Abstract. Two classes of nonlinear problems concerning wave and jet structures in steady flows of an in-homogeneous fluid are considered. One of them is related to the modeling of lee waves in atmospheric air-flows over uneven terrain. The other group of problems concerns the description of internal waves in deep-ocean currents. A common feature of both cases is the combined use of analytical approaches and com-puter algebra methods to construct asymptotic solutions of the equations of motion of a stratified fluid. The first approximate model is employed for the numerical calculation of jet-flow patterns and the nonlinear in-terference of lee waves over complex topography. The second model is used to characterize the qualitative properties and parameters of solitary internal waves in layered weakly stratified near-bottom flows.

Keywords: stratified fluid, internal waves, jet flows, normal modes, semianalytic methods

For citation: Makarenko N. I., Maltseva J. L. Semi-analytical modeling of wave and jet flows in a stratified fluid. Aerospace Engineering and Technology. 2025. Vol. 3, no. 4, pp. 563–583. https://doi.org/10.52467/2949-401X-2025-3-4-563-583; https://elibrary.ru/EYEVWU

References
1. Gutman L. N. Vvedeniye v nelineynuyu teoriyu mezo-meteorologicheskikh protsessov [Intro-duction to the nonlinear theory of meso-meteorological processes]. Leningrad: Hydrometeoizdat, 1969. 295 p. (In Russian).
2. Scorer R. Environmental Aerodynamics. New York: Halsted Press, 1978. 549.
3. Wurtele M. G., Sharman R. D., Datta A. Atmospheric lee waves. Annual Review on Fluid Mechanics. 1996. Vol. 28. P. 429-476. 1996. Vol. 28. Pp. 429–476. https://doi.org/10.1146/annurev.fl.28.010196.002241
4. Kozhevnikov V. N. Vozmushcheniya atmosfery pri obtekanii gor [Atmospheric disturbances over mountains]. Moscow: “Nauchny Mir”, 1999. 160 p. (In Russian). https://elibrary.ru/NGVELO
5. Kozhevnikov V. N. Oblaka kak proyavleniye volnovykh vozmushcheniy nad gornymi khrebtami [Clouds as manifestation of wave disturbance above mountain ridges]. Izvestiya RAS, At-mospheric and Oceanic Physics. 2022. Vol. 58, iss. 2, pp. 111–128. https://doi.org/10.31857/S0002351522020067; https://elibrary.ru/TPXMMR
6. Shelkovnikov M. S. Mezometeorologicheskiye protsessy v gornykh rayonakh i ikh vliyaniye na polety vozdushnykh sudov [Mesometeorological processes in mountainous areas and their impact on aircraft operations]. Leningrad: Hydrometeoizdat, 1985. 208 p. (In Russian).
7. Grubisic V., Lewis J. M. Sierra Wave Project revisited 50 Years later. Bulletin of American Meteorological Society. 2004. Vol. 85, no. 8, pp. 1127–1142. https://doi.org/10.1175/BAMS-85-8-112710.1175/BAMS-85-8-1127
8. Dorodnitsyn A. A. Vozmushcheniya vozdushnogo potoka. vyzyvayemyye nerovnostyami poverkhnosti Zemli [Perturbations of airflow caused by the terrain roughness]. Proceedings of the Central Geophysical Observatory. 1938. Is. 23, no. 6, pp. 3–17. (In Russian).
9. Lyra G. Theorie der stationaren Leewellen-stromung in freien Atmosphare. Zeitschrift an-gewandte Mathematik und Mechanik. 1943. Vol. 23, no. 1, pp. 1–23. (In German).
10. Long R.R. Some aspects of the flow of stratified fluids. III. Continuous density gradients. Tellus. 1955. Vol. 7, no. 3, pp. 341–357.
11. Long R. R. Finite amplitude disturbances in the flow of inviscid rotating and stratified flu-ids over an obstacle. Annual Review on Fluid Mechanics. 1972. No. 4, pp. 69–92. https://doi.org/10.1146/annurev.fl.04.010172.000441
12. Dubreil-Jacotin M.-L. Complement a une note anterieure sur les ondes de type permanent dans les liquides heterogenes. Rendiconti Accademia Nazionale Lincei. 1935. Vol. 21, no. 6, pp. 344–346. (In French).
13. Kozhevnikov V. N. Orograficheskiye vozmushcheniya v dvumernoy statsionarnoy zadache [Orographic disturbances in a two-dimensional stationary problem]. Izv. Akad. Nauk SSSR. 1968. Vol. 4, no. 1, pp. 33–52. (In Russian)
14. Miles J. W. Lee waves in a stratified flow. Part II. Semicircular obstacle. Journal of Fluid Mechanics. 1968. Vol. 33, no. 4, pp. 803–814. https://doi.org/10.1017/S0022112068001680
15. Lawrence G., Armi L. Stationary internal hydraulic jumps. Journal of Fluid Mechanics. 2022. Vol. 936. https://doi.org/10.1017/jfm.2022.74; https://elibrary.ru/KUVABF
16. Liapidevskii V. Yu., Chesnokov A. A. Sloy smesheniya v dvukhsloynykh sputnykh techeni-yakh stratifitsirovannoy zhidkosti [Mixing layer in two-layer cocurrent flows of stratified fluid]. Journal of Applied Mechanics and Technical Physics. 2022. Vol. 63, no. 6, pp. 1010–1021. https://doi.org/10.15372/PMTF20220614; https://elibrary.ru/LNJKVK (In Russian)
17. Liapidevskii V. Yu., Zuev O. A., Makarenko N. I., Morozov E. G., Frei D. I. Rasshchepleni-ye stratifitsirovannykh techeniy nad prepyatstviyami // Prikladnaya mekhanika i tekhnicheskaya fizika [Splitting of strati-fied flows over obstacles]. Journal of Applied Mechanics and Technical Physics. 2025. Vol. 66, no. 4, pp. 118–128.
18. Yakovenko S. N., Thomas T. G., Castro I. P. A turbulent patch arising from a breaking in-ternal wave. Journal of Fluid Mechanichs. 2011. Vol. 677. Pp. 103–133. https://doi.org/10.1017/jfm.2011.64; https://elibrary.ru/LSZGMP
19. Katagiri T., Watanabe T., Nagata K. Statistical properties of a model of a turbulent patch arising from a breaking internal wave. Physics of Fluids. 2021. Vol. 33. 055107 https://doi.org/10.1063/5.0046832; https://elibrary.ru/ZTUXHZ
20. Ovsyannikov L. V., Makarenko N. I., Nalimov V. I., et al. Nelineynyye problemy teorii pover-
khnostnykh i vnutrennikh voln [Nonlinear problems of the theory of surface and internal waves]. Novosibirsk: “Nauka”, 1985. 320 p. (In Russian).
21. Liapidevskii V. Yu., Teshukov V. M. Matematicheskiye modeli rasprostraneniya dlinnykh voln v neodnorodnoy zhidkosti [Mathematical models of long wave propagation in an inhomogene-ous fluid]. Novosibirsk: SB RAS Publ., 2000. 419 p. (In Russian).
22. Gavrilyuk S. L., Makarenko N. I., Sukhinin S. V. Waves in continuous media. Lectures Notes in Geosystem Mathematics and Computing. Cham, Switzerland: Birkhäuser, 2017. 149 p. https://elibrary.ru/DICZZD
23. Makarenko N. I., Maltseva J. L. Statsionarnyye volny v stratifitsirovannom techenii nad kombinirovannym prepyatstviyem [Steady waves in a stratified flow over a combined obstacle]. Journal of Applied Mechanics and Technical Physics. 2014. Vol. 55, no. 2, pp. 311–317. https://elibrary.ru/SMGNXR (In Russian)
24. Kochin N. E., Kibel’ I. A., Rose N. V. Teoreticheskaya gidromekhanika [Theoretical hydro-mechanics]. Part I. New York: Wiley, 1964. 584 p. (In Russian)
25. Makarenko N. I., Maltseva, J. L. An analytical model of large amplitude internal solitary waves. In: Extreme Ocean Waves. Eds.: E. Pelinovsky and Ch. Kharif. Dordrecht: Springer, 2016. Pp. 191–201. https://doi.org/10.1007/978-3-319-21575-4_11; https://elibrary.ru/WPSUYP
26. Chrgian A. Ch. Fizika atmosfery [Physics of atmosphere]. Мoscow: MSU Publ., 1986. 328 p. (In Russian).
27. Kozhevnikov V. N. K uchetu verkhnikh sloyev atmosfery pri modelirovanii orograficheskikh vozmushcheniy v troposfere [On taking into consideration the upper layers of the atmosphere to si-mu-late orographic disturbances in the troposphere]. Izvestiya, Atmospheric and Oceanic Physics. 2022. Vol. 57, iss. 2, pp. 142–147. https://doi.org/10.1134/S0001433821020079
28. Long R. R. On the Boussinesq approximation and its role in the theory of internal waves. Tellus. 1965. Vol. 17, no. 1, pp. 46–52.
29. Baines P. G. Topographic effects in stratified flows. New York: Cambridge University Press, 2022. 558 p.
30. Gyure B., Janosi I. M. Stratified flow over asymmetric and double bell-shaped obstacles. Dynamics Atmosphere and Oceans. 2003. Vol. 37. Pp. 155–170. https://doi.org/10.1016/S0377-0265(03)00030-7; https://elibrary.ru/ERPRGX
31. Makarenko N. I, Maltseva J. L., Cherevko A. A. Stratified flows over a complex relief. Journal of Physics: Conference Series. 2016. Vol. 722. P. 012027. https://doi.org/10.1088/1742-6596/722/1/012027
32. Stiperski I., Calaf M. Generalizing Monin – Obukhov similarity theory (1954) for complex atmospheric turbulence. Physical Review Letters. 2023. Vol. 130. P. 124001. https://doi.org/10.1103/PhysRevLett.130.124001; https://elibrary.ru/EJMDZM
33. Holmboe J., Klieforth H. Investigation of mountain lee waves and the air flow over the Sier-ra Nevada. Department of Meteorology, UCLA Final Report. Contract AF 19(604)–728. 1957. 283 p.
34. Kuettner J. P., Jenkins F. Flight aspects of the mountain wave. Air Force Surveys in Geo-physics. Air Force Cambridge Research Center, 1953. No. 35. 24 p.
35. Makarenko N. I., Maltseva J. L., Morozov E. G., Tarakanov R. Yu., Ivanova K. A. Vnutren-niye statsionarnyye volny v glubokovodnykh stratifitsirovannykh techeniyakh [Steady internal waves in deep stratified flows]. Journal of Applied Mechanics and Technical Physics. 2019. Vol. 60, no. 2, pp. 248–256. https://doi.org/10.15372/PMTF20190206; https://elibrary.ru/LCOPIL
36. Van Haren H., Gostiaux L., Morozov E., Tarakanov R. Extremely long Kelvin-Helmholtz billow trains in the Romanche Fracture Zone. Geophysical Research Letttrs. 2014. Vol. 44. Pp. 8445–8451. https://doi.org/10.1002/2014GL062421; https://elibrary.ru/UFWFQL
37. Morozov E.G., Tarakanov R.Y., Frey D.I. Bottom gravity currents and overflows in deep channels of the atlantic. Observations, Analysis, and Modeling. Dordrecht: Springer Nature. 2021. 483 p. https://doi.org/10.1007/978-3-030-83074-8
38. Tarakanov R. Yu., Morozov E. G., Demidova T. A., Van Haren H., Makarenko N. I. Struc-ture of the deep spillway in the western part of the Romanche Fracture Zone. Journal of Geo-physical Research: Oceans. 2018. Vol. 123, Issue 11, pp. 8508–8531. https://doi.org/10.1029/2018JC013961; https://elibrary.ru/GQJOME
39. Gavrilov N. V. Nepodvizhnyye v laboratornoy sisteme koordinat vnutrenniye uyedinennyye volny i plavnyye bory [Internal solitary waves and smooth bores stationary in the laboratory coordi-nate system]. Journal of Applied Mechanics and Technical Physics. 1994. Vol. 35, no. 1, pp. 29–34. https://elibrary.ru/USVFNV (in Russian)
40. Fructus D., Carr M., Grue J., Jensen A., Davies P.: Shear induced breaking of large internal solitary waves. Journal of Fluid Mechanics. 2009. Vol. 620. Pp. 1–29. https://doi.org/10.1017/S0022112008004898; https://elibrary.ru/LUQIQD
41. Vosper S. B. Three-dimensional numerical simulations of strongly stratified flow past conical orography. Journal of Atmospheric Sciences. 2000. Vol. 57, no. 22, pp. 3716–3739. https://doi.org/10.1175/1520-0469(2000)057<3716:TDNSOS>2.0.CO;2
42. Akylas T. R., Davis K. S. Three-dimensional aspects of nonlinear stratified flow over topog-raphy near the hydrostatic limit. Journal of Fluid Mechanics. 2001. Vol. 428. Pp. 81–105. https://elibrary.ru/ECMMJX
43. Smith R. B. Stratified flow over topography. In: Environmental Stratified Flows. Topics in Environmental Fluid Mechanics. Ed. R. Grimshaw. Springer, Boston, MA., Vol 3. Pp. 119–160. https://doi.org/10.1007/0-306-48024-7_5
44. Baker L. E., Mashayek A. The impact of representations of realistic topography on parame-terized oceanic lee wave energy flux. Journal of Geophysical Research: Oceans. 2022. Vol. 127. e2022JC018995. https://doi.org/10.1029/2022JC018995; https://elibrary.ru/VZDHQX
45. Hanazaki H. A numerical study of three-dimensional stratified flow past a sphere. Journal of Fluid Mechanics. 1988. Vol. 192. Pp. 393–419. https://doi.org/10.1017/S0022112088001910
46. Chashechkin Yu. D. Hydrodynamics of a sphere in a stratified fluid. Fluid Dynamics. 1989. Vol. 24. P. 1–7.
47. Voisin B. Lee waves from a sphere in a stratified flow. Journal of Fluid Mechanics. 2007. Vol. 574. Pp. 273–315. https://doi.org/10.1017/S0022112006004095; https://elibrary.ru/LPPDAP
48. Snyder W. H., Thompson R. S., Eskridge R. E., Lawson R. E., Castro I. P., Lee J. T., Hunt J. C. R., Ogawa Y. The structure of strongly stratified flow over hills: dividing streamline concept. Journal
of Fluid Mechanics. 1985. Vol. 152, Pp. 249–288. https://doi.org/10.1017/s0022112085000684