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\begin{document}
{\LARGE Fundamental Equations of Plasmas},\\
by E. Alec Johnson, March 28, 2007

\def\dotp{\cdot}
\def\div{\nabla\cdot}
\def\curl{\nabla\times}
\def\half{\frac{1}{2}}
\def\idtwo{\underline{\underline{\delta}}}
%\def\idtwo{\mathcal I}

\section{Full one-fluid plasma equations.}

\subsection{Definitions of Quantities.}

\def\fvel{\mathbf{v}}
\def\mfvel{v}
\def\bfield{\mathbf{B}}
\def\mbfield{B}
\def\energy{\mathcal{E}} %{\varepsilon}
\def\totenergy{\tilde\energy}
\def\pressure{p}
\def\temperature{T}
\def\heatflux{\mathbf{q}}
\def\resistivity{\eta}
\def\vstress{\underline{\underline{\sigma}}}
\def\R{\mathcal{R}}

\def\qdens{\sigma}
\def\cur{\mathbf{J}}
\def\mdens{\rho}
\def\mom{(\mdens\fvel)} %\mathbf{M}
\def\efield{\mathbf{E}}
\def\mefield{E}
\def\Poynting{\mathbf{S}}
\def\gasenergy{\energy}
\def\tenergy{\energy^t}
\def\kenergy{\energy^k}
\def\fieldenergy{\energy^f}
\def\stress{\underline{\underline{\tau}}}
\def\strain{\underline{\underline{e}}}
\def\fstress{\underline{\underline{T}}}
\def\light{c}
\def\gravity{\mathbf{g}}
\def\permTensor{\underline{\underline{\underline{\epsilon}}}}

%\def\Used{$\bullet$}
%\def\Needed{$\circ$}
%\def\Other{$\star$}

%\subsection{Key.}
%\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
%\item[\Used] denotes quantity directly used in the MHD equations.
%\item[\Star] denotes quantity indirectly used in the MHD equations.
%\item[\Circ] (Circle) denotes other quantities.
%\end{itemize}

\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $n$ = particle number density
\end{itemize}
%\nl All densities are by default per volume.

\subsubsection{Electromagnetic quantities.}

\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\bfield$ = magnetic field
\item[] $\efield$ = electric field
\item[] $\Poynting$ = Poynting vector
\item[] $\epsilon_0$ = permittivity of vacuum
\item[] $\mu_0$ = permeability of vacuum
\item[] $\light$ = speed of light
\item[] $\qdens$ = charge density
\item[] $\cur$ = current density (net charge flux)
\item[] $\resistivity$ = resistivity
\end{itemize}

\subsubsection{Mechanical quantities.}

\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\mdens$ = net mass density
\item[] $\fvel$ = fluid velocity
\item[] $\mom$ = momentum density (i.e. mass flux)
\item[] $\pressure$ = gas-dynamic pressure
\item[] $\vstress$ = viscous stress tensor
\item[] $\stress$ = total mechanical stress tensor
\item[] $\fstress$ = stress of electromagnetic field
\item[] $\strain$ = deformation rate (strain rate, even part)
\item[] $\mu$ = shear viscosity
\item[] $\lambda$ = ``balancing bulk viscosity''
\end{itemize}

\subsubsection{Thermodynamic quantities.}

\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\heatflux$ = heat flux
\item[] $\temperature$ = temperature
\item[] $\kappa$ = heat conductivity
\item[] $\gamma$ = ratio of specific heats
\item[] $\R$ = gas constant
\item[] $\totenergy$ = total energy density
\item[] $\gasenergy$ = gas-dynamic energy
\item[] $\tenergy$ = thermal energy
\item[] $\kenergy$ = kinetic energy
\item[] $\fieldenergy$ = electromagnetic {\bf f}ield energy
\end{itemize}

\subsection{Defining and Constituting Relationships.}

%\subsection{Key.}
%\def\Defn{$\circ$}
%\def\Cons{$\bullet$}
%\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
%\item[\Defn] marks a relationship that holds by definition.
%\item[\Cons] marks a constitutive relation.
%\end{itemize}

\subsubsection{Electromagnetic relations.}
\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\light^2\mu_0\epsilon_0 \equiv 1$
\item[] $\Poynting:=\frac{1}{\mu_0}\efield\times\bfield$
\item[] $\efield'=\efield+\fvel\times\bfield=\resistivity\cur$
\end{itemize}

\subsubsection{Mechanical relations.}
\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\strain := \half(\nabla \fvel + \nabla\fvel^T)$
\item[] $\vstress = \lambda\div\fvel\idtwo+2\mu\strain$
\item[] $\lambda = -\frac{2}{3}\mu$ (assuming trace$(\vstress)=0$)
\item[] $\stress = -\pressure\idtwo+\vstress$
\item[] $\fstress := \epsilon_0(\efield\efield-\half\mefield^2\idtwo)
        + \frac{1}{\mu_0}(\bfield\bfield-\half\mbfield^2\idtwo)$
%\item[] $\totalpressure := \pressure + \frac{1}{2\mu_0}\mbfield^2$
%        (+ electric pressure?)
\end{itemize}

\subsubsection{Thermodynamic relations.}
\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item[] $\pressure=\mdens \R\temperature$
%(So temperature is proportional to thermal energy per \emph{mass}.)
\item[] $\tenergy = \frac{\pressure}{\gamma-1}$
%(So pressure is proportional to thermal energy per \emph{volume}.)
\item[] $\kenergy := \half\mdens\mfvel^2$
\item[] $\fieldenergy := \epsilon_0\half\mefield^2 + \frac{1}{2\mu_0}\mbfield^2$
\item[] $\heatflux = -\kappa\nabla\temperature$
\item[] $\totenergy := \gasenergy + \fieldenergy$.
\item[] $\gasenergy := \tenergy + \kenergy$.
\end{itemize}

\subsection{Definitions of Symbols and Operators.}

\def\Dt{\partial_t}
\newcommand{\Dx}[1]{\ensuremath{\partial_{#1}}}
\def\dcv{d_t} % convected derivative
\def\deltabar{{\mathchar'26\mkern-9mu \delta}}
\def\dcs{\deltabar_t} % conservative derivative

\begin{itemize}[topsep=0pt, partopsep=0pt,itemsep=0pt,parsep=0pt]
\item $\permTensor$ = permutation tensor
\item $\Dt:=\frac{\partial}{\partial t}$
%\item $\Dx{j}:=\frac{\partial}{\partial x^j}$
% flow operators
\item $\dcv := \Dt + \fvel\dotp\nabla$ = {\bf convective derivative}. 
\item $\dcs := \alpha \mapsto
      (\Dt \alpha + \nabla\dotp(\fvel\alpha))$
      = {\bf conservative derivative}.
\end{itemize}

\subsection{Full One-fluid Plasma Balance Laws.}

The full one-fluid plasma equations are a system
of 11 equations which specify the
evolution of electromagnetic field, 
mass density, momentum, and energy.

\subsubsection{Electromagnetic evolution.}

\p Maxwell's 6 evolution equations with the constraints that
must be maintained by physical solutions are:
\begin{gather*}
 \Dt
   \begin{bmatrix}
     \bfield \\
     \efield
   \end{bmatrix}
 +\curl
   \begin{bmatrix}
     \efield \\
    -\light^2\bfield
   \end{bmatrix}
 = 
   \begin{bmatrix}
      0 \\
     -\frac{1}{\epsilon_0}\cur
   \end{bmatrix}
 \hbox{ and }
  \div
   \begin{bmatrix}
     \bfield \\
     \efield
   \end{bmatrix}
 = 
   \begin{bmatrix}
     0 \\
     \frac{1}{\epsilon_0}\qdens
   \end{bmatrix}
\end{gather*}

\subsubsection{Material balance laws (gas-dynamics)}

The material balance laws are simply statements of
conservation of mass, momentum, and energy.
See appendix~\ref{sec:conservationLawDerivation}
for a derivation of the electromagnetic part.
\begin{gather*}
 \dcs
   \begin{bmatrix}
       \mdens
    \\ \mom
    \\ \gasenergy
   \end{bmatrix}
 + \Dt
   \begin{bmatrix}
       0
    \\ \frac{1}{\light^2}\Poynting
    \\ \fieldenergy
   \end{bmatrix}
 + \div
   \begin{bmatrix}
       0
    \\ -\stress
    \\ -\stress\dotp\fvel + \heatflux
   \end{bmatrix}
 + \div
   \begin{bmatrix}
       0
    \\ -\fstress
    \\ \Poynting
   \end{bmatrix}
  = 0
\end{gather*}
\fbox{\begin{minipage}{3.80in}
\small
\begin{gather*}
 \hbox{i.e. } \dcs
   \begin{bmatrix}
       \mdens
    \\ \mom
    \\ \frac{\pressure}{\gamma-1}+\half\mdens\mfvel^2
   \end{bmatrix}
 + \Dt
   \begin{bmatrix}
       0
    \\ \epsilon_0\efield\times\bfield
    \\ \frac{\epsilon_0}{2}\mefield^2 + \frac{1}{2\mu_0}\mbfield^2
   \end{bmatrix}
 + \div
   \begin{bmatrix}
       0
    \\ \pressure\idtwo
    \\ \pressure\fvel
   \end{bmatrix}
 = \\
   \div
   \begin{bmatrix}
       0
    \\ \vstress
    \\ \vstress\dotp\fvel + \kappa\nabla\temperature
   \end{bmatrix}
 + \div
   \begin{bmatrix}
       0
    \\ \epsilon_0(\efield\efield-\frac{\mefield^2}{2}\idtwo)
        + \frac{1}{\mu_0}(\bfield\bfield-\frac{\mbfield^2}{2}\idtwo)
    \\ -\frac{1}{\mu_0}\efield\times\bfield
   \end{bmatrix}
\end{gather*}
\end{minipage}}
%% mass
%\nl $\dcs\mdens=0$
%% inertia
%\nl $\dcs\mom+\Dt(\frac{1}{\light^2}\Poynting)=\div\stress+\div\fstress$
%% energy
%\nl $\dcs\gasenergy+\Dt\fieldenergy + \div\Poynting
%  = \div(\stress\dotp\fvel)-\div\heatflux$

\newpage

\section{Conservation laws for MHD.}

The equations of MHD (Magnetohydrodynamics) are an
approximation to the full one-fluid plasma equations above.
The electric field $\efield$ is eliminated by discarding 
$\partial_t\efield$ (Ampere's magnetostatic approximation)
and quadratic order electric field terms.

\p
We will put each law in the form: \\
\fbox{\begin{minipage}{3.30in}
$\partial_t\hbox{(conserved quantity)}+\div\hbox{(hyperbolic flux)}\\
=\div\hbox{(parabolic flux)}$.
\end{minipage}}

\subsection{Magnetic field.}

The MHD equation for the evolution of $\bfield$ 
is obtained by using Ampere's law and Ohm's law
in Faraday's law to eliminate $\efield$ and $\cur$:

$\partial_t\bfield+\curl\efield=0$

$\partial_t\bfield+\curl(\bfield\times\fvel+\efield')=0$

\p \framebox{
$\partial_t\bfield+\div(\fvel\bfield-\bfield\fvel)
  =-\curl(\efield')=\div(\permTensor\cdot\efield').$
}
%\p \framebox{
%$\partial_t\bfield+\div(\fvel\bfield-\bfield\fvel)
%  =\div(\resistivity\frac{1}{\mu_0}(\nabla\bfield^T-\nabla\bfield))$
%}
\nl Note: %For $\efield'=\resistivity\cur$,
$-\curl(\resistivity\cur)
=-\curl(\resistivity\frac{1}{\mu_0}\curl\bfield) \\
=\div(\resistivity\frac{1}{\mu_0}(\nabla\bfield^T-\nabla\bfield))$.


\subsection{Mass balance.}
$\Dt\mdens+\div(\mdens\fvel)=0$.

\subsection{Momentum balance.}

\p For MHD we ignore second-order terms in the electric field.  This
means that we discard the momentum of the electromagnetic field
and retain only the magnetic terms in the electromagnetic stress
tensor.

\nl So the electromagnetic stress tensor is: \\
$\fstress=\frac{1}{\mu_0}(\bfield\bfield-\half\mbfield^2\idtwo)$

\np To see that we can discard the momentum of the electromagnetic field:
\nl $\partial_t(\efield\times\bfield)
   = (\partial_t \efield)\times\bfield+\efield\times(\partial_t\bfield)
\\ = (\partial_t \efield)\times\bfield-\efield\times\curl\efield
\approxeq 0$

\p Decompose the stress tensor into its diagonal component (pressure)
and its traceless component (viscous stress): 
$\stress=\pressure\idtwo+\vstress$.

\np Now substitute into the general momentum balance \\
$\dcs\mom+\Dt(\epsilon_0\efield\times\bfield)=\div\stress+\div\fstress$.

\p Splitting the stress tensor into hyperbolic
(pressure) and parabolic (viscous stress tensor) parts,
we express conservation of momentum as:

\framebox{
$\Dt\mdens\fvel+\div\Big(\mdens\fvel\fvel
  +(\pressure+\frac{1}{2\mu_0}\mbfield^2)\idtwo-\frac{1}{\mu_0}\bfield\bfield
  \Big)
 = \div\vstress$
}

\subsection{Energy balance.}

Using Ohm's law and Ampere's law we can express
the Poynting vector in terms of the magnetic field:
\np $\efield\times\bfield
   =(\efield'+\bfield\times\fvel)\times\bfield
\\ =\efield'\times\bfield +(\mbfield^2\fvel-\bfield\bfield\dotp\fvel)$

\np Again we discard the electric field term from the
electromagnetic energy since it is second-order: \\
$\fieldenergy= \frac{1}{2\mu_0}\mbfield^2$.

\np Invoke the relations
\\ $\stress = -\pressure\idtwo+\vstress$ and
\\ $\heatflux = -\kappa\nabla\temperature$.

\np Substituting into the general energy balance, \\
$\dcs\gasenergy+\Dt\fieldenergy+\div(\frac{1}{\mu_0}\efield\times\bfield)
  = \div(\stress\dotp\fvel)-\div\heatflux$,
\np
$\dcs\gasenergy+\frac{1}{2\mu_0}\Dt\mbfield^2
    + \div\frac{1}{\mu_0}(\mbfield^2\fvel-\bfield\bfield\dotp\fvel)
 + \div\Big(\efield'\times\bfield\Big)
\\ = \div(\vstress\dotp\fvel) - \div(\pressure\fvel)
 +\div(\kappa\nabla\temperature)$

\p Thus the energy balance with electric field expurgated and
hyperbolic and parabolic terms separated is: \\
\fbox{\begin{minipage}{3.30in}
$\Dt\totenergy
 + \div\Big((\totenergy+\pressure
 +\frac{1}{2\mu_0}\mbfield^2)\fvel-\frac{1}{\mu_0}\bfield\bfield\dotp\fvel\Big)=
 \\ \div(\vstress\dotp\fvel)+\div\Big(\kappa\nabla\temperature
  - \efield'\times\bfield\Big)$
\end{minipage}} \\
where $\totenergy=\tenergy+\half\mdens\mfvel^2+\frac{1}{2\mu_0}\mbfield^2$
is the total energy.
\nl Assuming the ideal gas law,
  $\tenergy=\frac{\pressure}{\gamma-1}$.  Note that:
\nl $-\resistivity\cur\times\bfield \approxeq
       -\resistivity\frac{1}{\mu_0}(\curl\bfield)\times\bfield
   = \resistivity\frac{1}{\mu_0}
       (\nabla(\half\mbfield^2))-\bfield\dotp\nabla\bfield
 \\  = \resistivity\frac{1}{\mu_0}
       \div(\half\mbfield^2\idtwo-\bfield\dotp\bfield).
$


\subsection{Full MHD system.}

Thus, the full system of viscous, resistive MHD equations 
for an ideal conducting gas is

\fbox{\begin{minipage}{3.50in}
\begin{gather*}
  \frac{\partial}{\partial t}
     \begin{bmatrix}
       \mdens \\ \mdens \fvel \\ \totenergy \\ \bfield
     \end{bmatrix}
 + \nabla \dotp
    \underbrace{
     \begin{bmatrix}
       \mdens \fvel \\ \mdens \fvel \fvel + \tilde{p} \,\idtwo
          - \frac{1}{\mu_0}\bfield \bfield
       \\ \fvel \bigl(\totenergy  + \tilde{p} \bigr)
          - \frac{1}{\mu_0}\bfield\bfield\dotp\fvel
       \\ \fvel \bfield - \bfield \fvel
     \end{bmatrix}
    }_{\hbox{hyperbolic flux}}
 \\ = \div
    \underbrace{
    \begin{bmatrix}
      0
     \\ 
        \vstress 
     \\ \vstress\dotp\fvel+\Big(\kappa\nabla\temperature
        -\efield'\times\bfield
        %+\resistivity\frac{1}{\mu_0^2}
        %\div\big(\half\mbfield^2\idtwo-\bfield\bfield\big)
        \Big)
     \\
        \permTensor \cdot\efield'
        %\resistivity\frac{1}{\mu_0}(\nabla\bfield^T-\nabla\bfield)
    \end{bmatrix}
    }_{\hbox{parabolic flux}}
    \\
    \hbox{and }\ \ \ \ \ \ \  \div \bfield = 0 \, ,
\end{gather*}
\end{minipage}}
where $\mdens$ is the mass density,
$\fvel$ is the fluid velocity
field, $\totenergy:=\gasenergy+\frac{1}{2\mu_0}\mbfield^2$ is the total energy
(gas-dynamic energy plus magnetic energy),
$\bfield$ is the magnetic
field, and
$\tilde{\pressure}:=\pressure+\frac{1}{2\mu_0}\mbfield^2$ is the total pressure
(gas-dynamic pressure plus magnetic pressure).
The gas-dynamic pressure is $p=(\gamma-1)(\gasenergy-\half\mdens\mfvel^2)$,
where $\gamma$ is the ratio of specific heats.

\section{Two-fluid plasma equations.}

The two-fluid plasma equations consist of 16 evolution equations
which specify balance laws for electromagnetic field and the
mass, momentum, and energy of each species of the plasma.
They model the plasma as a negatively
charged fluid of electrons and a positively charged fluid of ions
which occupy the same space and interact with the electromagnetic
field.  In the collisionless case, it is assumed that the
two fluids pass through one another freely with no direct
interaction, and therefore influence one another only by means
of their mutual interaction with the electromagnetic field.  In
more general models the two fluids may exert a drag force on one
another.

\p Our general two-fluid model consists simply of gas dynamics
for each of the two fluids, coupled to one another by drag force
and heat transfer and coupled to Maxwell's equations by means of
source terms consisting of the Lorentz force, the charge density,
and the current and displacement currents.

\def\drag{\mathbf{R}}
\p The 10 gas dynamics equations in generality are:
\begin{gather*}
 \Dt \begin{bmatrix}
  \mdens_s \\ \mdens_s \fvel_s \\ \gasenergy_s
  \end{bmatrix}
 + \div
     \underbrace{
      \begin{bmatrix}
     \mdens_s \fvel_s
     \\ \mdens_s \fvel_s \fvel_s
     \\ \gasenergy_s \fvel_s
    \end{bmatrix}
    }_{\hbox{advection}}
    = \div
    \begin{bmatrix}
     0 \\ \stress_s
     \\ \stress_s\dotp\fvel_s - \heatflux_s
    \end{bmatrix}
 \\ +
   \underbrace{
    \begin{bmatrix}
     0 \\ \drag_s 
     \\ \drag_s\dotp\fvel_s + Q_s 
    \end{bmatrix}
    }_{\hbox{interactive source}}
    +
    \underbrace{
    \begin{bmatrix}
     0 \\ \qdens_s\efield+\cur_s\times\bfield
     \\ \cur_s\dotp\efield
    \end{bmatrix} 
    }_{\hbox{electromagnetic source}},
\end{gather*}
where
$s$ is the species index ($i$ for ion, $e$ for electron),
$\mdens$ denotes mass density,
$\fvel$ is the fluid velocity,
$\gasenergy$ is the gas-dynamic energy,
$\stress$ is the stress,
$\heatflux$ is the heat flux,
$\qdens$ is the charge density,
$\cur$ is the current,
$\drag_s$ is the drag force on species $s$ from the other species, and
$Q_s$ is the heat transfer to species $s$ from the other species.

\p Maxwell's 6 evolution equations with constraints are:
\begin{gather*}
 \Dt
   \begin{bmatrix}
     \bfield \\
     \efield
   \end{bmatrix}
 +\curl
   \begin{bmatrix}
     \efield \\
    -\light^2\bfield
   \end{bmatrix}
 = 
   \begin{bmatrix}
      0 \\
     -\frac{1}{\epsilon_0}\cur
   \end{bmatrix}
 \hbox{ and }
  \div
   \begin{bmatrix}
     \bfield \\
     \efield
   \end{bmatrix}
 = 
   \begin{bmatrix}
     0 \\
     \frac{1}{\epsilon_0}\qdens
   \end{bmatrix},
\end{gather*}
where $\efield$ and $\bfield$ are the electric and magnetic fields,
$\qdens=\sum_s \qdens_s = \sum_s \frac{q_s}{m_s}\mdens_s$ is the
charge density, and $\cur=\sum_s \cur_s = \sum_s \frac{q_s}{m_s}\mdens_s\fvel_s$
is the current density.

\p We remark here that Maxwell's evolution equations can be viewed as
a conservation law for $\bfield$ and a balance law for
$\efield$ (with current providing a source term),
because a curl, like any spatial differential operator,
can be viewed as a divergence:
$\underline{\nabla}\times\underline{v}
=\Dx{j}\mathbf{e}_i\epsilon_{ijk}v_k
=-\underline{\nabla}\cdot(\underline{\underline{\underline{\epsilon}}}\dotp\underline{v})$.


\p The 10 gas dynamics equations expressed with hyperbolic and
parabolic flux terms and with interactive
and electromagnetic source terms are:
{\small
\fbox{\begin{minipage}{3.60in}
\begin{gather*}
 \partial_t \begin{bmatrix}
  \mdens_s \\ \mdens_s \fvel_s \\ \gasenergy_s
  \end{bmatrix}
 + \div
     \underbrace{
      \begin{bmatrix}
     \mdens_s \fvel_s
     \\ \mdens_s \fvel_s \fvel_s + p_s \, \idtwo
     \\ \fvel_s \bigl(\gasenergy_s  + p_s \bigr) 
    \end{bmatrix}
    }_{\hbox{hyperbolic flux}}
    = \div
    \underbrace{
    \begin{bmatrix}
     0 \\ \vstress_s
     \\ \vstress_s\dotp\fvel_s + \kappa_s\nabla\temperature_s
    \end{bmatrix}
    }_{\hbox{parabolic flux}}
    \\ +
    \underbrace{
    \begin{bmatrix}
     0 \\ \drag_s 
     \\ \drag_s\dotp\fvel_s + Q_s 
    \end{bmatrix}
    }_{\hbox{interactive source}}
    +
    \underbrace{
    \begin{bmatrix}
     0 \\ \frac{q_s}{m_s}\mdens_s(\efield+\fvel_s \times \bfield)
     \\ \frac{q_s}{m_s}\mdens_s\fvel_s\cdot\efield 
    \end{bmatrix} 
    }_{\hbox{electromagnetic source}}
\end{gather*}
\end{minipage}}
}
where
$\frac{q_s}{m_s}$ denotes charge-to-mass ratio, $p$ is the
pressure,
$\vstress$ is the viscous stress,
$\temperature$ is the temperature, and
$\kappa$ is the heat conductivity.

\p Typically $\drag_s$ is taken to be proportional to the
density of each species
and the difference in velocity between the two species.
$Q_s$ is similarly proportional to the density of each species and
the difference in temperature between them.

\p In the collisionless model, the interactive source
is assumed to be zero.  In the ideal model, the parabolic
flux is also assumed to be zero.  
% fix: is this true?
In the absence of shocks I think that we can then
replace energy conservation with entropy conservation: \\
% fix: need to define \dcv^s
$\dcv^s S_s=0$, where $S_s:=\ln(p_s \mdens_s^{-\gamma})$

\subsection{One-fluid from two-fluid.}

To obtain the full one-fluid model from the two-fluid model,
we simply sum the gas-dynamics balance laws over all species
for each conserved variable. \\
(So let $\mdens:=\sum_s\mdens_s$,
$\mom:=\sum_s\mom_s$,
$\gasenergy:=\sum_s\gasenergy_s$,
$\stress:=\sum_s\stress_s$,
$\heatflux:=\sum_s\heatflux_s$, and
$\cur:=\sum_s\cur_s$.)

\p Interactive source terms will cancel, since they simply serve
to exchange momentum and energy between species.  The species
index $s$ will effectively disappear, except for quadratic
deviations from the mean arising from the nonlinear term
labeled ``advection''; these nonlinearities can be absorbed
into the higher-order moments (the stress tensor in the case
of momentum conservation; the heat flux in the case of energy
conservation).  The full one-fluid model is only an approximation
to the two-fluid model, because it assumes that nice constitutive
relations for these higher-order moments still hold after
absorbing these nonlinearities.  [For details see my summary,
``A book-keeping derivation of 1-fluid equations from multi-fluid
plasma equations''.]

\appendix
\section{Derivation of basic laws.}
\label{sec:conservationLawDerivation}

\subsection{Conservation of momentum.}

\def\emforce{\mathbf{F}}
\def\otherforces{\mathbf{\tilde F}}
\def\momfield{\mathbf{M}^f}
The electromagnetic force on a particle of charge $q$ and
velocity $\fvel$ is
given by: $q(\efield+\fvel\times\bfield)$.
This means that
the electromagnetic force density on a continuum of net charge
density $\qdens$ and net current $\cur$ is given by
$\emforce=\qdens\efield+\cur\times\bfield$.
(To see this, let $n$ be the number density and $\fvel$ be
the velocity of a particular species of charge $q$.
Then the charge density of this species is $\qdens=nq$
and the current of this species is the charge flux,
$\cur=\qdens\fvel=nq\fvel$.)

\p
%We want to think of the force of the electromagnetic
%field as an exchange of momentum between the electromagnetic
%fields and the charges.
Conservation of momentum tells us that:

\p $\dcs\mom = \emforce + \div\stress$

%\fbox{\begin{minipage}{3.30in}
%   ($\dcs\mom:=$ rate of change of momentum of charges) \\
%    = ($\emforce=$ force of electromagnetic field on charges) \\
%    + ($\div\stress:=$ other forces on charges).
%\end{minipage}} \\
%If we can define a notion of the momentum of electromagnetic field,
%it will need to obey an equation of the form: \\
%\fbox{\begin{minipage}{3.30in}
%$\emforce+(\partial_t\momfield:=$ rate of change of field momentum) \\
%    = $\div$($\fstress:=$ outgoing flux of field momentum)
%\end{minipage}}
%Substituting for $\emforce$ in the first equation gives
%conservation of momentum: \\
%\framebox{$\dcs\mom + \partial_t\momfield = \div\fstress + \div\stress$}
%\fbox{\begin{minipage}{3.30in}
% ($\dcs\mom=$ rate of change of momentum of charges) \\
%  + ($\partial_t\momfield=$ rate of change of momentum of field) \\
%   + (flux of field momentum) \\
%  = ($\otherforces=$ other forces on charges)
%\end{minipage}}

\p We wish to write the force of the electromagnetic
field on the particles as the time derivative of
some function of electromagnetic field (which we will
regard as electromagnetic momentum) plus a spatial
derivative of another function of electromagnetic
field (which we will regard as flux of electromagnetic momentum).

\p To express the force purely in terms of electromagnetic
field quantities, use the nonhomogeneous Maxwell equations
to eliminate the charge density and the current:
$\emforce=\epsilon_0(\div\efield)\efield
  + (\frac{1}{\mu_0}\curl\bfield-\epsilon_0\Dt\efield)\times\bfield$.
\p Then use parts to get a time derivative of a single quantity.
$-(\Dt\efield)\times\bfield
  = -\Dt(\efield\times\bfield)+\efield\times\Dt\bfield$.
\p The quantity
$\epsilon_0\efield\times\bfield=\frac{1}{c^2}\Poynting$, %=:\momfield$,
where $\Poynting$ is the Poynting vector, is what we identify as
the momentum of the field.

\p Now we'll use the inhomogeneous equations to make everything
else look like the spatial derivative of a single quantity.
Faraday's law gives 
$\efield\times\Dt\bfield = -\efield\times(\curl\efield)$.
Now we try to write everything except the time derivative
as the divergence of some tensor.  For the electric field
terms we get: 
\nl $(\div\efield)\efield-\efield\times(\curl\efield)
\\=(\div\efield)\efield-(\nabla\efield)\dotp\efield+\efield\dotp(\nabla\efield)
\\=\div(\efield\efield)-\nabla(\half\mefield^2)
$
\nl For the magnetic field terms we get (since $\div\bfield=0$):
$(\curl\bfield)\times\bfield
  = \bfield\dotp\nabla\bfield-\nabla(\half\mbfield^2)
  = \div(\bfield\bfield-\half\mbfield^2\idtwo)$.

\np So the force of the field on the charges is \\
$\emforce=-\Dt(\frac{1}{c^2}\Poynting)
  + \div\fstress$, \\
where
$\fstress := \epsilon_0(\efield\efield-\half\mefield^2\idtwo)
        + \frac{1}{\mu_0}(\bfield\bfield-\half\mbfield^2\idtwo)$
\\ is the Maxwell stress tensor.

\framebox{$\dcs\mom + \Dt(\frac{1}{c^2}\Poynting) = \div\fstress + \div\stress$}

\subsection{Conservation of energy.}

%\def\emwork{\mathcal{W}}
\def\otherwork{\mathcal{\tilde W}}
\def\heatflow{\mathcal{\tilde Q}}
The power (rate of work) of an electromagnetic field
on a moving charged particle is
$\hbox{(force)}\dotp\hbox{(velocity)}
= q(\efield+\fvel\times\bfield)\dotp\fvel
= q\fvel\dotp\efield$.
This means that the power density on a net current
$\cur$ is given by $\cur\dotp\efield$.
(To see this, let $n$ be the number density and
$\fvel$ be the velocity of a particular species of
charge $q$.  Then the current of this species is
the charge flux, $\cur=nq\fvel$.)

\p
%We want to think of the work of the electromagnetic
%field as an exchange of energy between the electromagnetic
%fields and the charges.
Conservation of energy tells us that:

\p $\dcs\gasenergy = \cur\dotp\efield + \div(\stress\dotp\fvel)-\div\heatflux$

%\fbox{\begin{minipage}{3.30in}
%   ($\dcs\gasenergy:=$ rate of change of energy of charges) \\
%    = ($\emwork=$ work of electromagnetic field on charges) \\
%    + ($\div(\stress\dotp\fvel)=$ other work on charges) \\
%    - $\div(\heatflux:=$ heat flow out of charges),
%\end{minipage}}\\
%where $\stress$ is the mechanical stress tensor.
%If we can define a notion of energy of an electromagnetic field,
%it will need to obey an equation of the form: \\
%\fbox{\begin{minipage}{3.30in}
%\nl 0=($\partial_t\fieldenergy:=$ rate of change of field energy) \\
%    + $\div$($\Poynting:=$ field energy flux) + $\emwork$
%\end{minipage}}
%Substituting for $\emwork$ in the first equation gives
%the conservation of energy: \\

%\fbox{\begin{minipage}{3.30in}
% ($\dcs\gasenergy=$ rate of change of energy of charges) \\
%  + ($\partial_t\fieldenergy=$ rate of change of energy of field) \\
%  + $\div$($\Poynting=$ field energy flux) \\
%  = ($\otherwork=$ other work on charges) \\
%  + ($\heatflow=$ heat flow into charges).
%\end{minipage}}

\p We wish to write the work of the electromagnetic
field on the particles as the time derivative of
some function of electromagnetic field (which we will
regard as electromagnetic momentum) plus a spatial
derivative of another function of electromagnetic
field (which we will regard as flux of electromagnetic energy).

\p To express the work purely in terms of electromagnetic
field quantities, use the completed Ampere's law 
to eliminate the current, and then use parts and Faraday's law
to separate out a time and spatial derivative:

\np $-\cur\dotp\efield
   = \epsilon_0(\partial_t\efield-\light^2\curl\bfield)\dotp\efield
\\ = \epsilon_0(\partial_t(\half\mefield^2)-\light^2\efield\dotp\curl\bfield)
\\ = \epsilon_0\partial_t(\half\mefield^2)
  -\frac{1}{\mu_0}(\bfield\dotp\curl\efield-\div(\efield\times\bfield))
\\ = \epsilon_0\partial_t(\half\mefield^2)
  +\frac{1}{\mu_0}\partial_t(\half\mbfield^2)
  +\frac{1}{\mu_0}\div(\efield\times\bfield)
\\ = \partial_t\big(\underbrace{\epsilon_0(\half\mefield^2)
                    +\frac{1}{\mu_0}(\half\mbfield^2)}
                 _{\hbox{\small Call \large $\fieldenergy$}}\big)
  +\div\big(\underbrace{\frac{1}{\mu_0}\efield\times\bfield}
                 _{\hbox{\small Call \large $\Poynting$}}\big)$

\framebox{$\dcs\gasenergy+\Dt\fieldenergy + \div\Poynting
  = \div(\stress\dotp\fvel)-\div\heatflux$}

\section{Ohm's law.}

Ohm's law specifies the electric field
$\efield':=\efield+\fvel\times\bfield$
in the reference frame of the fluid.
(In the approximation of Galilean relativity,
the transformation of electromagnetic
field from the fixed reference frame to a reference frame
moving at velocity $\fvel$ is given by $\bfield \to \bfield$,
$\efield \to (\efield+\fvel\times\bfield$.))
%A generic version is $\efield'=\resistivity\cur+\efield_X$,
%where the proportionality constant $\resistivity$ is called
%the \emph{resistivity}, and $\efield_X$ serves to represent
%the additional terms used in extended MHD.
\def\w{\mathbf{w}}
Assuming quasineutrality and vanishing electron mass implies
that the electron velocity in the reference frame of the
ions is $\w_e:=-\cur/(en)$.
Then conservation of momentum for electrons yields
the generalized Ohm's law:
\nl \framebox{$\efield'=
    \resistivity\cur
  + \frac{\cur}{en}\times\bfield
  -\frac{\nabla p_e}{en} 
  + \frac{m_e}{e^2 n}[\partial_t\cur
     +\div(\cur\fvel+\fvel\cur-\frac{\cur\cur}{ne})]$}
\np Note that $\efield'-\frac{\cur}{en}\times\bfield$ is the
electric field in the reference frame of the electrons.
The final term represents the inertia.
%$[\partial_t(m_e n\fvel_e)+\div(m_e n\fvel_e\fvel_e)]
%=\frac{m_e}{e}[\partial_t \cur+\div(\cur\fvel+\fvel\cur-\frac{\cur\cur}{ne}]
%  + \frac{m_e}{m_i}[\partial_t(\mdens\fvel)+\div(\mdens\fvel\fvel]$.
%$\R_e=-\nu n^2 \w_e$

%Since ions are much heavier than electrons, the velocity of the
%fluid is essentially the velocity of the ions, and the current
%is proportional to the velocity of the electrons relative to the
%ions through which they pass.
%
%\p For an ideal plasma $\resistivity=0$.
%
%(Note that ions are much
%heavier than electrons, so the reference frame of the fluid
%is essentially the reference frame of the ions.  Current
%results from relative motion of the electrons through the ions.)

\section{The equations of electromagnetism.}

The fundamental equations of electromagnetism are
Maxwell's equations of electromagnetic fields
and the Lorentz force law.

\subsection{Lorentz force law.}

\def\vel{\mathbf{v}}
\def\force{\mathbf{F}}
\p The force $\force$ felt by a particle
of charge $q$ moving in the presence of an
electric field $\efield$ and a magnetic field
$\bfield$ with velocity $\vel$
is given by the Lorentz force law:
$$\force = q(\efield+\vel\times\bfield)$$

\subsection{Maxwell's equations.}

Maxwell's equations constrain and
govern the evolution of the electric
field $\efield$ and the magnetic field $\bfield$.

\subsubsection{Differential form.}

In their differential form, Maxwell's equations are
most commonly expressed as:
\smallskip
\\ $\bullet\ \div \bfield = 0$ \hfill (no monopoles)
\\ $\bullet\ \div \efield = \frac{\sigma}{\epsilon_0}$ \hfill (Gauss's law)
\\ $\bullet\ \curl\efield = -\Dt\bfield$ \hfill (Faraday's law)
\\ $\bullet\ \curl\bfield = \mu_0(\cur+\epsilon_0\Dt\efield)$
  \hfill (extended Ampere's law)

\p The last two equations can be read as evolution equations
for $\bfield$ and $\efield$.
The first two equations are constraints that continue to
hold if they hold at some initial time.
Taking the curl of the evolution equations and setting
the current to 0 show that the speed of light $c$ satisfies
$c^2\mu_0\epsilon_0=1$.

\subsubsection{Integral form of Maxwell's equations.}

To see their physical meaning, we put Maxwell's equations
in integral form.

\def\vectorfield{\mathbf{u}}
\def\normal{\mathbf{n}}
\def\tangent{\mathbf{\tau}}
\p Let $\vectorfield$ be a vector field,
let $V$ be a simple region of volume,
let $\partial V$ be the boundary of this volume,
let $S$ be a simple oriented surface in space,
and $\partial S$ be the boundary of this surface.
Let $\normal$ represent the outward normal or a
normal to an oriented surface,
and let $\tangent$ represent the oriented tangent to the
boundary of an oriented surface.
We can use the following versions of the fundamental theorem
of calculus to rewrite Maxwell's equations in integral form:
\begin{itemize}
\item $\int_V\div\vectorfield=\oint_{\partial V}\normal\dotp\vectorfield$
  \hfill (Divergence theorem)
\item $\int_S\normal\dotp\curl\vectorfield=\oint_{\partial S}\tangent\dotp\vectorfield$
  \hfill (Stokes' theorem)
\end{itemize}

We can put Maxwell's equations in their standard integral
form by integrating the divergence constraints over an
arbitrary volume and applying Gauss's law,
and taking a flux integral ($\int_S\normal\dotp$) of
the evolution equations over 
an arbitrary oriented surface $S$ and applying Stoke's law.
This gives:
\smallskip
\\ $\bullet\ \oint_{\partial V} \normal\dotp\bfield = 0$ \hfill (no monopoles)
\\ $\bullet\ \oint_{\partial V} \normal\dotp\efield
      = \int_V \frac{\sigma}{\epsilon_0}$ \hfill (Gauss's law)
\\ $\bullet\ \oint_{\partial S} \tangent\dotp\efield
     = -\frac{d}{dt}\int_S \normal\dotp\bfield$ \hfill (Faraday's law)
\\ $\bullet\ \oint_{\partial S} \tangent\dotp\bfield
     = \int_S \normal\dotp\mu_0(\cur+\epsilon_0\Dt\efield)$
  (extended Ampere)

\p So absence of monopoles says that the net flux of the
magnetic field out of any volume is zero,
Gauss's law says that the flux of the electric field
out of any volume is proportional to the total charge
inside,
Faraday's law says that the circulation of the
electric field around a loop is minus the rate of change of the flux
of the magnetic field through the loop,
and Ampere's law says that the circulation of the magnetic
field around a loop is proportional to the flux of the current plus
displacement current through the loop.

\subsubsection{Balance law form of Maxwell's equations.}

We can write the evolution equations as balance laws
with current as a source term for electromagnetic field:
\begin{gather*}
  \Dt \begin{bmatrix}
    \bfield \\ \efield
    \end{bmatrix} + \curl
        \begin{bmatrix}
    \efield \\ -c^2\bfield
    \end{bmatrix} = 
    \begin{bmatrix}
     0 \\ -\frac{1}{\epsilon_0}\cur
    \end{bmatrix}
\end{gather*}

These laws of electromagnetism, along with the additional
fundamental laws of conservation of mass, momentum balance (net force
equals rate of change of momentum), and Newton's
inverse square law for gravitation, constitute
the fundamental laws of classical mechanics.

\subsection{Taking Fundamental Constants to be Unity.}

You can take all fundamental constants to be unity
by writing the basic equations in the following form:
\begin{gather*}
 \partial_{(\light t)}
   \begin{bmatrix}
      (\light\bfield)
    \\ \efield
   \end{bmatrix}
 +\curl
   \begin{bmatrix}
        \efield
     \\ -(\light\bfield)
   \end{bmatrix}
 = 
   \begin{bmatrix}
        0
     \\ -\big(\frac{\cur}{\light\epsilon_0}\big)
   \end{bmatrix}
 \\
  \hbox{and } \ \ \ 
  \div
   \begin{bmatrix}
        (\light\bfield)
     \\ \efield
   \end{bmatrix}
  = 
   \begin{bmatrix}
       0
    \\ \big(\frac{\qdens}{\epsilon_0}\big)
   \end{bmatrix}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\end{gather*}

The electromagnetic field produces a Lorentz force
which (if it happens to stand alone) results in
a rate of change of momentum:
\begin{gather*}
   \Big(\frac{\mdens\light^2}{\epsilon_0}\Big)
    \frac{d}{d(t\light)}\Big(\frac{\fvel}{\light}\Big)
  = \Big(\frac{\force}{\epsilon_0}\Big)
  = \Big(\frac{\qdens}{\epsilon_0}\Big)\efield
   +\Big(\frac{\cur}{\light\epsilon_0}\Big)\times\big(\light\bfield\big).
\end{gather*}

Now replace each parenthesized quantity with the variable
it contains renamed with a tilde
and you can proceed as if all the fundamental constants
are unity.

% changes:
% - put Ohm's law stuff at end, add stuff about transforming
%   electromagnetic field under change in velocity.
%   make the part about taking fundamental constants to be
%   unity part of an appendix on Maxwell's equations

% Add:
% - index?
% - list of references

%\fbox{\begin{minipage}{3in}
%This multiline text is more flexible than 
%a tabular setting:
%\begin{itemize}
%\item it can contain any type of normal 
%LATEX typesetting;
%\item it can be any specified width;
%\item it can even have its own 
%footnotes\footnote{Like this}.
%\end{itemize}
%\end{minipage}}

\end{document}