True Gravity in Atmospheric Ekman Layer Dynamics

True gravity is a three-dimensional vector, g = igλ+jgφ+kgz, with (λ, φ,
z) the (longitude, latitude, height) and (i, j, k) the corresponding
unit vectors. The vertical direction is along g, not along k, which is
normal to the Earth spherical (or ellipsoidal) surface (called
deflected-vertical). Correspondingly, the spherical (or ellipsoidal)
surfaces are not horizontal surfaces (called deflected-horizontal
surfaces). In the (λ, φ, z) coordinates, the true gravity g has
longitudinal-latitudinal component, gh = igλ+jgφ, but it is neglected
completely in meteorology through using the standard gravity (-g0k, g0 =
9.81 m/s2) instead. Such simplification on the true gravity g has never
been challenged. This study uses the atmospheric Ekman layer as an
example to illustrate the importance of gh. The standard gravity (-g0k)
is replaced by the true gravity g in the classical atmospheric Ekman
layer equation with a constant eddy viscosity (K) and a
height-dependent-only density ρ(z) represented by an e-folding
stratification. New formulas for the Ekman spiral and Ekman pumping are
obtained. The second derivative of the gravity disturbance (T) versus z,
also causes the Ekman pumping, , in addition to the geostrophic
vorticity with DE the Ekman layer thickness and f the Coriolis
parameter. With data from the EIGEN-6C4 static gravity model, the global
mean strength of the Ekman pumping due to the true gravity is found to
be 4.0 cm s-1. Such evidently large value implies the urgency to include
the true gravity g into the atmospheric dynamics.

27 Oct 2021