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.
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