently. Quantities as defined in that original version are denoted by $\check{\cdot}$ where they differ from those defined here.

The thermal speed was defined using the parallel temperature

\[\begin{align} \check{v}_{Ts} &= \sqrt{\frac{2 T_{s\parallel}}{m_s}} \nonumber \\ &= \sqrt{\frac{2 (3 T_{s})}{m_s}} \nonumber \\ &= \sqrt{3} v_{Ts} \end{align}\]

with a corresponding change in the normalised velocity coordinate

\[\begin{align} \check{w}_\parallel &= \frac{v_\parallel - u_{s\parallel}}{\check{v}_{Ts}} \nonumber \\ &= \frac{v_\parallel - u_{s\parallel}}{\sqrt{3} v_{Ts}} \nonumber \\ &= \frac{w_\parallel}{\sqrt{3}} \end{align}\]

and in the shape function $\check F_s$ (it might be more consistent for the symbol for this shape function to have a $\bar{\cdot}$ as well as a $\check{\cdot}$, but that would look messy so the $\bar{\cdot}$ is omitted)

\[\begin{align} \check{F}_s(t,z,\check{w}_\parallel) &= \frac{\check{v}_{Ts}}{n_s} \bar{f}_s(t,z,v_\parallel) \nonumber \\ &= \sqrt{3} \frac{v_{Ts}}{n_s} \bar{f}_s(t,z,v_\parallel) \nonumber \\ &= \sqrt{3} \bar{F}_s(t,z,w_\parallel) \nonumber \\ \end{align}\]

Finally, the parallel heat flux was defined without the factor of $1/2$

\[\begin{align} \check{q}_{s\parallel} &= \int m_s (v_\parallel - u_{s\parallel})^3 \bar{f}_s dv_\parallel \nonumber \\ &= 2 q_{s\parallel} \end{align}\]

Note that the different definition of $\check w_\parallel$ compared to $w_\parallel$ and of $\check F_s$ compared to $\bar F_s$ can change form of the moment constraints - actually this only affects the energy constraint

\[\begin{align} 1 &= \int \bar{F}_s dw_\parallel = \int \frac{1}{\sqrt{3}} \check{F}_s \sqrt{3} d\check{w}_\parallel = \int \check{F}_s d\check{w}_\parallel \\ 0 &= \int w_\parallel \bar{F}_s dw_\parallel = \int \sqrt{3} \check{w}_\parallel \frac{1}{\sqrt{3}} \check{F}_s \sqrt{3} d\check{w}_\parallel = \sqrt{3} \int \check{w}_\parallel \check{F}_s d\check{w}_\parallel \\ \frac{3}{2} &= \int w_\parallel^2 \bar{F}_s dw_\parallel = \int 3 \check{w}_\parallel^2 \frac{1}{\sqrt{3}} \check{F}_s \sqrt{3} d\check{w}_\parallel = 3 \int \check{w}_\parallel^2 \check{F}_s d\check{w}_\parallel \nonumber \\ \Rightarrow \frac{1}{2} &= \int \check{w}_\parallel^2 \check{F}_s d\check{w}_\parallel \\ \end{align}\]

These conversions translate the equations as given above into the 1D1V forms in the Excalibur reports.

Dimensionless equations for code

To put the equations on a computer, we need to make them dimensionless. We choose a reference density $n_\mathrm{ref}$, temperature $T_\mathrm{ref}$, length $L_\mathrm{ref}$, and magnetic field $B_\mathrm{ref}$, and use the ion mass $m_i$ as the reference value (it might be useful to change this to 1 amu in future when we fully implement multi-species?). Denote dimensionless variables with a $\hat{\cdot}$ in this section.

We also define a reference speed, derived from the temperature and mass $c_\mathrm{ref} = \sqrt{T_\mathrm{ref}/m_i}$, chosen so that the relation

\[\begin{align} \frac{1}{2} m_i v_{Ts}^2 = T_s \end{align}\]

de-dimensionalises to

\[\begin{align} \frac{1}{2} \hat{v}_{Ts}^2 = \hat{T}_s \end{align}\]

The full set of dimensionless variables are related to the dimensional ones by

\[\begin{align} \hat{n}_s &= \frac{n_s}{n_\mathrm{ref}} \\ \hat{T}_s &= \frac{T_s}{T_\mathrm{ref}} \\ \hat{L}_z &= \frac{L_z}{L_\mathrm{ref}} \\ \hat{B} &= \frac{B}{B_\mathrm{ref}} \\ \hat{m}_i &= \frac{m_i}{m_i} = 1 \\ \hat{m}_e &= \frac{m_e}{m_i} \\ \hat{v}_{Ts} &= \frac{v_{Ts}}{c_\mathrm{ref}} \\ \hat{f}_s &= \frac{c_\mathrm{ref}^3}{n_\mathrm{ref}} f_s \\ \hat{F}_s &= F_s \\ \hat{\bar{f}}_s &= \frac{c_\mathrm{ref}}{n_\mathrm{ref}} \bar{f}_s \\ \hat{\bar{F}}_s &= \bar{F}_s \\ \hat{\phi} &= \frac{e \phi}{T_\mathrm{ref}} \\ \hat{E}_\parallel &= \frac{\partial \hat{\phi}}{\partial \hat z} = \frac{e L_\mathrm{ref}}{T_\mathrm{ref}} E_\parallel \\ \hat{t} &= \frac{c_\mathrm{ref} t}{L_\mathrm{ref}} \\ \frac{\partial}{\partial \hat{t}} &= \frac{L_\mathrm{ref}}{c_\mathrm{ref}} \frac{\partial}{\partial t} \\ \hat{z} &= \frac{z}{L_\mathrm{ref}} \\ \frac{\partial}{\partial \hat{z}} &= L_\mathrm{ref} \frac{\partial}{\partial z} \\ \hat{v}_\parallel &= \frac{v_\parallel}{c_\mathrm{ref}} \\ \frac{\partial}{\partial \hat{v}_\parallel} &= c_\mathrm{ref} \frac{\partial}{\partial v_\parallel} \\ \hat{v}_\perp &= \frac{v_\perp}{c_\mathrm{ref}} \\ \frac{\partial}{\partial \hat{v}_\perp} &= c_\mathrm{ref} \frac{\partial}{\partial v_\perp} \\ \hat{p}_s &= \hat{n}_s \hat{T_s} = \frac{p_s}{n_\mathrm{ref} T_\mathrm{ref}} \\ \hat{q}_{s\parallel} &= \frac{q_{s\parallel}}{m_i n_\mathrm{ref} c_\mathrm{ref}^3} = \frac{q_{s\parallel}}{n_\mathrm{ref} T_\mathrm{ref} c_\mathrm{ref}} \\ \hat{R}_\mathrm{CX} &= \frac{L_\mathrm{ref} n_\mathrm{ref} R_\mathrm{CX}}{c_\mathrm{ref}} \\ \hat{R}_\mathrm{ioniz} &= \frac{L_\mathrm{ref} n_\mathrm{ref} R_\mathrm{ioniz}}{c_\mathrm{ref}} \\ \hat{S}_s &= \frac{L_\mathrm{ref} c_\mathrm{ref}^2 S_s}{n_\mathrm{ref}} \\ \hat{\bar{S}}_s &= \frac{L_\mathrm{ref} \bar{S}_s}{n_\mathrm{ref}} \\ \hat{S}_{s,n} &= \frac{L_\mathrm{ref} S_{s,n}}{c_\mathrm{ref} n_\mathrm{ref}} \\ \hat{S}_{s,\mathrm{mom}} &= \frac{L_\mathrm{ref} S_{s,\mathrm{mom}}}{m_\mathrm{i} n_\mathrm{ref} c_\mathrm{ref}^2} \\ \hat{S}_{s,E} &= \frac{L_\mathrm{ref} S_{s,E}}{c_\mathrm{ref} m_\mathrm{i} n_\mathrm{ref} T_\mathrm{ref}} \\ \hat{S}_{s,p} &= \frac{L_\mathrm{ref} S_{s,p}}{c_\mathrm{ref} m_\mathrm{i} n_\mathrm{ref} T_\mathrm{ref}} \\ \hat{C}_{ii}(\hat{f_i}, \hat{f_i}) &= \frac{L_\mathrm{ref} c_\mathrm{ref}^2}{n_\mathrm{ref}} C_{ii}(f_i, f_i) \\ \hat{f}_\mathrm{Kw} &= c_\mathrm{ref}^4 f_\mathrm{Kw} \\ \hat{\bar{f}}_\mathrm{Kw} &= c_\mathrm{ref}^2 \bar{f}_\mathrm{Kw} \\ \end{align}\]

Using these definitions, the dimensionful equations above are converted to dimensionless equations just by putting a $\hat{\cdot}$ on all dimensionful quantities, and dropping the proton charge $e$ (as that is absorbed into $\hat \phi$). This may stop being true in 2D/3D where there are cross-field drifts where dimensionless parameters like $c_\mathrm{ref} m_i / L_\mathrm{ref} e B_\mathrm{ref}$ could enter. The dimensionless equations are written out in full below.

Definitions for any species

\[\begin{align} \hat{p}_{s\parallel} &= 2\pi\int_{-\infty}^\infty d\hat{v}_\parallel \int_0^\infty d\hat{v}_\perp \hat{v}_\perp \hat{m}_s \left( \hat{v}_\parallel - \hat{u}_{s\parallel} \right)^2 \hat{f}_s \nonumber \\ &= \hat{m}_s \hat{n}_s \hat{v}_{Ts}^2 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp w_\parallel^2 F_s \\ \hat{p}_{s\perp} &= 2\pi\int_{-\infty}^\infty d\hat{v}_\parallel \int_0^\infty d\hat{v}_\perp \hat{v}_\perp \hat{m}_s \frac{\hat{v}_\perp^2}{2} \hat{f}_s \\ &= \frac{1}{2} \hat{m}_s \hat{n}_s \hat{v}_{Ts}^2 2\pi \int_{-\infty}^\infty dw_\parallel \int_0^\infty dw_\perp w_\perp w_\perp^2 F_s \\ \hat{p}_s &= \hat{n}_s \hat{T}_s = \frac{\hat{m}_s}{2} \hat{n}_s \hat{v}_{Ts}^2 = \frac{1}{3}(\hat{p}_{s\parallel} + 2\hat{p}_{s\perp}) \\ \hat{T}_s &= \frac{1}{3}(\hat{T}_{s\parallel} + 2\hat{T}_{s\perp}) \\ \hat{v}_{Ts} &= \sqrt{\frac{2\hat{T}_s}{\hat{m}_s}}= \sqrt{\frac{2(\hat{T}_{s\parallel} + 2\hat{T}_{s\perp})}{3\hat{m}_s}} \\ \hat{q}_{s\parallel} &= \int \frac{\hat{m}_s}{2} \left( (\hat{v}_\parallel - \hat{u}_{s\parallel})^2 + \hat{v}_\perp^2 \right) (\hat{v}_\parallel - \hat{u}_{s\parallel}) \hat{f}_s d^3 \hat{v} \\ &= \hat{n}_s \hat{v}_{Ts}^3 \int \frac{\hat{m}_s}{2} \left( w_\parallel^2 + w_\perp^2 \right) w_\parallel F_s d^3 w \\ \end{align}\]

Dimensionless 1D2V ion equations

For convenience in future extension to multi-ion-species in future, we keep $\hat m_i$ in the equations below, even though it is equal to 1 in the conventions here.

\[\begin{align} & \frac{\partial \hat{n}_i}{\partial \hat{t}} + \frac{\partial}{\partial \hat{z}}\left( \hat{n}_i \hat{u}_{i\parallel} \right) = \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n + \hat{S}_{i,n} \\ & \hat{m}_i \frac{\partial}{\partial \hat{t}}(\hat{n}_i \hat{u}_{i\parallel}) + \hat{m}_i \frac{\partial}{\partial \hat{z}}(\hat{n}_i \hat{u}_{i\parallel}^2) + \frac{\partial \hat{p}_{i\parallel}}{\partial \hat{z}} + \hat{n}_i \frac{\partial \hat{\phi}}{\partial \hat{z}} \nonumber \\ &\quad= \hat{m}_i \hat{R}_\mathrm{CX} \hat{n}_i \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \hat{m}_i \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{u}_{n\parallel} + \hat{S}_{i,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial \hat{p}_i}{\partial \hat{t}} + \frac{\partial \hat{q}_{i\parallel}}{\partial \hat{z}} + \hat{p}_{i\parallel} \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{i\parallel} \frac{\partial \hat{p}_i}{\partial \hat{z}} + \frac{3}{2} \hat{p}_i \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= - \frac{1}{2} \hat{R}_\mathrm{CX} \hat{n}_i \hat{n}_n \left(3 \hat{T}_i - 3 \hat{T}_n - \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) + \frac{1}{2} \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \left(3 \hat{T}_n + \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} \hat{S}_{i,p} \\ \end{align}\]

Full kinetic equation

\[\begin{align} \frac{\partial \hat{f}_i}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{f}_i}{\partial \hat{z}} - \frac{1}{\hat{m}_i} \frac{\partial\hat{\phi}}{\partial \hat{z}} \frac{\partial \hat{f}_i}{\partial \hat{v}_\parallel} = \hat{C}_{ii}[\hat{f}_i, \hat{f}_i] - \hat{R}_\mathrm{CX} (\hat{n}_n \hat{f}_i - \hat{n}_i \hat{f}_n) + \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{f}_n + \hat{S}_i \end{align}\]

Separate $\hat n_i$

In this subsection $\hat F_s = c_\mathrm{ref}^3 f_s$.

\[\begin{align} & \frac{\partial \hat{F}_i}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{F}_i}{\partial \hat{z}} - \frac{1}{\hat{m}_i} \frac{\partial\hat{\phi}}{\partial \hat{z}} \frac{\partial \hat{F}_i}{\partial \hat{v}_\parallel} + \left( \frac{(\hat{v}_\parallel - \hat{u}_{i\parallel})}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} - \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} + \frac{1}{\hat{n}_i} \hat{S}_{i,n} \right) \hat{F}_i \nonumber \\ &\quad= \frac{1}{\hat{n}_i} \hat{C}_{ii}[\hat{n}_i \hat{F}_i, \hat{n}_i \hat{F}_i] - \hat{R}_\mathrm{CX} \hat{n}_n (\hat{F}_i - \hat{F}_n) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{F}_n + \frac{1}{\hat{n}_i} \hat{S}_i \\ \end{align}\]

Separate $\hat n_i$ and $\hat u_{i\parallel}$

In this subsection $\hat F_s = c_\mathrm{ref}^3 f_s$ and $\hat{\hat{w}}_\parallel = \hat{w}_\parallel / c_\mathrm{ref}$.

\[\begin{align} & \frac{\partial F_i}{\partial t} + (\hat{\hat{w}}_\parallel + \hat{u}_{i\parallel}) \frac{\partial \hat{F}_i}{\partial \hat{z}} \nonumber \\ &\quad- \left( \hat{\hat{w}}_\parallel \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} - \frac{1}{\hat{m}_i \hat{n}_i} \frac{\partial \hat{p}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{CX} \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{u}_{n\parallel} + \frac{1}{\hat{m}_i \hat{n}_i} \hat{S}_{i,\mathrm{mom}} \right) \frac{\partial \hat{F}_i}{\partial \hat{\hat{w}}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{\hat{w}}_\parallel}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} - \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} + \frac{1}{\hat{n}_i} \hat{S}_{i,n} \right) \hat{F}_i \nonumber \\ &\quad= \frac{1}{\hat{n}_i} \hat{C}_{ii}[\hat{n}_i \hat{F}_i, \hat{n}_i \hat{F}_i] - \hat{R}_\mathrm{CX} \hat{n}_n (\hat{F}_i - \hat{F}_n) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{F}_n + \frac{1}{\hat{n}_i} \hat{S}_i \\ \end{align}\]

Full moment-kinetics (separate $\hat n_i$, $\hat u_{i\parallel}$ and $\hat p_i$)

Note that we do not put a $\hat{\cdot}$ on the already dimensionless quantities $F_s$, $w_\parallel$, $w_\perp$.

\[\begin{align} &\frac{\partial F_i}{\partial \hat{t}} + \hat{\dot{z}} \frac{\partial F_i}{\partial \hat{z}} + \hat{\dot{w}}_\parallel \frac{\partial F_i}{\partial w_\parallel} + \hat{\dot{w}}_\perp \frac{\partial F_i}{\partial w_\perp} = \hat{\dot{F}}_i + \hat{\mathcal{C}}_i + \frac{\hat{v}_{Ti}^3}{\hat{n}_i} \hat{S}_i \\ &\hat{\dot{z}} = \hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel} \\ &\hat{\dot{w}}_\parallel = - \left( \frac{1}{\hat{v}_{Ti}} \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{i_\parallel}}{\hat{v}_{Ti}} \right) \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \frac{1}{\hat{m}_i \hat{v}_{Ti}} \frac{\partial\hat{\phi}}{\partial \hat{z}} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{t}} + w_\parallel \left( w_\parallel + \frac{\hat{u}_{i\parallel}}{\hat{v}_{Ti}} \right) \frac{\partial \hat{v}_{Ti}}{\partial \hat{z}} \right) \\ &\hat{\dot{w}}_\perp = -\left( \frac{w_\perp}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{i\parallel}}{\hat{v}_{Ti}} \right) w_\perp \frac{\partial \hat{v}_{Ti}}{\partial \hat{z}} \right) \\ &\frac{\hat{\dot{F}}_i}{F_i} = \frac{3}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{t}} + \frac{3 (\hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel})}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{z}} - \frac{1}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{t}} - \frac{(\hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel})}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} \\ &\hat{\mathcal{C}}_i = \frac{\hat{v}_{Ti}^3}{\hat{n}_i} \hat{C}_{ii}[\frac{\hat{n}_i F_i}{\hat{v}_{Ti}^3}, \frac{\hat{n}_i F_i}{\hat{v}_{Ti}^3}] - \hat{R}_\mathrm{CX} \hat{n}_n \left( F_i - \frac{\hat{v}_{Ti}^3}{\hat{v}_{Tn}^3} F_n \right) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \frac{\hat{v}_{Ti}^3}{\hat{v}_{Tn}^3} \hat{F}_n \\ \end{align}\]

Collision coefficient

The coefficient in front of the Fokker-Planck collision operator is

\[\begin{align} \gamma_{ss'} = \frac{2 \pi Z_s^2 Z_{s'}^2 e^4 \log\Lambda_{ss'}}{(4 \pi \epsilon_0)^2} \end{align}\]

and is made dimensionless as

\[\begin{align} \hat{\gamma}_{ss'} = \frac{n_\mathrm{ref} t_\mathrm{ref}}{m_\mathrm{ref}^2 c_\mathrm{ref}^3} \gamma_{ss'} = \frac{n_\mathrm{ref} L_\mathrm{ref}}{m_\mathrm{ref}^2 c_\mathrm{ref}^4} \gamma_{ss'} = \frac{1}{2} \frac{n_\mathrm{ref} Z_s^2 Z_{s'}^2 e^4 \log\Lambda_{ss'} L_\mathrm{ref}}{4 \pi \epsilon_0^2 m_\mathrm{ref}^2 c_\mathrm{ref}^4} \end{align}\]

\[\hat \gamma_{ss'}\]

is called nuii in moment_kinetics.fokker_planck.

nuii can also be set manually in the [fokker_planck_collisions] input section, which could be thought of as choosing $\log\Lambda$ to set the requested dimensionless nuii.

Dimensionless 1D2V neutral equations

\[\begin{align} & \frac{\partial \hat{n}_n}{\partial \hat{t}} + \frac{\partial}{\partial \hat{z}}\left( \hat{n}_n \hat{u}_{n\parallel} \right) = - \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n + \hat{S}_{n,n} \\ & \hat{m}_n \frac{\partial}{\partial \hat{t}}(\hat{n}_n \hat{u}_{n\parallel}) + \hat{m}_n \frac{\partial}{\partial \hat{z}}(\hat{n}_n \hat{u}_{n\parallel}^2) + \frac{\partial \hat{p}_{n\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{R}_\mathrm{CX} \hat{m}_n \hat{n}_i \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) - \hat{R}_\mathrm{ioniz} \hat{m}_n \hat{n}_e \hat{n}_n \hat{u}_{n\parallel} + \hat{S}_{n,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial \hat{p}_n}{\partial \hat{t}} + \frac{\partial \hat{q}_{n\parallel}}{\partial \hat{z}} + \hat{p}_{n\parallel} \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{n\parallel} \frac{\partial \hat{p}_n}{\partial \hat{z}} + \frac{3}{2} \hat{p}_n \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= \frac{1}{2} \hat{R}_\mathrm{CX} \hat{n}_i \hat{n}_n \left(3 \hat{T}_i - 3 \hat{T}_n + \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) - \frac{3}{2} \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{T}_n \nonumber \\ &\qquad+ \frac{3}{2} \hat{S}_{n,p} \\ \end{align}\]

Full kinetic equation

\[\begin{align} & \frac{\partial \hat{f}_n}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{f}_n}{\partial \hat{t}} = \hat{R}_\mathrm{CX} (\hat{n}_n \hat{f}_i - \hat{n}_i \hat{f}_n) - \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{f}_n + \hat{S}_n \\ \end{align}\]

Separate $\hat n_n$

In this subsection $\hat F_s = c_\mathrm{ref}^3 f_s$.

\[\begin{align} & \frac{\partial \hat{F}_n}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{F}_n}{\partial \hat{z}} + \left( \frac{(\hat{v}_\parallel - \hat{u}_{n\parallel})}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} - \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{1}{\hat{n}_n} \hat{S}_{n,n} \right) \hat{F}_n \nonumber \\ &\quad= \hat{R}_\mathrm{CX} \hat{n}_i (\hat{F}_i - \hat{F}_n) + \frac{1}{\hat{n}_n} \hat{S}_n \\ \end{align}\]

Separate $\hat n_n$ and $\hat u_{n\parallel}$

In this subsection $\hat F_s = c_\mathrm{ref}^3 f_s$ and $\hat{\hat{w}}_\parallel = \hat{w}_\parallel / c_\mathrm{ref}$.

\[\begin{align} & \frac{\partial \hat{F}_n}{\partial \hat{t}} + (\hat{\hat{w}}_\parallel + \hat{u}_{n\parallel}) \frac{\partial \hat{F}_n}{\partial \hat{z}} \nonumber \\ &\quad- \left( \hat{\hat{w}}_\parallel \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} - \frac{1}{\hat{m}_n \hat{n}_n} \frac{\partial \hat{p}_{n\parallel}}{\partial \hat{z}} - \hat{R}_\mathrm{CX} \hat{n}_i (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \frac{1}{\hat{m}_n \hat{n}_n} \hat{S}_{n,\mathrm{mom}} \right) \frac{\partial \hat{F}_n}{\partial \hat{\hat{w}}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{\hat{w}}_\parallel}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} - \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{1}{\hat{n}_n} \hat{S}_{n,n} \right) \hat{F}_n \nonumber \\ &\quad= \hat{R}_\mathrm{CX} \hat{n}_i (\hat{F}_i - \hat{F}_n) + \frac{1}{\hat{n}_n} \hat{S}_n \\ \end{align}\]

Full moment-kinetics (separate $\hat n_n$, $\hat u_{n\parallel}$ and $\hat p_{n}$)

Note that we do not put a $\hat{\cdot}$ on the already dimensionless quantities $F_s$, $w_\parallel$, $w_\perp$.

\[\begin{align} &\frac{\partial F_n}{\partial \hat{t}} + \hat{\dot{z}} \frac{\partial F_n}{\partial \hat{z}} + \hat{\dot{w}}_\parallel \frac{\partial F_n}{\partial w_\parallel} + \hat{\dot{w}}_\perp \frac{\partial F_n}{\partial w_\perp} = \hat{\dot{F}}_n + \hat{\mathcal{C}}_n + \frac{\hat{v}_{Tn}^3}{\hat{n}_n} \hat{S}_n \\ &\hat{\dot{z}} = \hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel} \\ &\hat{\dot{w}}_\parallel = - \left( \frac{1}{\hat{v}_{Tn}} \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{n_\parallel}}{\hat{v}_{Tn}} \right) \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{t}} + w_\parallel \left( w_\parallel + \frac{\hat{u}_{n\parallel}}{\hat{v}_{Tn}} \right) \frac{\partial \hat{v}_{Tn}}{\partial \hat{z}} \right) \\ &\hat{\dot{w}}_\perp = -\left( \frac{w_\perp}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{n\parallel}}{\hat{v}_{Tn}} \right) w_\perp \frac{\partial \hat{v}_{Tn}}{\partial \hat{z}} \right) \\ &\frac{\hat{\dot{F}}_n}{F_n} = \frac{3}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{t}} + \frac{3 (\hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel})}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{z}} - \frac{1}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{t}} - \frac{(\hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel})}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} \\ \hat{\mathcal{C}}_n &= \hat{R}_\mathrm{CX} \hat{n}_i \left( \frac{\hat{v}_{Tn}^3}{\hat{v}_{Ti}^3} F_i - F_n \right) \\ \end{align}\]

Dimensionless 1D1V ion equations

\[\begin{align} &\frac{\partial \hat{n}_i}{\partial \hat{t}} + \frac{\partial}{\partial \hat{z}}(\hat{n}_i \hat{u}_{i\parallel}) = \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n + \hat{S}_{i,n} \\ & \hat{m}_i \frac{\partial}{\partial \hat{t}}(\hat{n}_i \hat{u}_{i\parallel}) + \hat{m}_i \frac{\partial}{\partial \hat{z}}(\hat{n}_i \hat{u}_{i_\parallel}^2) + \frac{\partial \hat{p}_{i\parallel}}{\partial \hat{z}} + \hat{n}_i \frac{\partial \hat{\phi}}{\partial \hat{z}} \nonumber \\ &\quad = \hat{R}_\mathrm{CX} \hat{m}_i \hat{n}_i \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \hat{R}_\mathrm{ioniz} \hat{m}_i \hat{n}_e \hat{n}_n \hat{u}_{n\parallel} + \hat{S}_{i,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial \hat{p}_i}{\partial \hat{t}} + \frac{\partial \hat{q}_{i\parallel}}{\partial \hat{z}} + \hat{p}_{i\parallel} \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{i\parallel} \frac{\partial \hat{p}_i}{\partial \hat{z}} + \frac{3}{2} \hat{p}_i \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= - \frac{1}{2} \hat{R}_\mathrm{CX} \hat{n}_i \hat{n}_n \left(3 \hat{T}_i - 3 \hat{T}_n - \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) + \frac{1}{2} \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \left(3 \hat{T}_n + \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) \nonumber \\ &\qquad+ \frac{3}{2} \hat{S}_{i,p} \\ \end{align}\]

Full kinetic equation

\[\begin{align} \frac{\partial \hat{\bar{f}}_i}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{\bar{f}}_i}{\partial \hat{z}} - \frac{1}{\hat{m}_i} \frac{\partial\hat{\phi}}{\partial \hat{z}} \frac{\partial \hat{\bar{f}}_i}{\partial \hat{v}_\parallel} = \hat{\bar{C}}_{ii}[\hat{\bar{f}}_i, \hat{\bar{f}}_i] - \hat{R}_\mathrm{CX} (\hat{n}_n \hat{\bar{f}}_i - \hat{n}_i \hat{\bar{f}}_n) + \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{\bar{f}}_n + \hat{\bar{S}}_i \end{align}\]

Separate $\hat n_i$

In this subsection $\hat{\bar{F}}_s = c_\mathrm{ref} \bar f_s$.

\[\begin{align} & \frac{\partial \hat{\bar{F}}_i}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{\bar{F}}_i}{\partial \hat{z}} - \frac{1}{\hat{m}_i} \frac{\partial\hat{\phi}}{\partial \hat{z}} \frac{\partial \hat{\bar{F}}_i}{\partial \hat{v}_\parallel} + \left( \frac{(\hat{v}_\parallel - \hat{u}_{i\parallel})}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} - \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} + \frac{1}{\hat{n}_i} \hat{S}_{i,n} \right) \hat{\bar{F}}_i \nonumber \\ &\quad= \frac{1}{\hat{n}_i} \int \hat{C}_{ii}[\hat{n}_i \hat{F}_i, \hat{n}_i \hat{F}_i] d^2 \hat{v}_\perp - \hat{R}_\mathrm{CX} \hat{n}_n (\hat{\bar{F}}_i - \hat{\bar{F}}_n) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{\bar{F}}_n + \frac{1}{\hat{n}_i} \hat{\bar{S}}_i \\ \end{align}\]

Separate $\hat n_i$ and $\hat u_{i\parallel}$

In this subsection $\hat{\bar{F}}_s = c_\mathrm{ref} \bar f_s$ and $\hat{\hat{w}}_\parallel = \hat{w}_\parallel / c_\mathrm{ref}$.

\[\begin{align} & \frac{\partial \hat{\bar{F}}_i}{\partial \hat{t}} + (\hat{\hat{w}}_\parallel + \hat{u}_{i\parallel}) \frac{\partial \hat{\bar{F}}_i}{\partial \hat{z}} \nonumber \\ &\quad- \left( \hat{w}_\parallel \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} - \frac{1}{\hat{m}_i \hat{n}_i} \frac{\partial \hat{p}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{CX} \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{u}_{n\parallel} + \frac{1}{\hat{m}_i \hat{n}_i} \hat{S}_{i,\mathrm{mom}} \right) \frac{\partial \hat{\bar{F}}_i}{\partial \hat{\hat{w}}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{\hat{w}}_\parallel}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} - \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} + \frac{1}{\hat{n}_i} \hat{S}_{i,n} \right) \hat{\bar{F}}_i \nonumber \\ &\quad= \frac{1}{\hat{n}_i} \int \hat{C}_{ii}[\hat{n}_i \hat{F}_i, \hat{n}_i \hat{F}_i] d^2 \hat{v}_\perp - \hat{R}_\mathrm{CX} \hat{n}_n (\hat{\bar{F}}_i - \hat{\bar{F}}_n) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \hat{\bar{F}}_n + \frac{1}{\hat{n}_i} \hat{\bar{S}}_i \\ \end{align}\]

Full moment-kinetics (separate $\hat n_i$, $\hat u_{i\parallel}$ and $\hat p_i$)

\[\begin{align} &\frac{\partial \bar{F}_i}{\partial \hat{t}} + \hat{\dot{z}} \frac{\partial \bar{F}_i}{\partial \hat{z}} + \hat{\dot{w}}_\parallel \frac{\partial \bar{F}_i}{\partial w_\parallel} + \hat{\dot{w}}_\perp \frac{\partial \bar{F}_i}{\partial w_\perp} = \hat{\dot{\bar{F}}}_i + \hat{\bar{\mathcal{C}}}_i + \frac{\hat{v}_{Ti}}{\hat{n}_i} \hat{\bar{S}}_i \\ &\hat{\dot{z}} = \hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel} \\ &\hat{\dot{w}}_\parallel = - \left( \frac{1}{\hat{v}_{Ti}} \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{i_\parallel}}{\hat{v}_{Ti}} \right) \frac{\partial \hat{u}_{i\parallel}}{\partial \hat{z}} + \frac{1}{\hat{m}_i \hat{v}_{Ti}} \frac{\partial\hat{\phi}}{\partial \hat{z}} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{t}} + w_\parallel \left( w_\parallel + \frac{\hat{u}_{i\parallel}}{\hat{v}_{Ti}} \right) \frac{\partial \hat{v}_{Ti}}{\partial \hat{z}} \right) \\ &\frac{\hat{\dot{\bar{F}}}_i}{\bar{F}_i} = \frac{1}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{t}} + \frac{(\hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel})}{\hat{v}_{Ti}} \frac{\partial \hat{v}_{Ti}}{\partial \hat{z}} - \frac{1}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{t}} - \frac{(\hat{v}_{Ti} w_\parallel + \hat{u}_{i\parallel})}{\hat{n}_i} \frac{\partial \hat{n}_i}{\partial \hat{z}} \\ &\hat{\bar{\mathcal{C}}}_i = \frac{\hat{v}_{Ti}}{\hat{n}_i} \int \hat{C}_{ii}\left[\frac{\hat{n}_i}{\hat{v}_{Ti}^3} F_i, \frac{\hat{n}_i}{\hat{v}_{Ti}^3} F_i\right] d^2 \hat{v}_\perp - \hat{R}_\mathrm{CX} \hat{n}_n \left( \bar{F}_i - \frac{\hat{v}_{Ti}}{\hat{v}_{Tn}} \bar{F}_n \right) + \hat{R}_\mathrm{ioniz} \frac{\hat{n}_e \hat{n}_n}{\hat{n}_i} \frac{\hat{v}_{Ti}}{\hat{v}_{Tn}} \bar{F}_n \\ \end{align}\]

Dimensionless 1D1V neutral equations

\[\begin{align} &\frac{\partial \hat{n}_n}{\partial \hat{t}} + \frac{\partial}{\partial \hat{z}}\left( \hat{n}_n \hat{u}_{n\parallel} \right) = - \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n + \hat{S}_{n,n} \\ & \hat{m}_n \frac{\partial}{\partial \hat{t}}(\hat{n}_n \hat{u}_{n\parallel}) + \hat{m}_n \frac{\partial}{\partial \hat{z}}(\hat{n}_n \hat{u}_{n\parallel}^2) + \frac{\partial \hat{p}_{n\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= -\hat{R}_\mathrm{CX} \hat{m}_n \hat{n}_i \hat{n}_n (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) - \hat{R}_\mathrm{ioniz} \hat{m}_n \hat{n}_e \hat{n}_n \hat{u}_{n\parallel} + \hat{S}_{n,\mathrm{mom}} \\ & \frac{3}{2} \frac{\partial \hat{p}_n}{\partial \hat{t}} + \frac{\partial \hat{q}_{n\parallel}}{\partial \hat{z}} + \hat{p}_{n\parallel} \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{3}{2} \hat{u}_{n\parallel} \frac{\partial \hat{p}_n}{\partial \hat{z}} + \frac{3}{2} \hat{p}_n \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} \nonumber \\ &\quad= \frac{1}{2} \hat{R}_\mathrm{CX} \hat{n}_i \hat{n}_n \left(3 \hat{T}_i - 3 \hat{T}_n + \hat{m}_i (\hat{u}_{i\parallel} - \hat{u}_{n\parallel})^2 \right) - \frac{3}{2} \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{n}_n \hat{T}_n \nonumber \\ &\qquad+ \frac{3}{2} \hat{S}_{n,p} \\ \end{align}\]

Full kinetic equation

\[\begin{align} & \frac{\partial \hat{\bar{f}}_n}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{\bar{f}}_n}{\partial \hat{z}} = \hat{R}_\mathrm{CX} (\hat{n}_n \hat{\bar{f}}_i - \hat{n}_i \hat{\bar{f}}_n) - \hat{R}_\mathrm{ioniz} \hat{n}_e \hat{\bar{f}}_n + \hat{\bar{S}}_n \\ \end{align}\]

Separate $\hat n_n$

In this subsection $\hat{\bar{F}}_s = c_\mathrm{ref} \bar f_s$.

\[\begin{align} & \frac{\partial \hat{\bar{F}}_n}{\partial \hat{t}} + \hat{v}_\parallel \frac{\partial \hat{\bar{F}}_n}{\partial \hat{z}} + \left( \frac{(\hat{v}_\parallel - \hat{u}_{n\parallel})}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} - \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{1}{\hat{n}_n} \hat{S}_{n,n} \right) \hat{\bar{F}}_n \nonumber \\ &\quad= \hat{R}_\mathrm{CX} \hat{n}_i (\hat{\bar{F}}_i - \hat{\bar{F}}_n) + \frac{1}{\hat{n}_n} \hat{\bar{S}}_n \\ \end{align}\]

Separate $\hat n_n$ and $\hat u_{n\parallel}$

In this subsection $\hat{\bar{F}}_s = c_\mathrm{ref} \bar f_s$ and $\hat{\hat{w}}_\parallel = \hat{w}_\parallel / c_\mathrm{ref}$.

\[\begin{align} & \frac{\partial \hat{\bar{F}}_n}{\partial \hat{t}} + (\hat{\hat{w}}_\parallel + \hat{u}_{n\parallel}) \frac{\partial \hat{\bar{F}}_n}{\partial \hat{z}} \nonumber \\ &\quad- \left( \hat{\hat{w}}_\parallel \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} - \frac{1}{\hat{m}_n \hat{n}_n} \frac{\partial \hat{p}_{n\parallel}}{\partial \hat{z}} - \hat{R}_\mathrm{CX} \hat{n}_i (\hat{u}_{n\parallel} - \hat{u}_{i\parallel}) + \frac{1}{\hat{m}_n \hat{n}_n} \hat{S}_{n,\mathrm{mom}} \right) \frac{\partial \hat{\bar{F}}_n}{\partial \hat{\hat{w}}_\parallel} \nonumber \\ &\quad+ \left( \frac{\hat{\hat{w}}_\parallel}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} - \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} + \frac{1}{\hat{n}_n} \hat{S}_{n,n} \right) \hat{\bar{F}}_n \nonumber \\ &\quad= \hat{R}_\mathrm{CX} \hat{n}_i (\hat{\bar{F}}_i - \hat{\bar{F}}_n) + \frac{1}{\hat{n}_n} \hat{\bar{S}}_n \\ \end{align}\]

Full moment-kinetics (separate $\hat n_n$, $\hat u_{n\parallel}$ and $\hat p_{n}$)

Note that we do not put a $\hat{\cdot}$ on the already dimensionless quantities $\bar F_s$, $w_\parallel$, $w_\perp$.

\[\begin{align} &\frac{\partial \bar{F}_n}{\partial \hat{t}} + \hat{\dot{z}} \frac{\partial \bar{F}_n}{\partial \hat{z}} + \hat{\dot{w}}_\parallel \frac{\partial \bar{F}_n}{\partial w_\parallel} = \hat{\dot{\bar{F}}}_n + \hat{\bar{\mathcal{C}}}_n + \frac{\hat{v}_{Tn}}{\hat{n}_n} \hat{\bar{S}}_n \\ &\hat{\dot{z}} = \hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel} \\ &\hat{\dot{w}}_\parallel = - \left( \frac{1}{\hat{v}_{Tn}} \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{t}} + \left( w_\parallel + \frac{\hat{u}_{n_\parallel}}{\hat{v}_{Tn}} \right) \frac{\partial \hat{u}_{n\parallel}}{\partial \hat{z}} \right. \nonumber \\ &\qquad\quad \left. + \frac{w_\parallel}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{t}} + w_\parallel \left( w_\parallel + \frac{\hat{u}_{n\parallel}}{\hat{v}_{Tn}} \right) \frac{\partial \hat{v}_{Tn}}{\partial \hat{z}} \right) \\ &\frac{\hat{\dot{\bar{F}}}_n}{\bar{F}_n} = \frac{1}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{t}} + \frac{(\hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel})}{\hat{v}_{Tn}} \frac{\partial \hat{v}_{Tn}}{\partial \hat{z}} - \frac{1}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{t}} - \frac{(\hat{v}_{Tn} w_\parallel + \hat{u}_{n\parallel})}{\hat{n}_n} \frac{\partial \hat{n}_n}{\partial \hat{z}} \\ &\hat{\bar{\mathcal{C}}}_n = \hat{R}_\mathrm{CX} \hat{n}_i \left( \frac{\hat{v}_{Tn}}{\hat{v}_{Ti}} \bar{F}_i - \bar{F}_n \right) - R_\mathrm{ioniz} n_e \bar{F}_n \\ \end{align}\]

Conversion to old dimensionless equations

In the 1D1V Excalibur reports, and the original version of the code, a different set of dimensionless variables were chosen, where $\check c_\mathrm{ref} = \sqrt{2 \check T_\mathrm{ref} / m_i}$ was used as the primary reference variable, along with a reference density $\check n_\mathrm{ref}$, length $\check L_\mathrm{ref}$, and magnetic field $\check B_\mathrm{ref}$. In the Excalibur reports where a Boltzmann electron response, which implies constant electron temperature, was used the reference temperature was taken to be $\check T_\mathrm{ref} = T_e$, but this was later generalised in the code to an arbitrary $\check T_\mathrm{ref}$ to allow non-constant electron temperature (with Braginskii or kinetic electrons) or for varying the constant electron temperature of the Boltzmann response without re-scaling the rest of the dimensionless variables. Temperatures were de-dimensionalised using $\check T_\mathrm{ref}$, not $\check m_\mathrm{ref} \check c_\mathrm{ref}^2$.

In this section, denote dimensionless variables of the original version's conventions with a $\mathring{\cdot}$.

The 2V distribution function $f_s$ is de-dimensionalised so that a Maxwellian $\check f_{Ms} = \frac{n_s}{(\pi)^{3/2} \check v_{Ts}} \exp\left(-v_\parallel^2 / \check v_{Ts}^2 \right)$ would have a maximum value of $\mathring n_s / \mathring v_{Ts}^3$.

The 1V, marginalised distribution function $\bar f_s$ is de-dimensionalised so that a 1V Maxwellian $\check{\bar{f}}_{Ms} = \frac{n_s}{\sqrt{\pi} \check v_{Ts}} \exp\left(-v_\parallel^2 / \check v_{Ts}^2 \right)$ would have a maximum value of $\mathring n_s / \mathring v_{Ts}$, and its shape function $\check F_s$ would have a maximum value of 1.

Important to note, in the original version, the dimensionless electrostatic potential was defined using $\check T_\mathrm{ref}$, as $\mathring \phi = e \phi / \check T_\mathrm{ref}$, not using $m_i \check c_\mathrm{ref}^2$ that was used for temperatures.

In the original, for 1D1V runs the input temperatures were $T_{s\parallel}$ not $T_s$, whereas in the updated version the input temperatures are always $T_s$, which for 1D1V is $T_s = T_{s\parallel}/3$. However the electron temperature $T_e$, which is used when calculating the electrostatic potential using the Boltzmann response, is constant and is effectively always $T_{e\parallel}$ which is assumed equal to $T_e$ for Boltzmann response (i.e. do not assume $T_{e\perp}=0$ for Boltzmann electrons in 1D1V).

Using these definitions, the old dimensionless variables in terms of the physical variables and of the current dimensionless variables (where we assume for the conversion that $\check n_\mathrm{ref} = n_\mathrm{ref}$, $\check T_\mathrm{ref} = T_\mathrm{ref}$, and $\check B_\mathrm{ref} = B_\mathrm{ref}$) are, listing the ones with differences first

\[\begin{alignat}{2} \mathring{v}_{Ts} &= \frac{\check{v}_{Ts}}{\check{c}_\mathrm{ref}} = \frac{\sqrt{3} v_{Ts}}{\sqrt{2} c_\mathrm{ref}} = \sqrt{\frac{3}{2}} \hat{v}_{Ts} && \hat{v}_{Ts} = \sqrt{\frac{2}{3}} \mathring{v}_{Ts} \\ \mathring{f}_s &= \frac{\pi^{3/2} \check{c}_\mathrm{ref}^3}{\check{n}_\mathrm{ref}} f_s = \frac{(2 \pi)^{3/2} c_\mathrm{ref}^3}{n_\mathrm{ref}} f_s = (2 \pi)^{3/2} \hat{f}_s && \hat{f}_s = \frac{1}{(2 \pi)^{3/2}} \mathring{f}_s \\ \mathring{\bar{f}}_s &= \frac{\sqrt{\pi} \check{c}_\mathrm{ref}}{\check{n}_\mathrm{ref}} \bar{f}_s = \frac{\sqrt{2 \pi} c_\mathrm{ref}}{n_\mathrm{ref}} \bar{f}_s = \sqrt{2 \pi} \hat{\bar{f}}_s && \hat{\bar{f}}_s = \frac{1}{\sqrt{2 \pi}} \mathring{\bar{f}}_s \\ \mathring{F}_s &= \sqrt{\pi} \check{F}_s = \sqrt{\pi} \sqrt{3} \bar{F}_s && \bar{F}_s = \frac{1}{\sqrt{\pi} \sqrt{3}} \mathring{\bar{F}}_s \\ \mathring{f}_\mathrm{Kw} &= \frac{\pi^{3/2} \check{c}_\mathrm{ref}^4}{\check{n}_\mathrm{ref}} f_\mathrm{Kw} = 4\pi^{3/2} \frac{c_\mathrm{ref}^4}{n_\mathrm{ref}} f_\mathrm{Kw} = 4\pi^{3/2} \hat{f}_\mathrm{Kw} && \hat{f}_\mathrm{Kw} = \frac{1}{4\pi^{3/2}} \mathring{f}_\mathrm{Kw} \\ \mathring{\bar{f}}_\mathrm{Kw} &= \sqrt{\pi} \check{c}_\mathrm{ref}^2 \bar{f}_\mathrm{Kw} = 2\sqrt{\pi} c_\mathrm{ref} \bar{f}_\mathrm{Kw} = 2\sqrt{\pi} \hat{\bar{f}}_\mathrm{Kw} && \hat{\bar{f}}_\mathrm{Kw} = \frac{1}{2\sqrt{\pi}} \mathring{\bar{f}}_\mathrm{Kw} \\ \mathring{t} &= \frac{\check{c}_\mathrm{ref} t}{\check{L}_\mathrm{ref}} = \sqrt{2} \frac{c_\mathrm{ref} t}{L_\mathrm{ref}} = \sqrt{2} \hat{t} && \hat{t} = \frac{1}{\sqrt{2}} \mathring{t} \\ \frac{\partial}{\partial \mathring{t}} &= \frac{\check{L}_\mathrm{ref}}{\check{c}_\mathrm{ref}} \frac{\partial}{\partial t} = \frac{L_\mathrm{ref}}{\sqrt{2} c_\mathrm{ref}} \frac{\partial}{\partial \hat{t}} = \frac{1}{\sqrt{2}} \frac{\partial}{\partial \hat{t}} && \frac{\partial}{\partial \hat{t}} = \sqrt{2} \frac{\partial}{\partial \mathring{t}} \\ \mathring{v}_\parallel &= \frac{v_\parallel}{\check{c}_\mathrm{ref}} = \frac{v_\parallel}{\sqrt{2} c_\mathrm{ref}} = \frac{\hat{v}_\parallel}{\sqrt{2}} && \hat{v}_\parallel = \sqrt{2} \mathring{v}_\parallel \\ \frac{\partial}{\partial \mathring{v}_\parallel} &= \check{c}_\mathrm{ref} \frac{\partial}{\partial v_\parallel} = \sqrt{2} c_\mathrm{ref} \frac{\partial}{\partial v_\parallel} = \sqrt{2} \frac{\partial}{\partial \hat{v}_\parallel} && \frac{\partial}{\partial \hat{v}_\parallel} = \frac{1}{\sqrt{2}} \frac{\partial}{\partial \mathring{v}_\parallel} \\ \mathring{v}_\perp &= \frac{v_\perp}{\check{c}_\mathrm{ref}} = \frac{v_\perp}{\sqrt{2} c_\mathrm{ref}} = \frac{\hat{v}_\perp}{\sqrt{2}} && \hat{v}_\perp = \sqrt{2} \mathring{v}_\perp \\ \frac{\partial}{\partial \mathring{v}_\perp} &= \check{c}_\mathrm{ref} \frac{\partial}{\partial v_\perp} = \sqrt{2} c_\mathrm{ref} \frac{\partial}{\partial v_\perp} = \sqrt{2} \frac{\partial}{\partial \hat{v}_\perp} && \frac{\partial}{\partial \hat{v}_\perp} = \frac{1}{\sqrt{2}} \frac{\partial}{\partial \mathring{v}_\perp} \\ \mathring{w}_\parallel &= \frac{\mathring{v}_\parallel}{\mathring{v}_{Ts}} = \frac{v_\parallel}{\check{v}_{Ts}} = \frac{v_\parallel}{\sqrt{3}v_{Ts}} = \frac{w_\parallel}{\sqrt{3}} && w_\parallel = \sqrt{3} \mathring{w}_\parallel \\ \mathring{u}_{s\parallel} &= \frac{u_{s\parallel}}{\check{c}_\mathrm{ref}} = \frac{u_{s\parallel}}{\sqrt{2} c_\mathrm{ref}} = \frac{\hat{u}_{s\parallel}}{\sqrt{2}} && \hat{u}_{s\parallel} = \sqrt{2} \mathring{u}_{s\parallel} \\ \mathring{p}_s &= \frac{p_s}{\check{n}_\mathrm{ref} m_i \check{c}_\mathrm{ref}^2} = \frac{p_s}{\check{n}_\mathrm{ref} m_i 2 \check{T}_\mathrm{ref}} = \frac{\hat{p}_s}{2} && \hat{p}_s = 2 \mathring{p}_s \\ \mathring{q}_{s\parallel} &= \frac{\check{q}_{s\parallel}}{m_i \check{n}_\mathrm{ref} \check{c}_\mathrm{ref}^3} = \frac{2 q_{s\parallel}}{m_i n_\mathrm{ref} 2^{3/2} c_\mathrm{ref}^3} = \frac{\hat{q}_{s\parallel}}{\sqrt{2}} && \hat{q}_{s\parallel} = \sqrt{2}\mathring{q}_{s\parallel} \\ \mathring{R}_\mathrm{CX} &= \frac{\check{L}_\mathrm{ref} \check{n}_\mathrm{ref} R_\mathrm{CX}}{\check{c}_\mathrm{ref}} = \frac{L_\mathrm{ref} n_\mathrm{ref} R_\mathrm{CX}}{\sqrt{2} c_\mathrm{ref}} = \frac{1}{\sqrt{2}} \hat{R}_\mathrm{CX} && \hat{R}_\mathrm{CX} = \sqrt{2} \mathring{R}_\mathrm{CX} \\ \mathring{R}_\mathrm{ioniz} &= \frac{\check{L}_\mathrm{ref} \check{n}_\mathrm{ref} R_\mathrm{ioniz}}{\check{c}_\mathrm{ref}} = \frac{L_\mathrm{ref} n_\mathrm{ref} R_\mathrm{ioniz}}{\sqrt{2} c_\mathrm{ref}} = \frac{1}{\sqrt{2}} \hat{R}_\mathrm{ioniz} \quad && \hat{R}_\mathrm{ioniz} = \sqrt{2} \mathring{R}_\mathrm{ioniz} \\ \mathring{S}_s &= \frac{\pi^{3/2} \check{L}_\mathrm{ref} \check{c}_\mathrm{ref}^2 S_s}{\check{n}_\mathrm{ref}} = \frac{\pi^{3/2} L_\mathrm{ref} 2 c_\mathrm{ref}^2 S_s}{n_\mathrm{ref}} = 2 \pi^{3/2} \hat{S}_s && \hat{S}_s = \frac{1}{2 \pi^{3/2}} \mathring{S}_s \\ \mathring{\bar{S}}_s &= \frac{\sqrt{\pi} \check{L}_\mathrm{ref} \bar{S}_s}{\check{n}_\mathrm{ref}} = \sqrt{\pi} \hat{\bar{S}}_s && \hat{\bar{S}}_s = \frac{1}{\sqrt{\pi}} \mathring{\bar{S}}_s \\ \mathring{S}_{s,n} &= \frac{\check{L}_\mathrm{ref} S_{s,n}}{\check{c}_\mathrm{ref} \check{n}_\mathrm{ref}} = \frac{L_\mathrm{ref} S_{s,n}}{\sqrt{2} c_\mathrm{ref} n_\mathrm{ref}} = \frac{1}{\sqrt{2}} \hat{S}_{s,n} && \hat{S}_{s,n} = \sqrt{2} \mathring{S}_{s,n} \\ \mathring{S}_{s,\mathrm{mom}} &= \frac{\check{L}_\mathrm{ref} S_{s,\mathrm{mom}}}{\check{m}_\mathrm{ref} \check{n}_\mathrm{ref} \check{c}_\mathrm{ref}^2} = \frac{L_\mathrm{ref} S_{s,\mathrm{mom}}}{2 m_\mathrm{ref} n_\mathrm{ref} c_\mathrm{ref}^2} = \frac{1}{2} \hat{S}_{s,\mathrm{mom}} && \hat{S}_{s,\mathrm{mom}} = 2 \mathring{S}_{s,\mathrm{mom}} \\ \mathring{S}_{s,E} &= \frac{\check{L}_\mathrm{ref} S_{s,E}}{\check{m}_\mathrm{ref} \check{n}_\mathrm{ref} \check{c}_\mathrm{ref}^3} = \frac{L_\mathrm{ref} S_{s,E}}{2^{3/2} m_\mathrm{ref} n_\mathrm{ref} c_\mathrm{ref}^2} = \frac{1}{2 \sqrt{2}} \hat{S}_{s,E} && \hat{S}_{s,E} = 2 \sqrt{2} \mathring{S}_{s,E} \\ \mathring{S}_{s,p} &= \frac{\check{L}_\mathrm{ref} S_{s,p}}{\check{m}_\mathrm{ref} \check{n}_\mathrm{ref} \check{c}_\mathrm{ref}^3} = \frac{L_\mathrm{ref} S_{s,p}}{2^{3/2} m_\mathrm{ref} n_\mathrm{ref} c_\mathrm{ref}^2} = \frac{1}{2 \sqrt{2}} \hat{S}_{s,p} && \hat{S}_{s,p} = 2 \sqrt{2} \mathring{S}_{s,p} \\ \mathring{S}_{s,p_\parallel} &= \frac{1}{2 \sqrt{2}} \hat{S}_{s,p_\parallel} = \frac{3}{2 \sqrt{2}} \hat{S}_{s,p} && \hat{S}_{s,p} = \frac{2 \sqrt{2}}{3} \mathring{S}_{s,p_\parallel} \\ \mathring{\gamma}_{ss'} &= \frac{\check{n}_\mathrm{ref} \check{L}_\mathrm{ref}}{\check{m}_\mathrm{ref}^2 \check{c}_\mathrm{ref}^4} \gamma_{ss'} = \frac{n_\mathrm{ref} L_\mathrm{ref}}{4 m_\mathrm{ref}^2 c_\mathrm{ref}^4} \gamma_{ss'} = \frac{1}{4} \hat{\gamma}_{ss'} && \hat{\gamma}_{ss'} = 4 \mathring{\gamma}_{ss'} \\ \mathring{n}_s &= \frac{n_s}{\check{n}_\mathrm{ref}} = \hat{n}_s && \hat{n}_s = \mathring{n}_s \\ \mathring{T}_s &= \frac{T_s}{\check{T}_\mathrm{ref}} = \frac{T_s}{\check{T}_\mathrm{ref}} = \hat{T}_s && \hat{T}_s = \mathring{T}_s \\ \mathring{L}_z &= \frac{L_z}{\check{L}_\mathrm{ref}} = \hat{L}_z && \hat{L}_z = \mathring{L}_z \\ \mathring{B} &= \frac{B}{\check{B}_\mathrm{ref}} = \hat{B} && \hat{B} = \mathring{B} \\ \mathring{m}_i &= \frac{m_i}{m_i} = 1 = \hat{m}_i && \hat{m}_i = \mathring{m}_i \\ \mathring{m}_e &= \frac{m_e}{m_i} = \hat{m}_e && \hat{m}_e = \mathring{m_e} \\ \mathring{\phi} &= \frac{e \phi}{\check{T}_\mathrm{ref}} = \hat{\phi} && \hat{\phi} = \mathring{\phi} \\ \mathring{E}_\parallel &= \frac{e \check{L}_\mathrm{ref} E_\parallel}{\check{T}_\mathrm{ref}} = \hat{E}_\parallel && \hat{E}_\parallel = \mathring{E}_\parallel \\ \mathring{z} &= \frac{z}{\check{L}_\mathrm{ref}} = \hat{z} && \hat{z} = \mathring{z} \\ \frac{\partial}{\partial \mathring{z}} &= \check{L}_\mathrm{ref} \frac{\partial}{\partial z} = \frac{\partial}{\partial \hat{z}} && \frac{\partial}{\partial \hat{z}} = \frac{\partial}{\partial \mathring{z}} \\ \end{alignat}\]

In the 'old' equations a parallel-pressure source term $S_{s,p_\parallel} = \int m_s (v_\parallel - u_{s\parallel})^2 \bar S_s dv_\parallel$ was used in the parallel pressure equation. In the 1D1V limit the pressure source is $S_{s,p} = \int \frac{1}{3} m_s (v_\parallel - u_{s\parallel})^2 \bar S_s dv_\parallel = S_{s,p_\parallel}/3$.

Old 1D1V moment kinetic equations

These were the definitions and dimensionless variables used before PR #322, April 2025.