Doppler and Non-Doppler Shifts in Dispersion Relations for Rossby Waves and Galilean Invariance

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.