[WIP] Starting plasma physics documentation

mkdocs.yml

@@ -20,6 +20,7 @@ nav:
   - 'Example Codes': examples.md
   - 'Electromagnetics': electromagnetics.md
   - 'Fluid Dynamics': fluids.md
+  - 'Plasma Physics': plasma.md
   - 'Meshing': meshing-miniapps.md
   - 'NURBS Discretization': nurbs.md
   - 'Automatic Differentiation': autodiff.md

src/getting-started.md

@@ -25,6 +25,7 @@ We recommend that new users start with these articles:
 
 - [Electromagnetics Miniapps](electromagnetics.md)
 - [Fluid Dynamics Miniapp](fluids.md)
+- [Plasma Physics Miniapps](plasma.md)
 - [Meshing Miniapps](meshing-miniapps.md)
 - [AD Miniapps](autodiff.md)
 

src/plasma.md

@@ -0,0 +1,175 @@
+tag-gettingstarted:
+
+# Plasma Physics Mini Applications
+
+$\newcommand{\A}{\vec{A}}\newcommand{\B}{\vec{B}}
+\newcommand{\D}{\vec{D}}\newcommand{\E}{\vec{E}}
+\newcommand{\H}{\vec{H}}\newcommand{\J}{\vec{J}}
+\newcommand{\M}{\vec{M}}\newcommand{\P}{\vec{P}}
+\newcommand{\F}{\vec{F}}
+\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}}
+\newcommand{\cross}{\times}\newcommand{\inner}{\cdot}
+\newcommand{\div}{\nabla\cdot}\newcommand{\curl}{\nabla\times}
+\newcommand{\grad}{\nabla}$
+
+The `miniapps/plasma` directory contains a collection of plasma
+physics miniapps based on MFEM.
+
+Plasma Physics is a broad and fascinating field. Our mini applications
+will tread a narrow path through this field but hopefully point the
+way towards a more thorough exploration using MFEM. We will confine
+ourselves to simple examples of magnetized cold plasmas. The existance
+of a background magnetic field will create anisotropies in otherwise
+familiar partial differential equations. The "cold" plasma limitation
+means that thermal motions of electrons and ions are small enough as
+to be negligible which simplifies many interactions and effective
+material coefficients.
+
+Compared to the [example codes](examples.md), the miniapps are more complex,
+demonstrating more advanced usage of the library. They are intended to be more
+representative of MFEM-based application codes. We recommend that new users
+start with the example codes before moving to the miniapps.
+
+The current (and future) plasma miniapps are described below.
+
+## Electromagnetic Wave Propagation in Magnetized Cold Plasma
+
+The equations describing electromagnetic phenomena are known
+collectively as the Maxwell Equations. For a brief overview see the
+discussion in
+[Electromagnetics](electromagnetics.md#Electromagnetics). For harmonic
+fields with time dependence of the form $e^{-i\omega t}$ the Maxwell
+equations can be written as:
+
+  $$\begin{align}
+    \curl\H + i\omega\D & = \J    \label{ampere}  \\\\
+    \curl\E - i\omega\B & = 0     \label{faraday} \\\\
+                 \div\D & = \rho  \label{gauss}   \\\\
+                 \div\B & = 0     \label{divb}
+  \end{align}$$
+
+
+In magnetized cold plasmas the Maxwell equations still hold
+unaltered. However, the background magnetic field, $\B_0$, restricts
+the motion of electrons and ions which move freely along the field but
+follow helical paths when accelerated in other directions. This leads
+to an effective dielectric tensor, $\epsilon$, which can be highly
+anisotropic.  This coefficient appears in the constitutive relation
+$D = \epsilon\E$. The form of this tensor can be written as:
+
+$$
+\epsilon/\epsilon_0 = (I - \hat\{b}\otimes\hat\{b})\, S + \hat\{b}\otimes\hat\{b}\, P + i\,\[\hat\{b}\]_\times\, D
+$$
+
+Where $\hat{b}\equiv\frac{\B_0}{\|\B_0\|}$, $\[\hat{b}]_\times$ is the
+operator which computes the cross product with $\hat\{b}$. As an
+example consider a background magnetic flux directed along the
+$\hat{z}$ direction. In this case the dielectric tensor becomes:
+
+$$\epsilon = \epsilon_0\,\left(
+\begin{array}{3}
+   S & -i\,D & 0\\\\
+i\,D &     S & 0\\\\
+   0 &     0 & P
+\end{array}
+\right)$$
+
+$S$, $D$, and $P$ are called Stix coefficients (named after [Thomas H.
+Stix](https://en.wikipedia.org/wiki/Thomas_H._Stix) from Princeton
+University) given by:
+
+$$\begin{align\*}
+S & = 1 - \sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2} \\\\
+D & = \sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2} \\\\
+P & = 1 - \sum_s\frac{\omega_{ps}^2}{\omega^2} \\\\
+\end{align\*}$$
+
+Where the sums are taken over ion species, $\omega=2\pi f$ and $f$ is
+the wave frequency, $\omega_{ps}$ is the plasma frequency related to
+mass, charge, and density of the plasma species, and $\Omega_s$ is the
+cyclotron frequency related to the background magnetic field intensity
+and the mass and charge of the ion species.
+
+Clearly, this anisotropy will give rise to effects related to the
+polarization of the electric field and the direction of wave
+propagation. In addition to this it must be noted that the complicated
+dependence of the Stix coefficients on the background magnetic field
+strength and the electron and ion densities can produce drastic
+changes in the character of the dielectric tensor throughout the
+domain with its eigenvalues varying by several orders of magnitude. In
+particular we should point out that this effective dielectric tensor
+may have strongly negative eigenvalues. These characteristics of the
+dielectric tensor significantly increase the numerical difficulties
+already present in seeking solutions to the Maxwell equations.
+
+### Stix 1D Mini Application
+
+The Stix 1D mini application implemented in `stix_r1d_eb.cpp` solves
+for all three components of the electric field on a 1D domain. The
+purpose of this relatively simple 1D simulation code is to validate
+the underlying cold-plasma-dielectric Maxwell solver using some of the
+non-trivial plane wave solutions which arise in homogeneous plasmas.
+
+<img class="floatright" src="../img/examples/stix-r1d-eb.gif">
+
+The equation being solved is the second order frequency domain version
+of the Maxwell equations which can be written as:
+
+$$
+\curl\left(\frac{1}{\mu}\curl\E\right) - \omega^2 \epsilon \E = i \omega \J
+$$
+
+In 1D this becomes:
+
+$$
+\begin{align\*}
+-\omega^2\left(\epsilon\E\right)_x &= i\omega J_x\\\\
+-\frac{1}{\mu}E''_y-\omega^2\left(\epsilon\E\right)_y &= i\omega J_y\\\\
+-\frac{1}{\mu}E''_z-\omega^2\left(\epsilon\E\right)_z &= i\omega J_z
+\end{align\*}
+$$
+
+Where the anisotropic $\epsilon$ provides the coupling between these
+three equations.
+
+The four most simple solutions in 1D occur when the background
+magnetic field is either parallel or perpendicular to the 1D
+domain. In addition to the direction of propagation each configuration
+supports two distinct polarizations. In each case the plane wave
+solution can be written as:
+
+$$\E = \E_p e^{i(\vec{k}_p\cdot\vec{x} - \omega t)}$$
+
+and
+
+$$\B = \frac{1}{\omega}\left(\vec{k}_p\times\E_p\right) e^{i(\vec{k}_p\cdot\vec{x} - \omega t)}$$
+
+These four plane wave solutions are often referred to by single
+character abbreviations:
+
+| Abbreviation | $\E_p$ | $\frac\{\vec{k}_p}\{\omega\sqrt\{\epsilon_0\mu_0}}$ | $\vec\{k}_p$ vs. $\B_0$ | $\hat\{b}$ | Description |
+|-------|--------|-------|--------|----|----------------------------|
+| L | $\|\E_L\|\frac\{(\hat\{y}-i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S-D}\,\hat\{x}$ | $\vec\{k}_L \parallel \B_0$ | $\hat\{x}$ | Left Circularly Polarized  |
+| R | $\|\E_R\|\frac\{(\hat\{y}+i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S+D}\,\hat\{x}$ | $\vec\{k}_R \parallel \B_0$ | $\hat\{x}$ | Right Circularly Polarized |
+| O | $\|\E_O\|\,\hat\{y}$ | $\sqrt\{P}\,\hat\{x}$ | $\vec\{k}_O \perp \B_0$ | $\hat\{y}$ | Ordinary Wave |
+| X | $\frac\{\|\E_X\|}\{\sqrt\{D^2-S^2}}\left(D\,\hat\{x}+i\,S\,\hat\{z}\right)$ | $\sqrt\{S-\frac\{D^2}\{S}}\,\hat\{x}$ | $\vec\{k}_X \perp \B_0$ | $\hat\{y}$ | Extraordinary Wave |
+
+The first three types of waves could be realized in isotropic media
+although, in that case, their wave vectors would be of the same
+magnitude. The fourth type has no analogue in isotropic media.
+
+As an example of the challenge of accurately modeling electromagnetic
+waves in a magnetized plasma consider ionized hydrogen gas with a
+density of $10^{19}$ particles$/m^3$ in a background 1 Tesla magnetic
+field. In this case the wavelengths of the $R$ and $X$ waves are
+roughly 80cm and 60cm respectively. Such waves can be accurately
+captured with mesh spacings of a few centimeters. However, the $L$ and
+$O$ waves decay on the length scale of a few millimeters requiring a
+mesh spacing hundreds of times smaller.
+
+#### Mini Application Features
+
+## Anisotropic Thermal Diffusion in Magnetized Plasma
+
+<script type="text/x-mathjax-config">MathJax.Hub.Config({TeX: {equationNumbers: {autoNumber: "all"}}, tex2jax: {inlineMath: [['$','$']]}});</script>
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