True gravity in ocean dynamics Part 1 Ekman transport

The direction normal to the Earth spherical (or ellipsoidal) surface is not vertical (called deflected vertical) since the vertical direction is along the true gravity g (= ig_λ+ jg_φ+ kg_z). Here, (λ, φ, z) are (longitude, latitude, depth), and (i, j, k) are the corresponding unit vectors. The spherical (or ellipsoidal) surfaces are not horizontal surfaces (called deflected-horizontal surfaces). The most important body force g (true gravity) has been greatly simplified without justification in oceanography to the standard gravity (-g₀k) with g₀ = 9.81 m/s². Impact of such simplification on ocean dynamics is investigated in this paper using the Ekman layer model. In the classical Ekman layer dynamic equation, the standard gravity (-g₀k) is replaced by the true gravity g(λ, φ, z) with a constant eddy viscosity and a depth-dependent-only density ρ(z) represented by an e-folding near-inertial buoyancy frequency. New Ekman spiral and in turn new formulae for the Ekman transport are obtained for ocean with and without bottom. With the gravity data from the global static gravity model EIGEN-6C4 and the surface wind stress data from the Comprehensive Ocean-Atmosphere Data Set (COADS), large difference is found in the Ekman transport using the true gravity and standard gravity.