From ec45b08c27774d88aa38cdc5e66c14b9f3ea84d0 Mon Sep 17 00:00:00 2001 From: "Stowell, Mark L." Date: Fri, 26 Sep 2025 15:57:19 -0700 Subject: [PATCH 1/2] Starting plasma physics documentation --- mkdocs.yml | 1 + src/getting-started.md | 1 + src/img/examples/stix-r1d-eb.gif | Bin 0 -> 995204 bytes src/plasma.md | 146 +++++++++++++++++++++++++++++++ 4 files changed, 148 insertions(+) create mode 100644 src/img/examples/stix-r1d-eb.gif create mode 100644 src/plasma.md diff --git a/mkdocs.yml b/mkdocs.yml index 61a9bb5b..54de9deb 100644 --- a/mkdocs.yml +++ b/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 diff --git a/src/getting-started.md b/src/getting-started.md index 1363b619..da4f13a2 100644 --- a/src/getting-started.md +++ b/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) diff --git a/src/img/examples/stix-r1d-eb.gif b/src/img/examples/stix-r1d-eb.gif new file mode 100644 index 0000000000000000000000000000000000000000..ed68be95f1a83381f84663d8c8a7d03004fc4a39 GIT binary patch literal 995204 zcmXt;bxhpN7w*3cEbi_sTD-WlK+)n_tax#kBBkg;ad#+Iytuo&YjJm2v{-T2`~L3D zJ^##PGILJyB$Lc}<|8ZrQAqg9D_9%h?g{XuK*jSy_WA^<1&jee z9{@~{1HELx5+*Ro4GeJs^Sro5C!o>^nEnW?DgZN|fmI`5SQ+?Z3#>Q- zb6&uz7clG${B;8Ussld(qZz<@6>t~^%ufJ!!We)x6+o{6gqi{N3IHf%|B?y(zj`DS z0C7Kny$G;R0<8Uj=rHhM8sORlI2QrlGXR(o23Fq#LjqVw$dXHd`V64G27DR=O#c8@ zYrxkzz;hXJ9tUI(0lh;&`2|q826P?)dl+E)47fc2-p3#yTOU|-0ajgsF+1R&ZdELh zlMT#=0jr_FVhXU54h+WvmvO*S9N`I{nV9&p_O zz8wR$Pk{R?$m}E9N;_| z2wn!Frh$erAb1CeJONU7fQ%y``5DN71M&ZWm}{VT2PirPnsUO8MG?kwoTCr3NFlg=48)W}}(%bwv{yLaxVaBXz&0eu@7;r&6dd 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+\newcommand{\div}{\nabla\cdot}\newcommand{\curl}{\nabla\times} +\newcommand{\grad}{\nabla}$ + +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. + + + +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. $\B_0$ | Description | +|-------|--------|-------|--------|----------------------------| +| L | $\|\E_L\|\frac\{(\hat\{y}-i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S-D}\,\hat\{x}$ | $\vec\{k}_L \parallel \B_0$ | Left Circularly Polarized | +| R | $\|\E_R\|\frac\{(\hat\{y}+i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S+D}\,\hat\{x}$ | $\vec\{k}_R \parallel \B_0$ | Right Circularly Polarized | +| O | $\|\E_O\|\,\hat\{y}$ | $\sqrt\{P}\,\hat\{x}$ | $\vec\{k}_O \perp \B_0$ | Ordinary Wave | +| X | $\frac\{\|\E_X\|}\{\sqrt\{D^2-S^2}}\left(D\,\hat\{x}+i\,S\,\hat\{z}\right)$ | $\sqrt\{S-\frac\{D^2}\{S}}\,\hat\{x}$ | $\vec\{k}_X \perp \B_0$ | 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. + +#### Mini Application Features + +## Anisotropic Thermal Diffusion in Magnetized Plasma + + + From b72ea07817f444dfd0013faee9024f380268b9bd Mon Sep 17 00:00:00 2001 From: "Stowell, Mark L." Date: Fri, 26 Sep 2025 16:52:05 -0700 Subject: [PATCH 2/2] Finished first draft of stix1d motivation --- src/plasma.md | 41 +++++++++++++++++++++++++++++++++++------ 1 file changed, 35 insertions(+), 6 deletions(-) diff --git a/src/plasma.md b/src/plasma.md index 211fedac..b33227cc 100644 --- a/src/plasma.md +++ b/src/plasma.md @@ -112,6 +112,26 @@ non-trivial plane wave solutions which arise in homogeneous plasmas. +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 @@ -127,17 +147,26 @@ $$\B = \frac{1}{\omega}\left(\vec{k}_p\times\E_p\right) e^{i(\vec{k}_p\cdot\vec{ 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. $\B_0$ | Description | -|-------|--------|-------|--------|----------------------------| -| L | $\|\E_L\|\frac\{(\hat\{y}-i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S-D}\,\hat\{x}$ | $\vec\{k}_L \parallel \B_0$ | Left Circularly Polarized | -| R | $\|\E_R\|\frac\{(\hat\{y}+i\,\hat\{z})}\{\sqrt\{2}}$ | $\sqrt\{S+D}\,\hat\{x}$ | $\vec\{k}_R \parallel \B_0$ | Right Circularly Polarized | -| O | $\|\E_O\|\,\hat\{y}$ | $\sqrt\{P}\,\hat\{x}$ | $\vec\{k}_O \perp \B_0$ | Ordinary Wave | -| X | $\frac\{\|\E_X\|}\{\sqrt\{D^2-S^2}}\left(D\,\hat\{x}+i\,S\,\hat\{z}\right)$ | $\sqrt\{S-\frac\{D^2}\{S}}\,\hat\{x}$ | $\vec\{k}_X \perp \B_0$ | Extraordinary Wave | +| 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