{ "cells": [ { "cell_type": "markdown", "id": "06f7f570", "metadata": {}, "source": [ "# Stix Dispersion Solver\n", "\n", "[stix]: ../../api_static/plasmapy.dispersion.analytical.stix_.rst\n", "[bellan2012]: https://doi.org/10.1029/2012ja017856\n", "[stix1992]: https://link.springer.com/book/9780883188590\n", "\n", "This notebook details the functionality of the [stix()][stix] function. This is an analytical solution of equation 8 in [Bellan 2012][bellan2012], the function is defined by [Stix 1992][stix1992] in §1.2 to be:\n", "\n", "$$\n", " (S \\sin^2(θ) + P \\cos^2(θ)) \\left ( \\frac{ck}{ω} \\right)^4\n", " - [\n", " RL \\sin^2(θ) + PS (1 + \\cos^2(θ))\n", " ] \\left ( \\frac{ck}{ω} \\right)^2 + PRL = 0\n", "$$\n", "\n", "where,\n", "\n", "$$\n", " \\mathbf{B}_0 = B_0 \\mathbf{\\hat{z}}\n", " \\cos θ = \\frac{k_z}{k} \\\\\n", " \\mathbf{k} = k_{\\rm x} \\hat{x} + k_{\\rm z} \\hat{z}\n", "$$\n", "\n", "$$\n", " S = 1 - \\sum_s \\frac{ω^2_{p,s}}{ω^2 -\n", " ω^2_{c,s}}\\hspace{2.5cm}\n", " P = 1 - \\sum_s \\frac{ω^2_{p,s}}{ω^2}\\hspace{2.5cm}\n", " D = \\sum_s\n", " \\frac{ω_{c,s}}{ω}\n", " \\frac{ω^2_{p,s}}{ω^2 - ω_{c,s}^2}\n", "$$\n", "\n", "$$\n", " R = S + D \\hspace{1cm} L = S - D\n", "$$\n", "\n", "$ω$ is the wave frequency, $k$ is the wavenumber, $θ$ is the wave propagation angle with respect to the background magnetic field $\\mathbf{B}_0$, $s$ corresponds to plasma species, $ω_{p,s}$ is the plasma frequency of species and $ω_{c,s}$ is the gyrofrequency of species $s$." ] }, { "cell_type": "markdown", "id": "013394df-d0eb-4be1-8c9f-b1d2f484ae7d", "metadata": {}, "source": [ "