Description
- Abstract:
- Abstract of Geophysical Fluid Dynamics and the Calculus of Moving Space, by Joel Feske, Ph.D., Brown University, May 2026. Geophysical fluid dynamics (GFD) is the study of rotating, stratified flow over the rough topography of a nearly spheroidal Earth in the Newtonian limit of gravity and velocity. While most fields of fluid dynamics quickly introduce an Eulerian description---the fluid moving through a stationary coordinate frame, and a Lagrangian description---the coordinate frame moving and deforming with the fluid, GFD often takes place in semi-Lagrangian coordinates that change in time as the fluid moves, but are not carried along with it---for example following surfaces of constant density or bottom topography. In this work, we use and extend tools from tensor analysis and differential geometry to build a coordinate-agnostic description of the core GFD equations. This framework expresses the equations of motion for geophysical fluids in totally generalized, time-varying, curvilinear coordinates, allowing both straight-forward algebraic manipulation and a strong connection to geometric meaning. The second chapter defines the “setting velocity,” the velocity field specifying moving coordinates throughout space and time, and defines what it means to take a time derivative while traveling along an arbitrary path through such moving coordinates. We separate conceptually the changes due to fluid motion and the changes due to coordinate motion and classify the apparent forces that arise in deforming coordinate systems. The third chapter lays out the full GFD equations, specifies basis vectors and metrics for all common GFD coordinates, and introduces common approximations and parameterizations using the standard projection operators of tensor analysis. The final chapter discusses the appearance of integral theorems in this framework and presents a curious result absent from more standard treatments of differential geometry and exterior calculus. This work lays the foundation for more generalized and modular approaches to oceanic and atmospheric modeling, allowing models written in different coordinates to share methods. Generalized coordinate expressions for exact equations, upon which approximations may be layered, allow the rigorous integration of higher-order numerical schemes, hybrid-coordinates, and Arbitrary Lagrangian-Eulerian methods.
- Notes:
- Thesis (Ph. D.)--Brown University, 2026
Citation
Feske, Joel Simon,
"Geophysical Fluid Dynamics and the Calculus of Moving Space"
(2026).
Earth, Environmental and Planetary Sciences Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:2dqc8q29/
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Earth, Environmental and Planetary Sciences Theses and Dissertations
Theses and Dissertations for the Earth, Environmental and Planetary Sciences department....