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Dispersion relations in cold magnetized plasmas

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

    Mathematics Subject Classification: Primary: 35Q60, 35Q83, 82D10; Secondary: 35F21.


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  • Figure 1.  Spherical coordinates of $p\in {{\mathbb{R}}^{3}}$ after straigntening

    Figure 2.  Spherical coordinates of $ \xi \in \mathbb{R}^3 $ after straigntening

    Figure 3.  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)

    Figure 6.  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

    Figure 7.  Parallel propagation ($\varpi=0$). Mixing of $\text{V}_{\! x} (\mathbf{x}, 0) \, $ and $\text{V}_{\! o} (\mathbf{x}, 0) \, $

    Figure 8.  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 $

    Figure 9.  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})$

    Figure 5.  Graph of $ g_+ (\mathbf{x}, \varpi, \cdot) $

    Figure 4.  Graph of $ g_- (\mathbf{x}, \varpi, \cdot) $

    Figure 10.  Ordinary resonance cone. $ {{\mathcal{V}}_{o}}(\mathbf{x},\tau )\text{ for}\ \tau \in [0,\min ({{\mathbf{b}}_{e}}(\mathbf{x}),\kappa (\mathbf{x}))]$

    Figure 11.  Extraordinary resonance cone. $ \mathcal{V} _{\! x} (\mathbf{x}, \tau)$ for $\tau \in [\max(\mathbf{b}_e(\mathbf{x}), \kappa(\mathbf{x})), \sqrt{\mathbf{b}_e(\mathbf{x})^2+\kappa(\mathbf{x})^2}]$

    Figure 12.  The extraordinary sphere nested into the ordinary sphere, for $\tau \gtrsim \tau_0^+ (\mathbf{x}) $

    Figure 13.  Asymptotic merge of the ordinary and extraordinary spheres, for $\tau \gg \tau_0^+ (\mathbf{x}) $

    Figure 14.  Parallel propagation ($\varpi=0$) with $\tau$ written as a function of $r$. Mixing of $\text{V}_{\! x} (\mathbf{x}, 0) $ and $\text{V}_{\! o} (\mathbf{x}, 0)$

    Figure 15.  Spectrogram of chorus emission

    Figure 16.  The characteristic variety as a graph of functions depending on $ \tau $

    Figure 17.  Whistler-mode chorus: Spectrogram of the electromagnetic field [34]

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