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
1. Bulatov, V. V., Yu. V. Vladimirov (2012) , Wave Dynamics of Stratified Mediums, 584 pp., Nauka, Moscow EDN: https://elibrary.ru/QKKWAB
2. 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 EDN: https://elibrary.ru/BVHGER
3. Bulatov, V. V., Yu. V. Vladimirov (2019) , A General Approach to Ocean Wave Dynamics Research: Modelling, Asymptotics, Measurements, 587 pp., Onto Print Publishers, Moscow EDN: https://elibrary.ru/YZXBTN
4. 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 EDN: https://elibrary.ru/BURRMS
5. 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 EDN: https://elibrary.ru/HLJQYQ
6. 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
7. 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 EDN: https://elibrary.ru/YMUXGX
8. 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
9. 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 EDN: https://elibrary.ru/UDCWIE
10. 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
11. 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 EDN: https://elibrary.ru/YHOSZN
12. 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
13. 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
14. 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 EDN: https://elibrary.ru/YCLNOH
15. 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 EDN: https://elibrary.ru/LHVOAL
16. 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 EDN: https://elibrary.ru/LLMFGT
17. 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 EDN: https://elibrary.ru/YVDNIP
18. Pedlosky, J. (2010) , Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics, 260 pp., Springer, Berlin
19. 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 EDN: https://elibrary.ru/SKQVHR
20. Sutherland, B. R. (2010) , Internal Gravity Waves, 394 pp., Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511780318
21. 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
22. 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 EDN: https://elibrary.ru/MTMIKB
23. Watson, G. N. (1995) , A Treatise on the Theory of Bessel Functions, 804 pp., Cambridge University Press, Cambridge
