General coordinate ocean models have been proposed to avoid the problem of "spurious" diapycnal diffusion in fixed grid models. However, conventional regridding and remapping methods may not be accurate enough to reduce the spurious mixing to insignificant levels. The most widely used reconstruction method is PPM (piecewise parabolic method) which is notionally third order accurate. We introduce the piecewise quartic method (PQM) which is a natural extension of PPM and is shown to be significantly more accurate. The careful design of limiters is also key to avoid unnecessary diffusion when remapping. The use of limiters typically resorts to low order reconstructions near physical boundaries. We find it imperative to use well-designed extrapolation (in this case, using rational functions) to avoid near-boundary spurious diffusion in the remapping step. Regridding (solving for new grids) is also sensitive to such algorithmic choices.
The most adiabatic ocean model formulation is the layered isopycnal model. Ironically, these layered models use the lowest order representation possible in the vertical, namely piecewise constant density within each layer. In the adiabatic limit, layered models have no remapping step. In contrast, vertical advection in fixed-grid models can be interpreted as remapping of the water column and it is widely accepted that high-order methods naturally lead to improvements in accuracy and fidelity. The use of high-order reconstruction is at odds with the usual isopycnal interpretation and compromises the strict adiabatic properties of the stacked shallow water approach. With the intent of finding an accurate, adiabatic general coordinate formulation we adopt a continuous isopycnal interpretation which defines isopycnal coordinates at interfaces rather than by layer. Relaxing the piecewise constant by layer representation impacts many terms in the model, particularly the pressure gradient. A comparison between a traditional isopycnal model, our continuous isopycnal model and two Eulerian coordinate models validates the approach and provides convincing evidence that fixed-grid models still exhibit excessive spurious diffusion even when employing very high-order methods.