Revealing horizontal gravity force in geopotential coordinates via metric tensors

Abstract. Earth gravity force g is represented by geopotential Φ, g =∇Φ , with the three-dimensional gradient operator. True gravity g has horizontal component. Oceanographic and meteorological communities use two approaches to eliminate horizontal gravity force. The first one is to replace Φ by geopotential for uniform Earth mass density (Φ_uniform) representing no horizontal gravity force, such as spherical, spheroidal, or recently developed Realistic, Ellipsoidal, Analytically Tractable (GREAT) Φ_GREAT on the base of |Φ - Φ_uniform | <<| Φ_uniform |, however, it is not sufficient to justify because we still need to compare ∇(Φ - Φ_uniform) to other forces such as Coriolis force and pressure gradient force. The second approach is to use Φ to establish geopotential coordinates. Consider local Cartesian coordinates (ξ, η, ζ) with unit vectors (ξ^, η^, ζ^). The geopotential coordinates (x, y, Z) and corresponding unit vectors (x̂, ŷ, Ẑ) are defined by x = Z = -Φ/g₀, g₀= 9.81 m/s². From such a relationship, metric tensors between (ξ, η, ζ) and (x, y, Z), and in turn the horizontal gradient operator are obtained. The pressure gradient is ∇_C p = (∂_ξp)ξ^ + (∂_ηp)η^ + (∂_ζp)ζ^ in the local Cartesian coordinates, and ∇_G p = (∂_xp + ∂_xN∂_zp)x̂ + (∂_yp + ∂_yN∂_zp)ŷ + (∂_Zp)Ẑ in the geopotential coordinates with N the geoidal undulation. The horizontal gravity force exists in horizontal momentum equation with the geopotential coordinates. Importance of the horizontal gravity force is verified using two publicly available datasets. Five concerns with PNAS-e2416636121 are presented in Appendix. The major equation in PNAS-e2416636121, ∇p = (∂_ξp)ξ^ + (∂_ηp)η^ + (∂_ζp)ζ^ = (∂_xp)x̂ + (∂_yp)ŷ +(|∇Z|∂_Zp)Ẑ, is valid only for the gravity with no horizontal component, i.e., for the geopotential coordinates coincident with the local Cartesian coordinates.