The calculation of the fine structure in two-dimensional periodic flows in a compressible atmosphere
- Authors: Ochirov A.A.1, Trifonova U.O.2, Chashechkin Y.D.1
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Affiliations:
- Ishlinsky Institute for problems in mechanics RAS
- Demidov Yaroslavl State University
- Issue: Vol 89, No 3 (2025)
- Pages: 494-511
- Section: Articles
- URL: https://freezetech.ru/0032-8235/article/view/688826
- DOI: https://doi.org/10.31857/S0032823525030101
- EDN: https://elibrary.ru/JLPGVB
- ID: 688826
Cite item
Abstract
Based on a linearized system of fundamental equations for the mechanics of compressible and heterogeneous fluids and gases, including an equation of state for the medium, methods from the theory of singular perturbation theory are used to compute complete dispersion relations for periodic flows. Regular components of the solution describe waves and, in limiting transitions, are reduced to known relationships from linear wave theory. Singular solutions inherent to all wave types – acoustic and gravitational – characterize ligaments that form the fine-scale structure of a heterogeneous medium. These singularities are lost as one moves towards idealized environments.
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About the authors
A. A. Ochirov
Ishlinsky Institute for problems in mechanics RAS
Author for correspondence.
Email: otchirov@mail.ru
Russian Federation, Moscow
U. O. Trifonova
Demidov Yaroslavl State University
Email: otchirov@mail.ru
Russian Federation, Yaroslavl
Yu. D. Chashechkin
Ishlinsky Institute for problems in mechanics RAS
Email: yulidch@gmail.com
Russian Federation, Moscow
References
- Darrigol O. Worlds of Flow. A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford: Univ. Press, 2005. 356 p.
- Krasilnikov V.A. Introduction to Acoustics: Textbook. Moscow: MSU, 1992. 152 p. (in Russian)
- Brekhovskikh L.M., Godin O.A. Acoustics of Inhomogeneous Media. Moscow: Nauka, 2007. (in Russian)
- Sretensky L.N. Theory of Wave Motions of a Liquid. 1936. 304 p. (in Russian)
- LeBlond P.H.; Mysak L.A. Waves in the Ocean. Elsevier Oceanogr. Ser.; Elsevier Science, 1981. 602 p.
- Slunyaev A.V., Pelinovsky D.E., Pelinovsky E.N. Rogue waves in the sea: observations, physics, and mathematics // Phys. Usp., 2023, vol. 66, no. 2, pp. 148–172.
- Rudenko O.V. Nonlinear waves: some biomedical applications // Phys. Usp., 2007, vol. 50, no. 4, рр. 359.
- Whitham G.B. Linear and Nonlinear Waves. Wiley, 2011.
- Pinault J.-L. A review of the role of the oceanic Rossby waves in climate variability // J. Mar. Sci. Eng., 2022, vol. 10, no. 4, pp. 493.
- Tolstoy I., Clay C.S. Ocean Acoustics: Theory and Experiment in Underwater Sound. McGraw-Hill, UK, 1966. 293 p.
- Fedorov K.N. The Thermohaline Finestructure of the Ocean. Pergamon, 1978. 179 p.
- Chunchuzov I.P., Kulichkov S.P. Propagation of Infrasound Waves in an Anisotropic Fluctuating Atmosphere. Moscow: Geos, 2020. 260 p. (in Russian)
- Landau L.D., Lifshitz E.M. Fluid Mechanics. V. 6. Course of Theoretical Physics. Oxford (UK): Pergamon, 1987. 560 p.
- Müller P. The Equations of Oceanic Motions. Cambridge: Univ. Press, 2006.
- Vallis G.K. Atmospheric and Oceanic Fluid Dynamics. Cambridge: Univ. Press, 2017.
- Nayfeh A.H. Introduction to Perturbation Technique. N.Y.: Wiley, 1993. 536 p.
- Chashechkin Yu.D. Conventional partial and new complete solutions of the fundamental equations of fluid mechanics in the problem of periodic internal waves with accompanying ligaments generation // Mathematics, 2021, vol. 9, no. 6, art. no. 586.
- Сhashechkin Yu.D., Ochirov A.A. Periodic waves and ligaments on the surface of a viscous exponentially stratified fluid in a uniform gravity field // Axioms, 2022, vol. 11, no. 8, art. no. 402.
- Chashechkin Y.D. Foundations of engineering mathematics applied for fluid flows // Axioms, 2021, vol. 10, no. 4, art. no. 286.
- US Standard Atmosphere 1976. NOAA-S/T-76-1562. NASA-TM-X-74335. Accession No. 77N16482. https://ntrs.nasa.gov/citations/19770009539
- GOST 4401-81. Interstate standard: The atmosphere is standard. Parameters. Date of introduction: 07/01. 1982. IPK. Standards Publ. House, 2004. Updated 08/20/2023. (in Russian) https://standartgost.ru/g/٪D0٪93٪D0٪9E٪D0٪A1٪D0٪A2_4401-81
- Rayleigh L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density // Proc. of the London Math. Soc., 1882, vol. 1, no. 1, pp. 170–177.
- Smirnov S.A., Chashechkin Yu.D., Il'inykh Yu.S. High-accuracy method for measuring profiles of buoyancy periods // Measur. Techn., 1998, vol. 41, no. 6, pp. 514–519.
- Teoh S.G., Ivey G.N., Imberger J. Laboratory study of the interaction between two internal wave rays // J. of Fluid Mech., 1997, vol. 336, pp. 91–122.
- Chashechkin Yu.D. Singularly perturbed components of flows – linear precursors of shock waves // Math. Model. Nat. Phenom., 2018, vol. 13, no. 2, pp. 1–29.
- Chashechkin Yu.D., Kistovich Yu.V. Problem of generation of monochromatic internal waves: Exacts solution and a model of power sources // Dokl. RAN, 1997, vol. 355, no. 1, pp. 54–57.
- Kistovich Yu.V., Chashechkin Yu.D. An exact solution of a linearized problem of the radiation of monochromatic internal waves in a viscous fluid // JAMM, 1999, vol. 63, iss. 4, pp. 587–594.
- Kistovich A.V., Chashechkin Yu.D. Regular and singular components of periodic flows in the fluid interior // JAMM, 2007, vol. 71, no. 5, pp. 762–777.
- Kistovich A.V., Chashechkin Y.D. Dynamics of gravity-capillary waves on the surface of a nonuniformly heated fluid // Izv. Atmos.&Ocean. Phys., 2007, vol. 43, pp. 95–102.
- Ochirov A.A., Chashechkin Yu.D. Two-dimensional surface periodic flows of an incompressible fluid in various models of the medium // Izv. Atmos.&Ocean. Phys., 2024, vol. 60, no. 1, pp. 1–14. https://doi.org/10.1134/S0001433824700087. ISSN 0001-4338
- Chashechkin Yu.D. The laws of the matter distribution in a colored free-falling drop in a transparent target fluid (Review) // Fluid Dyn., 2024, vol. 59, no. 6, pp. 1693–1734.
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