On Instability of Dynamic Equilibrium States of Vlasov-Poisson Plasma

Yuriy G. Gubarev^{1, 2,}

¹Laboratory for Fluid and Gas Vortical Motions, Lavrentyev Institute for Hydrodynamics, Novosibirsk, Russian Federation

²Department for Differential Equations, Novosibirsk State University, Novosibirsk, Russian Federation

Cite this paper:
Yuriy G. Gubarev. On Instability of Dynamic Equilibrium States of Vlasov-Poisson Plasma. Applied Mathematics and Physics. 2016; 4(1):1-8. doi: 10.12691/amp-4-1-1

Abstract

The problem on linear stability of one–dimensional (1D) states of dynamic equilibrium boundless electrically neutral collisionless plasma in electrostatic approximation (the Vlasov–Poisson plasma) is studied. It is proved by the direct Lyapunov method that these equilibrium states are absolutely unstable with respect to small 1D perturbations in the case when the Vlasov–Poisson plasma contains electrons with stationary distribution function, which is constant over the physical space and variable in velocities, and one variety of ions whose distribution function is constant over the phase space as a whole. In addition, sufficient conditions for linear practical instability are obtained, the a priori exponential lower estimate is constructed, and initial data for perturbations, growing in time, are described. Finally, the illustrative analytical example of considered 1D states of dynamic equilibrium and superimposed small 1D perturbations, which grow on time in accordance with the obtained estimate, is constructed.

Keywords:
the Vlasov-Poisson Plasma Dynamic Equilibrium States the Direct Lyapunov Method Instability

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References:

[1]	Krall, N.A. and Trivelpiece, A.W., Principles of Plasma Physics, McGraw–Hill Book Company, Inc., New York, 1973.
[2]	Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A., “Nonlinear stability of fluid and plasma equilibria,” Phys. Reports, 123 (1 & 2), 1-116, 1985.
[3]	Zakharov, V.E. and Kuznetsov, E.A., “Hamiltonian formalism for nonlinear waves,” Phys. Usp., 40, 1087-1116, 1997.
[4]	Pankavich, S. and Allen, R., “Instability conditions for some periodic BGK waves in the Vlasov–Poisson system,” Eur. Phys. J. D, 68 (12), 363-369, 2014.
[5]	Han–Kwan, D. and Hauray, M. “Stability issues in the quasineutral limit of the one–dimensional Vlasov–Poisson equation,” Comm. Math. Phys., 334 (2), 1101-1152, 2015.
[6]	Esenturk, E. and Hwang, H.–J., “Linear stability of the Vlasov–Poisson system with boundary conditions”, Nonlinear Anal.–Theor., 139, 75-105, 2016.
[7]	Bernstein, I.B., “Waves in a plasma in a magnetic field,” Phys. Rev., 109, 10-21, 1958.
[8]	Gardner, C.S., “Bound on the energy available from a plasma,” Phys. Fluids, 6, 839-840, 1963.
[9]	Rosenbluth, M.N., “Topics in microinstabilities,” in: Advanced Plasma Theory, Academic Press, New York, 137-158.
[10]	Gubarev, Yu.G. and Gubkin, A.A., “On the stability for a class of 1D states of dynamical equilibrium of the Vlasov–Poisson plasma,” in: Differential Equations. Functional Spaces. Theory of Approximations, Institute for Mathematics SB RAS, Novosibirsk, 122-122 (in Russian).
[11]	Gubarev, Yu.G. and Gubkin, A.A., “On the stability for a class of 1–D states of dynamical equilibrium of the Vlasov–Poisson plasma,” in: Advanced Mathematics, Computations, and Applications, Academizdat, Novosibirsk, 79-80.
[12]	Zakharov, V.E. and Kuznetsov, E.A., Hamiltonian Formalism for Systems of Hydrodynamic Type, Preprint No. 186, Institute for Automation and Electrometry SB AS USSR, Novosibirsk, 1982 (in Russian).
[13]	Zakharov, V.E., “Benney's equations and quasiclassical approximation in the inverse problem method,” Funktsional. Anal. i Prilozgen., 14, 1980, 15-24 (in Russian).
[14]	Gubarev, Yu.G., “On an analogy between the Benney's equations and the Vlasov–Poisson's equations,” Dinamika Sploshn. Sredy, 110, 1995, 78-90 (in Russian).
[15]	Yakubovich, V.A. and Starzginskiyi, V.M., Linear Differential Equations with Periodical Coefficients and Its Applications, Nauka, Moscow, 1972 (in Russian).
[16]	Lyapunov, A.M., The general problem of the stability of motion, Taylor & Francis, London, 1992.
[17]	Chetaev, N.G., Stability of motion, Nauka, Moscow, 1990 (in Russian).
[18]	Gubarev, Yu.G., The direct Lyapunov method. The stability of quiescent states and steady–state flows of fluids and gases, Palmarium Academic Publishing, Saarbrücken, 2012 (in Russian).
[19]	Karacharov, K.A. and Pilyutik, A.G., Introduction in technical theory of motion stability, Fizmatgiz, Moscow, 1962 (in Russian).
[20]	La Salle, J. and Lefschetz, S., Stability by Liapunov's direct method with applications, Academic Press, New York, 1961.
[21]	Chandrasekhar, S., Ellipsoidal figures of equilibrium, Yale University Press, New Haven, 1969.
[22]	Gubarev, Yu.G., “Linear stability criterion for steady screw magnetohydrodynamic flows of ideal fluid,” T & A, 16 (3), 407-418, 2009.
[23]	Gavrilieva, A.A. and Gubarev, Yu.G., “Stability of steady–state plane–parallel shear flows of an ideal stratified fluid in the gravity field,” Vestnik of the NEFU named after M.K. Ammosov, 9 (3), 15-21, 2012 (in Russian).
[24]	Gubarev, Yu.G., “The problem of adequate mathematical modeling for liquids fluidity,” AJFD, 3 (3), 67-74, 2013.
[25]	Gavrilieva, A.A., Gubarev, Yu.G., Lebedev, M.P., “Rapid approach to resolving the adequacy problem of mathematical modeling of physical phenomena by the example of solving one problem of hydrodynamic instability,” IJTMP, 3 (4), 123-129, 2013.
[26]	Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, Clarendon Press, Oxford, 1961.
[27]	Godunov, S.K., Equations of mathematical physics, Nauka, Moscow, 1979 (in Russian).