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The Cold Plasma Model

  • Thomas H. Otway
Chapter
  • 683 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2043)

Abstract

Because a plasma is a fluid, its evolution must satisfy the equations of fluid dynamics. But because the fluid is composed of electrons and one or more species of ions, the charges on these particles act as sources of an electromagnetic field, which is governed by Maxwell’s equations. The presence of this intrinsic field leads to highly nonlinear behavior; and in fact, the dominance of long-range electromagnetic interactions over the short-range interatomic or intermolecular forces is often cited as the defining characteristic of the plasma state. In order to construct a mathematically rigorous model for the plasma which is also accessible to analysis, hypotheses must be imposed which control these nonlinearities. In Sect. 3.6 we assumed that the pressure on the plasma was zero and that magnetic forces dominated over other forces. Those hypotheses reduced the governing equations to the Beltrami equations (3.62), (3.65). In this section we impose a similar physical hypothesis: that the temperature of the plasma is zero.

Keywords

Weak Solution Dirichlet Problem Cold Plasma Resonance Curve Beltrami Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Allis, W.P: Waves in a plasma, Mass. Inst. Technol. Research Lab. Electronics Quart. Progr. Rep. 54(5) (1959)Google Scholar
  2. 2.
    Astrom, E.O.: Waves in an ionized gas. Arkiv. Fysik 2, 443 (1950)MathSciNetGoogle Scholar
  3. 3.
    Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)zbMATHGoogle Scholar
  4. 4.
    Cibrario, M.: Intorno ad una equazione lineare alle derivate parziali del secondo ordine di tipe misto iperbolico-ellittica. Ann. Sc. Norm. Sup. Pisa, Cl. Sci., Ser. 2 3(3, 4), 255–285 (1934)Google Scholar
  5. 5.
    Czechowski, A., Grzedzielski, S.: A cold plasma layer at the heliopause. Adv. Space Res. 16, 321–325 (1995)CrossRefGoogle Scholar
  6. 6.
    Didenko, V.P.: On the generalized solvability of the Tricomi problem. Ukrain. Math. J. 25, 10–18 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Diff. Equations 7, 77–116 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)zbMATHGoogle Scholar
  9. 9.
    Grossman, W., Weitzner, H.: A reformulation of lower-hybrid wave propagation and absorption. Phys. Fluids, 27, 1699–1703 (1984)CrossRefGoogle Scholar
  10. 10.
    Gu, C.: On partial differential equations of mixed type in n independent variables. Commun. Pure Appl. Math. 34, 333–345 (1981)CrossRefzbMATHGoogle Scholar
  11. 11.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory and Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)Google Scholar
  12. 12.
    Killian, T.C.,Pattard, T., Pohl, T., Rost, J.M.: Ultracold neutral plasmas. Phys. Rep. 449, 77–130 (2007)Google Scholar
  13. 13.
    W. S. Kurth, Waves in space plasmas, e–note (n.d.).http://www-pw.physics.uiowa.edu/plasma-wave/tutorial/waves.html. Cited 2 Aug 2011
  14. 14.
    Lazzaro, E., Maroli, C.: Lower hybrid resonance in an inhomogeneous cold and collisionless plasma slab. Nuovo Cim. 16B, 44–54 (1973)CrossRefGoogle Scholar
  15. 15.
    Lupo, D., Morawetz, C.S., Payne, K.R.: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math. 60, 1319–1348 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lupo, D., Morawetz, C.S., Payne, K.R.: Erratum: “On closed boundary value problems for equations of mixed elliptic-hyperbolic type,” [Commun. Pure Appl. Math. 60, 1319–1348 (2007)]. Commun. Pure Appl. Math. 61, 594 (2008)Google Scholar
  17. 17.
    Lupo, D., Payne, K.R.: A dual variational approach to a class of nonlocal semilinear Tricomi problems. Nonlinear Differential Equations Appl. 6, 247–266 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lupo, D., Payne, K.R.: Critical exponents for semilinear equations of mixed elliptic-hyperbolic and degenerate types. Commun. Pure Appl. Math. 56, 403–424 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    McDonald, K.T.: An electrostatic wave. arXiv:physics/0312025v1Google Scholar
  20. 20.
    Morawetz, C.S.: Mixed equations and transonic flow. Rend. Mat. 25, 1–28 (1966)Google Scholar
  21. 21.
    Morawetz, C.S., Stevens, D.C., Weitzner, H.: A numerical experiment on a second-order partial differential equation of mixed type. Commun. Pure Appl. Math. 44, 1091–1106 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Olver, F.W.J.: Asymptotics and Special Functions. A K Peters, Natick (1997)zbMATHGoogle Scholar
  23. 23.
    Otway, T.H.: A boundary-value problem for cold plasma dynamics. J. Appl. Math. 3, 17–33 (2003)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Otway, T.H.: Energy inequalities for a model of wave propagation in cold plasma. Publ. Mat. 52, 195–234 (2008)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Otway, T.H.: Mathematical aspects of the cold plasma model. In: Duan, J., Fu, X., Yang, Y. (eds.) Perspectives in Mathematical Sciences, pp. 181–210. World Scientific Press, Singapore (2010)CrossRefGoogle Scholar
  26. 26.
    Otway, T.H.: Unique solutions to boundary value problems in the cold plasma model. SIAM J. Math. Anal. 42, 3045–3053 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Piliya, A.D., Fedorov, V.I.: Singularities of the field of an electromagnetic wave in a cold anisotropic plasma with two-dimensional inhomogeneity. Sov. Phys. JETP 33, 210–215 (1971)Google Scholar
  28. 28.
    Sitenko, A.G., Stepanov, K.N.: On the oscillations of an electron plasma in a magnetic field. Z. Eksp. Teoret. Fiz. [in Russian] 31, 642 (1956) [Sov. Phys. JETP, 4, 512 (1957)]Google Scholar
  29. 29.
    Tonks, L., Langmuir, I.: Oscillations of ionized gases. Phys. Rev. 33, 195–210 (1929)Google Scholar
  30. 30.
    Weitzner, H.: “Wave propagation in a plasma based on the cold plasma model.” Courant Inst. Math. Sci. Magneto-Fluid Dynamics Div. Report MF–103, August, 1984Google Scholar
  31. 31.
    Weitzner, H.: Lower hybrid waves in the cold plasma model. Commun. Pure Appl. Math. 38, 919–932 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Yamamoto, Y.: Existence and uniqueness of a generalized solution for a system of equations of mixed type. Ph.D. thesis, Polytechnic University of New York (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas H. Otway
    • 1
  1. 1.Department of Mathematical SciencesYeshiva UniversityNew YorkUSA

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