Explore the dynamics of planetary-scale fluid flows in this 43-minute conference talk delivered at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the effects of Earth's rotation on ocean and atmospheric motion, focusing on the crucial role of the Coriolis force in large-scale flow dynamics. Examine the behavior of fluid systems near a shear flow under the beta-plane approximation, comparing and contrasting it with non-Coriolis cases. Gain insights into the richer dynamics introduced by the Coriolis effect and understand how the spectral properties of the linearized operator around the shear flow contribute to these complex behaviors. This talk, part of the Thematic Programme on "The Dynamics of Planetary-scale Fluid Flows," offers a deep dive into the mathematical and physical aspects of geophysical fluid dynamics.