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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 and Related Models, 2017, 10 (2) : 373-421. doi: 10.3934/krm.2017015
References:
[1]

E. V. Appleton, Wireless studies of the ionosphere, J. Inst. Electr. Eng, 71 (1932), 642-650. 

[2]

A. BackT. HattoriS. LabrunieJ.-R. Roche and P. Bertrand, Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition, ESAIM Math. Model. Numer. Anal., 49 (2015), 1239-1260.  doi: 10.1051/m2an/2015009.

[3]

J. BortnikR. M. Thorne and N. P. Meredith, The unexpected origin of plasmaspheric hiss from discrete chorus emissions, Nature, 452 (2008), 62-66.  doi: 10.1038/nature06741.

[4]

M. Bostan and C. Negulescu, Mathematical models for strongly magnetized plasmas with mass disparate particles, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 513-544.  doi: 10.3934/dcdsb.2011.15.513.

[5]

M. Braun, Mathematical remarks on the Van Allen radiation belt: A survey of old and new results, SIAM Rev., 23 (1981), 61-93.  doi: 10.1137/1023005.

[6]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics Volume 32 of Oxford Lecture Series in Mathematics and its Applications The Clarendon Press Oxford University Press, Oxford, 2006. An introduction to rotating fluids and the Navier-Stokes equations.

[7]

C. Cheverry, Can One Hear Whistler Waves?, Comm. Math. Phys, 338 (2015), 641-703.  doi: 10.1007/s00220-015-2389-6.

[8]

R. Ciurea-BorciaG. MatthieussentE. Le BelF. Simonet and J. Solomon, Oblique whistler waves generated in cold plasma by relativistic electron beams, Physics of plasmas, 7 (2000), 359-370.  doi: 10.1063/1.873804.

[9]

M. H. Denton, J. E. Borovsky, and T. E. Cayton, A density temperature description of the outer electron radiation belt during geomagnetic storms Journal of Geophysical Research: Space Physics 115 (2010). doi: 10.1029/2009JA014183.

[10]

B. Després L.-M. Imbert-Gérard and R. Weder, Hybrid resonance of Maxwell's equations in slab geometry, J. Math. Pures Appl.(9), 101 (2014), 623-659.  doi: 10.1016/j.matpur.2013.10.001.

[11]

L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[12]

R. Fitzpatrick, Plasma Physics: An Introduction Hardcover, 2014.

[13]

E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.

[14]

P. GhendrihM. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725.  doi: 10.3934/krm.2009.2.707.

[15]

C. Gillmor, Wilhelm Altar, Edward Appleton, and the Magneto-Ionic Theory, Proceedings of the American Philosophical Society, 126 (1982), 395-440. 

[16]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[17]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817.  doi: 10.1016/S0021-7824(99)00021-5.

[18]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714.  doi: 10.1142/S0218202503002647.

[19]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. Ecole Norm. Sup. (2015).

[20]

D. R. Hartree, The propagation of electromagnetic waves in a stratified medium, Mathematical Proceedings of the Cambridge Philosophical Society, 25 (1929), 97-120.  doi: 10.1017/S0305004100018600.

[21]

R. A. Helliwell, Whistlers and Related Ionospheric Phenomena Stanford University Press, 1965.

[22]

C. F. Kennel, Low frequency whistler mode, Journal of Plasma Physics, 71 (1966), 1-28.  doi: 10.1063/1.1761588.

[23]

C. F. Kennel and H. E. Petschek, Limit on stably trapped particles, Journal of Geophysical Research, 9 (1966), 2190-2202. 

[24]

E. Le Bel, Etude Physique et Numérique de la Saturation Des Ceintures de Van Allen Ph. D thesis, Paris 11 Orsay, 2001.

[25]

W. Li, J. Bortnik, R. M. Thorne, Y. Nishimura, V. Angelopoulos and L. Chen, Modulation of whistler mode chorus waves: 2. Role of density variations Journal of Geophysical Research 116 (2011). doi: 10.1029/2010JA016313.

[26]

Z. Lin and W. A. Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60 (2007), 724-787.  doi: 10.1002/cpa.20158.

[27]

K. Liu, S. P. Gary and D. Winske, Excitation of banded whistler waves in the magnetosphere Geophysical Research Letters 38 (2011). doi: 10.1029/2011GL048375.

[28]

J. D. MeniettiJ. S. PickettD. A. Gurnett and J. D. Scudder, Electrostatic electron cyclotron waves observed by the plasma wave instrument on board Polar, Journal of Geophysical Research, 106 (2001), 6043-6057.  doi: 10.1029/2000JA003016.

[29]

G. Métivier, The mathematics of nonlinear optics, Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 169-313.  doi: 10.1016/S1874-5717(08)00210-7.

[30]

D. C. Montgomery and D. A. Tidman, Plasma Kinetic Theory McGraw-Hill Book Company, 1964.

[31]

M. E. OakesR. B. MichieK. H. Tsui and J. E. Copeland, Cold plasma dispersion surfaces, Journal of Plasma Physics, 21 (1979), 205-224.  doi: 10.1017/S0022377800021784.

[32]

Y. Omura, M. Hikishima, Y. Katoh, D. Summers and S. Yagitani, Nonlinear mechanisms of lower-band and upper-band VLF chorus emissions in the magnetosphere Journal of Geophysical Research 114 (2009). doi: 10.1029/2009JA014206.

[33]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics Graduate Studies in Mathematics, American Mathematical Society, 2012. doi: 10.1090/gsm/133.

[34]

O. Santolik, D. A. Gurnett, J. S. Pickett, J. M. Parrot and N. Cornilleau-Wehrlin, Spatio-temporal structure of storm-time chorus Journal of Geophysical Research 108 (2003), SMP 7-1. doi: 10.1029/2002JA009791.

[35]

S. S. Sazhin and M. Hayakawa, Magnetospheric chorus emissions: A review, Planetary and Space Science, 40 (1992), 681-697.  doi: 10.1016/0032-0633(92)90009-D.

[36]

D. Sidhu and H. Unz, The magneto-ionic theory for drifting plasma: The whistler mode, Transactions of the Kansas Academy of Science, 70 (1967), 432-450.  doi: 10.2307/3627595.

[37]

T. H. Stix, Waves in Plasma American Institute of Physics, 1992.

[38]

V. Harid, Coherent Interactions between Whistler Mode Waves and Energetic Electrons in the Earth's Radiation Belts Ph. D thesis, Stanford, 2015.

[39]

R. Woollett, Electromagnetic Wave Propagation in a Cold, Collisionless Atomic Hydrogen Plasma Nasa technical note, Nasa TN D-2071,1964.

[40]

F. XiaoR. M. Thorne and D. Summers, Intsability of electromagnetic R-mode waves in a relativistic plasma, Physics of Plasmas, 5 (1998), 2489-2497.  doi: 10.1063/1.872932.

[41]

K. YamaguchiT. MatsumuroY. Omura and D. Nunn, Ray tracing of whistler-mode chorus elements, Ann. Geophys., 31 (2013), 665-673. 

show all references

References:
[1]

E. V. Appleton, Wireless studies of the ionosphere, J. Inst. Electr. Eng, 71 (1932), 642-650. 

[2]

A. BackT. HattoriS. LabrunieJ.-R. Roche and P. Bertrand, Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition, ESAIM Math. Model. Numer. Anal., 49 (2015), 1239-1260.  doi: 10.1051/m2an/2015009.

[3]

J. BortnikR. M. Thorne and N. P. Meredith, The unexpected origin of plasmaspheric hiss from discrete chorus emissions, Nature, 452 (2008), 62-66.  doi: 10.1038/nature06741.

[4]

M. Bostan and C. Negulescu, Mathematical models for strongly magnetized plasmas with mass disparate particles, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 513-544.  doi: 10.3934/dcdsb.2011.15.513.

[5]

M. Braun, Mathematical remarks on the Van Allen radiation belt: A survey of old and new results, SIAM Rev., 23 (1981), 61-93.  doi: 10.1137/1023005.

[6]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics Volume 32 of Oxford Lecture Series in Mathematics and its Applications The Clarendon Press Oxford University Press, Oxford, 2006. An introduction to rotating fluids and the Navier-Stokes equations.

[7]

C. Cheverry, Can One Hear Whistler Waves?, Comm. Math. Phys, 338 (2015), 641-703.  doi: 10.1007/s00220-015-2389-6.

[8]

R. Ciurea-BorciaG. MatthieussentE. Le BelF. Simonet and J. Solomon, Oblique whistler waves generated in cold plasma by relativistic electron beams, Physics of plasmas, 7 (2000), 359-370.  doi: 10.1063/1.873804.

[9]

M. H. Denton, J. E. Borovsky, and T. E. Cayton, A density temperature description of the outer electron radiation belt during geomagnetic storms Journal of Geophysical Research: Space Physics 115 (2010). doi: 10.1029/2009JA014183.

[10]

B. Després L.-M. Imbert-Gérard and R. Weder, Hybrid resonance of Maxwell's equations in slab geometry, J. Math. Pures Appl.(9), 101 (2014), 623-659.  doi: 10.1016/j.matpur.2013.10.001.

[11]

L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[12]

R. Fitzpatrick, Plasma Physics: An Introduction Hardcover, 2014.

[13]

E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.

[14]

P. GhendrihM. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725.  doi: 10.3934/krm.2009.2.707.

[15]

C. Gillmor, Wilhelm Altar, Edward Appleton, and the Magneto-Ionic Theory, Proceedings of the American Philosophical Society, 126 (1982), 395-440. 

[16]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[17]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817.  doi: 10.1016/S0021-7824(99)00021-5.

[18]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714.  doi: 10.1142/S0218202503002647.

[19]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. Ecole Norm. Sup. (2015).

[20]

D. R. Hartree, The propagation of electromagnetic waves in a stratified medium, Mathematical Proceedings of the Cambridge Philosophical Society, 25 (1929), 97-120.  doi: 10.1017/S0305004100018600.

[21]

R. A. Helliwell, Whistlers and Related Ionospheric Phenomena Stanford University Press, 1965.

[22]

C. F. Kennel, Low frequency whistler mode, Journal of Plasma Physics, 71 (1966), 1-28.  doi: 10.1063/1.1761588.

[23]

C. F. Kennel and H. E. Petschek, Limit on stably trapped particles, Journal of Geophysical Research, 9 (1966), 2190-2202. 

[24]

E. Le Bel, Etude Physique et Numérique de la Saturation Des Ceintures de Van Allen Ph. D thesis, Paris 11 Orsay, 2001.

[25]

W. Li, J. Bortnik, R. M. Thorne, Y. Nishimura, V. Angelopoulos and L. Chen, Modulation of whistler mode chorus waves: 2. Role of density variations Journal of Geophysical Research 116 (2011). doi: 10.1029/2010JA016313.

[26]

Z. Lin and W. A. Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60 (2007), 724-787.  doi: 10.1002/cpa.20158.

[27]

K. Liu, S. P. Gary and D. Winske, Excitation of banded whistler waves in the magnetosphere Geophysical Research Letters 38 (2011). doi: 10.1029/2011GL048375.

[28]

J. D. MeniettiJ. S. PickettD. A. Gurnett and J. D. Scudder, Electrostatic electron cyclotron waves observed by the plasma wave instrument on board Polar, Journal of Geophysical Research, 106 (2001), 6043-6057.  doi: 10.1029/2000JA003016.

[29]

G. Métivier, The mathematics of nonlinear optics, Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 169-313.  doi: 10.1016/S1874-5717(08)00210-7.

[30]

D. C. Montgomery and D. A. Tidman, Plasma Kinetic Theory McGraw-Hill Book Company, 1964.

[31]

M. E. OakesR. B. MichieK. H. Tsui and J. E. Copeland, Cold plasma dispersion surfaces, Journal of Plasma Physics, 21 (1979), 205-224.  doi: 10.1017/S0022377800021784.

[32]

Y. Omura, M. Hikishima, Y. Katoh, D. Summers and S. Yagitani, Nonlinear mechanisms of lower-band and upper-band VLF chorus emissions in the magnetosphere Journal of Geophysical Research 114 (2009). doi: 10.1029/2009JA014206.

[33]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics Graduate Studies in Mathematics, American Mathematical Society, 2012. doi: 10.1090/gsm/133.

[34]

O. Santolik, D. A. Gurnett, J. S. Pickett, J. M. Parrot and N. Cornilleau-Wehrlin, Spatio-temporal structure of storm-time chorus Journal of Geophysical Research 108 (2003), SMP 7-1. doi: 10.1029/2002JA009791.

[35]

S. S. Sazhin and M. Hayakawa, Magnetospheric chorus emissions: A review, Planetary and Space Science, 40 (1992), 681-697.  doi: 10.1016/0032-0633(92)90009-D.

[36]

D. Sidhu and H. Unz, The magneto-ionic theory for drifting plasma: The whistler mode, Transactions of the Kansas Academy of Science, 70 (1967), 432-450.  doi: 10.2307/3627595.

[37]

T. H. Stix, Waves in Plasma American Institute of Physics, 1992.

[38]

V. Harid, Coherent Interactions between Whistler Mode Waves and Energetic Electrons in the Earth's Radiation Belts Ph. D thesis, Stanford, 2015.

[39]

R. Woollett, Electromagnetic Wave Propagation in a Cold, Collisionless Atomic Hydrogen Plasma Nasa technical note, Nasa TN D-2071,1964.

[40]

F. XiaoR. M. Thorne and D. Summers, Intsability of electromagnetic R-mode waves in a relativistic plasma, Physics of Plasmas, 5 (1998), 2489-2497.  doi: 10.1063/1.872932.

[41]

K. YamaguchiT. MatsumuroY. Omura and D. Nunn, Ray tracing of whistler-mode chorus elements, Ann. Geophys., 31 (2013), 665-673. 

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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