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June  2017, 10(2): 373-421. doi: 10.3934/krm.2017015

## Dispersion relations in cold magnetized plasmas

 Institut Mathématique de Rennes, Campus de Beaulieu, 263 avenue du Général Leclerc CS 74205,35042 Rennes Cedex, France

Received  December 2015 Revised  June 2016 Published  November 2016

Fund Project: This work received the support of the Agence Nationale de la Recherche: projet NOSEVOL (ANR 2011 BS01019 01) and ANR blanc DYFICOLTI

Starting from kinetic models of cold magnetized collisionless plasmas, we provide a complete description of the characteristic variety sustaining electromagnetic wave propagation. As in [4, 13, 17], the analysis is based on some asymptotic calculus exploiting the presence at the level of dimensionless relativistic Vlasov-Maxwell equations of a large parameter: the electron gyrofrequency. The method is inspired from geometric optics [29, 33]. It allows to unify all the preceding results [8, 12, 38, 31, 37, 40], while incorporating new aspects. Specifically, the non trivial effects [5, 9, 10, 24] of the spatial variations of the background density, temperature and magnetic field are exhibited. In this way, a comprehensive overview of the dispersion relations becomes available, with important possible applications in plasma physics [7, 28, 30].

Citation: Christophe Cheverry, Adrien Fontaine. Dispersion relations in cold magnetized plasmas. Kinetic & Related Models, 2017, 10 (2) : 373-421. doi: 10.3934/krm.2017015
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##### References:
Spherical coordinates of $p\in {{\mathbb{R}}^{3}}$ after straigntening
Spherical coordinates of $\xi \in \mathbb{R}^3$ after straigntening
Parallel propagation ($\varpi=0$). Standard nomenclature. $\text{V}_{\! r} (\mathbf{x}, 0)$ (in orange), ${{\text{V}}_{l}}(\mathbf{x},0)$ (in blue-green) and $\text{V}_{\! 0} (\mathbf{x}, 0)$ (in black)
Oblique propagation for angles $\varpi \in \, ] 0, \pi / 2] \,$: -Ordinary waves ${{\text{V}}_{o}}(\mathbf{x},\varpi )=\text{V}_{o}^{-}(\mathbf{x},\varpi )\sqcup \text{V}_{\text{o}}^{\text{+}}(\mathbf{x},\varpi )$ in blue; -Extraordinary waves ${{\text{V}}_{x}}(\mathbf{x},\varpi )=\text{V}_{x}^{-}(\mathbf{x},\varpi )\sqcup \text{V}_{\text{x}}^{\text{+}}(\mathbf{x},\varpi )$ in red
Parallel propagation ($\varpi=0$). Mixing of $\text{V}_{\! x} (\mathbf{x}, 0) \,$ and $\text{V}_{\! o} (\mathbf{x}, 0) \,$
Evolution of ${{\text{V}}_{x}}(\mathbf{x},\varpi )\text{ and }{{\text{V}}_{o}}(\mathbf{x},\varpi )$ in function of $\varpi$. Asymptotic behaviour when $\varpi \rightarrow 0$ (see $\varpi =0.1 \sim 0$) and $\varpi \rightarrow \pi / 2$ (see $\varpi=1.47 \sim \pi / 2$
The characteristic variety in the variables $(\tau, \varpi, r)$. From left to right: $\mathcal{V}_{\text{o}}^{\text{-}}(\mathbf{x}),\text{ }\mathcal{V}_{\text{x}}^{\text{-}}(\mathbf{x}),\text{ }\mathcal{V}_{\text{o}}^{\text{+}}(\mathbf{x})$ and $\mathcal{V}_{\text{x}}^{\text{+}}(\mathbf{x})$
Graph of $g_+ (\mathbf{x}, \varpi, \cdot)$
Graph of $g_- (\mathbf{x}, \varpi, \cdot)$
Ordinary resonance cone. ${{\mathcal{V}}_{o}}(\mathbf{x},\tau )\text{ for}\ \tau \in [0,\min ({{\mathbf{b}}_{e}}(\mathbf{x}),\kappa (\mathbf{x}))]$
Extraordinary resonance cone. $\mathcal{V} _{\! x} (\mathbf{x}, \tau)$ for $\tau \in [\max(\mathbf{b}_e(\mathbf{x}), \kappa(\mathbf{x})), \sqrt{\mathbf{b}_e(\mathbf{x})^2+\kappa(\mathbf{x})^2}]$
The extraordinary sphere nested into the ordinary sphere, for $\tau \gtrsim \tau_0^+ (\mathbf{x})$
Asymptotic merge of the ordinary and extraordinary spheres, for $\tau \gg \tau_0^+ (\mathbf{x})$
Parallel propagation ($\varpi=0$) with $\tau$ written as a function of $r$. Mixing of $\text{V}_{\! x} (\mathbf{x}, 0)$ and $\text{V}_{\! o} (\mathbf{x}, 0)$
Spectrogram of chorus emission
The characteristic variety as a graph of functions depending on $\tau$
Whistler-mode chorus: Spectrogram of the electromagnetic field [34]
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