normalisations.lyx

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\index Index
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\begin_body

\begin_layout Standard
Normalise:
\end_layout

\begin_layout Itemize
times to the electron-electron collision time 
\begin_inset Formula $\tau_{ref}=\tau_{ee}(n_{ref},T_{ref})$
\end_inset


\end_layout

\begin_layout Itemize
densities to reference density 
\begin_inset Formula $n_{ref}=10^{19}m^{-3}$
\end_inset


\end_layout

\begin_layout Itemize
mass to electron mass 
\begin_inset Formula $m_{e}$
\end_inset


\end_layout

\begin_layout Itemize
energy to 
\begin_inset Formula $T_{ref}$
\end_inset


\end_layout

\begin_layout Itemize
pressure to 
\begin_inset Formula $n_{ref}T_{ref}$
\end_inset


\end_layout

\begin_layout Itemize
electric potential to 
\begin_inset Formula $T_{ref}/e$
\end_inset


\end_layout

\begin_layout Standard
Derived quantities:
\end_layout

\begin_layout Itemize
velocity to electron 
\begin_inset Quotes bld
\end_inset

thermal speed
\begin_inset Quotes brd
\end_inset

 
\begin_inset Formula $v_{ref}=\sqrt{T_{ref}/m_{e}}$
\end_inset


\end_layout

\begin_layout Itemize
length to mean free path 
\begin_inset Formula $L_{ref}=\lambda_{ref}=v_{ref}\tau_{ee}=\tau_{ee}\sqrt{T_{ref}/m_{e}}$
\end_inset


\end_layout

\begin_layout Section
Unnormalised equations
\end_layout

\begin_layout Standard
Cut-down version of Chodura 1971, dropping perpendicular gradients and also
 (temporarily?) friction terms.
 We will replace electron continuity equation with quasineutrality, and
 neglect electron inertia to solve electron parallel momentum equation for
 
\begin_inset Formula $E_{\|}$
\end_inset

, assuming ambipolarity 
\begin_inset Formula $V_{e}=V_{i}$
\end_inset

.
 Here for brevity 
\begin_inset Formula $V_{s}\equiv V_{\|s}$
\end_inset

, 
\begin_inset Formula $\nabla=\nabla_{\|}$
\end_inset

.
 Assume a straight, uniform magnetic field to simplify the geometric quantities
 [
\begin_inset Formula $h_{\|}=1,h_{\perp}=0,\tau_{\|\|}=1,\tau_{\alpha\neq\|,\beta}=0,n_{\|\|}=0,n_{\|\perp}=n_{\perp\parallel}=0,n_{\perp\perp}=1$
\end_inset

].
 Equation numbers refer to Chodura 1971
\begin_inset Newline newline
\end_inset

(1)
\begin_inset Formula 
\begin{align*}
\frac{\partial\rho_{i}}{\partial t}+\nabla\left(\rho_{i}V_{i}\right) & =0
\end{align*}

\end_inset

(2)
\begin_inset Formula 
\begin{align*}
\rho_{i}\frac{d_{i}V_{i}}{dt}+\nabla p_{i\|}-\rho_{i}\frac{e}{m_{i}}E_{\|} & =\cancel{\sum_{r}I_{\|}^{ir}}\\
\nabla p_{e\|}+\rho_{e}\frac{e}{m_{e}}E_{\|} & =0
\end{align*}

\end_inset

(3)
\begin_inset Formula 
\begin{align*}
\frac{d_{s}p_{s\perp}}{dt}+\left(\nabla V_{s}\right)\left[p_{s\perp}+\pi_{s\|\perp}\right]+\nabla S_{s\|}^{\perp} & =\sum_{r}J_{sr\perp\perp}
\end{align*}

\end_inset

(4) 
\begin_inset Formula 
\[
\frac{d_{s}p_{s\|}}{dt}+\left(\nabla V_{s}\right)\left[3p_{s\|}+2\pi_{s\|\|}\right]+\nabla S_{s\|}^{\|}=\sum_{r}J_{sr\|\|}
\]

\end_inset

(17) for us 
\begin_inset Formula $d^{sr}$
\end_inset

 vanishes as we have no 
\begin_inset Formula $V_{\perp}$
\end_inset

 and we assume the parallel velocities are equal 
\begin_inset Formula 
\[
d_{\alpha}=d_{\alpha}^{sr}=V_{s\alpha}-V_{r\alpha}=0
\]

\end_inset

(15), (24)
\begin_inset Formula 
\[
\beta_{s\perp}=\frac{m_{s}}{T_{s\perp}},\quad\beta_{s\|}=\frac{m_{s}}{T_{s\|}},\quad b_{s\perp}=\frac{\beta_{s\perp}}{\beta_{s\perp}+\beta_{r\perp}},\quad b_{s\|}=\frac{\beta_{s\|}}{\beta_{s\|}+\beta_{r\|}}
\]

\end_inset

(29)
\begin_inset Formula 
\begin{align*}
J_{sr\perp\perp} & =\frac{4\rho_{s}\nu_{sr}}{m_{r}+m_{s}}\left[2K_{200}^{sr}\left(T_{r\perp}-T_{s\perp}\right)+\frac{m_{r}}{\beta_{\perp rs}}\left(K_{002}^{sr}-K_{200}^{sr}\right)\right]
\end{align*}

\end_inset

(21), (20)
\begin_inset Formula 
\[
\beta_{sr\|}=\frac{\beta_{s\|}\beta_{r\|}}{\left(\beta_{s\|}+\beta_{r\|}\right)},\quad\beta_{sr\perp}=\frac{\beta_{s\perp}\beta_{r\perp}}{\left(\beta_{s\perp}+\beta_{r\perp}\right)},\quad\alpha_{sr}=\frac{\beta_{sr\|}}{\beta_{sr\perp}}=\frac{\beta_{s\|}\beta_{r\|}\left(\beta_{s\perp}+\beta_{r\perp}\right)}{\left(\beta_{s\|}+\beta_{r\|}\right)\beta_{s\perp}\beta_{r\perp}}
\]

\end_inset

(30)
\begin_inset Formula 
\[
J_{sr\|\|}=\frac{8\rho_{s}\nu_{sr}}{m_{r}+m_{s}}\alpha_{sr}\left[K_{002}^{sr}\left(T_{r\|}-T_{s\|}\right)+\frac{m_{r}}{\beta_{rs\|}}\left(K_{200}^{sr}-K_{002}^{sr}\right)\right]
\]

\end_inset

text, section 4
\begin_inset Formula 
\[
\pi_{e}\approx0
\]

\end_inset

(62), (63)
\begin_inset Formula 
\[
\pi_{i\perp\perp}=\pi_{i\perp\|}=0
\]

\end_inset


\begin_inset Formula $\pi_{i\|\|}$
\end_inset

 does not seem to be given, so set (?)
\begin_inset Formula 
\[
\pi_{i\|\|}=0
\]

\end_inset

(64)
\begin_inset Formula 
\[
S_{s\|}^{\perp}=\frac{1}{\left(c_{\perp}e_{\|}-c_{\|}e_{\perp}\right)}\frac{p_{s\|}}{m_{s}}\left\{ e_{\|}\left(\nabla T_{s\perp}-\cancel{k_{s\|}^{\perp}}\right)-3c_{\|}\left(\nabla T_{s\|}-\cancel{k_{s\|}^{\|}}\right)\right\} 
\]

\end_inset

(65)
\begin_inset Formula 
\[
S_{s\|}^{\|}=\frac{1}{\left(c_{\|}e_{\perp}-c_{\perp}e_{\|}\right)}\frac{p_{s\|}}{m_{s}}\left\{ e_{\perp}\left(\nabla T_{s\perp}-\cancel{k_{s\|}^{\perp}}\right)-3c_{\perp}\left(\nabla T_{s\|}-\cancel{k_{s\|}^{\|}}\right)\right\} 
\]

\end_inset

(21)
\begin_inset Formula 
\[
\alpha_{sr}=\frac{\beta_{sr\|}}{\beta_{sr\perp}},\quad X_{sr}=\alpha_{sr}-1
\]

\end_inset

(36), (37)
\begin_inset Formula 
\[
\psi_{s}=\left(\alpha_{s}\right)^{2}\left(K_{004}^{s}-K_{202}^{s}\right)+\frac{1}{2}\alpha_{s}\left(K_{200}^{s}-K_{002}^{s}\right),\quad\phi_{s}=4K_{220}^{s}-2K_{202}^{s}-K_{200}^{s}+K_{002}^{s}
\]

\end_inset

definitions on pp.
 657-658 - for electrons
\begin_inset Formula 
\begin{align*}
c_{e\perp} & =-4\nu_{ee}\left(6\alpha K_{202}+\frac{\phi_{e}}{2}\right)+4\alpha\nu_{ei}\left(-32K_{222}+4K_{204}+10K_{202}-2K_{004}-K_{002}\right)\\
c_{e\|} & =2\nu_{ee}\psi_{e}+4\alpha^{2}\nu_{ei}\left(-\frac{8}{3}\alpha K_{204}+\frac{2}{3}\alpha K_{006}+4K_{202}-\left(1-\frac{\alpha}{3}\right)K_{004}-\frac{1}{2}K_{002}\right)\\
e_{e\perp} & =12\nu_{ee}\phi_{e}+12\nu_{ei}\left(16\alpha K_{222}-4\alpha K_{204}-2\left(2\alpha-1\right)K_{202}+2\alpha K_{004}-K_{002}\right)\\
e_{e\|} & =-12\nu_{ee}\psi_{e}+4\nu_{ei}\alpha\left(4\alpha^{2}K_{204}-2\alpha^{2}K_{006}-6\alpha K_{202}+4\alpha K_{004}-\frac{3}{2}K_{002}\right)\\
\boldsymbol{k}_{e}^{\perp} & \propto\boldsymbol{d}_{\|}=0\\
\boldsymbol{k}_{e}^{\|} & \propto\boldsymbol{d}_{\|}=0
\end{align*}

\end_inset

and for ions
\begin_inset Formula 
\begin{align*}
c_{i\perp} & =-4\nu_{ii}\left(6\alpha_{i}K_{202}^{i}+\frac{1}{2}\phi_{i}\right)-4\nu_{ie}\left(2K_{002}^{e}+\alpha_{e}K_{002}^{e}\right)\\
c_{i\|} & =2\psi_{i}\nu_{ii}\\
e_{i\perp} & =12\phi_{i}\nu_{ii}\\
e_{i\|} & =-12\psi_{i}\nu_{ii}-12\nu_{ie}\alpha_{e}K_{002}^{e}\\
\boldsymbol{k}_{i}^{\perp} & =\boldsymbol{k}_{i}^{\|}=0
\end{align*}

\end_inset


\end_layout

\begin_layout Section
Normalised equations
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\frac{\partial\hat{n}_{i}}{\partial\hat{t}}+\hat{\nabla}\left(\hat{n}_{i}\hat{V}_{i}\right) & =0
\end{align*}

\end_inset

(2)
\begin_inset Formula 
\begin{align*}
\frac{n_{ref}v_{ref}}{\tau_{ref}}\hat{n}_{i}\left(\frac{\partial\hat{V}_{i}}{\partial\hat{t}}+\hat{V}_{i}\hat{\nabla}\hat{V}_{i}\right)+\frac{n_{ref}T_{ref}}{m_{e}\lambda_{ref}}\frac{1}{\hat{m}_{i}}\hat{\nabla}\hat{p}_{i\|}-\frac{e}{m_{e}}n_{ref}\frac{T_{ref}}{e}\frac{1}{\lambda_{ref}}\hat{n}_{i}\frac{1}{\hat{m}_{i}}\hat{E}_{\|} & =0\\
\frac{n_{ref}T_{ref}}{m_{e}\lambda_{ref}}\hat{n}_{i}\left(\frac{\partial\hat{V}_{i}}{\partial\hat{t}}+\hat{V}_{i}\hat{\nabla}\hat{V}_{i}\right)+\frac{n_{ref}T_{ref}}{m_{e}\lambda_{ref}}\frac{1}{\hat{m}_{i}}\hat{\nabla}\hat{p}_{i\|}-\frac{n_{ref}T_{ref}}{m_{e}\lambda_{ref}}\hat{n}_{i}\frac{1}{\hat{m}_{i}}\hat{E}_{\|} & =0\\
\frac{\partial\hat{V}_{i}}{\partial\hat{t}}+\hat{V}_{i}\hat{\nabla}\hat{V}_{i}+\frac{1}{\hat{m}_{i}\hat{n}_{i}}\hat{\nabla}\hat{p}_{i\|}-\frac{1}{\hat{m}_{i}}\hat{E}_{\|} & =0
\end{align*}

\end_inset


\begin_inset Formula 
\begin{align*}
\frac{n_{ref}T_{ref}}{\lambda_{ref}}\hat{\nabla}\hat{p}_{e\|}+m_{e}\frac{n_{ref}T_{ref}}{e\lambda_{ref}}\hat{n}_{i}\frac{e}{m_{e}}\hat{E}_{\|} & =0\\
\hat{E}_{\|} & =-\frac{1}{\hat{n}_{i}}\hat{\nabla}\hat{p}_{e\|}
\end{align*}

\end_inset

(3)
\begin_inset Formula 
\begin{align*}
\frac{n_{ref}T_{ref}}{\tau_{ref}}\left(\frac{\partial\hat{p}_{s\perp}}{\partial\hat{t}}+\hat{V}_{s}\hat{\nabla}\hat{p}_{s\perp}\right)+\frac{v_{ref}n_{ref}T_{ref}}{\lambda_{ref}}\left(\hat{\nabla}\hat{V}_{s}\right)\left[\hat{p}_{s\perp}\right]+\frac{S_{ref}}{\lambda_{ref}}\hat{\nabla}\hat{S}_{s\|}^{\perp} & =J_{ref}\sum_{r}\hat{J}_{sr\perp\perp}\\
\left(\frac{\partial\hat{p}_{s\perp}}{\partial\hat{t}}+\hat{V}_{s}\hat{\nabla}\hat{p}_{s\perp}\right)+\left(\hat{\nabla}\hat{V}_{s}\right)\left[\hat{p}_{s\perp}\right]+\hat{\nabla}\hat{S}_{s\|}^{\perp} & =\sum_{r}\hat{J}_{sr\perp\perp}
\end{align*}

\end_inset

with 
\begin_inset Formula $S_{ref}=n_{ref}v_{ref}T_{ref}$
\end_inset

 and 
\begin_inset Formula $J_{ref}=n_{ref}T_{ref}/\tau_{ref}$
\end_inset


\begin_inset Newline newline
\end_inset

(4) is similar to (3) dimension-wise - 
\begin_inset Formula $S^{\|}$
\end_inset

 is normalised like 
\begin_inset Formula $S^{\perp}$
\end_inset

 and 
\begin_inset Formula $J_{\|\|}$
\end_inset

 like 
\begin_inset Formula $J_{\perp\perp}$
\end_inset


\begin_inset Formula 
\[
\left(\frac{\partial\hat{p}_{s\|}}{\partial\hat{t}}+\hat{V}_{s}\hat{\nabla}\hat{p}_{s\|}\right)+\left(\hat{\nabla}\hat{V}_{s}\right)\left[3\hat{p}_{s\|}\right]+\hat{\nabla}\hat{S}_{s\|}^{\|}=\sum_{r}\hat{J}_{sr\|\|}
\]

\end_inset

Noting that the 
\begin_inset Formula $K$
\end_inset

's are dimensionless, as are 
\begin_inset Formula $\alpha_{sr}$
\end_inset

, 
\begin_inset Formula $\psi_{s}$
\end_inset

, 
\begin_inset Formula $\phi_{s}$
\end_inset

, we find that 
\begin_inset Formula $c_{e\perp}=\frac{1}{\tau_{ref}}\hat{c}_{e\perp}$
\end_inset

, 
\begin_inset Formula $c_{e\|}=\frac{1}{\tau_{ref}}\hat{c}_{e\|}$
\end_inset

, 
\begin_inset Formula $e_{e\perp}=\frac{1}{\tau_{ref}}\hat{e}_{e\perp}$
\end_inset

, 
\begin_inset Formula $e_{e\|}=\frac{1}{\tau_{ref}}\hat{e}_{e\|}$
\end_inset

.
 
\begin_inset Formula $\beta_{s\perp}=\frac{1}{v_{ref}^{2}}\hat{\beta}_{s\perp}$
\end_inset

, 
\begin_inset Formula $\beta_{s\|}=\frac{1}{v_{ref}^{2}}\hat{\beta}_{s\|}$
\end_inset

, 
\begin_inset Formula $\beta_{sr\perp}=\frac{1}{v_{ref}^{2}}\hat{\beta}_{sr\perp}$
\end_inset

, and 
\begin_inset Formula $\beta_{sr\|}=\frac{1}{v_{ref}^{2}}\hat{\beta}_{sr\|}$
\end_inset

.
 Using these, we can check that the expressions above for 
\begin_inset Formula $S_{s\|}^{\perp}$
\end_inset

 and 
\begin_inset Formula $S_{s\|}^{\|}$
\end_inset

 should have dimensions
\begin_inset Formula 
\[
\tau_{ref}^{2}n_{ref}T_{ref}\frac{1}{m_{ref}}\frac{1}{\tau_{ref}}\frac{1}{\lambda_{ref}}T_{ref}=\frac{\tau_{ref}n_{ref}T_{ref}^{2}}{m_{ref}\lambda_{ref}}=\frac{n_{ref}T_{ref}^{2}}{m_{ref}v_{ref}}=\frac{n_{ref}T_{ref}m_{ref}v_{ref}^{2}}{m_{ref}v_{ref}}=n_{ref}T_{ref}v_{ref}
\]

\end_inset

confirming our 
\begin_inset Formula $S_{ref}$
\end_inset

 above, and 
\begin_inset Formula $J_{\perp\perp}$
\end_inset

, 
\begin_inset Formula $J_{\|\|}$
\end_inset

 should have dimensions
\begin_inset Formula 
\[
m_{e}n_{ref}\frac{1}{\tau_{ref}}\frac{1}{m_{e}}T_{ref}=\frac{n_{ref}T_{ref}}{\tau_{ref}}.
\]

\end_inset

Finally note that (neglecting any difference between 
\begin_inset Formula $\ln\Lambda_{ee}$
\end_inset

, 
\begin_inset Formula $\ln\Lambda_{ei}$
\end_inset

 and 
\begin_inset Formula $\ln\Lambda_{ii}$
\end_inset

)
\begin_inset Formula 
\begin{align*}
\hat{\nu}_{ee} & =\hat{\nu}_{ei}=\frac{\hat{n}_{i}}{\hat{T}_{e}^{3/2}}\\
\hat{\nu}_{ii} & =\frac{1}{\sqrt{\hat{m}_{i}}}\frac{\hat{n}_{i}}{\hat{T}_{i}^{3/2}}\\
\hat{\nu}_{ie} & =\frac{1}{\hat{m}_{i}}\frac{\hat{n}_{i}}{\hat{T}_{e}^{3/2}}
\end{align*}

\end_inset


\end_layout

\end_body
\end_document