Toward the design of a generalized vertical coordinate to properly represent vertical modes in an ocean model

The choice of an appropriate vertical coordinate system is crucial and remains a difficult problem in the design of an oceanic model. Vertical motion can be represented in a purely Eulerian way, where the grid is fixed, or in a purely Lagrangian way, where the grid follows the motion of the fluid particles (e.g. Griffies et al., 2020). An alternative is the vertical-ALE (Arbitrary Lagrangian-Eulerian) method in which the motion of the target grid is not typically Lagrangian. In this study, the focus is on the V-ALE method, with a particular emphasis on specifying the target grid. 

The proposed approach aims to define the target grid for an ""optimal"" representation of the vertical normal modes of linearized equations of motion which are the eigenfunctions of a Sturm-Liouville (SL) problem (e.g. Gill 1982). The density profile is a key element influencing the vertical structure of modes. In this presentation, we will discuss the formulation of the discrete SL problem compatible with the discretization of the model, as well as the appropriate definition of the optimization problem to be solved as a function of the number of eigenvalues we wish to correctly approximate.
We will illustrate the results obtained using our approach for numerical tests in idealized cases and using density profiles from climatological data.

Co-authors : L. Debreu (Inria), F. Lemarié (Inria), J. Chanut (MOI)