Models and simulations for the laser-plasma interaction and the three-wave coupling problem

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April 2012, 5(2): 329-343. doi: 10.3934/dcdss.2012.5.329

Models and simulations for the laser-plasma interaction and the three-wave coupling problem

Remi Sentis ^1,

1.	CEA, DAM, DIF. Bruyeres, 91297 Arpajon, France

Received September 2009 Revised March 2010 Published September 2011

Abstract
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We are concerned here with the modelling of the laser propagation and its interaction with a plasma. In a first part, we recall first some features related to the paraxial approximation of the solution of the wave equation and the coupling model between the plasma hydrodynamics and the laser propagation. In a second part, we consider the coupling with the ion acoustic waves which has to be accounted to model the Brillouin instability. It leads to the three-wave coupling problem which is a crucial in plasma physics. We give some mathematical properties of this system specially in the case when the speed of light is assumed to be infinite. We also propose a numerical method based on a implicit time discretization. It is illustrated on test cases.

Keywords: Numerical analysis., Laser-Plasma Interaction, Three-wave coupling.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Remi Sentis. Models and simulations for the laser-plasma interaction and the three-wave coupling problem. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 329-343. doi: 10.3934/dcdss.2012.5.329

References:

[1]	Ph. Ballereau, M. Casanova, F. Duboc, D. Dureau, H. Jourdren, P. Loiseau, J. Metral, O. Morice and R. Sentis, Simulation of the paraxial laser propagation coupled with hydrodynamics in 3D geometry, J. Sci. Comp., 33 (2007), 1-24. doi: 10.1007/s10915-007-9135-y.
[2]	P. G. Baines, The stability of planetary waves on a sphere, J. Fluid Mech., 73 (1976), 193-213. doi: 10.1017/S0022112076001341.
[3]	T. J. M. Boyd and J. G. Turner, Lagrangian studies of plasma wave interaction, 5 (1972), 881-896.
[4]	M. Casanova, et al., Self-generated loss of coherency in Brillouin scattering and reduction of reflectivity, Phys. Review Letters, 54 (1985), 2230-2233. doi: 10.1103/PhysRevLett.54.2230.
[5]	M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interaction, Differential Integral Equations, 17 (2004), 297-330.
[6]	S. Desroziers, F. Nataf and R. Sentis, Simulation of laser propagation in a plasma with a frequency wave equation, J. Comp. Physics, 227 (2008), 2610-2625. doi: 10.1016/j.jcp.2007.11.008.
[7]	M. Doumic, F. Duboc, F. Golse and R. Sentis, Simulation of laser beam propagation with a paraxial model in a tilted frame, J. Comp. Physics, 228 (2009), 861-880. doi: 10.1016/j.jcp.2008.10.009.
[8]	L. Divol and R. L. Berger, et al., Three-dimensional modeling of laser-plasma interaction: Benchmarking our predictive modeling tools versus experiments, Phys. of Plasmas, 15 (2008), 056313.
[9]	B. B. Kadomtsev, "Plasma Turbulence," translated from the Russian, Academic Press, 1965.
[10]	J. B. Keller and R. M. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations, in "Surveys in Applied Mathematics, Vol.1" (eds. J. B. Keller, W. McLaughlin and G. C. Papanicolaou), Plenum, New York, (1995), 1-82.
[11]	W. L. Kruer, "The Physics of Laser-Plasma Interaction," Addison-Wesley, New York, 1988.
[12]	P. Loiseau, et al., Laser-beam smoothing induced by stimulated Brillouin scattering in an inhomogeneous plasma, Physical Review Letters, 97 (2006), 205001. doi: 10.1103/PhysRevLett.97.205001.
[13]	G. Metivier and R. Sentis, On the Boyd-Kadomtsev System and its asymptotic limit, Comm. Math. Physics, to be published (2011).
[14]	Ph. Mounaix, D. Pesme and M. Casanova, Nonlinear reflectivity of an inhomogeneous plasma, Phys. Review E, 55 (1997), 4653. doi: 10.1103/PhysRevE.55.4653.
[15]	S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ and V. E. Zakharov, "Theory of Solitons. The Inverse Scattering Method," Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984.
[16]	D. Pesme, Interaction collisionnelle et collective, in "La Fusion par Confinement Inertiel I. Interaction Laser-Matière" (ed. R. Dautray-Watteau), Eyrolles, Paris, 1995.
[17]	R. Sentis, On the Boyd-Kadomtsev system for the three-wave coupling problem, C. R. Math. Ac. Sciences Paris, 347 (2009), 933-938.
[18]	R. Sentis, Mathematical models for laser-plasma interaction, M2AN Math. Modelling Num. Analysis, 39 (2005), 275-318.

show all references

References:

[1]	Ph. Ballereau, M. Casanova, F. Duboc, D. Dureau, H. Jourdren, P. Loiseau, J. Metral, O. Morice and R. Sentis, Simulation of the paraxial laser propagation coupled with hydrodynamics in 3D geometry, J. Sci. Comp., 33 (2007), 1-24. doi: 10.1007/s10915-007-9135-y.
[2]	P. G. Baines, The stability of planetary waves on a sphere, J. Fluid Mech., 73 (1976), 193-213. doi: 10.1017/S0022112076001341.
[3]	T. J. M. Boyd and J. G. Turner, Lagrangian studies of plasma wave interaction, 5 (1972), 881-896.
[4]	M. Casanova, et al., Self-generated loss of coherency in Brillouin scattering and reduction of reflectivity, Phys. Review Letters, 54 (1985), 2230-2233. doi: 10.1103/PhysRevLett.54.2230.
[5]	M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interaction, Differential Integral Equations, 17 (2004), 297-330.
[6]	S. Desroziers, F. Nataf and R. Sentis, Simulation of laser propagation in a plasma with a frequency wave equation, J. Comp. Physics, 227 (2008), 2610-2625. doi: 10.1016/j.jcp.2007.11.008.
[7]	M. Doumic, F. Duboc, F. Golse and R. Sentis, Simulation of laser beam propagation with a paraxial model in a tilted frame, J. Comp. Physics, 228 (2009), 861-880. doi: 10.1016/j.jcp.2008.10.009.
[8]	L. Divol and R. L. Berger, et al., Three-dimensional modeling of laser-plasma interaction: Benchmarking our predictive modeling tools versus experiments, Phys. of Plasmas, 15 (2008), 056313.
[9]	B. B. Kadomtsev, "Plasma Turbulence," translated from the Russian, Academic Press, 1965.
[10]	J. B. Keller and R. M. Lewis, Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations, in "Surveys in Applied Mathematics, Vol.1" (eds. J. B. Keller, W. McLaughlin and G. C. Papanicolaou), Plenum, New York, (1995), 1-82.
[11]	W. L. Kruer, "The Physics of Laser-Plasma Interaction," Addison-Wesley, New York, 1988.
[12]	P. Loiseau, et al., Laser-beam smoothing induced by stimulated Brillouin scattering in an inhomogeneous plasma, Physical Review Letters, 97 (2006), 205001. doi: 10.1103/PhysRevLett.97.205001.
[13]	G. Metivier and R. Sentis, On the Boyd-Kadomtsev System and its asymptotic limit, Comm. Math. Physics, to be published (2011).
[14]	Ph. Mounaix, D. Pesme and M. Casanova, Nonlinear reflectivity of an inhomogeneous plasma, Phys. Review E, 55 (1997), 4653. doi: 10.1103/PhysRevE.55.4653.
[15]	S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ and V. E. Zakharov, "Theory of Solitons. The Inverse Scattering Method," Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984.
[16]	D. Pesme, Interaction collisionnelle et collective, in "La Fusion par Confinement Inertiel I. Interaction Laser-Matière" (ed. R. Dautray-Watteau), Eyrolles, Paris, 1995.
[17]	R. Sentis, On the Boyd-Kadomtsev system for the three-wave coupling problem, C. R. Math. Ac. Sciences Paris, 347 (2009), 933-938.
[18]	R. Sentis, Mathematical models for laser-plasma interaction, M2AN Math. Modelling Num. Analysis, 39 (2005), 275-318.

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