Abstract
In this paper, we solve the problem of constructing uniform asymptotics of the far fields of internal gravitational waves from the initial perturbation of lines of equal density with radial symmetry. A constant model distribution of the buoyancy frequency is considered, and an analytical solution to the problem is obtained as the sum of wave modes using the Fourier–Hankel transform. Uniform asymptotics of solutions describing the spatiotemporal characteristics of the elevation of isopycnals (equal density lines) and vertical and horizontal (radial) velocity components are obtained. The asymptotics of an individual wave mode of the main wave field components are expressed as the squared Airy function and its derivatives. The exact and asymptotic results are compared. At times on the order of ten or more buoyancy periods, we show that the uniform asymptotics make it possible to efficiently calculate the far fields of waves.
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REFERENCES
Konyaev, K.V. and Sabinin, K.V., Volny vnutri okeana (Waves Inside the Ocean), St. Petersburg: Gidrometeoizdat, 1992.
Pedlosky, J., Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics, Berlin, Heildelberg: Springer, 2010.
Sutherland, B.R., Internal Gravity Waves, Cambridge: Cambridge Univ. Press, 2010.
Morozov, E.G., Oceanic Internal Tides. Observations, Analysis and Modeling, Berlin: Springer, 2018.
The Ocean in Motion, Velarde, M.G., Tarakanov, R.Yu., and Marchenko, A.V., Eds., Springer, 2018.
Bulatov, V.V. and Vladimirov, Yu.V., Volny v stratifitsirovannykh sredakh (Waves in Stratified Mediums), Moscow: Nauka, 2015.
Brekhovskikh, L.M. and Godin, O.A., Akustika neodnorodnykh sred (v 2 tomakh) (Acoustic of Inhomogeneous Mediums. In 2 Vols.), Vol. 1: Osnovy teorii otrazheniya i rasprostraneniya zvuka (Fundamentals of the Theory of Sound Reflection and Propagation), Vol. 2: Zvukovye polya v sloistykh i trekhmerno-neodnorodnykh sredakh (Sound Fields in Layered and Three-Dimensionally Inhomogeneous Mediums), Moscow: Nauka, 2007, 2009.
Gushchin, V.A. and Matyushin, P.V., Simulation and study of stratified flows around finite bodies, Comput. Math. Math. Phys., 2016, vol. 56, no. 6, pp. 1034–1048.
Matyushin, P.V., Process of the formation of internal waves initiated by the start of motion of a body in a stratified viscous fluid, Fluid Dyn., 2019, vol. 54, pp. 374–388.
Voelker, G.S., Myers, P.G., Walter, M., and Sutherland, B.R., Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance, Dyn. Atm. Oceans, 2019, vol. 86, pp. 116–133.
Wang, J., Wang, S., Chen, X., Wang, W., and Xu, Y., Three-dimensional evolution of internal waves rejected from a submarine seamount, Phys. Fluids, 2017, vol. 29, p. 106601.
Bulatov, V.V. and Vladimirov, Yu.V., Analytical solutions of the equation describing internal gravity waves generated by a moving nonlocal source of perturbations, Comput. Math. Math. Phys., 2021, vol. 61. no. 4, pp. 556–563.
Bulatov, V. and Vladimirov, Yu., Generation of internal gravity waves far from moving non-local source, Symmetry, 2020, vol. 12, no. 11, p. 1899.
Haney, S. and Young, W.R., Radiation of internal waves from groups of surface gravity waves, J. Fluid Mech., 2017, vol. 829, pp. 280–303.
Broutman, D., Brandt, L., Rottman, J., and Taylor, C.A., WKB derivation for internal waves generated by a horizontally moving body in a thermocline, Wave Motion, 2021, vol. 105, p. 102759.
Svirkunov, P.N. and Kalashnik, M.V., Phase patterns of dispersive waves from moving localized sources, Phys. Usp., 2014, vol. 57, pp. 80–91.
Belyaev, M.Yu., Desinov, L.V., Krikalev, S.K., Kumakshev, S.A., and Sekerzh-Zen’kovich, S.Ya., Identification of a system of oceanic waves based on space imagery, J. Comput. Syst. Sci. Int., 2009, vol. 48, pp. 110–120.
Morozov, E.G., Tarakanov, R.Yu., Frey, D.I., Demidova, T.A., and Makarenko, N.I., Bottom water flows in the tropical fractures of the Northern Mid-Atlantic Ridge, J. Oceanogr., 2018, vol. 74, no. 2, pp. 147–167.
Khimchenko, E.E., Frey, D.I., and Morozov, E.G., Tidal internal waves in the Bransfield Strait, Antarctica, Russ. J. Earth. Sci., 2020, vol. 20, p. ES2006.
Bulatov, V.V. and Vladimirov, Yu.V., Asymptotics of the far fields of internal gravity waves excited by a source of radial symmetry, Fluid Dyn., 2021, vol. 56, no. 5, pp. 672–677.
Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge: Univ. Press, 1995.
Kravtsov, Y. and Orlov, Y., Caustics, Catastrophes, and Wave Fields, Berlin: Springer, 1999.
Froman, N. and Froman, P., Physical Problems Solved by the Phase-Integral Method, Cambridge: Univ. Press, 2002.
Bulatov, V.V., Vladimirov, Yu.V., and Vladimirov, I.Yu., Uniform and nonuniform asymptotics of far surface fields from a flashed localized source, Fluid Dyn., 2021, vol. 56, no. 7, pp. 975–980.
Funding
The work was carried out within a state task, topic nos. AAAA-A20-120011690131-7, 0128-2021-0002 and was partially supported by the Russian Foundation for Basic Research, project no. 20-01-00111A.
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Translated by A. Ivanov
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Bulatov, V.V., Vladimirov, I.Y. Uniform Asymptotics of Internal Gravitational Wave Fields from an Initial Radially Symmetric Perturbation. Fluid Dyn 56, 1112–1118 (2021). https://doi.org/10.1134/S0015462821080103
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DOI: https://doi.org/10.1134/S0015462821080103