Dynamically consistent shallow-water equation sets in non-spherical geometry with latitudinal variation of gravity

The shallow-water equations in spherical geometry have proven to be an invaluable prototypical tool to advance geophysical fluid dynamics. Many of the fundamental terms and properties, including dynamical consistency, needed to model Earth's atmosphere and oceans are embodied within them. Limitations to their wider use include the following: representation of the Earth as a sphere, rather than as an ellipsoid; gravity not varying latitudinally, but being held constant; the Coriolis force being incompletely represented; and no representation of vertical acceleration. Recent work has addressed the first three limitations, but not simultaneously. The present work addresses all four simultaneously by endowing the equations selectively with supplementary terms, whilst respecting dynamical consistency. This is accomplished using Hamilton's principle of least action, but could be done otherwise. It leads to a switch-controlled quartet of shallow-water equation sets, all of which include topography. Potential future applications include the following: further sensitivity tests to assess the possible impact of latitudinal variation of gravity; development of improved numerical methods with good conservation properties; examination of the stability of discretizations of the Coriolis terms; and sensitivity tests for the possible impact of vertical acceleration as a function of decreasing horizontal scale. Sensitivity experimentation would, however, necessitate scale analysis of flow regimes to ensure correct interpretation of results.