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Dynamically consistent shallow‐water equation sets in non‐spherical geometry with latitudinal variation of gravity
Author(s) -
Staniforth Andrew
Publication year - 2015
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.2533
Subject(s) - shallow water equations , representation (politics) , acceleration , sensitivity (control systems) , consistency (knowledge bases) , scale (ratio) , stability (learning theory) , flow (mathematics) , ellipsoid , gravitational acceleration , geology , gravitation , mathematics , classical mechanics , physics , geometry , geodesy , mathematical analysis , computer science , quantum mechanics , electronic engineering , machine learning , law , engineering , politics , political science
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.