  The development of a set of high-order accurate finite-volume formulations
for evaluation of the pressure gradient force in layered ocean models is
described. A pair of new schemes are presented, both based on an integration of
the contact pressure force about the perimeter of an associated momentum
control-volume. The two proposed methods differ in their choice of
control-volume geometries. High-order accurate numerical integration techniques
are employed in both schemes to account for non-linearities in the underlying
equation-of-state definitions and thermodynamic profiles, and details of an
associated vertical interpolation and quadrature scheme are discussed in
detail. Numerical experiments are used to confirm the consistency of the two
formulations, and it is demonstrated that the new methods maintain hydrostatic
and thermobaric equilibrium in the presence of strongly-sloping layer-wise
geometry, non-linear equation-of-state definitions and non-uniform vertical
stratification profiles. Additionally, one scheme is shown to maintain high
levels of consistency in the presence of non-linear thermodynamic
stratification. Use of the new pressure gradient force formulations for hybrid
vertical coordinate and/or terrain-following general circulation models is
discussed.
