## Anomalous Behavior of the Vertical Structure of Rossby Waves on Non-Zonal Shear Flow in the Vicinity of the Focus

*V. G. Gnevyshev ^{1}, T. V. Belonenko^{2, ✉}*

^{1} P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russian Federation

^{2} St Petersburg University, St Petersburg, Russian Federation

^{✉} e-mail: btvlisab@yandex.ru

### Abstract

*Purpose*. The work aims to study the behavior of vertical barotropic-baroclinic modes of Rossby waves on a non-zonal shear flow in the vicinity of the focus.

*Methods and Results*. Inferred from the reference equation, we consider some variants of the behavior of eigenfunctions in the vicinity of the focus. It is shown that the number of possible variants for non-zonal flows increases compared with the zonal case. This means that qualitatively new additional scenarios appear in the case of a non-zonal flow compared with the problem for internal waves when the behavior of Rossby waves in the vicinity of the localization level qualitatively coincides with the behavior for the zonal case, herewith the coefficient of a passage through the focus is always exponentially small. The solution becomes extremely sensitive to the initial parameters of the wave incident on the non-zonal focus. Another important point is that the second, additional anomalous focus appears on the non-zonal flow. When a wave falls on this focus on one side, it behaves like a classic focus with a classic wave adhering. And when falling from the opposite side, the Rossby wave does not notice the focus and passes it without a short-wave transformation. In the problem, abnormal scenarios appear with the passage of the focus without difficulty with a coefficient of passage equal to one in addition to the scenario with an infinitely long time adhering to the focus and an exponentially small coefficient of passage.

*Conclusions*. The anomalous behavior of Rossby waves in the horizontal plane on non-zonal flows is accompanied by anomalous behavior of the vertical mode, in contrast to the strictly zonal case of flow with different kinematics. The eigenvalues of the Sturm-Liouville problem change abruptly during the transition from the non-zonal to the zonal case. As a consequence, there is no limit transition from a weakly non-zonal case to a strictly zonal one. Such an extremely ambiguous analytical behavior of Rossby waves in the vicinity of the focus on baroclinic non-zonal flows rather indicates the absence of analytical prediction and the need for a deeper and more detailed analysis using numerical methods.

### Keywords

Rossby waves, non-zonal flow, focus, Sturm-Liouville problem

### Acknowledgements

The study was carried out with the support of the RSF grant No. 22-27-00004 and within the framework of the state assignment on theme 0128-2021-0003.

### Original russian text

Original Russian Text © V. G. Gnevyshev, T. V. Belonenko, 2022, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 38, Iss. 6, pp. 585-604 (2022)

### For citation

Gnevyshev, V.G. and Belonenko, T.V., 2022. Anomalous Behavior of the Vertical Structure Rossby Waves on Non-Zonal Shear Flow in the Vicinity of the Focus. *Physical Oceanography*, 29(6), pp. 567-586. doi:10.22449/1573-160X-2022-6-567-586

### DOI

10.22449/1573-160X-2022-6-567-586

### References

- LaCasce, J.H., 2017. The Prevalence of Oceanic Surface Modes.
*Geophysical Research Letters*, 44(21), pp. 11097-11105. doi:10.1002/2017gl075430 - Bulatov, V. and Vladimirov, Yu., 2020. Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows.
*Symmetry*, 12(11), 1865. doi:10.3390/sym12111865 - Gnevyshev, V.G., Badulin, S.I. and Belonenko, T.V., 2020. Rossby Waves on Non-Zonal Currents: Structural Stability of Critical Layer Effects.
*Pure and Applied Geophysics*, 177, pp. 5585-5598. doi:10.1007/s00024-020-02567-0 - Gnevyshev, V.G., Badulin, S.I., Koldunov, A.V. and Belonenko, T.V., 2021. Rossby Waves on Non-Zonal Flows: Vertical Focusing and Effect of the Current Stratification.
*Pure and Applied Geophysics*, 178, pp. 3247-3261. doi:10.1007/s00024-021-02799-8 - Shi, Y., Yang, D., Feng, X., Qi, J., Yang, H. and Yin, B., 2018. One Possible Mechanism for Eddy Distribution in Zonal Current with Meridional Shear.
*Scientific Reports*, 8, 10106. doi:10.1038/s41598-018-28465-z - Reznik, G.M. and Kravtsov, S.V., 2020. Singular Vortices on a Beta-Plane: A Brief Review and Recent Results.
*Physical Oceanography*, 27(6), pp. 659-676. doi:10.22449/1573-160X2020-6-659-676 - Kravtsov, S. and Reznik, G., 2020. Monopoles in a Uniform Zonal Flow on a QuasiGeostrophic β-Plane: Effects of the Galilean Non-Invariance of the Rotating Shallow-Water Equations.
*Journal of Fluid Mechanics*, 909, A23. doi:10.1017/jfm.2020.906 - Sutyrin, G.G., 2020. How Oceanic Vortices Can Be Super Long-Lived.
*Physical Oceanography*, 27(6), pp. 677-691. doi:10.22449/1573-160X-2020-6-677-691 - Gnevyshev, V.G. and Belonenko, T.V., 2021. Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis.
*Physical Oceanography*, 28(5), pp. 486-498. doi:10.22449/1573-160X-2021-5-486-498 - Gnevyshev, V.G. and Belonenko, T.V., 2021. Parabolic Traps of Rossby Waves in the Ocean.
*Fundamental and Applied Hydrophysics*, 14(4), pp. 14-24. https://doi.org/10.7868/S207366732104002X (in Russian). - Gnevyshev, V.G. and Belonenko, T.V., 2022. Analytical Solution of the Ray Equations of Hamilton for Rossby Waves on Stationary Shear Flows.
*Fundamental and Applied Hydrophysics*, 15(2), pp. 8-18. doi:10.48612/fpg/4eh4-83zr-r1fm - Bulatov, V.V. and Vladimirov, Yu.V., 2013. Fields of Internal Gravity Waves in Heterogeneous and Non-Stationary Stratified Media.
*Fundamental and Applied Hydrophysics*, 6(2), 55-70 (in Russian). - Gnevyshev, V.G. and Shrira, V.I., 1989. Dynamics of Rossby Wave Packets in the Vicinity of the Zonal Critical Layer Taking into Account Viscosity.
*Izvestiya Akademii Nauk SSSR, Fizika Atmosfery i Okeana*, 25(10), 1064-1074 (in Russian). - Gnevyshev, V.G. and Shrira, V.I., 1989. Kinematics of Rossby Waves on Non-Uniform Meridional Current.
*Okeanologiya*, 29(4), 543-548 (in Russian). - Badulin, S.I. and Shrira, V.I., 1993. On the Irreversibility of Internal-Wave Dynamics due to Wave Trapping by Mean Flow Inhomogeneities. Part 1. Local Analysis.
*Journal of Fluid Mechanics*, 251, pp. 21-53. doi:10.1017/S0022112093003325 - Tulloch, R., Marshall, J. and Smith, K.S., 2009. Interpretation of the Propagation of Surface Altimetric Observations in Terms of Planetary Waves and Geostrophic Turbulence.
*Journal of Geophysical Research: Oceans*, 114(C2), C02005. doi:10.1029/2008jc005055 - Killworth, P.D. and Blundell, J.R., 2003. 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*, 33(4), pp. 784-801. https://doi.org/10.1175/1520-0485(2003)33%3C784:LEPWPI%3E2.0.CO;2 - Killworth, P.D. and Blundell, J.R., 2005. The Dispersion Relation for Planetary Waves in the Presence of Mean Flow and Topography. Part II: Two-Dimensional Examples and Global Results.
*Journal of Physical Oceanography*, 35(11), pp. 2110-2133. doi:10.1175/JPO2817.1 - LeBlond, P.H. and Mysak, L.A., 1978.
*Waves in the Ocean*. Elsevier Oceanography Series, vol. 20. Amsterdam: Elsevier, 602 p. - Erokhin, N.S. and Sagdeev, R.Z., 1985. To the Theory of Anomalous Focusing of Internal Waves in a Two-Dimensional Non-Uniform Fluid. Part I: A Stationary Problem.
*Morskoy Gidrofizicheskiy Zhurnal*, (2), pp. 15-27 (in Russian). - Erokhin, N.S. and Sagdeev, R.Z., 1985. On the Theory of Anomalous Focus of Internal Waves in Horizontally-Inhomogeneous Fluid. Part 2. Precise Solution of Two-Dimensional Problem with Regard for Viscosity and Non-Stationarity.
*Morskoy Gidrofizicheskiy Zhurnal*, (4), pp. 3-10 (in Russian). - Yamagata, T., 1976. On the Propagation of Rossby Waves in a Weak Shear Flow.
*Journal of the Meteorological Society of Japan. Ser. II*, 54(2), pp. 126-128. https://doi.org/10.2151/jmsj1965.54.2_126 - Yamagata, T., 1976. On Trajectories of Rossby Wave-Packets Released in a Lateral Shear Flow.
*Journal of the Oceanographic Society of Japan*, 32(4), pp. 162-168. https://doi.org/10.1007/BF02107270 - Stepanyants, Yu.A. and Fabrikant, A.L., 1989. Propagation of Waves in Hydrodynamic Shear Flows.
*Soviet Physics Uspekhi*, 32(9), pp. 783-805. doi:10.1070/PU1989v032n09ABEH002757