Analytical Solution of the Ray Equations of Hamilton for Rossby Waves on Stationary Shear Flows
https://doi.org/10.59887/fpg/4eh4-83zr-r1fm
Abstract
The asymptotic behavior of Rossby waves in the ocean interacting with a shear stationary flow is considered. It is shown that there is a qualitative difference between the problems for the zonal and non-zonal background flow. Whereas only one critical layer arises for a zonal flow, then several critical layers can exist for a non-zonal flow. It is established that the integrated ray equations of Hamilton are equivalent to the asymptotic behavior of the Cauchy problem solution. Explicit analytical solutions are obtained for the tracks of Rossby waves as a function of time and initial parameters of the wave disturbance, as well as the magnitude of the shear and angle of inclination of the flow to the zonal direction. The ray equations of Hamilton are analytically integrated for Rossby waves on a shear flow. The obtained explicit expressions make it possible to calculate in real-time the Rossby wave tracks for any initial wave direction and any shear current inclination angle. It is shown qualitatively that these tracks for a non-zonal flow are strongly anisotropic.
Keywords
Rossby waves,
shear flow,
zonal,
non-zonal,
Hermitian operators,
Non-Hermitian operators,
ray equations of Hamilton
Supporting agencies: The publication was funded by the Russian Science Foundation, project No 22-27-00004. The work of V.G.G. was carried out within the State Task for the Shirshov Institute of Oceanology RAS, project No 0128-2021-0003.
About the Authors
V. G. Gnevyshev
Shirshov Institute of Oceanology, Russian Academy of Sciences
Russian Federation
Russian Federation
117997, Nahimovsky Pr., 36, Moscow
T. V. Belonenko
St. Petersburg State University
Russian Federation
Russian Federation
199034, 7–9, Universitetskaya Emb., St. Petersburg
References
1. Rossby C.G. et al. Relation between variations in the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. Journal of Marine Research. 1939, 2, 38–55. doi:10.1357/002224039806649023
2. Fu L.-L., Cazenave A. Satellite altimetry and Earth sciences: a handbook of techniques and applications. San Diego, 2000. CA: Academic Press.
3. Gnevyshev V.G., Badulin S.I., Belonenko T.V. Rossby waves on non-zonal currents: structural stability of critical layer effects. Pure and Applied Geophysics. 2020, 177(11), 5585–5598. doi:10.1007/s00024-020-02567-0
4. Gnevyshev V.G., Badulin S.I., Koldunov A.V., Belonenko T.V. Rossby waves on non-zonal flows: vertical focusing and effect of the current stratification. Pure and Applied Geophysics. 2020, 178(8), 3247–3261. doi:10.1007/s00024-021-02799-8
5. LeBlond P.H., Mysak L.A. Waves in the ocean. Elsevier Scientific Publishing Company. 1978, 602. doi:10.1016/s0422-9894(08) x7037-0
6. Bulatov V.V., Vladimirov Yu.V. Analytical solutions of the internal gravity wave equation in a stratified medium with shear flows. Computational Mathematics and Mathematical Physics. 2019, 59, 7, 1121–1130. doi:10.1134/S0965542519070030
7. Bulatov V.V. New problems of mathematical modeling of wave dynamics of stratified media. 2021. Moscow, Publishing House “OntoPrint”, 277 p. (In Russian).
8. Bulatov V.V., Vladimirov Yu.V. Dynamics of internal gravity waves in the ocean with shear flows. Russian Journal of Earth Sciences. 2020, 20, ES4004. doi:10.2205/2020ES000732
9. Gottwald G.A., Pelinovsky D.E. On the impossibility of solitary Rossby waves in meridionally unbounded domains. Physics of Fluids. 2018, 30(11), 116601. doi:10.1063/1.5052191
10. Fabrikant A.L. Reflection of Rossby waves from the surface of the tangential velocity discontinuity. Izvestiya Akademii Nauk SSSR, Fizika Atmosfery i Okeana. 1987, 23, 106–109 (In Russian).
11. Fabrikant A.L., Stepanyants Yu.A. Propagation of waves in shear flows. 1998. World Scientific, Singapore, 287 p.
12. Stepanyants Y.A., Fabrikant A.L. Propagation of waves in hydrodynamic shear flows. Uspekhi Fizicheskih Nauk. 1989, 32 (9), 783–805, doi:10.1070/PU1989v032n09ABEH002757
13. Gnevyshev V.G., Shrira V.I. On the evaluation of barotropic-baroclinic instability parameters of zonal flows on a betaplane. Journal of Fluid Mechanics. 1990. 221, 161–181. doi:10.1017/S0022112090003524
14. Ahmed B.M., Eltaeb I.A. On the propagation, reflexion, transmission and stability of atmospheric Rossby-gravity waves on a beta-plane in the presence of latitudinally sheared zonal flows. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1980, 298 (1435), 45–85. doi:10.1098/rsta.1980.0240
15. Duba C.T., Doyle T.B., McKenzie J.F. Rossby wave patterns in zonal and meridional winds. Geophysical & Astrophysical Fluid Dynamics. 2014, 108 (3), 237–257. doi:10.1080/03091929.2013.867604
16. Yamagata T. On the propagation of Rossby waves in a weak shear flow. Journal of the Meteorological Society of Japan. 1976, 54 (2), 126–128. doi: 10.2151/jmsj1965.54.2_126
17. Yamagata T. On trajectories of Rossby wave-packets released in a lateral shear flow. Journal of the Oceanographic Society of Japan. 1976, 32, 162–168. doi:10.1007/BF02107270
18. Salmon R. Lectures on geophysical fluid dynamics. Oxford University Press. 1st ed. 1998, 392 p.
19. Killworth P.D., Blundell J.R. Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part I. The local problem. Journal of Physical Oceanography. 2003, 33 (4), 784–801. doi:10.1175/1520–0485(2003)33<
20. Gnevyshev V.G., Frolova A.V., Kubryakov A.A., Sobko Yu.V., Belonenko T.V. Interaction between Rossby waves and a jet flow: basic equations and verification for the antarctic circumpolar current. Izvestiya, Atmospheric and Oceanic Physics. 2019, 55 (5), 412–422. doi:10.1134/S0001433819050074
21. Gnevyshev V.G., Frolova A.V., Koldunov A.V., Belonenko T.V. Topographic effect for Rossby waves on a zonal shear flow. Fundamental and Applied Hydrophysics. 2021, 14, 1, 4–14. doi:10.7868/S2073667321010019 (in Russian).
Review
For citations:
Gnevyshev V.G., Belonenko T.V. Analytical Solution of the Ray Equations of Hamilton for Rossby Waves on Stationary Shear Flows. Fundamental and Applied Hydrophysics. 2022;15(2):8-18. https://doi.org/10.59887/fpg/4eh4-83zr-r1fm
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