Russian Journal of Earth Sciences Russian Journal of Earth Sciences Russian Journal of Earth Sciences 1681-1208 46528 10.2205/2020ES000732 ОРИГИНАЛЬНЫЕ СТАТЬИ ORIGINAL ARTICLES ОРИГИНАЛЬНЫЕ СТАТЬИ Dynamics of internal gravity waves in the ocean with shear flows Dynamics of internal gravity waves in the ocean with shear flows Bulatov V V Bulatov V V internalwave@mail.ru Vladimirov Yu V Vladimirov Yu V Ishlinsky Institute for Problems in Mechanics RAS ru Ishlinsky Institute for Problems in Mechanics RAS ru Ishlinsky Institute for Problems in Mechanics RAS ru Ishlinsky Institute for Problems in Mechanics RAS ru 20 4 29 10 2021 https://rjes.ru/en/nauka/article/46528/view

The problem of the harmonic internal gravity wave dynamics in a stratified ocean of finite depth with shear flows is solved. Stratification with constant distribution of the buoyancy frequency and various linear dependences of the shear flow on depth were used for the analytical solution of the problem. Dispersion dependences were obtained, which are expressed through a modified Bessel function of an imaginary index. The Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions under the Miles stability condition and large Richardson numbers. The asymptotic properties of the dispersion equation are studied. The main analytical properties of dispersion curves are investigated. The results of numerical calculations of the fields of phase structures of the generated internal gravity waves for various models of wave generation are presented.

The problem of the harmonic internal gravity wave dynamics in a stratified ocean of finite depth with shear flows is solved. Stratification with constant distribution of the buoyancy frequency and various linear dependences of the shear flow on depth were used for the analytical solution of the problem. Dispersion dependences were obtained, which are expressed through a modified Bessel function of an imaginary index. The Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions under the Miles stability condition and large Richardson numbers. The asymptotic properties of the dispersion equation are studied. The main analytical properties of dispersion curves are investigated. The results of numerical calculations of the fields of phase structures of the generated internal gravity waves for various models of wave generation are presented.

 Internal gravity waves stratified ocean shear flows asymptotics modified Bessel function dispersion relations Internal gravity waves stratified ocean shear flows asymptotics modified Bessel function dispersion relations The work was supported by the Russian Foundation for Basic Research (grant No. 20-01-00111A).

Bulatov, V. V., Yu. V. Vladimirov (2012) , Wave Dynamics of Stratified Mediums, 584 pp., Nauka, Moscow Bulatov, V. V., Yu. V. Vladimirov (2012) , Wave Dynamics of Stratified Mediums, 584 pp., Nauka, Moscow Bulatov, V. V., Yu. V. Vladimirov (2018) , Far fields of internal gravity waves from a nonstationary source, Oceanology, 58, no. 6, p. 796-801, https://doi.org/10.1134/S0001437018060036 Bulatov, V. V., Yu. V. Vladimirov (2018) , Far fields of internal gravity waves from a nonstationary source, Oceanology, 58, no. 6, p. 796-801, https://doi.org/10.1134/S0001437018060036 Bulatov, V. V., Yu. V. Vladimirov (2019) , A General Approach to Ocean Wave Dynamics Research: Modelling, Asymptotics, Measurements, 587 pp., Onto Print Publishers, Moscow Bulatov, V. V., Yu. V. Vladimirov (2019) , A General Approach to Ocean Wave Dynamics Research: Modelling, Asymptotics, Measurements, 587 pp., Onto Print Publishers, Moscow Bulatov, V. V., Yu. V. Vladimirov, I. Yu. Vladimirov (2019) , Far fields of internal gravity waves from a source moving in the ocean with an arbitrary buoyancy frequency distribution, Russian J. Earth Sciences, 19, p. ES5003, https://doi.org/10.2205/2019ES000667 Bulatov, V. V., Yu. V. Vladimirov, I. Yu. Vladimirov (2019) , Far fields of internal gravity waves from a source moving in the ocean with an arbitrary buoyancy frequency distribution, Russian J. Earth Sciences, 19, p. ES5003, https://doi.org/10.2205/2019ES000667 Fabrikant, A. L., Yu. A. Stepanyants (1998) , Propagation of Waves in Shear Flows, 304 pp., World Scientific Publishing, London, https://doi.org/10.1142/2557 Fabrikant, A. L., Yu. A. Stepanyants (1998) , Propagation of Waves in Shear Flows, 304 pp., World Scientific Publishing, London, https://doi.org/10.1142/2557 Fraternale, F., L. Domenicale, G. Staffilan, et al. (2018) , Internal waves in sheared flows: lower bound of the vorticity growth and propagation discontinuities in the parameter space, Phys. Review, 97, no. 6, p. 063102, https://doi.org/10.1103/PhysRevE.97.063102 Fraternale, F., L. Domenicale, G. Staffilan, et al. (2018) , Internal waves in sheared flows: lower bound of the vorticity growth and propagation discontinuities in the parameter space, Phys. Review, 97, no. 6, p. 063102, https://doi.org/10.1103/PhysRevE.97.063102 Frey, D. I., A. N. Novigatsky, M. D. Kravchishina, et al. (2017) , Water structure and currents in the Bear Island Trough in July-August 2017, Russian J. Earth Sciences, 17, p. ES3003, https://doi.org/10.2205/2017ES000602 Frey, D. I., A. N. Novigatsky, M. D. Kravchishina, et al. (2017) , Water structure and currents in the Bear Island Trough in July-August 2017, Russian J. Earth Sciences, 17, p. ES3003, https://doi.org/10.2205/2017ES000602 Furuichi, N., T. Hibiya, Y. Niwa (2008) , Model predicted distribution of wind-induced internal wave energy in the world's oceans, J. Geophys. Research: Oceans, 113, p. C09034, https://doi.org/10.1029/2008JC004768 Furuichi, N., T. Hibiya, Y. Niwa (2008) , Model predicted distribution of wind-induced internal wave energy in the world's oceans, J. Geophys. Research: Oceans, 113, p. C09034, https://doi.org/10.1029/2008JC004768 Gavrileva, A. A., Yu. G. Gubarev, M. P. Lebedev (2019) , The Miles theorem and the first boundary value problem for the Taylor-Goldstein equation, J. Applied and Industrial Mathematics, 13, no. 3, p. 460-471, https://doi.org/10.1134/S1990478919030074 Gavrileva, A. A., Yu. G. Gubarev, M. P. Lebedev (2019) , The Miles theorem and the first boundary value problem for the Taylor-Goldstein equation, J. Applied and Industrial Mathematics, 13, no. 3, p. 460-471, https://doi.org/10.1134/S1990478919030074 Massel, S. R. (2015) , Internal Gravity Waves in the Shallow Seas, 163 pp., Springer, Berlin, https://doi.org/10.1007/978-3-319-18908-6_7 Massel, S. R. (2015) , Internal Gravity Waves in the Shallow Seas, 163 pp., Springer, Berlin, https://doi.org/10.1007/978-3-319-18908-6_7 Mei, C. C., M. Stiassnie, D. K.-P. Yue (2017) , Theory and Applications of Ocean Surface Waves. Advanced series of ocean engineering, V. 42, 1500 pp., World Scientific Publishing, London, https://doi.org/10.1142/10212 Mei, C. C., M. Stiassnie, D. K.-P. Yue (2017) , Theory and Applications of Ocean Surface Waves. Advanced series of ocean engineering, V. 42, 1500 pp., World Scientific Publishing, London, https://doi.org/10.1142/10212 Miles, J. W. (1961) , On the stability of heterogeneous shear flow, J. Fluid Mechanics, 10, no. 4, p. 495-509, https://doi.org/10.1017/S0022112061000305 Miles, J. W. (1961) , On the stability of heterogeneous shear flow, J. Fluid Mechanics, 10, no. 4, p. 495-509, https://doi.org/10.1017/S0022112061000305 Miropolsky, Y. Z. (2001) , Dynamics of Internal Gravity Waves in the Ocean, Shishkina O. (ed.), 406 pp., Springer, Berlin, https://doi.org/10.1007/978-94-017-1325-2 Miropolsky, Y. Z. (2001) , Dynamics of Internal Gravity Waves in the Ocean, Shishkina O. (ed.), 406 pp., Springer, Berlin, https://doi.org/10.1007/978-94-017-1325-2 Morozov, E. G. (2018) , Oceanic Internal Tides. Observations, Analysis and Modeling: A Global View, 366 pp., Springer, Dordrecht, https://doi.org/10.1007/978-3-319-73159-9 Morozov, E. G. (2018) , Oceanic Internal Tides. Observations, Analysis and Modeling: A Global View, 366 pp., Springer, Dordrecht, https://doi.org/10.1007/978-3-319-73159-9 Morozov, E. G., G. Parrilla-Barrera, M. G. Velarde, et al. (2003) , The Straits of Gibraltar and Kara Gates: A comparison of internal tides, Oceanologica Acta, 26, no. 3, p. 231-241, https://doi.org/10.1016/S0399-1784(03)00023-9 Morozov, E. G., G. Parrilla-Barrera, M. G. Velarde, et al. (2003) , The Straits of Gibraltar and Kara Gates: A comparison of internal tides, Oceanologica Acta, 26, no. 3, p. 231-241, https://doi.org/10.1016/S0399-1784(03)00023-9 Morozov, E. G., V. T. Paka, V. V. Bakhanov (2008) , Strong internal tides in the Kara Gates Strait, Geoph. Research Letters, 35, p. L16603, https://doi.org/10.1029/2008GL033804 Morozov, E. G., V. T. Paka, V. V. Bakhanov (2008) , Strong internal tides in the Kara Gates Strait, Geoph. Research Letters, 35, p. L16603, https://doi.org/10.1029/2008GL033804 Morozov, E. G., I. E. Kozlov, S. A. Shchuka, et al. (2017) , Internal tide in the Kara Gates Strait, Oceanology, 57, p. 8-18, https://doi.org/10.1134/S0001437017010106 Morozov, E. G., I. E. Kozlov, S. A. Shchuka, et al. (2017) , Internal tide in the Kara Gates Strait, Oceanology, 57, p. 8-18, https://doi.org/10.1134/S0001437017010106 Pedlosky, J. (2010) , Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics, 260 pp., Springer, Berlin Pedlosky, J. (2010) , Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics, 260 pp., Springer, Berlin Svirkunov, P. N., M. V. Kalashnik (2014) , Phase patterns of dispersive waves from moving localized sources, Phys.-Usp., 57, no. 1, p. 80-91, https://doi.org/10.3367/UFNe.0184.201401d.0089 Svirkunov, P. N., M. V. Kalashnik (2014) , Phase patterns of dispersive waves from moving localized sources, Phys.-Usp., 57, no. 1, p. 80-91, https://doi.org/10.3367/UFNe.0184.201401d.0089 Sutherland, B. R. (2010) , Internal Gravity Waves, 394 pp., Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511780318 Sutherland, B. R. (2010) , Internal Gravity Waves, 394 pp., Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511780318 Velarde, M. G., R. Yu. Tarakanov, A. V. Marchenko (2018) , The Ocean in Motion, 625 pp., Springer Oceanography, Springer International Publishing AG, Berlin, https://doi.org/10.1007/978-3-319-71934-4 Velarde, M. G., R. Yu. Tarakanov, A. V. Marchenko (2018) , The Ocean in Motion, 625 pp., Springer Oceanography, Springer International Publishing AG, Berlin, https://doi.org/10.1007/978-3-319-71934-4 Voelker, G. S., P. G. Myers, et al. (2019) , Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance, Dyn. Atmosphere Oceans, 86, p. 116-133, https://doi.org/10.1016/j.dynatmoce.2019.03.005 Voelker, G. S., P. G. Myers, et al. (2019) , Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance, Dyn. Atmosphere Oceans, 86, p. 116-133, https://doi.org/10.1016/j.dynatmoce.2019.03.005 Watson, G. N. (1995) , A Treatise on the Theory of Bessel Functions, 804 pp., Cambridge University Press, Cambridge Watson, G. N. (1995) , A Treatise on the Theory of Bessel Functions, 804 pp., Cambridge University Press, Cambridge