Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3301810.22363/2658-4670-2022-30-4-374-378Research ArticleApproximation of radial structure of unstable ion-sound modes in rotating magnetized plasma column by eikonal equationMarusovNikita A.<p>Candidate of Sciences in Physics and Mathematics, Senior Researcher of Department of Plasma Theory of Kurchatov Institute; Senior Lecturer of Institute of Physical Research and Technology of Peoples’ Friendship University of Russia</p>marusov-na@rudn.ruhttps://orcid.org/0000-0002-0763-1505Kurchatov InstitutePeoples’ Friendship University of Russia (RUDN University)2612202230437437826122022Copyright © 2022, Marusov N.A.2022<p style="text-align: justify;">The problem of the correct asymptotic construction of the radial structure of linearly unstable ion-sound electrostatic eigenmodes is studied. The eigenvalue problem with boundary conditions of the first and second kind (electrodynamic and hydrodynamic types) for the oscillations that propagate in a uniform cylindrical column of magnetized plasma along an axial homogeneous magnetic field is formulated. A method for constructing a discrete spectrum of small-scale unstable oscillations of the system based on the basic principles of geometric optics is proposed. The main idea of the method is an explicit idea of the type of boundary conditions - the conductivity and absorbing properties of the wall bounding the plasma cylinder. A dispersion relation for unstable small-scale modes destabilized due to the effects of differential rotation is derived from the Eikonal equation. For the correct construction instability growth rates spectra an universal recipe for the selection of radial wave numbers of small-scale eigenmodes in accordance with any of the types of boundary conditions is proposed.</p>plasma wavesplasma instabilitiesgeometrical opticsnormal modesволны в плазменеустойчивости плазмыгеометрическая оптикасобственные колебания[A. B. Mikhailovskii, Theory of plasma instabilities: Volume 1: Instabilities of a homogeneous plasma. New York: Consultants Bureau, 1974.][A. B. Mikhailovskii, Theory of plasma instabilities: Volume 2: Instabilities of a homogeneous plasma. New York: Consultants Bureau, 1974.][E. P. Vedenov, R. Z. Velikhov, and R. Z. Zagdeev, “Stability of plasma,” Sov. Phys. Usp., vol. 4, pp. 332-369, 1961.][A. V. Timofeev, “Oscillations of inhomogeneous flows of plasma and liquids,” Sov. Phys. Usp., vol. 13, pp. 632-646, 1971.][A. B. Mikhailovskii et al., “High-frequency extensions of magnetorotational instability in astrophysical plasmas,” Plasma Physics Reports, vol. 34, no. 8, pp. 678-687, 2008.][D. A. Shalybkov, “Hydrodynamic and hydromagnetic stability of the couette flow,” Physics-Uspekhi, vol. 52, no. 9, pp. 915-935, 2009.][N. A. Marusov et al., “Stability of electrostatic axisymmetric perturbations in rotating Hall plasmas,” in Proc. 47th EPS Conference on Plasma Physics, EPS 2021, Sitges, Spain, 2021, p. 16190.][L. D. Landau and E. M. Lifshitz, Fluid Mechanics. Course of Theoretical Physics. Vol. 6. New York: Pergamon Press, 1987.][S. Borowitz, Fundamentals of quantum mechanics: Particles, waves, and wave mechanics. New York: Pergamon Press, 1967.][A. V. Timofeev, “Geometrical optics and the diffraction phenomenon,” Phys. Usp., vol. 48, pp. 609-613, 2005.]