An ALE approach to reduce spurious numerical mixing through variational minimizers: application to internal waves

Spurious numerical mixing is a frequent phenomenon in ocean models. In this paper, we present an efficient and robust methodology that defines the vertical grid motion so that this mixing is reduced. This motion is defined as the solution of an optimization problem that -- using the ideas of the calculus of variations -- results in an elliptic partial differential equation, which is straightforward to analyze and discretize. This framework is generally applicable to any ocean model that uses an Arbitrary Lagrangian-Eulerian (ALE) vertical coordinate and can be tuned to fit the modeler's specific needs based on the guidelines presented herein. The method is applied to the nonhydrostatic solver presented by the authors in [Alexandris-Galanopoulos et al., 2024]. While the majority of spurious numerical mixing studies focus on large-scale processes, herein the proposed method is applied and tested in small-scale nonhydrostatic phenomena. Specifically, the effectiveness of the method in capturing fully nonlinear internal waves is investigated for the test cases of wave propagation, breaking and overturning. Overturning serves as a demanding test for the proposed scheme as it induces rapid vertical accelerations and thus the mesh-moving algorithm must incorporate this motion with the goal of reducing numerical mixing, while not suppressing physically relevant vertical mass transfer. These numerical benchmarks show the ability of the method to reduce spurious mixing, while attaining the physical relevance of the results.