Coupling hydrostatic and non-hydrostatic ocean circulation models. Vertical modes perspective.

Large-scale ocean modeling relies on the so-called primitive equations, which are a simplified form of the Navier-Stokes equations based on common assumptions. These assumptions include the Boussinesq approximation, the hydrostatic assumption and the incompressibility hypothesis. The hydrostatic assumption is yet no longer valid in the framework of high-resolution regional ocean modeling and is thus to be removed, leading to a significant increase in the computational load of the model. This gives rise to consideration of model coupling, where the hydrostatic assumption is selectively lifted within specific regions of the computational domain. This is the subject of this work. One of the primary challenges involves accurately representing the various types of waves that propagate in the ocean. Waves are studied by analyzing the dispersion relations after linearizing the equations. A key distinction between hydrostatic and non-hydrostatic modeling lies in the dispersive nature of the wave propragation in the latter. We initially explore an idealized scenario by coupling a transport equation with a Korteweg-de Vries equation. To manage dispersive effects, we employ Perfectly Matched Layers (PML), typically used for replicating absorbing boundary conditions, they can also be regarded as a buffer zone for filtering dispersive effects. Expanding this approach into a three-dimensional framework involves projecting the vertical ocean structure onto vertical modes associated with the ocean's background vertical stratification. We show that this projection allows us to reduce the problem to the previously studied case.