Moment kinetic equations

The following are partial notes on the derivation of the equations being solved by moment_kinetics. It would be useful to expand them with more details from the Excalibur/Neptune reports. Equation references give the report number and equation number, e.g. (TN-04;1) is equation (1) from report TN-04.pdf.

The drift kinetic equation (DKE), marginalised over $v_{\perp}$, for ions is, adding ionization and a source term to the form in (TN-04;1),

\[\begin{align} \frac{\partial f_{i}}{\partial t} +v_{\|}\frac{\partial f_{i}}{\partial z} -\frac{e}{m}\frac{\partial\phi}{\partial z}\frac{\partial f_{i}}{\partial v_{\|}} &= -R_{\mathrm{in}}\left(n_{n}f_{i}-n_{i}f_{n}\right)+R_{\mathrm{ion}}n_{i}f_{n} + S_i, \end{align}\]

and for neutrals, adding ionization and a source term to (TN-04;2)

\[\begin{align} \frac{\partial f_{n}}{\partial t} +v_{\|}\frac{\partial f_{n}}{\partial z} &= -R_{\mathrm{in}}\left(n_{i}f_{n}-n_{n}f_{i}\right)-R_{\mathrm{ion}}n_{i}f_{n} + S_n. \end{align}\]

Using the normalizations (TN04;5-11)

\[\begin{align} \tilde{f}_{s} & \doteq f_{s}\frac{c_{s}\sqrt{\pi}}{N_{e}}\\ \tilde{t} & \doteq t\frac{c_{s}}{L_{z}}\\ \tilde{z} & \doteq\frac{z}{L_{z}}\\ \tilde{v}_{\|} & \doteq\frac{v_{\|}}{c_{s}}\\ \tilde{n}_{s} & \doteq\frac{n_{s}}{N_{e}}\\ \tilde{\phi} & \doteq\frac{e\phi}{T_{e}}\\ \tilde{R}_{\mathrm{in}} & \doteq R_{\mathrm{in}}\frac{N_{e}L_{z}}{c_{s}}\\ \tilde{R}_{\mathrm{ion}} & \doteq R_{\mathrm{ion}}\frac{N_{e}L_{z}}{c_{s}} \tilde{S}_i = S_i \frac{c_s\sqrt{\pi}}{N_e} \frac{L_z}{c_s} = S_i \frac{L_z\sqrt{\pi}}{N_e} \end{align}\]

with $c_{s}\doteq\sqrt{2T_{e}/m_{s}}$ where $L_{z}$, $N_{e}$ and $T_{e}$ are constant reference parameters, the ion DKE is

\[\begin{align} \frac{\partial\tilde{f}_{i}}{\partial\tilde{t}} + \tilde{v}_{\|}\frac{\partial\tilde{f}_{i}}{\partial\tilde{z}} - \frac{1}{2}\frac{\partial\tilde{\phi}}{\partial\tilde{z}} \frac{\partial\tilde{f}_{i}}{\partial\tilde{v}_{\|}} &= -\tilde{R}_{in}\left(\tilde{n}_{n}\tilde{f}_{i}-\tilde{n}_{i}\tilde{f}_{n}\right) + \tilde{R}_{\mathrm{ion}}\tilde{n}_{i}\tilde{f}_{n} + \tilde{S}_i \end{align}\]

and the neutral DKE is

\[\begin{align} \frac{\partial\tilde{f}_{n}}{\partial\tilde{t}} + v_{\|}\frac{\partial\tilde{f}_{n}}{\partial\tilde{z}} &= -\tilde{R}_{in}\left(\tilde{n}_{i}\tilde{f}_{n}-\tilde{n}_{n}\tilde{f}_{i}\right) - \tilde{R}_{\mathrm{ion}}\tilde{n}_{i}\tilde{f}_{n} + \tilde{S}_n. \end{align}\]

Moment equations

Recalling the definitions (TN-04;15,29,63-66), but writing the integral in the energy equation over $\tilde{v}_{\|}$ instead of $w_{\|}$,

\[\begin{align} \tilde{n}_{s} & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{f}_{s}\\ % \tilde{n}_{s}\tilde{u}_{s} & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{v}_{\|}\tilde{f}_{s}\\ % \tilde{p}_{\|,s} & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\left(\tilde{v}_{\|} - \tilde{u}_{s}\right)^{2}\tilde{f}_{s} = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{v}_{\|}^{2}\tilde{f}_{s} - \tilde{n}_{s}\tilde{u}_{s}^{2}\\ % \tilde{q}_{\|,s} & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|} \left(\tilde{v}_{\|}-\tilde{u}_{s}\right)^{3}\tilde{f}_{s} \end{align}\]

[ intermediate steps ]

\[\begin{align*} \tilde{q}_{\|,s} & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{v}_{\|}^{3}\tilde{f}_{s} - 3\tilde{u}_{s}\frac{1}{\sqrt{\pi}}\int dv_{\|}v_{\|}^{2}f_{s} + 3u_{s}^{2}\frac{1}{\sqrt{\pi}}\int dv_{\|}v_{\|}f_{s} - u_{s}^{3}\frac{1}{\sqrt{\pi}}\int dv_{\|}f_{s} \\ % & = \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{v}_{\|}^{3}\tilde{f}_{s} - 3\tilde{u}_{s}\left(\tilde{p}_{\|,s}+\tilde{n}_{s}\tilde{u}_{s}^{2}\right) + 3\tilde{u}_{s}^{2}\tilde{n}_{s}\tilde{u}_{s}-\tilde{u}_{s}^{3}\tilde{n}_{s} \end{align*}\]

\[\begin{align} \tilde{q}_{\|,s} &= \frac{1}{\sqrt{\pi}}\int d\tilde{v}_{\|}\tilde{v}_{\|}^{3}\tilde{f}_{s} - 3\tilde{u}_{s}\tilde{p}_{\|,s} - \tilde{n}_{s}\tilde{u}_{s}^{3} \end{align}\]

we can take moments of the ion DKE to give ion moment equations (dropping tildes from here on)

\[\begin{align} \frac{\partial n_{i}}{\partial t}+\frac{\partial\left(n_{i}u_{i}\right)}{\partial z} & = -R_{in}\left(n_{n}n_{i}-n_{i}n_{n}\right)+R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_i\\ % & = R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_i \end{align}\]

\[\begin{align} \frac{\partial\left(n_{i}u_{i}\right)}{\partial t} + \frac{\partial\left(p_{\|,i} + n_{i}u_{i}^{2}\right)}{\partial z} + \frac{1}{2}\frac{\partial\phi}{\partial z}n_{i} &= -R_{in}\left(n_{n}n_{i}u_{i} - n_{i}n_{n}u_{n}\right) + R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} n_{i}\frac{\partial u_{i}}{\partial t} + u_{i}\frac{\partial n_{i}}{\partial t} + \frac{\partial p_{\|,i}}{\partial z} + u_{i}\frac{\partial\left(n_{i}u_{i}\right)}{\partial z} + n_{i}u_{i}\frac{\partial u_{i}}{\partial z} + \frac{1}{2}\frac{\partial\phi}{\partial z}n_{i} & = -R_{in}\left(n_{n}n_{i}u_{i} - n_{i}n_{n}u_{n}\right) + R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ % n_{i}\frac{\partial u_{i}}{\partial t} + u_{i}\left(R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{i}\right) + \frac{\partial p_{\|,i}}{\partial z} + n_{i}u_{i}\frac{\partial u_{i}}{\partial z} + \frac{1}{2}\frac{\partial\phi}{\partial z}n_{i} & = -R_{in}\left(n_{n}n_{i}u_{i} - n_{i}n_{n}u_{n}\right) + R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ \end{align*}\]

\[\begin{align} \frac{\partial u_{i}}{\partial t} + \frac{1}{n_{i}}\frac{\partial p_{\|,i}}{\partial z} + u_{i}\frac{\partial u_{i}}{\partial z} + \frac{1}{2}\frac{\partial\phi}{\partial z} &= -R_{in}n_{n}\left(u_{i}-u_{n}\right) + R_{\mathrm{ion}}n_{i}n_{n}\left(u_{n}-u_{i}\right) - \frac{u_{i}}{n_{i}} \int dv_\parallel S_{i} \end{align}\]

\[\begin{align} & \frac{\partial\left(p_{\|,i} + n_{i}u_{i}^{2}\right)}{\partial t} + \frac{\partial\left(q_{\|,i} + 3u_{i}p_{\|,i} + n_{i}u_{i}^{3}\right)}{\partial z} + \frac{\partial\phi}{\partial z}n_{i}u_{i} \\ & = -R_{in}\left(n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right)\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n}+n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{i} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} \frac{\partial p_{\|,i}}{\partial t} + \frac{1}{n_{i}}\frac{\partial\left(n_{i}u_{i}\right)^{2}}{\partial t} - \frac{\left(n_{i}u_{i}\right)^{2}}{n_{i}^{2}}\frac{\partial n_{i}}{\partial t} + \frac{\partial\left(q_{\|,i} + 3u_{i}p_{\|,i} + n_{i}u_{i}^{3}\right)}{\partial z} + \frac{\partial\phi}{\partial z}n_{i}u_{i} & = -R_{in}\left(n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right)\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{i} \\ % \frac{p_{\|,i}}{\partial t} + 2u_{i}\frac{\partial n_{i}u_{i}}{\partial t} - u_{i}^{2}\frac{\partial n_{i}}{\partial t} + \frac{\partial\left(q_{\|,i} + 3u_{i}p_{\|,i} + n_{i}u_{i}^{3}\right)}{\partial z} + \frac{\partial\phi}{\partial z}n_{i}u_{i} & = -R_{in}\left(n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right)\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{i} \\ % \frac{\partial p_{\|,i}}{\partial t} + 2u_{i}\left(-\frac{\partial p_{\|,i}}{\partial z} - \frac{\partial\left(n_{i}u_{i}^{2}\right)}{\partial z} - \frac{1}{2}\frac{\partial\phi}{\partial z}n_{i} - R_{in}\left(n_{n}n_{i}u_{i} - n_{i}n_{n}u_{n}\right) + R_{\mathrm{ion}}n_{i}n_{n}u_{n}\right) \\ -u_{i}^{2}\left(-\frac{\partial\left(n_{i}u_{i}\right)}{\partial z} + R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{i}\right) + \frac{\partial q_{\|,i}}{\partial z} + \frac{\partial\left(3u_{i}p_{\|,i}\right)}{\partial z} + \frac{\partial\left(n_{i}u_{i}^{3}\right)}{\partial z} + \frac{\partial\phi}{\partial z}n_{i}u_{i} & = -R_{in}\left(n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right)\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{i} \\ % \frac{\partial p_{\|,i}}{\partial t} + u_{i}\frac{\partial p_{\|,i}}{\partial z} + 3p_{\|,i}\frac{\partial u_{i}}{\partial z} + \frac{\partial q_{\|,i}}{\partial z} & = -R_{in}\left(n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - 2u_{i}\left(n_{n}n_{i}u_{i} - n_{i}n_{n}u_{n}\right)\right) \\ & \quad + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2} + n_{n}u_{i}^{2} - 2n_{n}u_{i}u_{n}\right) + \int dv_\parallel v_\parallel^2 S_{i} + u_{i}^2 \int dv_\parallel S_{i} \\ % \frac{\partial p_{\|,i}}{\partial t} + u_{i}\frac{\partial p_{\|,i}}{\partial z} + 3p_{\|,i}\frac{\partial u_{i}}{\partial z} + \frac{\partial q_{\|,i}}{\partial z} & = -R_{in}\left(n_{n}p_{\|,i} - n_{i}p_{\|,n} - n_{i}n_{n}\left(u_{i}^{2} + u_{n}^{2} - 2u_{i}u_{n}\right)\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}\left(u_{n} - u_{i}\right)^{2}\right) \\ & \quad + \int dv_\parallel v_\parallel^2 S_{i} + u_{i}^2 \int dv_\parallel S_{i} \\ \end{align*}\]

\[\begin{align} & \frac{\partial p_{\|,i}}{\partial t} + u_{i}\frac{\partial p_{\|,i}}{\partial z} + 3p_{\|,i}\frac{\partial u_{i}}{\partial z} + \frac{\partial q_{\|,i}}{\partial z} \\ & = -R_{in}\left(n_{n}p_{\|,i} - n_{i}p_{\|,n} - n_{i}n_{n}\left(u_{i} - u_{n}\right)^{2}\right) + R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}\left(u_{n} - u_{i}\right)^{2}\right) \\ & \quad + \int dv_\parallel v_\parallel^2 S_{i} + u_{i}^2 \int dv_\parallel S_{i} \\ \end{align}\]

and of the neutral DKE to give neutral moment equations

\[\begin{align} \frac{\partial n_{n}}{\partial t} + \frac{\partial\left(n_{n}u_{n}\right)}{\partial z} & = -R_{i}\left(n_{i}n_{n} - n_{n}n_{i}\right) - R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{n} \\ % & =-R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{n} \end{align}\]

\[\begin{align} \frac{\partial\left(n_{n}u_{n}\right)}{\partial t} + \frac{\partial\left(p_{\|,n} + n_{n}u_{n}^{2}\right)}{\partial z} &= -R_{in}\left(n_{i}n_{n}u_{n} - n_{n}n_{i}u_{i}\right) - R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align} n_{n}\frac{\partial u_{n}}{\partial t} + u_{n}\frac{\partial n_{n}}{\partial t} + \frac{\partial p_{\|,n}}{\partial z} + u_{n}\frac{\partial\left(n_{n}u_{n}\right)}{\partial z} + n_{n}u_{n}\frac{\partial u_{n}}{\partial z} & = -R_{in}\left(n_{i}n_{n}u_{n} - n_{n}n_{i}u_{i}\right) - R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ % n_{n}\frac{\partial u_{n}}{\partial t} + u_{n}\left(-R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{n}\right) + \frac{\partial p_{\|,n}}{\partial z} + n_{n}u_{s}\frac{\partial u_{n}}{\partial z} & = -R_{in}\left(n_{i}n_{n}u_{n} - n_{n}n_{i}u_{i}\right) - R_{\mathrm{ion}}n_{i}n_{n}u_{n} \\ \end{align}\]

\[\begin{align} \frac{\partial u_{n}}{\partial t} + \frac{1}{n_{n}}\frac{\partial p_{\|,n}}{\partial z} + u_{n}\frac{\partial u_{n}}{\partial z} &= -R_{in}n_{i}\left(u_{n} - u_{i}\right) - \frac{u_{n}}{n_{n}} \int dv_\parallel S_{n} \end{align}\]

\[\begin{align} & \frac{\partial\left(p_{\|,n} + n_{n}u_{n}^{2}\right)}{\partial t} + \frac{\partial\left(q_{\|,n} + 3u_{n}p_{\|,n} + n_{n}u_{n}^{3}\right)}{\partial z} + q_{n}\frac{\partial\phi}{\partial z}n_{n}u_{n} \\ & = -R_{in}\left(n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right)\right) - R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{n} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} \frac{\partial p_{\|,n}}{\partial t} + \frac{1}{n_{n}}\frac{\partial\left(n_{n}u_{n}\right)^{2}}{\partial t} - \frac{\left(n_{n}u_{n}\right)^{2}}{n_{n}^{2}}\frac{\partial n_{n}}{\partial t} + \frac{\partial\left(q_{\|,n} + 3u_{n}p_{\|,n} + n_{n}u_{n}^{3}\right)}{\partial z} + q_{n}\frac{\partial\phi}{\partial z}n_{n}u_{n} & =-R_{in}\left(n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right)\right) - R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{n} \\ % \frac{\partial p_{\|,n}}{\partial t} + 2u_{n}\frac{\partial n_{n}u_{n}}{\partial t} - u_{n}^{2}\frac{\partial n_{n}}{\partial t} + \frac{\partial\left(q_{\|,n} + 3u_{n}p_{\|,n} + n_{n}u_{n}^{3}\right)}{\partial z} + q_{n}\frac{\partial\phi}{\partial z}n_{n}u_{n} & = -R_{in}\left(n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right)\right) - R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{n} \\ % \frac{\partial p_{\|,n}}{\partial t} + 2u_{n}\left(-\frac{\partial p_{\|,n}}{\partial z} - \frac{\partial\left(n_{n}u_{n}^{2}\right)}{\partial z} - \frac{q_{n}}{2}\frac{\partial\phi}{\partial z}n_{n} - R_{in}\left(n_{i}n_{n}u_{n} - n_{n}n_{i}u_{i}\right) - R_{\mathrm{ion}}n_{i}n_{n}u_{n}\right) \\ - u_{n}^{2}\left(-\frac{\partial\left(n_{n}u_{n}\right)}{\partial z} - R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{n}\right) + \frac{\partial q_{\|,n}}{\partial z} + \frac{\partial\left(3u_{n}p_{\|,n}\right)}{\partial z} + \frac{\partial\left(n_{n}u_{n}^{3}\right)}{\partial z} & = -R_{in}\left(n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right)\right) - R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) + \int dv_\parallel v_\parallel^2 S_{n} \\ % \frac{\partial p_{\|,n}}{\partial t} + u_{n}\frac{\partial p_{\|,n}}{\partial z} + 3p_{\|,n}\frac{\partial u_{n}}{\partial z} + \frac{\partial q_{\|,n}}{\partial z} & = -R_{in}\left(n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2}\right) - n_{n}\left(p_{\|,i} + n_{i}u_{i}^{2}\right) - 2u_{n}\left(n_{i}n_{n}u_{n} - n_{n}n_{i}u_{i}\right)\right) - R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + n_{n}u_{n}^{2} + n_{n}u_{n}^{2} - 2n_{n}u_{n}u_{n}\right) + \int dv_\parallel v_\parallel^2 S_{n} + u_{n}^2\int dv_\parallel S_{n} \\ % \frac{\partial p_{\|,n}}{\partial t} + u_{n}\frac{\partial p_{\|,n}}{\partial z} + 3p_{\|,n}\frac{\partial u_{n}}{\partial z} + \frac{\partial q_{\|,n}}{\partial z} & = -R_{in}\left(n_{i}p_{\|,n} - n_{n}p_{\|,i} - n_{n}n_{i}\left(u_{n}^{2} + u_{i}^{2} - 2u_{n}u_{i}\right)\right) - R_{\mathrm{ion}}n_{i}p_{\|,n} + \int dv_\parallel v_\parallel^2 S_{n} + u_{n}^2\int dv_\parallel S_{n} \\ \end{align*}\]

\[\begin{align} & \frac{\partial p_{\|,n}}{\partial t} + u_{n}\frac{\partial p_{\|,n}}{\partial z} + 3p_{\|,n}\frac{\partial u_{n}}{\partial z} + \frac{\partial q_{\|,n}}{\partial z} \\ & = -R_{in}\left(n_{i}p_{\|,n} - n_{n}p_{\|,i} - n_{n}n_{i}\left(u_{n} - u_{i}\right)^{2}\right) - R_{\mathrm{ion}}n_{i}p_{\|,n} \\ & \quad + \int dv_\parallel v_\parallel^2 S_{n} + u_{n}^2\int dv_\parallel S_{n} \\ \end{align}\]

Kinetic equation

For the moment-kinetic equation for the normalized distribution function

\[\begin{align} g_{s}(w_{\|,s}) &= \frac{v_{\mathrm{th},s}}{n_{s}}f_{s}(v_{\|}(w_{\|,s})) \end{align}\]

we transform to the normalized velocity coordinate

\[\begin{align} w_{\|,s} &= \frac{v_{\|} - u_{s}}{v_{\mathrm{th},s}} \end{align}\]

The derivatives transform as

We use an energy equation that evolves $p_{\|,s}$ not $v_{\mathrm{th},s}$, so use

\[\begin{align} v_{\mathrm{th},s}^{2} & = 2\frac{p_{\|,s}}{n_{s}} \\ % \Rightarrow v_{\mathrm{th},s}\frac{\partial v_{\mathrm{th},s}}{\partial t} & = \frac{1}{n_{s}}\frac{\partial p_{\|,s}}{\partial t} - \frac{p_{\|,s}}{n_{s}^{2}}\frac{\partial n_{s}}{\partial t}\\ % v_{\mathrm{th},s}\frac{\partial v_{\mathrm{th},s}}{\partial z} & = \frac{1}{n_{s}}\frac{\partial p_{\|,s}}{\partial z} - \frac{p_{\|,s}}{n_{s}^{2}}\frac{\partial n_{s}}{\partial z} \end{align}\]

to convert the transformations above to

\[\begin{align} \left.\frac{\partial f_{s}}{\partial t}\right|_{z,v\|} & \rightarrow\left.\frac{\partial f_{s}}{\partial t}\right|_{z,w\|} - \frac{1}{v_{\mathrm{th},s}}\frac{\partial u_{s}}{\partial t}\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|} - \frac{w_{\|,s}}{v_{\mathrm{th},s}^{2}}\left(\frac{1}{n_{s}}\frac{\partial p_{\|,s}}{\partial t} - \frac{p_{\|,s}}{n_{s}^{2}}\frac{\partial n_{s}}{\partial t}\right)\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|}\\ % & = \left.\frac{\partial f_{s}}{\partial t}\right|_{z,w\|} - \frac{1}{v_{\mathrm{th},s}}\frac{\partial u_{s}}{\partial t}\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|} - \frac{w_{\|,s}}{2}\left(\frac{1}{p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} - \frac{1}{n_{s}}\frac{\partial n_{s}}{\partial t}\right)\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|}\\ % \left.\frac{\partial f_{s}}{\partial z}\right|_{z,v\|} & \rightarrow\left.\frac{\partial f_{s}}{\partial z}\right|_{z,w\|} - \frac{1}{v_{\mathrm{th},s}}\frac{\partial u_{s}}{\partial z}\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|} - \frac{w_{\|,s}}{v_{\mathrm{th},s}^{2}}\left(\frac{1}{n_{s}}\frac{\partial p_{\|,s}}{\partial z} - \frac{p_{\|,s}}{n_{s}^{2}}\frac{\partial n_{s}}{\partial z}\right)\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|}\\ % & = \left.\frac{\partial f_{s}}{\partial z}\right|_{z,w\|} - \frac{1}{v_{\mathrm{th},s}}\frac{\partial u_{s}}{\partial z}\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|} - \frac{w_{\|,s}}{2}\left(\frac{1}{p_{\|,s}}\frac{\partial p_{\|,s}}{\partial z} - \frac{1}{n_{s}}\frac{\partial n_{s}}{\partial z}\right)\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|}\\ % \left.\frac{\partial f_{s}}{\partial v_{\|}}\right|_{z,v\|} & \rightarrow\frac{1}{v_{\mathrm{th},s}}\left.\frac{\partial f_{s}}{\partial w_{\|,s}}\right|_{z,w\|} \end{align}\]

Using these transformations gives the ion DKE in a form similar to (TN-04;55) (but writing out $\dot{w}_{\|}$ in full here, and not using the moment equations for the moment)

\[\begin{align} & \frac{\partial f_{i}}{\partial t} - \frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial t}\frac{\partial f_{i}}{\partial w_{\|,i}} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial t} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial t}\right)\frac{\partial f_{i}}{\partial w_{\|,i}} \\ & + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\left(\frac{\partial f_{i}}{\partial z} - \frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial z}\frac{\partial f_{i}}{\partial w_{\|,i}} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial z} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial z}\right)\frac{\partial f_{i}}{\partial w_{\|,i}}\right) \\ & - \frac{1}{2v_{\mathrm{th},i}}\frac{\partial\phi}{\partial z}\frac{\partial f_{i}}{\partial w_{\|,i}} \\ & = -R_{in}\left(n_{n}f_{i} - n_{i}f_{n}\right) + R_{\mathrm{ion}}n_{i}f_{n} + S_{i} \end{align}\]

[ intermediate steps ]

\[\begin{align*} \frac{\partial f_{i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial f_{i}}{\partial z} - \frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial t}\frac{\partial f_{i}}{\partial w_{\|,i}} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial t} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial t}\right)\frac{\partial f_{i}}{\partial w_{\|,i}} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\left(-\frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial z}\frac{\partial f_{i}}{\partial w_{\|,i}} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial z} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial z}\right)\frac{\partial f_{i}}{\partial w_{\|,i}}\right) - \frac{1}{2v_{\mathrm{th},i}}\frac{\partial\phi}{\partial z}\frac{\partial f_{i}}{\partial w_{\|,i}} & = -R_{in}\left(n_{n}f_{i} - n_{i}f_{n}\right) + R_{\mathrm{ion}}n_{i}f_{n} + S_{i} \\ % \frac{\partial f_{i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial f_{i}}{\partial z} + \left[-\frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial t} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial t} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial t}\right) + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\left(-\frac{1}{v_{\mathrm{th},i}}\frac{\partial u_{i}}{\partial z} - \frac{w_{\|,i}}{2}\left(\frac{1}{p_{\|,i}}\frac{\partial p_{\|,i}}{\partial z} - \frac{1}{n_{i}}\frac{\partial n_{i}}{\partial z}\right)\right) - \frac{1}{2v_{\mathrm{th},i}}\frac{\partial\phi}{\partial z}\right]\frac{\partial f_{i}}{\partial w_{\|,i}} & = -R_{in}\left(n_{n}f_{i} - n_{i}f_{n}\right) + R_{\mathrm{ion}}n_{i}f_{n} + S_{i} \\ \end{align*}\]

\[\begin{align} & \frac{\partial f_{i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial f_{i}}{\partial z} \\ & + \left[-\frac{1}{v_{\mathrm{th},i}}\left(\frac{\partial u_{i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial u_{i}}{\partial z} + \frac{1}{2}\frac{\partial\phi}{\partial z}\right)\right. \\ & \qquad - \frac{w_{\|,i}}{2}\frac{1}{p_{\|,i}}\left(\frac{\partial p_{\|,i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial p_{\|,i}}{\partial z}\right) \\ & \qquad + \frac{w_{\|,i}}{2}\frac{1}{n_{i}}\left(\frac{\partial n_{i}}{\partial t} + \left(v_{\mathrm{th},i}w_{\|,i} + \left.u_{i}\right)\frac{\partial n_{i}}{\partial z}\right)\right]\frac{\partial f_{i}}{\partial w_{\|,i}} \\ & = -R_{in}\left(n_{n}f_{i} - n_{i}f_{n}\right) + R_{\mathrm{ion}}n_{i}f_{n} + S_{i} \end{align}\]

and the neutral DKE

\[\begin{align} & \frac{\partial f_{n}}{\partial t} - \frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial t}\frac{\partial f_{n}}{\partial w_{\|,n}} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial t} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial t}\right)\frac{\partial f_{n}}{\partial w_{\|,n}} \\ & + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\left(\frac{\partial f_{n}}{\partial z} - \frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial z}\frac{\partial f_{n}}{\partial w_{\|,n}} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial z} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial z}\right)\frac{\partial f_{n}}{\partial w_{\|,n}}\right) \\ & = -R_{in}\left(n_{i}f_{n} - n_{n}f_{i}\right) - R_{\mathrm{ion}}n_{i}f_{n} + S_{n} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} \frac{\partial f_{n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial f_{n}}{\partial z} - \frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial t}\frac{\partial f_{n}}{\partial w_{\|,n}} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial t} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial t}\right)\frac{\partial f_{n}}{\partial w_{\|,n}} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\left(-\frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial z}\frac{\partial f_{n}}{\partial w_{\|,n}} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial z} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial z}\right)\frac{\partial f_{n}}{\partial w_{\|,n}}\right) & = -R_{in}\left(n_{i}f_{n} - n_{n}f_{i}\right) - R_{\mathrm{ion}}n_{i}f_{n} + S_{n} \\ % \frac{\partial f_{n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial f_{n}}{\partial z} + \left[-\frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial t} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial t} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial t}\right) + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\left(-\frac{1}{v_{\mathrm{th},n}}\frac{\partial u_{n}}{\partial z} - \frac{w_{\|,n}}{2}\left(\frac{1}{p_{\|,n}}\frac{\partial p_{\|,n}}{\partial z} - \frac{1}{n_{n}}\frac{\partial n_{n}}{\partial z}\right)\right)\right]\frac{\partial f_{n}}{\partial w_{\|,n}} & = -R_{in}\left(n_{i}f_{n} - n_{n}f_{i}\right) - R_{\mathrm{ion}}n_{i}f_{n} + S_{n} \\ \end{align*}\]

\[\begin{align} & \frac{\partial f_{n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial f_{n}}{\partial z} \\ & + \left[-\frac{1}{v_{\mathrm{th},n}}\left(\frac{\partial u_{n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n}+u_{n}\right)\frac{\partial u_{n}}{\partial z}\right)\right. \\ & \qquad - \frac{w_{\|,n}}{2}\frac{1}{p_{\|,n}}\left(\frac{\partial p_{\|,n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial p_{\|,n}}{\partial z}\right) \\ & \qquad + \left.\frac{w_{\|,n}}{2}\frac{1}{n_{n}}\left(\frac{\partial n_{n}}{\partial t} + \left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial n_{n}}{\partial z}\right)\right]\frac{\partial f_{n}}{\partial w_{\|,n}} \\ & = -R_{in}\left(n_{i}f_{n} - n_{n}f_{i}\right) - R_{\mathrm{ion}}n_{i}f_{n} + S_{n} \end{align}\]

We also normalise $f$ and write the DKEs for

\[\begin{align} g_{s} & =\frac{v_{\mathrm{th,s}}}{n_{s}}f_{s} \\ % \Rightarrow\frac{\partial f_{s}}{\partial t} & = \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial t} + \frac{g_{s}}{v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} - \frac{n_{s}g_{s}}{v_{\mathrm{th},s}^{2}}\frac{\partial v_{\mathrm{th},s}}{\partial t} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} \frac{\partial f_{s}}{\partial t} & = \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial t} + \frac{g_{s}}{v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} - \frac{n_{s}g_{s}}{v_{\mathrm{th},s}^{3}}\left(\frac{1}{n_{s}}\frac{\partial p_{\|,s}}{\partial t} - \frac{p_{\|,s}}{n_{s}^{2}}\frac{\partial n_{s}}{\partial t}\right) \\ % & = \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial t} + \frac{g_{s}}{v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}n_{s}}{2v_{\mathrm{th},s}p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} + \frac{g_{s}}{2v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} \\ \end{align*}\]

\[\begin{align} \frac{\partial f_{s}}{\partial t} & = \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial t} + \frac{3g_{s}}{2v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}n_{s}}{2v_{\mathrm{th},s}p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} \\ % \frac{\partial f_{s}}{\partial w_{\|,s}} & = \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial w_{\|,s}}, \end{align}\]

For brevity, do the following manipulations for $g_{s}$ rather than for ions and neutrals separately by using $q_{i}=1$, $q_{n}=0$ and with the $+$'ve sign for the ion DKE and $-$'ve sign for the neutral DKE.

\[\begin{align} & \frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial t} + \frac{3g_{s}}{2v_{\mathrm{th},s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}n_{s}}{2v_{\mathrm{th},s}p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} \\ & + \left[-\frac{1}{v_{\mathrm{th},s}}\left(\frac{\partial u_{s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial u_{s}}{\partial z} + \frac{q_{s}}{2}\frac{\partial\phi}{\partial z}\right)\right. \\ & \qquad - \frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(\frac{\partial p_{\|,s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \qquad + \left.\frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\frac{\partial n_{s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial n_{s}}{\partial z}\right)\right]\frac{n_{s}}{v_{\mathrm{th},s}}\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}\left(n_{s'}\frac{n_{s}}{v_{\mathrm{th},s}}g_{s} - n_{s}\frac{n_{s'}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + S_{s} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align} \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \frac{3g_{s}}{2n_{s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}}{2p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} \\ & + \left[-\frac{1}{v_{\mathrm{th},s}}\left(\frac{\partial u_{s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial u_{s}}{\partial z} + \frac{q_{s}}{2}\frac{\partial\phi}{\partial z}\right) - \frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(\frac{\partial p_{\|,s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial p_{\|,s}}{\partial z}\right) + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\frac{\partial n_{s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \\ % \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \frac{3g_{s}}{2n_{s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}}{2p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} \\ & + \left[-\frac{1}{v_{\mathrm{th},s}}\left(\frac{n_{s}}{n_{s}}\frac{\partial u_{s}}{\partial t} + \frac{n_{s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial u_{s}}{\partial z} + \frac{u_{s}}{n_{s}}\left(\frac{\partial n}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial n}{\partial z}\right) + \frac{q_{s}}{2}\frac{\partial\phi}{\partial z}\right) + \frac{u_{s}}{n_{s}v_{\mathrm{th},s}}\left(\frac{\partial n}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial n}{\partial z}\right) - \frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(\frac{\partial p_{\|,s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial p_{\|,s}}{\partial z}\right) + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\frac{\partial n_{s}}{\partial t} + \left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \\ \end{align}\]

So then if we use the moment equations we can rewrite the DKE as

\[\begin{align} & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \frac{3g_{s}}{2n_{s}}\frac{\partial n_{s}}{\partial t} - \frac{g_{s}}{2p_{\|,s}}\frac{\partial p_{\|,s}}{\partial t} \\ & + \left[-\frac{1}{n_{s}v_{\mathrm{th},s}}\left(\frac{\partial n_{s}u_{s}}{\partial t} + u_{s}\left(n_{s}\frac{\partial u_{s}}{\partial z} + u_{s}\frac{\partial n_{s}}{\partial z}\right) - \frac{1}{2}n_{s}E_{\|} + v_{\mathrm{th},s}w_{\|,s}\left(n_{s}\frac{\partial u_{s}}{\partial z} + u_{s}\frac{\partial n_{s}}{\partial z}\right)\right)\right. \\ & \qquad + \frac{u_{s}}{n_{s}v_{\mathrm{th},s}}\left(\frac{\partial n_{s}}{\partial t} + u_{s}\frac{\partial n_{s}}{\partial z} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}\right) \\ & \qquad-\frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(\frac{\partial p_{\|,s}}{\partial t} + u_{s}\frac{\partial p_{\|,s}}{\partial z} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \qquad\left. + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\frac{\partial n_{s}}{\partial t} + u_{s}\frac{\partial n_{s}}{\partial z} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \\ \end{align}\]

[ intermediate steps ]

\[\begin{align*} \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \frac{3g_{s}}{2n_{s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s} - u_{s}\frac{\partial n_{s}}{\partial z} - n_{s}\frac{\partial u_{s}}{\partial z}\right) \\ & -\frac{g_{s}}{2p_{\|,s}}\left(-u_{s}\frac{\partial p_{\|,s}}{\partial z} - \frac{\partial q_{\|,s}}{\partial z} - 3p_{\|,s}\frac{\partial u_{s}}{\partial z} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - m_{s}n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \pm R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + m_{s}n_{n}\left(u_{n} - u_{s}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s} \right) \\ & + \left[-\frac{1}{n_{s}v_{\mathrm{th},s}}\left(-\underbrace{\cancel{n_{s}u_{s}\frac{\partial u_{s}}{\partial z}}}_{A} - \frac{\partial p_{\|,s}}{\partial z} + R_{ss'}n_{s}n_{s'}\left(u_{s'} - u_{s}\right) \pm R_{\mathrm{ion}}n_{i}n_{n}u_{n} + v_{\mathrm{th},s}w_{\|,s}\left(\underbrace{\cancel{n_{s}\frac{\partial u_{s}}{\partial z}}}_{B} + \underbrace{\cancel{u_{s}\frac{\partial n_{s}}{\partial z}}}_{C}\right)\right)\right. \\ & \quad + \frac{u_{s}}{n_{s}v_{\mathrm{th},s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s} - \underbrace{\cancel{n_{s}\frac{\partial u_{s}}{\partial z}}}_{A} + \underbrace{\cancel{v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}}}_{C}\right) \\ & \quad-\frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(-\frac{\partial q_{\|,s}}{\partial z} - \underbrace{\cancel{3p_{\|,s}\frac{\partial u_{s}}{\partial z}}}_{B} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - m_{s}n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \pm R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + m_{s}n_{n}\left(u_{n} - u_{s}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \quad\left. + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s} - \underbrace{\cancel{n_{s}\frac{\partial u_{s}}{\partial z}}}_{B} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \\ % \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \frac{3g_{s}}{2n_{s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s} - u_{s}\frac{\partial n_{s}}{\partial z} - n_{s}\frac{\partial u_{s}}{\partial z}\right) \\ & -\frac{g_{s}}{2p_{\|,s}}\left(-u_{s}\frac{\partial p_{\|,s}}{\partial z} - \frac{\partial q_{\|,s}}{\partial z} - 3p_{\|,s}\frac{\partial u_{s}}{\partial z} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - m_{s}n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \pm R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + m_{s}n_{n}\left(u_{n} - u_{s}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s}\right) \\ & + \left[-\frac{1}{n_{s}v_{\mathrm{th},s}}\left(-\frac{\partial p_{\|,s}}{\partial z} + R_{ss'}n_{s}n_{s'}\left(u_{s'} - u_{s}\right)\pm R_{\mathrm{ion}}n_{i}n_{n}u_{n}\right)\right. \\ & \quad + \frac{u_{s}}{n_{s}v_{\mathrm{th},s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s}\right) \\ & \quad-\frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(-\frac{\partial q_{\|,s}}{\partial z} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - m_{s}n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \pm R_{\mathrm{ion}}n_{i}\left(p_{\|,n} + m_{s}n_{n}\left(u_{n} - u_{s}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \quad\left. + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\pm R_{\mathrm{ion}}n_{i}n_{n} + \int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s}\\ % \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + g_{s}\left(\pm\frac{3}{2}R_{\mathrm{ion}}n_{i}\frac{n_{n}}{n_{s}} + \frac{3}{2n_{s}}\int dv_\parallel S_{s} - \frac{3u_{s}}{2n_{s}}\frac{\partial n_{s}}{\partial z}\right) \\ & + g_{s}\left(\frac{u_{s}}{2p_{\|,s}}\frac{\partial p_{\|,s}}{\partial z} + \frac{1}{2p_{\|,s}}\frac{\partial q_{\|,s}}{\partial z} + \frac{1}{2p_{\|,s}}R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \mp\frac{1}{2}R_{\mathrm{ion}}\frac{n_{i}}{p_{\|,s}}\left(p_{\|,n} + n_{n}\left(u_{n} - u_{s}\right)^{2}\right) - \frac{1}{2p_{\parallel,s}}\int dv_\parallel v_\parallel^2 S_{s} - \frac{u_{s}^2}{2p_{\parallel,s}}\int dv_\parallel S_{s}\right) \\ & + \left[-\frac{1}{n_{s}v_{\mathrm{th},s}}\left(-\frac{\partial p_{\|,s}}{\partial z} + R_{ss'}n_{s}n_{s'}\left(u_{s'} - u_{s}\right) \pm R_{\mathrm{ion}}n_{i}n_{n}\left(u_{n} - u_{s}\right) - u_{s}\int dv_\parallel S_{s}\right)\right. \\ & \quad-\frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(-\frac{\partial q_{\|,s}}{\partial z} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \quad\mp\frac{w_{\|,s}}{2}R_{\mathrm{ion}}n_{i}\left(\frac{p_{\|,n}}{p_{\|,s}} - \frac{n_{n}}{n_{s}} + \frac{n_{n}}{p_{\|,s}}\left(u_{n} - u_{s}\right)^{2}\right) \\ & \quad\left. + \frac{w_{\|,s}}{2}\frac{1}{n_{s}}\left(\int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial n_{s}}{\partial z}\right)\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \end{align*}\]

and finally using

\[\begin{align*} \frac{u_{s}}{v_{\mathrm{th},s}}\frac{\partial v_{\mathrm{th},s}}{\partial z} & =u_{s}\sqrt{\frac{n_{s}}{p_{\|,s}}}\frac{\partial}{\partial z}\sqrt{\frac{p_{\|,s}}{n_{s}}} \\ & = \frac{u_{s}}{2}\left(\frac{1}{p_{\|,s}}\frac{\partial p_{\|,s}}{\partial z} - \frac{1}{n_{s}}\frac{\partial n_{s}}{\partial z}\right) \end{align*}\]

gives

\[\begin{align} \Rightarrow & \frac{\partial g_{s}}{\partial t} + \frac{v_{\mathrm{th},s}}{n_{s}}\left(v_{\mathrm{th},s}w_{\|,s} + u_{s}\right)\frac{\partial f_{s}}{\partial z} + \left(\pm\frac{3}{2}R_{\mathrm{ion}}n_{i}\frac{n_{n}}{n_{s}} + \frac{3}{2n_{s}} \int dv_\parallel S_{s} - \frac{u_{s}}{n_{s}}\frac{\partial n_{s}}{\partial z}\right)g_{s} \\ & + \left(\frac{u_{s}}{v_{\mathrm{th},s}}\frac{\partial v_{\mathrm{th},s}}{\partial z} + \frac{1}{2p_{\|,s}}\frac{\partial q_{\|,s}}{\partial z}\right. \\ & \qquad + \frac{1}{2p_{\|,s}}R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) \\ & \qquad \left.\mp\frac{1}{2}R_{\mathrm{ion}}\frac{n_{i}}{p_{\|,s}}\left(p_{\|,n} + n_{n}\left(u_{n} - u_{s}\right)^{2}\right) - \frac{1}{2p_{\parallel,s}}\int dv_\parallel v_\parallel^2 S_{s} - \frac{u_{s}^2}{2p_{\parallel,s}}\int dv_\parallel S_{s}\right)g_{s} \\ & + \left[-\frac{1}{n_{s}v_{\mathrm{th},s}}\left(-\frac{\partial p_{\|,s}}{\partial z} + R_{ss'}n_{s}n_{s'}\left(u_{s'} - u_{s}\right) \pm R_{\mathrm{ion}}n_{i}n_{n}\left(u_{n} - u_{s}\right) - u_{s}\int dv_\parallel S_{s}\right)\right. \\ & \qquad-\frac{w_{\|,s}}{2}\frac{1}{p_{\|,s}}\left(-\frac{\partial q_{\|,s}}{\partial z} - R_{ss'}\left(n_{s'}p_{\|,s} - n_{s}p_{\|,s'} - n_{s}n_{s'}\left(u_{s} - u_{s'}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{s} + u_{s}^2 \int dv_\parallel S_{s} + v_{\mathrm{th},s}w_{\|,s}\frac{\partial p_{\|,s}}{\partial z}\right) \\ & \qquad\mp\frac{w_{\|,s}}{2}R_{\mathrm{ion}}n_{i}\left(\frac{p_{\|,n}}{p_{\|,s}} - \frac{n_{n}}{n_{s}} + \frac{n_{n}}{p_{\|,s}}\left(u_{n} - u_{s}\right)^{2}\right) \\ & \qquad\left. + \frac{w_{\parallel,s}}{2}\frac{1}{n_{s}}\int dv_\parallel S_{s} + \frac{w_{\|,s}^{2}}{2}\frac{v_{\mathrm{th},s}}{n_{s}}\frac{\partial n_{s}}{\partial z}\right]\frac{\partial g_{s}}{\partial w_{\|,s}} \\ & = -R_{ss'}n_{s'}\left(g_{s} - \frac{v_{\mathrm{th},s}}{v_{\mathrm{th},s'}}g_{s'}\right) \pm R_{\mathrm{ion}}\frac{v_{\mathrm{th},s}}{n_{s}}n_{i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},s}}{n_{s}} S_{s} \end{align}\]

Writing out the final result fully for ions

\[\begin{align} & \frac{\partial g_{i}}{\partial t} + \frac{v_{\mathrm{th},i}}{n_{i}}\left(v_{\mathrm{th},i}w_{\|,i} + u_{i}\right)\frac{\partial f_{i}}{\partial z} + \left(\frac{3}{2}R_{\mathrm{ion}}n_{n} + \frac{3}{2n_{i}}\int dv_\parallel S_{i} - \frac{u_{i}}{n_{i}}\frac{\partial n_{i}}{\partial z}\right)g_{i} \\ & + \left(\frac{u_{i}}{v_{\mathrm{th},i}}\frac{\partial v_{\mathrm{th},i}}{\partial z} + \frac{1}{2p_{\|,i}}\frac{\partial q_{\|,i}}{\partial z}\right. \\ & \qquad + \frac{1}{2p_{\|,i}}R_{in}\left(n_{n}p_{\|,i} - n_{i}p_{\|,n} - n_{i}n_{n}\left(u_{i} - u_{n}\right)^{2}\right) \\ & \qquad \left. - \frac{1}{2}R_{\mathrm{ion}}\frac{n_{i}}{p_{\|,i}}\left(p_{\|,n} + n_{n}\left(u_{n} - u_{i}\right)^{2}\right) - \frac{1}{2p_{\parallel,i}}\int dv_\parallel v_\parallel^2 S_{i} - \frac{u_{i}^2}{2p_{\parallel,i}}\int dv_\parallel S_{i}\right)g_{i} \\ & + \left[-\frac{1}{n_{i}v_{\mathrm{th},i}}\left(-\frac{\partial p_{\|,i}}{\partial z} + R_{in}n_{i}n_{n}\left(u_{n} - u_{i}\right) + R_{\mathrm{ion}}n_{i}n_{n}\left(u_{n} - u_{i}\right) - u_{i}\int dv_\parallel S_{i}\right)\right. \\ & \qquad-\frac{w_{\|,i}}{2}\frac{1}{p_{\|,i}}\left(-\frac{\partial q_{\|,i}}{\partial z} - R_{in}\left(n_{n}p_{\|,i} - n_{i}p_{\|,n} - n_{i}n_{n}\left(u_{i} - u_{n}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{i} + u_{i}^2 \int dv_\parallel S_{i} + v_{\mathrm{th},i}w_{\|,i}\frac{\partial p_{\|,i}}{\partial z}\right) \\ & \qquad - \frac{w_{\|,i}}{2}R_{\mathrm{ion}}n_{i}\left(\frac{p_{\|,n}}{p_{\|,i}} - \frac{n_{n}}{n_{i}} + \frac{n_{n}}{p_{\|,i}}\left(u_{n} - u_{i}\right)^{2}\right) \\ & \qquad\left. + \frac{w_{\parallel,i}}{2} \frac{1}{n_{i}}\int dv_\parallel S_{i} + \frac{w_{\|,i}^{2}}{2}\frac{v_{\mathrm{th},i}}{n_{i}}\frac{\partial n_{i}}{\partial z}\right]\frac{\partial g_{i}}{\partial w_{\|,i}} \\ & = -R_{in}n_{n}\left(g_{i} - \frac{v_{\mathrm{th},i}}{v_{\mathrm{th},n}}g_{n}\right) + R_{\mathrm{ion}}v_{\mathrm{th},i}\frac{n_{n}}{v_{\mathrm{th},n}}g_{n} + \frac{v_{\mathrm{th},i}}{n_{i}} S_{i} \end{align}\]

and for neutrals where several of the ionization terms cancel

\[\begin{align} \Rightarrow & \frac{\partial g_{n}}{\partial t} + \frac{v_{\mathrm{th},n}}{n_{n}}\left(v_{\mathrm{th},n}w_{\|,n} + u_{n}\right)\frac{\partial f_{n}}{\partial z} + \left(-\frac{3}{2}R_{\mathrm{ion}}n_{i} + \frac{3}{2n_{n}}\int dv_\parallel S_{n} - \frac{u_{n}}{n_{n}}\frac{\partial n_{n}}{\partial z}\right)g_{n} \\ & + \left(\frac{u_{n}}{v_{\mathrm{th},n}}\frac{\partial v_{\mathrm{th},n}}{\partial z} + \frac{1}{2p_{\|,n}}\frac{\partial q_{\|,n}}{\partial z}\right. \\ & \qquad \left. + \frac{1}{2p_{\|,n}}R_{in}\left(n_{i}p_{\|,n} - n_{n}p_{\|,i} - n_{n}n_{i}\left(u_{n} - u_{i}\right)^{2}\right) + \frac{1}{2}R_{\mathrm{ion}}n_{i} - \frac{1}{2p_{\parallel,n}}\int dv_\parallel v_\parallel^2 S_{n} - \frac{u_{n}^2}{2p_{\parallel,n}}\int dv_\parallel S_{n}\right)g_{n} \\ & + \left[-\frac{1}{n_{n}v_{\mathrm{th},n}}\left(-\frac{\partial p_{\|,n}}{\partial z} + R_{in}n_{n}n_{i}\left(u_{i} - u_{n}\right) - u_{n}\int dv_\parallel S_{n}\right)\right. \\ & \qquad-\frac{w_{\|,n}}{2}\frac{1}{p_{\|,n}}\left(-\frac{\partial q_{\|,n}}{\partial z} - R_{in}\left(n_{i}p_{\|,n} - n_{n}p_{\|,i} - n_{n}n_{i}\left(u_{n} - u_{i}\right)^{2}\right) + \int dv_\parallel v_\parallel^2 S_{n} + u_{n}^2\int dv_\parallel S_{n} + v_{\mathrm{th},n}w_{\|,n}\frac{\partial p_{\|,n}}{\partial z}\right) \\ & \qquad\left. + \frac{w_{\parallel,n}}{2}\frac{1}{n_{n}}\int dv_\parallel S_{n} + \frac{w_{\|,n}^{2}}{2}\frac{v_{\mathrm{th},n}}{n_{n}}\frac{\partial n_{n}}{\partial z}\right]\frac{\partial g_{n}}{\partial w_{\|,n}} \\ & = -R_{in}n_{i}\left(g_{n} - \frac{v_{\mathrm{th},n}}{v_{\mathrm{th},i}}g_{i}\right) - R_{\mathrm{ion}}n_{i}g_{n} + \frac{v_{\mathrm{th},n}}{n_{n}} S_{n} \end{align}\]