The Planetary Geostrophic Equations (PGEs) play a central role
in large-scale ocean circulation theory. There is only one dynamic
equation for the temperature field and the 3-D velocity field is
determined by the planetary geostrophic balance, hydrostatic balance,
and the incompressibility condition. The system is reformulated such
that all the velocity profiles can be explicitly represented as
functionals of temperature gradient, by utilizing the special form
of the Coriolis parameter. As a result, the PDE system is shown to
be well-posed and the corresponding numerical method can be
efficiently proposed. The 3-D MAC (marker and cell) scheme, which gives
values of physical variables on staggered mesh grid points, are chosen
as spatial discretization. The usage of such a staggered grid assures
the computed velocity field satisfies the divergence-free property
in a discrete level. Furthermore, applying long stencil and compact
difference approximations on the grid leads to an efficient fourth
order scheme, which is a widely accepted way to improve the accuracy
within the limited resolution due to the enormous large scale of
the three-dimensional setting. Convergence analysis for the numerical
schemes is given, and some computational results of large-scale oceanic
circulation are also presented.
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