The rotation of the planet, the stratification of density, and the generation of friction all play important roles in the motion of geophysical fluids. These mechanisms are always present in our ocean and atmosphere, as well as that of other planets. They are collectively responsible for many well-known phenomena that we observe in nature, e.g., jet streams, zonal jets, the El Niño cycle, and Jupiter's Great Red Spot, to name only a few. These mechanisms typically serve to constrain the motion of the fluid in a very particular way. For instance, it is observed that in a rapidly rotating fluid in three-dimensions, particles that are aligned along a common vertical parallel to the axis of rotation move nearly in unison, thus rendering the overall motion of the flow to be essentially two-dimensional. Despite many experimental and computational efforts to understand the precise development of such phenomena, the mathematical justification for them, that is, from directly studying the equations of motion themselves, remains largely open. This project will systematically address such concerns in various geophysical settings. This project will also provide research and mentorship opportunities for students at the undergraduate and graduate levels, as well as postdoctoral scholars. An overarching goal of this project is to understand various manifestations of finite-dimensionality and its interconnections with the mechanisms of dissipation, rotation, and stratification. The main