True Gravity in Ocean Circulation

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 =
i*g*_{λ}+j*g*_{φ}+k*g*_{z},
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 (-g_{0}k,
g_{0} = 9.81 ^{} m/s^{2}). 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, g_{h} =
i*g*_{λ}+j*g*_{φ} (or =
i*g*_{x}+j*g*_{y}), which is
neglected completely in oceanography. This study uses the classic ocean
circulation theories to illustrate the importance of
g_{h} in the vertical-deflected coordinates. The
standard gravity (-g_{0}k) 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 g_{h} cannot be neglected against the
Coriolis force, density gradient forcing, and wind stress curl.