Design of a generalized vertical coordinate to properly represent the structure of normal modes in an oceanic model

In ocean modeling, the choice of a suitable vertical coordinate system is crucial due to the complex vertical dynamics of the ocean, playing a significant role in climate considerations. The objective of this research is to develop a generalized vertical coordinate system applicable across various scales, from global to coastal applications. 

Three primary algorithms for solving generalized vertical coordinate ocean equations are considered: quasi-Eulerian (QE) coordinate, Vertical Lagrangian Remapping (VLR), and the Arbitrary Lagrangian-Eulerian (V-ALE) method. The V-ALE method offers an alternative between QE and VLR, incorporating both Lagrangian and Eulerian components in the movement of the target grid. In this study, the focus is on the V-ALE method, with a particular emphasis on specifying the target grid. 

The proposed approach consists in defining the target grid to solve a propagation problem in an inhomogeneous environment. The stratification state, a key element of this environment, has a significant influence on the mode structure.

From a discrete point of view, the challenge is to identify the optimum grid point positions for correctly representing vertical modes. The aim is to understand the errors introduced by discretization and minimize them to find the optimal point positions. A variational method is also proposed, leading to a non-uniform grid that better represents vertical modes. We illustrate the results obtained with these methods using stratification profiles derived from climatological data.