Doppler effect and Rossby waves in the ocean: A brief history and new approaches

The review is devoted to Rossby waves in the ocean. Currently, there are many monographs and scientific articles on this topic, in which the authors approach the presentation of the material in different ways. For researchers who have not delved too deeply into this topic, the analysis of these sources is often perceived as a collection of disparate and often contradictory information that does not allow an adequate understanding of the subject. The review is based on the analysis of the main publications on this topic, and it systematizes the main ideas in various aspects. We also give the readers a comparison of different approaches in this area. Particular attention is paid to the review of the dispersion relations of Rossby waves in the presence of a background flow with an emphasis on the presence or absence of a Doppler additive to the frequency. Although the problem statements and variance ratios under consideration are generally well-known, however, they are presented in significantly different ways by many authors, which often leads to misunderstanding and confusion. We draw the reader’s attention to the key controversial points and bring various approaches into a single coherent system. If long Rossby waves do not “feel” the flow, then this is true for the “shallow water” model and is a consequence of the Galilean non-invariance of the dispersion relation. Considering various approaches, we show that there is no strict dispersion relation for the Galilean-non-invariant dispersion relation. Some not quite strict assumptions and assumptions are always added, such as the formal existence of vertical boundaries or the dependence of the barotropic radius on the variable transverse coordinate. The derivation of the dispersion relation with the Doppler shift also contains some asymptotic expansions, accompanied by an analysis of the theory of dimensions. Using common terminology, we combine the main analytical results on the topic and present them in a single logic.