Plane internal gravity waves with arbitrary amplitude

Heading: 
Space Physics
1Cheremnykh, OK, 1Cheremnykh, SO, Lashkin, VM, 1Fedorenko, AK
1Space Research Institute under NAS and National Space Agency of Ukraine, Kyiv, Ukraine
Kinemat. fiz. nebesnyh tel (Online) 2024, 40(5):73-83
https://doi.org/10.15407/kfnt2024.05.073
Language: Ukrainian
Abstract: 

The non-linear equations called the Stenflo equations are usually used for the analytical description of the propagation of internal gravity waves in the Earth’s upper atmosphere. The solutions in the form of dipole vortices, tripole vortices and vortex chains were previously obtained with the help of these equations. The Stenflo equations also describe rogue waves, breathers, and dark solitons. It is known that when disturbances cease to be small, their profiles are usually deformed and it is assumed that they cannot be represented in the form of plane waves. This paper shows that this is not always the case for internal gravity waves and that even at large amplitudes these waves can propagate as plane waves. An exact solution of the system of nonlinear Stenflo equations for internal gravity waves containing nonlinear terms in the form of Poisson brackets is presented. The solution is obtained in the form of plane waves with an arbitrary amplitude. To find a solution, the original system of equations was transformed. We split it into equations for the stream functions and vorticity functions, as well as equations for the perturbed density. To solve the obtained equations, the procedure of successive zeroing of Poisson brackets was applied. As a result, linear equations were obtained, that allow finding the accurate analytical solutions for internal gravity waves in the form of plane waves with arbitrary amplitude. By solving these linear equations in two different ways, we analytically found expressions for the perturbed quantities and the dispersion equation. The nonlinear equations we obtained for the current, vorticity, and perturbed density functions can be used to find other nonlinear solutions. The presented solutions in the form of plane waves with an arbitrary amplitude may be of interest for the analysis of the propagation of internal gravity waves in the Earth’s atmosphere and the interpretation of experimental data.
Keywords: internal gravitational wave, stream function, system of nonlinear equations

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References: 

1. Cheremnykh O. K., Cheremnykh S. O., Vlasov D. I. (2022).The influence of the Earth's atmosphere rotation on the spectrum of acoustic-gravity waves. Kinematics and Phys. Celestial Bodies. 38, 121-131.
https://doi.org/10.3103/S0884591322030023

2. Beer T. (1973). Supersonic generation of atmospheric waves. Nature. 242(5392) 34-34. doi:10.1038/242034a0.
https://doi.org/10.1038/242034a0

3. Cheremnykh O. K., Fedorenko A. K., Selivanov Y. A., Cheremnykh S. O. (2021). Continuous spectrum of evanescent acoustic-gravity waves in an isothermal atmosphere. Mon. Notic. Roy. Astron. Soc. 503. 5545-5553. DOI: 10.1093/mnras/stab845.
https://doi.org/10.1093/mnras/stab845

4. Cheremnykh O., Kaladze T., Selivanov Yu., Cheremnykh S. (2022). Evanescent acoustic-gravity waves in a rotating stratified atmosphere. Adv. in Space Res. 69. 1272-1280.
https://doi.org/10.1016/j.asr.2021.10.050

5. Dong B., Yeh K. C. (1998). Resonant and nonresonant wave-wave interactions in an isothermal atmosphere. Geophys. Res. 93. 3729-3744.
https://doi.org/10.1029/JD093iD04p03729

6. Francis S. H. (1975). Global propagation of atmospheric gravity waves: A review. J. Atmos. Sol.-Terr. Phys. 37. 1011-1054.
https://doi.org/10.1016/0021-9169(75)90012-4

7. Fritts D. C., Alexander M. J. (2003). Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41. 1003-1062.
https://doi.org/10.1029/2001RG000106

8. Fritts D. C., Laughman B., Lund T. S., Snively J. B. (2015). Self-acceleration and instability of gravity wave packets: 1. Effects of temporal localization. J. Geophys. Res. Atmos. 120. 8783-8803.
https://doi.org/10.1002/2015JD023363

9. Fritts D. C., Sun S., Wang D.-Y. (1992). Wave-wave interactions in a compressible atmosphere 1. A general formulation including rotation and wind shear. J. Geophys. Res. 97. 9975-9988.
https://doi.org/10.1029/92JD00347

10. Gossard E. E., Hooke W. H. Waves in the Atmosphere: Atmospheric Infrasound and Gravity Waves: Their Generation and Propagation (Elsevier Scientific Publishing Company, 1975).

11. Hines C. O. (1960). Internal atmospheric gravity waves at ionospheric heights. Can. J. Phys. 38. 1441-1481.
https://doi.org/10.1029/GM018p0248

12. Horton W., Kaladze T. D., Van Dam J. W., Garner T. W. (2008). Zonal flow generation by internal gravity waves in the atmosphere. J. Geophys. Res. 13, A08312.
https://doi.org/10.1029/2007JA012952

13. Huang C. S., Li J. (1991). Weak nonlinear theory of the ionospheric response to atmospheric gravity waves in the F-region. J. Atmos. and Terr. Phys. 53. 903-908.
https://doi.org/10.1016/0021-9169(91)90003-P

14. Huang C. S., Li J. (1992). Interaction of atmospheric gravity solitary waves with ion acoustic solitary waves in the ionospheric F-region. J. Atmos. and Terr. Phys. 54. 951-956.
https://doi.org/10.1016/0021-9169(92)90060-X

15. Huang K. M., Zhang S. D., Yi F., Huang C. M., Gan Q., Gong Y., Zhang Y. H. (2014). Nonlinear interaction of gravity waves in a nonisothermal and dissipative atmosphere. Ann. Geophys. 32. 263-275.
https://doi.org/10.5194/angeo-32-263-2014

16. Jovanovi D., Stenflo L., Shukla P. K. (2001). Acoustic gravity tripolar vortices. Phys. Lett. A. 279. 70-74.
https://doi.org/10.1016/S0375-9601(00)00796-9

17. Jovanovi D., Stenflo L., Shukla P. K. (2002). Acoustic-gravity nonlinear structures. Nonlin. Proc. Geophys. 9. 333-339.
https://doi.org/10.5194/npg-9-333-2002

18. Kaladze T. D., Misra A. P., Roy A., Chatterjee D. (2022). Nonlinear evolution of internal gravity waves in the Earth's ionosphere: Analytical and numerical approach. Adv. Space Res. 69. 3374-3385.
https://doi.org/10.1016/j.asr.2022.02.014

19. Kaladze T. D., Pokhotelov O. A., Shan H. A., Shan M. I., Stenflo L. (2008). Acoustic-gravity waves in the Earth's ionosphere. J. Atmos. Sol.-Terr. Phys. 70. 1607- 1616.
https://doi.org/10.1016/j.jastp.2008.06.009

20. Kshevetskii S. P., Gavrilov N. M. (2005). Vertical propagation, breaking and effects of nonlinear gravity waves in the atmosphere. J. Atmos. Sol.-Terr. Phys. 67. 1014- 1030.
https://doi.org/10.1016/j.jastp.2005.02.013

21. Lashkin V. M., Cheremnykh O. K. (2023). Acoustic-gravity waves in quasiisothermal atmospheres with a random vertical temperature profile. Wave Motion. 119. 103140.
https://doi.org/10.1016/j.wavemoti.2023.103140

22. Lashkin V. M., Cheremnykh O. K. (2024). Nonlinear internal gravity waves in the atmosphere: Rogue waves, breathers and dark solitons. Communications in Nonlinear Science and Numerical Simulation. 130. 107757.
https://doi.org/10.1016/j.cnsns.2023.107757

23. Misra A. P., Roy A., Chatterjee D., Kaladze T. D. (2022). Internal gravity waves in the Earth's ionosphere. IEEE Trans. Plasma Sci. 50. 2603-2608.
https://doi.org/10.1109/TPS.2022.3178133

24. Mixa T., Fritts D., Lund T., Laughman B., Wang L., Kantha L. (2019). Numerical simulations of high-frequency gravity wave propagation through fine structures in the mesosphere. J. Geophys. Res. Atmos. 124. 9372-9390.
https://doi.org/10.1029/2018JD029746

25. Nekrasov A. K. (1994). Nonlinear saturation of atmospheric gravity waves. J. Atmos. and Terr. Phys. 56. 931-937.
https://doi.org/10.1016/0021-9169(94)90154-6

26. Nekrasov A. K., Erokhin N. S. (2005). Self-influence of the collapsing internal gravity wave in the inhomogeneous atmosphere. Phys. Lett. A. 335. 417-423.
https://doi.org/10.1016/j.physleta.2004.11.062

27. Onishchenko O., Pokhotelov O., Fedun V. (2013). Convective cells of internal gravity waves in the Earth's atmosphere with finite temperature gradient. Ann. Geophys. 31. 459-462.
https://doi.org/10.5194/angeo-31-459-2013

28. Shukla P. K., Shaikh A. A. (1998). Dust-acoustic gravity vortices in a nonuniform dusty atmosphere, Phys. Scripta. T75. 247-248.
https://doi.org/10.1238/Physica.Topical.075a00247

29. Snively J. B. (2017). Nonlinear gravity wave forcing as a source of acoustic waves in the mesosphere, thermosphere, and ionosphere. Geophys. Res. Lett. 44. 12020- 12027.
https://doi.org/10.1002/2017GL075360

30. Stenflo L. (1987). Acoustic solitary waves. Phys. Fluids. 30. P. 3297-3299.
https://doi.org/10.1063/1.866458

31. Stenflo L. (1990). Acoustic gravity vortices. Phys. Scripta. 41. 641-642.
https://doi.org/10.1088/0031-8949/41/5/001

32. Stenflo L., Shukla P. K. (2009). Nonlinear acoustic-gravity waves. J. Plasma Phys. 75. 841-847.
https://doi.org/10.1017/S0022377809007892

33. Sutherland B. R. Internal Gravity Waves (Cambridge University Press, Cambridge, 2015).

34. Tolstoy I. (1967). Long-period gravity waves in the atmosphere. J. Geophys. Res. 72. 4605-4610.
https://doi.org/10.1029/JZ072i018p04605

35. Yeh K. C., Liu C. H. (1974). Acoustic-gravity waves in the upper atmosphere. Rev. Geophys. Space Phys. 12. 193-216.
https://doi.org/10.1029/RG012i002p00193