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Analytic Properties of Solutions to the Equation of Internal Gravity Waves with Flows for Critical Modes of Wave Generation

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Abstract

Issues related to the statement of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows in critical modes of wave generation are considered. Model physical statements of problems in which critical levels may arise are discussed in the two-dimensional case. Analytic properties of the solutions near critical levels are studied. A system describing a flow of a stratified medium incident on an obstacle behind which outgoing waves may arise is discussed, in which case a singularity at the critical level is formed far away from the obstacle. Asymptotics of the solutions near the critical level are constructed and expressed in terms of the incomplete gamma function.

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References

  1. A. Alias, R. H. J. Grimshaw, and K. R. Khusnutdinova, “Coupled Ostrovsky equations for internal waves in a shear flow,” Phys. Fluids 26 (12), 126603 (2014).

    Article  Google Scholar 

  2. A. Ya. Basovich and L. Sh. Tsimring, “Internal waves in a horizontally inhomogeneous flow,” J. Fluid Mech. 142, 233–249 (1984).

    Article  MathSciNet  Google Scholar 

  3. V. A. Borovikov, Uniform Stationary Phase Method (Inst. Electr. Eng., London, 1994), IEE Electromagn. Waves Ser. 40.

    Google Scholar 

  4. P. Bouruet-Aubertot and S. A. Thorpe, “Numerical experiments on internal gravity waves in an accelerating shear flow,” Dyn. Atmos. Oceans 29 (1), 41–63 (1999).

    Article  Google Scholar 

  5. F. P. Bretherton, “The propagation of groups of internal gravity waves in a shear flow,” Q. J. R. Metereol. Soc. 92 (394), 466–480 (1966).

    Article  Google Scholar 

  6. D. Broutman, L. Brandt, J. W. Rottman, and C. K. Taylor, “A WKB derivation for internal waves generated by a horizontally moving body in a thermocline,” Wave Motion 105, 102759 (2021).

    Article  MathSciNet  Google Scholar 

  7. D. Broutman and J. W. Rottman, “A simplified Fourier method for computing the internal wavefield generated by an oscillating source in a horizontally moving, depth-dependent background,” Phys. Fluids 16 (10), 3682–3689 (2004).

    Article  Google Scholar 

  8. V. V. Bulatov, New Problems of Mathematical Modeling of Wave Dynamics of Stratified Media (OntoPrint, Moscow, 2021) [in Russian].

    Google Scholar 

  9. V. V. Bulatov, “Analytic properties of the Green’s function for the equation of internal gravitational waves in a stratified medium with shear flows,” Theor. Math. Phys. 211 (2), 611–624 (2022) [transl. from Teor. Mat. Fiz. 211 (2), 200–215 (2022)].

    Article  MathSciNet  Google Scholar 

  10. V. V. Bulatov and Yu. V. Vladimirov, Waves in Stratified Media (Nauka, Moscow, 2015) [in Russian].

    Google Scholar 

  11. V. V. Bulatov and Yu. V. Vladimirov, “Dynamics of internal gravity waves in the ocean with shear flows,” Russ. J. Earth Sci. 20, ES4004 (2020).

    Article  Google Scholar 

  12. V. Bulatov and Yu. Vladimirov, “Analytical approximations of dispersion relations for internal gravity waves equation with shear flows,” Symmetry 12 (11), 1865 (2020).

    Article  Google Scholar 

  13. V. V. Bulatov, Yu. V. Vladimirov, and I. Yu. Vladimirov, “Internal gravity waves from an oscillating source in the ocean,” Izv., Atmos. Ocean. Phys. 57 (3), 321–328 (2021) [transl. from Izv. Ross. Akad. Nauk, Fiz. Atmos. Okeana 57 (3), 362–371 (2021)].

    Article  Google Scholar 

  14. V. V. Bulatov, Yu. V. Vladimirov, and I. Yu. Vladimirov, “Phase structure of internal gravity waves in the ocean with shear flows,” Phys. Oceanogr. 28 (4), 438–453 (2021) [transl. from Morskoi Gidrofiz. Zh. 37 (4), 473–489 (2021)].

    Article  Google Scholar 

  15. J. R. Carpenter, N. J. Balmforth, and G. A. Lawrence, “Identifying unstable modes in stratified shear layers,” Phys. Fluids 22 (5), 054104 (2010).

    Article  Google Scholar 

  16. S. Churilov, “On the stability analysis of sharply stratified shear flows,” Ocean Dyn. 68 (7), 867–884 (2018).

    Article  Google Scholar 

  17. A. L. Fabrikant and Yu. A. Stepanyants, Propagation of Waves in Shear Flows (World Scientific, Singapore, 1998).

    Book  Google Scholar 

  18. F. Fraternale, L. Domenicale, G. Staffilani, and D. Tordella, “Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space,” Phys. Rev. E 97 (6), 063102 (2018).

    Article  Google Scholar 

  19. D. I. Frey, A. N. Novigatsky, M. D. Kravchishina, and E. G. Morozov, “Water structure and currents in the Bear Island Trough in July–August 2017,” Russ. J. Earth Sci. 17, ES3003 (2017).

    Article  Google Scholar 

  20. A. A. Gavrilieva, Yu. G. Gubarev, and M. P. Lebedev, “The Miles theorem and new particular solutions to the Taylor–Goldstein equation,” Lobachevskii J. Math. 38 (3), 560–570 (2017) [transl. from Uchen. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 158 (2), 156–171 (2016)].

    Article  MathSciNet  Google Scholar 

  21. A. A. Gavril’eva, Yu. G. Gubarev, and M. P. Lebedev, “The Miles theorem and the first boundary value problem for the Taylor–Goldstein equation,” J. Appl. Ind. Math. 13 (3), 460–471 (2019).

    Article  MathSciNet  Google Scholar 

  22. V. Gnevyshev and S. Badulin, “Wave patterns of gravity–capillary waves from moving localized sources,” Fluids 5 (4), 219 (2020).

    Article  Google Scholar 

  23. M. Hirota and P. J. Morrison, “Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows,” Phys. Lett. A 380 (21), 1856–1860 (2016).

    Article  Google Scholar 

  24. C. J. Howland, J. R. Taylor, and C. P. Caulfield, “Shear-induced breaking of internal gravity waves,” J. Fluid Mech. 921, A24 (2021).

    Article  MathSciNet  Google Scholar 

  25. E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. I: Gewöhnliche Differentialgleichungen (Teubner, Stuttgart, 1977).

    Book  Google Scholar 

  26. E. E. Khimchenko, D. I. Frey, and E. G. Morozov, “Tidal internal waves in the Bransfield Strait, Antarctica,” Russ. J. Earth Sci. 20, ES2006 (2020).

    Article  Google Scholar 

  27. Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, Berlin, 1999).

    Google Scholar 

  28. M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  29. J. Lighthill, Waves in Fluids (Cambridge Univ. Press, Cambridge, 2001).

    Google Scholar 

  30. P. Meunier, S. Le Dizès, L. Redekopp, and G. R. Spedding, “Internal waves generated by a stratified wake: Experiment and theory,” J. Fluid Mech. 846, 752–788 (2018).

    Article  Google Scholar 

  31. J. W. Miles, “On the stability of heterogeneous shear flows,” J. Fluid Mech. 10 (4), 496–508 (1961).

    Article  MathSciNet  Google Scholar 

  32. Yu. Z. Miropol’sky, Dynamics of Internal Gravity Waves in the Ocean (Kluwer, Dordrecht, 2001).

    Book  Google Scholar 

  33. E. G. Morozov, Oceanic Internal Tides: Observations, Analysis and Modeling: A Global View (Springer, Cham, 2018).

    Book  Google Scholar 

  34. E. G. Morozov, G. Parrilla-Barrera, M. G. Velarde, and A. D. Scherbinin, “The Straits of Gibraltar and Kara Gates: A comparison of internal tides,” Oceanol. Acta 26 (3), 231–241 (2003).

    Article  Google Scholar 

  35. E. G. Morozov, R. Yu. Tarakanov, and D. I. Frey, Bottom Gravity Currents and Overflows in Deep Channels of the Atlantic Ocean: Observations, Analysis, and Modeling (Springer, Cham, 2021).

    Book  Google Scholar 

  36. E. G. Morozov, R. Yu. Tarakanov, D. I. Frey, T. A. Demidova, and N. I. Makarenko, “Bottom water flows in the tropical fractures of the Northern Mid-Atlantic Ridge,” J. Oceanogr. 74 (2), 147–167 (2018).

    Article  Google Scholar 

  37. A. H. Nayfeh, Introduction to Perturbation Techniques (J. Wiley & Sons, New York, 1981).

    Google Scholar 

  38. J. Pedlosky, Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics (Springer, Berlin, 2010).

    Google Scholar 

  39. I. Shugan and Y.-Y. Chen, “Kinematics of the ship’s wake in the presence of a shear flow,” J. Mar. Sci. Eng. 9 (1), 7 (2021).

    Article  Google Scholar 

  40. Yu. V. Sidorov, M. V. Fedoryuk, and M. I. Shabunin, Lectures on the Theory of Functions of a Complex Variable (Mir, Moscow, 1985) [transl. from Russian (Nauka, Moscow, 1982)].

    Google Scholar 

  41. A. A. Slepyshev and N. V. Laktionova, “Vertical transport of momentum by internal waves in a shear current,” Izv., Atmos. Ocean. Phys. 55 (6), 662–668 (2019) [transl. from Izv. Ross. Akad. Nauk, Fiz. Atmos. Okeana 55 (6), 194–200 (2019)].

    Article  Google Scholar 

  42. A. A. Slepyshev and D. I. Vorotnikov, “Generation of vertical fine structure by internal waves in a shear flow,” Open J. Fluid Mech. 9 (2), 140–157 (2019).

    Google Scholar 

  43. B. R. Sutherland, Internal Gravity Waves (Cambridge Univ. Press, Cambridge, 2010).

    Book  Google Scholar 

  44. P. N. Svirkunov and M. V. Kalashnik, “Phase patterns of dispersive waves from moving localized sources,” Phys. Usp. 57 (1), 80–91 (2014) [transl. from Usp. Fiz. Nauk 184 (1), 89–100 (2014)].

    Article  Google Scholar 

  45. The Ocean in Motion: Circulation, Waves, Polar Oceanography, Ed. by M. G. Velarde, R. Yu. Tarakanov, and A. V. Marchenko (Springer, Cham, 2018), Springer Oceanogr.

    Google Scholar 

  46. V. Vlasenko, N. Stashchuk, and K. Hutter, Baroclinic Tides: Theoretical Modeling and Observational Evidence (Cambridge Univ. Press, New York, 2005).

    Book  Google Scholar 

  47. N. N. Vorob’ev, Theory of Series (Lan’, St. Petersburg, 2002) [in Russian].

    Google Scholar 

  48. G. B. Whitham, Linear and Nonlinear Waves (J. Wiley & Sons, New York, 1974).

    Google Scholar 

  49. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1996).

    Book  Google Scholar 

  50. W. R. Young, P. B. Rhines, and C. J. R. Garrett, “Shear-flow dispersion, internal waves and horizontal mixing in the ocean,” J. Phys. Oceanogr. 12 (6), 515–527 (1982).

    Article  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation under grant no. 23-21-00194, https://rscf.ru/project/23-21-00194/.

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  1. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia

    V. V. Bulatov

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  1. V. V. Bulatov
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Correspondence to V. V. Bulatov.

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Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 322, pp. 71–82 https://doi.org/10.4213/tm4347.

Translated by I. Nikitin

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Bulatov, V.V. Analytic Properties of Solutions to the Equation of Internal Gravity Waves with Flows for Critical Modes of Wave Generation. Proc. Steklov Inst. Math. 322, 65–76 (2023). https://doi.org/10.1134/S0081543823040065

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  • DOI: https://doi.org/10.1134/S0081543823040065

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