Why We Can Approximate Spheroidal Geopotential Surfaces as Spherical but Can't Approximate True Geoptential Surfaces as Spheroidal in NWP

Three types of gravity and corresponding geopotentials exist in the atmosphere due to mass density and rotation of the solid Earth. The standard gravity g_s is radial and perpendicular to the spherical geopotential (Φ_s) surfaces for the solid Earth with homogeneous mass density and without rotation. The apparent (effective) gravity g_a is non-radial and perpendicular to the spheroidal geopotential (Φ_a) surfaces for the solid Earth with homogeneous mass density and rotation. The true gravity g_t is also non-radial and perpendicular to the true geopotential (Φ_t) surfaces for the solid Earth with inhomogeneous mass density and rotation.

The spherical geopotential (Φ_s) is the simplest and adopted in almost all models in NWP after using two approximations: (1) spheroidal (ellipsoidal) geopotential approximation (EGA) which is to approximate the true geopotential surfaces as spheroidal, and (2) spherical geopotential approximation (SGA) which is to approximate the spheroidal geopotential surfaces as spherical. The EGA and SGA involve metric (geometrical) and horizontal pressure gradient (dynamical) errors. The metric errors in EGA and SGA are negligible (< 0.3%). The horizontal pressure gradient error is negligible in SGA (~ 0.3%) but is non-negligible and equals the horizontal gravity disturbance vector δg (≡ g_t - g_a) in EGA.

Several non-dimensional numbers are proposed to identify the magnitude of horizontal gravity disturbance vector δg versus traditional horizontal forcing terms such as pressure gradient force, Coriolis force, and surface wind stress. These non-dimensional numbers are calculated from publicly available and independent datasets such as the geoid undulation (N) from the static gravity field model EIGEN-6C4, long-term mean geopotential height (Z), wind velocity (u, v), temperature (T) at 12 pressure levels in troposphere from the NCEP/NCAR reanalyzed climatology. The results demonstrate the EGA error is non-negligible in geostrophic wind, geostrophic vorticity, Q vector, Omega equation, and Ekman layer dynamics. Thus, EGA is invalid. The horizontal gravity disturbance vector δg should be added to NWP numerical models.

References

Chu, P. C. (2021). True gravity in ocean dynamics Part-1 Ekman transport. Dynamics of Atmospheres and Oceans, 96, 101268. https://www.sciencedirect.com/science/article/pii/S0377026521000634?via%3Dihub

Chu, P. C. (2023). Horizontal gravity disturbance vector in atmospheric dynamics. Dynamics of Atmospheres and Oceans,102, 101369. https://www.sciencedirect.com/science/article/pii/S0377026523000209

Chu, P.C. (2024). Invalid spheroidal geopotential approximation and non-decomposable centrifugal acceleration from gravity. Dynamics of Atmospheres and Oceans, 105, 101450, https://www.sciencedirect.com/science/article/abs/pii/S0377026524000186.