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\begin{document}

\title{True Gravity in Ocean Circulation}



\author[1]{Peter C Chu}%
\affil[1]{Naval Postgraduate School}%


\vspace{-1em}



  
  \date{June 16, 2024}


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\selectlanguage{english}
\begin{abstract}
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\selectlanguage{greek}\emph{g\textsubscript{λ}}\selectlanguage{english}+j\selectlanguage{greek}\emph{g\textsubscript{φ}}\selectlanguage{english}+k\emph{g\textsubscript{z}},
with (\selectlanguage{greek}\emph{λ}, \selectlanguage{english}\selectlanguage{greek}\emph{φ}, \selectlanguage{english}\emph{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\textsubscript{0}k,
g\textsubscript{0} = 9.81 \textsuperscript{} m/s\textsuperscript{2}). In
the spherical/ellipsoidal coordinate (\selectlanguage{greek}\emph{λ}, \selectlanguage{english}\selectlanguage{greek}\emph{φ}, \selectlanguage{english}\emph{z}) and
local coordinate (\emph{x}, \emph{y}, \emph{z}), the \emph{z}-direction
is along k (positive upward). The spherical/ellipsoidal surface and
(\emph{x}, \emph{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\emph{\textsubscript{h}} =
i\selectlanguage{greek}\emph{g\textsubscript{λ}}\selectlanguage{english}+j\selectlanguage{greek}\emph{g\textsubscript{φ}} \selectlanguage{english}(or =
i\emph{g\textsubscript{x}}+j\emph{g\textsubscript{y}}), which is
neglected completely in oceanography. This study uses the classic ocean
circulation theories to illustrate the importance of
g\emph{\textsubscript{h}} in the vertical-deflected coordinates. The
standard gravity (-g\textsubscript{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 (\emph{C}, \emph{D},
\emph{F}) numbers are calculated from four publicly available datasets
to prove that g\emph{\textsubscript{h}} cannot be neglected against the
Coriolis force, density gradient forcing, and wind stress curl.%
\end{abstract}%



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