CGD Special Seminar- Why We Can Approximate Spheroidal Geopotential Surfaces as Spherical but Can’t Approximate True Geopotential Surfaces as Spheroidal in Atmospheric and Oceanic Dynamics

Spherical, spheroidal, and true geopotentials exist in the atmosphere and oceans. The spherical geopotential (Φs) coordinate system [λ, φ, r] s associated with the standard gravity gs is for the Earth with homogeneous mass density and without rotation. The spheroidal geopotential (Φa) coordinate system [λ, φ, z]a associated with the apparent gravity ga is for the Earth with homogeneous mass density and rotation. Here, λ is the longitude, φ the geocentric latitude, r the radial, and z the spheroidal (ellipsoidal) height. The true geopotential (Φt) following coordinate system [λ, φ, zt]t associated with the true gravity gt for the Earth with inhomogeneous mass density and rotation. The orthometric height zt depends on (λ, φ), but (r, z) do not. The spherical geopotential is used in almost all atmospheric and oceanic models after two approximations: (1) spheroidal 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 two approximations involve errors in metric terms and horizontal (i.e., on geopotential surfaces) pressure gradient. The metric-term errors are negligible (< 0.003) in both approximations. However, the horizontal pressure gradient force error is negligible in the SGA (~ 0.003) but non-negligible and equals the horizontal gravity disturbance vector g0∇N in the EGA due to dependence of zt on (λ, φ). Here, g0 = 9.81 m/s2 , N is the geoid height. Four publicly available datasets are used to confirm the rejection of the EGA: (a) the global static gravity field model EIGEN-6C4 for the geoid N (from http://icgem.gfz-potsdam.de/home), (b) NCAR/NCEP reanalyzed monthly long-term mean geopotential height (Z) and wind velocity (u, v), at 12 pressure levels 1,000, 925, 850, 700, 600, 500, 400, 300, 250, 200, 150, and 100 hPa (from https://psl.noaa.gov/data/gridded/data.ncep.reanalysis.derived.html), (c) the climatological annual mean temperature and salinity from the NCEI WOA23 for the sea water density (ρ) data (from https://www.ncei.noaa.gov/products/world-ocean-atlas) and (d) the climatological annual mean surface wind stress (τλ, τφ) from the Atlas of Surface Marine Data 1994 (from https://iridl.ldeo.columbia.edu/SOURCES/.DASILVA/.) The magnitude of g0∇N is comparable to the Coriolis force with the ratio from 0.62 on 1,000 hPa to 0.18 on 100 hPa, and to the horizontal pressure gradient force with the ratio of 0.41 on 1,000 hPa to 0.19 on 100 hPa. With such evidence, it is urgent to include the horizontal gravity disturbance vector (g0∇N) in any analytical or numerical oceanic models.