On the Theory of Symmetric Instability of Time-Periodic Flows with a Complete Account for the Coriolis Force

Inertial instability of the time-periodic Kolmogorov flow in a rotating fluid with the full account of the Coriolis force

“…Consider the motion of the atmosphere or ocean on a generalized f-plane, where the Coriolis force is accounted of fully. The basic flow, whose stability is studied, constitutes an angle φ with zonal direction (figure 1); see also Kurgansky (2022).…”

“…This approach is convenient theoretically and is more justified for flows in the deep ocean or in the stellar interior far from horizontal boundaries that bear the imprint of gravity effects. A similar transformation (at φ = 0) was performed in Kurgansky (1980) when solving the geostrophic adjustment problem with the full account of the Coriolis force; see also Tort et al (2016) and Zeitlin (2016) where this transformation was made when studying inertial instability on the f plane with complete Coriolis force. In the notation δ = f cos φ/f, we have more concisely (η = y − δ z, ζ = z) and the spatial derivatives transform according to the rules…”

“…Throughout the work we will consider that the Kolmogorov flow is not steady, unlike in Kurgansky (2021), but that its amplitude is modulated over time according to a periodic law, for example, due to the action of tidal forces or longitudinal libration, as the latter occurs in a laboratory experiment or on other planets (and their satellites) (Noir et al 2009, Ghasemi et al 2016. We note that the study of inertial (symmetric) instability of time-periodic flows having a linear velocity profile, with an exact account of the Coriolis force and the arbitrary orientation of the flow with respect to the zonal direction, was carried out in Kurgansky (2022). In general, time-periodic flows constitute an important special class of fluid motions, and their stability has long been an object of study in both geophysical and classical fluid dynamics (Davis 1976, Samelson 2009).…”