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December  2016, 9(4): 621-656. doi: 10.3934/krm.2016010

A minimization formulation of a bi-kinetic sheath

Mehdi Badsi 1, , Martin Campos Pinto 1, and  Bruno Després 1,

1. 

UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05, France, France, France

Received  October 2014 Revised  July 2015 Published  September 2016

The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampère boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.
Keywords: Vlasov-Poisson system., plasma sheath, Bohm criterion, ambipolarity, Kinetic equation.
Mathematics Subject Classification: 35B35, 35L60, 82D10, 35B3.
Citation: Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010
References:
[1]

J. Apell, The superposition operator in function spaces. A survey, Expositiones Mathematicae, 1988. Google Scholar

[2]

A. Ambroso, X. Fleury, B. Lucquin-Desreux and P. A. Raviart, Some remarks on a stationary Vlasov-Poisson system with source term arising in ion beam neutralization, Transport Theory Statist. Phys., 30 (2001), 587-616. doi: 10.1081/TT-100107418.  Google Scholar

[3]

S. Baalrud and C. Hegna, Kinetic theory of the presheath and the Bohm criterion, Plasma Sources Science and Technology, 20 (2011). Google Scholar

[4]

H. Beresticky and T. Lachant-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Math. Nachr., 266 (2004), 3-19. doi: 10.1002/mana.200310139.  Google Scholar

[5]

D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, McGraw Hill, New York, 1949. Google Scholar

[6]

R. Chalise and R. Khanal, A Kinetic Trajectory Simulation for Magnetized Plasma Sheath, Institute of Physics Publishing, 2012. Google Scholar

[7]

F. Chen, Introduction to Plasma Physics, Plenum Press, 1974. Google Scholar

[8]

R. J. Diperna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

M. Feldman, S. Y. HA and M. Slemrod, A Geometric level-set formulation of a plasma sheath interface, Arch. Rat. Mech. Anal., 178 (2005), 81-123. doi: 10.1007/s00205-005-0368-3.  Google Scholar

[11]

Y. Güçlü, The plasma sheath for 1D-1V Vlasov-Poisson solvers, Invited talk at the Numkin 2013 Conference, Garching, Germany, http://www.ipp.mpg.de/1525371/Gueclue.pdf. Google Scholar

[12]

Y. Guo, C.-H. Shu and T. Zie, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635 (electronic). doi: 10.1137/S0036141003421133.  Google Scholar

[13]

O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques, Springer-Verlage, 1993.  Google Scholar

[14]

H. Kohno, J. R. Myra and D. A. D'Ippolito, Radio-frequency sheath-plasma interactions with magnetic field tangency points along the sheath surface, Physics of Plasmas, 20 (2013), 082514. Google Scholar

[15]

J. G. Laframboise, Theory of spherical and cylindrical Langmuir probes in a collision less, Maxwellian plasma at rest, Institute for Aerospace Studies, University of Toronto, Report No. 100, 1966. Google Scholar

[16]

P.-H. Maire, Établissement et Comparaison de Modèles Fluides Pour un Plasma Faiblement Ionisé Quasi-neutre. Détermination des Conditions aux Limites à la Paroi, Thèse de Doctorat (in French) de l'Université Paris 6, 1996. Google Scholar

[17]

G. Manfredi and S. Devaux, Magnetized Plasma-Wall Transition. Consequences for Wall Sputtering and Erosion, Institute of Physics Publishing, 2008. Google Scholar

[18]

R. E. Marshak, The variational method for asymptotic neutron densities, Physical Review, 71 (1947), 688-693.  Google Scholar

[19]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, National Institute of Standards and Technology, Cambridge University Press, 2010.  Google Scholar

[20]

P.-A. Raviart and C. Greengard, A Boundary-Value problem for the stationary Vlasov-Poisson equations: The Plane Diode, Comm. Pure Appl. Math., 43 (1990), 473-507. doi: 10.1002/cpa.3160430404.  Google Scholar

[21]

K.-U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 493. doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[22]

T. E. Sheridan, Solution of the Plasma-Sheath Equation with a Cool Maxwellian Ion Source, AIP Publishing, 2001. Google Scholar

[23]

G. Stampachia and D. Kinderlehrer, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.  Google Scholar

[24]

P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. Google Scholar

[25]

F. Valsaque and G. Manfredi, Numerical study of plasma wall transition in an oblique magnetic field, Journal of Nuclear Materials, 290-293 (2001), 763-767. doi: 10.1016/S0022-3115(00)00454-2.  Google Scholar

show all references

References:
[1]

J. Apell, The superposition operator in function spaces. A survey, Expositiones Mathematicae, 1988. Google Scholar

[2]

A. Ambroso, X. Fleury, B. Lucquin-Desreux and P. A. Raviart, Some remarks on a stationary Vlasov-Poisson system with source term arising in ion beam neutralization, Transport Theory Statist. Phys., 30 (2001), 587-616. doi: 10.1081/TT-100107418.  Google Scholar

[3]

S. Baalrud and C. Hegna, Kinetic theory of the presheath and the Bohm criterion, Plasma Sources Science and Technology, 20 (2011). Google Scholar

[4]

H. Beresticky and T. Lachant-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Math. Nachr., 266 (2004), 3-19. doi: 10.1002/mana.200310139.  Google Scholar

[5]

D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, McGraw Hill, New York, 1949. Google Scholar

[6]

R. Chalise and R. Khanal, A Kinetic Trajectory Simulation for Magnetized Plasma Sheath, Institute of Physics Publishing, 2012. Google Scholar

[7]

F. Chen, Introduction to Plasma Physics, Plenum Press, 1974. Google Scholar

[8]

R. J. Diperna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

M. Feldman, S. Y. HA and M. Slemrod, A Geometric level-set formulation of a plasma sheath interface, Arch. Rat. Mech. Anal., 178 (2005), 81-123. doi: 10.1007/s00205-005-0368-3.  Google Scholar

[11]

Y. Güçlü, The plasma sheath for 1D-1V Vlasov-Poisson solvers, Invited talk at the Numkin 2013 Conference, Garching, Germany, http://www.ipp.mpg.de/1525371/Gueclue.pdf. Google Scholar

[12]

Y. Guo, C.-H. Shu and T. Zie, The dynamics of a plane diode, SIAM J. Math. Anal., 35 (2004), 1617-1635 (electronic). doi: 10.1137/S0036141003421133.  Google Scholar

[13]

O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques, Springer-Verlage, 1993.  Google Scholar

[14]

H. Kohno, J. R. Myra and D. A. D'Ippolito, Radio-frequency sheath-plasma interactions with magnetic field tangency points along the sheath surface, Physics of Plasmas, 20 (2013), 082514. Google Scholar

[15]

J. G. Laframboise, Theory of spherical and cylindrical Langmuir probes in a collision less, Maxwellian plasma at rest, Institute for Aerospace Studies, University of Toronto, Report No. 100, 1966. Google Scholar

[16]

P.-H. Maire, Établissement et Comparaison de Modèles Fluides Pour un Plasma Faiblement Ionisé Quasi-neutre. Détermination des Conditions aux Limites à la Paroi, Thèse de Doctorat (in French) de l'Université Paris 6, 1996. Google Scholar

[17]

G. Manfredi and S. Devaux, Magnetized Plasma-Wall Transition. Consequences for Wall Sputtering and Erosion, Institute of Physics Publishing, 2008. Google Scholar

[18]

R. E. Marshak, The variational method for asymptotic neutron densities, Physical Review, 71 (1947), 688-693.  Google Scholar

[19]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, National Institute of Standards and Technology, Cambridge University Press, 2010.  Google Scholar

[20]

P.-A. Raviart and C. Greengard, A Boundary-Value problem for the stationary Vlasov-Poisson equations: The Plane Diode, Comm. Pure Appl. Math., 43 (1990), 473-507. doi: 10.1002/cpa.3160430404.  Google Scholar

[21]

K.-U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 493. doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[22]

T. E. Sheridan, Solution of the Plasma-Sheath Equation with a Cool Maxwellian Ion Source, AIP Publishing, 2001. Google Scholar

[23]

G. Stampachia and D. Kinderlehrer, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.  Google Scholar

[24]

P. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, 2000. Google Scholar

[25]

F. Valsaque and G. Manfredi, Numerical study of plasma wall transition in an oblique magnetic field, Journal of Nuclear Materials, 290-293 (2001), 763-767. doi: 10.1016/S0022-3115(00)00454-2.  Google Scholar

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