Journal Article

Almost Everything You Always Wanted to Know About Representing Gravity in Global Models but Were Afraid to Ask

Staniforth, Andrew, White, Andy

2025

Overview

For over 60 years, an approximation involving spherical geopotentials has underlain the representation of gravity in global numerical models of Earth's atmosphere and oceans. This article explores how departures from sphericity can be allowed by assuming spheroidal geopotentials instead. A route to more accurate model formulation is indicated, and the theoretical basis of the classical spherical‐geopotential approximation is illuminated too. An overview—from the time of Newton to the present—is given of the development of zeroth‐ and first‐order approximations to the geopotential field in terms of two small non‐dimensional parameters. The importance of using geopotential coordinate systems in atmospheric and oceanic models is emphasized. Early suggestions for such systems using non‐spherical coordinates involved qualitatively inappropriate choices of ellipsoids derived from families of confocal ellipses. Specific examples of appropriate ellipsoid choices are considered before presentation of the recently developed Geophysically Realistic, Ellipsoidal, Analytically Tractable (GREAT) system. This is based on a suitably constructed geopotential field approximation, first order accurate, from which—without further approximation—may be analytically derived equations for geopotential surfaces and surfaces orthogonal to them. GREAT coordinates satisfy stated desiderata for geopotential coordinate systems and are applicable both above and below Earth's geoid (assumed to coincide with the WGS 84 [2004, https://gis‐lab.info/docs/nima‐tr8350.2‐wgs84fin.pdf] reference ellipsoid to an excellent approximation). GREAT‐coordinate analysis provides justification for the classical spherical‐geopotential approximation: It is revealed as a mathematical limit, but not a physically realizable one. Attention is drawn to a certain partially spherical limit that is realizable physically. Plain Language Summary Gravity is by far the dominant external force in the equations of motion for Earth's atmosphere and oceans. It is crucially important that it be adequately represented in global atmospheric and oceanic models for climate and weather prediction. In principle, one simply applies Newton's inverse square law of gravitational attraction to represent gravity in the equations of motion. In practice, this is far too complicated to do exactly—due principally to the rotating Earth being closer to spheroidal than spherical in shape, with an inhomogeneous mass distribution—thereby necessitating approximation. The dominant nature of gravity means that forecast accuracy is enhanced if one constructs an orthogonal coordinate system to integrate the equations of motion, whereby gravity only acts in the vertical and not in the horizontal. This is termed a geopotential coordinate system and, to further complicate matters, its construction is intrinsically coupled to a suitable approximation of gravity. We first review the basic concepts to represent gravity in global atmospheric and oceanic models. Next, we discuss the importance and principles of geopotential coordinates for modeling purposes. Various geopotential coordinate systems of varying accuracy are then compared. Finally, we outline some possible developments for representing gravity in future models. Key Points We review the derivation from first principles of geopotential approximations to represent gravity in global atmospheric and oceanic models Various geopotential coordinate systems of varying accuracy are compared, leading to one that satisfies all desiderata for such a system The low‐order, ubiquitous, classical, spherical‐geopotential approximation is shown to be the asymptotic limit of a more accurate one

• Publisher: John Wiley & Sons, Inc,American Geophysical Union (AGU)
• Subject: Approximation / Atmosphere / Coordinate systems / Dynamic height / Earth / Geographical information systems / Geoid / Geopotential / geopotential approximation / geopotential coordinates / geopotential theory and determination / Global positioning systems / GPS / gravitational fields / Gravity / GREAT coordinates / hydrodynamic modeling / lunar and planetary geodesy and gravity / Mathematical models / Numerical models / Ocean models / Oceans / reference systems / spherical geopotential approximation / spheroidal geopotential approximation