Russian Journal of Earth Sciences Russian Journal of Earth Sciences Russian Journal of Earth Sciences 1681-1208 95700 10.2205/2025ES001012 cfncgt ОРИГИНАЛЬНЫЕ СТАТЬИ ORIGINAL ARTICLES ОРИГИНАЛЬНЫЕ СТАТЬИ Doppler and Non-Doppler Shifts in Dispersion Relations for Rossby Waves and Galilean Invariance Doppler and Non-Doppler Shifts in Dispersion Relations for Rossby Waves and Galilean Invariance https://orcid.org/0000-0001-6654-5570 Гневышев Владимир Григорьевич Gnevyshev Vladimir Grigor'evich Белоненко Татьяна Васильевна Belonenko Tatyana Vasil'evna btvlisab@yandex.ru Институт океанологии им. П. П. Ширшова РАН Россия P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences Russian Federation Санкт-Петербургский государственный университет St Petersburg Россия St Petersburg University St Petersburg Russian Federation Санкт-Петербургский государственный университет Санкт-Петербург Россия Saint Petersburg State University Saint Petersburg Russian Federation СПбГУ Санкт-Петербург Россия SPbU Saint Petersburg Russian Federation 07 07 2025 07 07 2025 25 4 ES4006 03 03 2025 09 04 2025 https://rjes.ru/en/nauka/article/95700/view

The purpose of this work is to draw the reader's attention to a paradoxical fact – the existence of two different dispersion relations for linear Rossby waves: with Doppler and nonDoppler shifts. This paper highlights aspects arising from studying the interaction of Rossby waves and large-scale stationary flows within the framework of the linear wave approximation. The methods used in the work consist of the analysis of dispersion relations obtained by different authors. They are subordinated to the main task of the study – to establish where and when a non-Doppler shift appears in the system of two-dimensional linear equations of Rossby waves. Assuming that the flow is homogeneous, additional terms appear in the dispersion relation of Rossby waves for the solution in a plane wave, which can have both Doppler and non-Doppler effects. The paper shows that the non-Doppler character of the dispersion relation of Rossby waves on the current appears due to an additional assumption about the slope of the free surface, or the slope of the interface in a two-layer model (pycnocline for the ocean, and tropopause for the atmosphere). It is established that to derive some of these relations, excessive requirements for boundary conditions or separate terms in the equation for potential vorticity were previously applied. It is shown that to deduce the dispersion relation of Rossby waves with a non-Doppler shift, it is not necessary to throw out the topographic term in the boundary condition or abandon the hydrostatic approximation.

The purpose of this work is to draw the reader's attention to a paradoxical fact – the existence of two different dispersion relations for linear Rossby waves: with Doppler and nonDoppler shifts. This paper highlights aspects arising from studying the interaction of Rossby waves and large-scale stationary flows within the framework of the linear wave approximation. The methods used in the work consist of the analysis of dispersion relations obtained by different authors. They are subordinated to the main task of the study – to establish where and when a non-Doppler shift appears in the system of two-dimensional linear equations of Rossby waves. Assuming that the flow is homogeneous, additional terms appear in the dispersion relation of Rossby waves for the solution in a plane wave, which can have both Doppler and non-Doppler effects. The paper shows that the non-Doppler character of the dispersion relation of Rossby waves on the current appears due to an additional assumption about the slope of the free surface, or the slope of the interface in a two-layer model (pycnocline for the ocean, and tropopause for the atmosphere). It is established that to derive some of these relations, excessive requirements for boundary conditions or separate terms in the equation for potential vorticity were previously applied. It is shown that to deduce the dispersion relation of Rossby waves with a non-Doppler shift, it is not necessary to throw out the topographic term in the boundary condition or abandon the hydrostatic approximation.

 Rossby waves dispersion relation doppler shift non-doppler Galilean invariance Rossby waves dispersion relation doppler shift non-doppler Galilean invariance The research was supported by St. Petersburg University (grant no. 129659573), the Russian Science Foundation (RSF, grant no. 25-17-00021). V. G. Gnevyshev was received within the framework of the state assignment of Ministry of Science and Higher Education of the Russian Federation through Grant no. FMWE-2024-0017. The research was supported by St. Petersburg University (grant no. 129659573), the Russian Science Foundation (RSF, grant no. 25-17-00021). V. G. Gnevyshev was received within the framework of the state assignment of Ministry of Science and Higher Education of the Russian Federation through Grant no. FMWE-2024-0017.

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