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True Gravity in Ocean Circulation
  • Peter C Chu
Peter C Chu
Naval Postgraduate School

Corresponding Author:[email protected]

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Two related issues in oceanography are addressed: (1) the unit vector (k) normal to the Earth spherical/ellipsoidal surface is not vertical (called deflected-vertical) since the vertical is in the direction of the true gravity, g = igλ+jgφ+kgz, with (λ, φ, z) the (longitude, latitude, depth) and (i, j, k) the corresponding unit vectors; and (2) the true gravity g is replaced by the standard gravity (-g0k, g0 = 9.81 m/s2). In the spherical/ellipsoidal coordinate (λ, φ, z) and local coordinate (x, y, z), the z-direction is along k (positive upward). The spherical/ellipsoidal surface and (x, y) plane are perpendicular to k, and therefore they are not horizontal (called deflected-horizontal) since the horizontal surfaces are perpendicular to the true gravity g such as the geoid surface. In the vertical-deflected coordinates, the true gravity g has deflected-horizontal component, gh = igλ+jgφ (or = igx+jgy), which is neglected completely in oceanography. This study uses the classic ocean circulation theories to illustrate the importance of gh in the vertical-deflected coordinates. The standard gravity (-g0k) is replaced by the true gravity g in the existing equations for geostrophic current, thermal wind relation, and Sverdrup-Stommel-Munk wind driven circulation to obtain updated formulas. The proposed non-dimensional (C, D, F) numbers are calculated from four publicly available datasets to prove that gh cannot be neglected against the Coriolis force, density gradient forcing, and wind stress curl.