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

TL;DR

Spurious numerical mixing is a persistent artifact in ocean modeling, particularly in nonhydrostatic, stratified flows. The authors develop a variational ALE vertical mesh movement that defines layer positions by solving an elliptic (Helmholtz-type) problem for the vertical correction velocity, guided by a cost functional with a Lagrangian term, smoothing terms, and a monitor function. Integrated into the SLS solver within a Generalized Vertical Coordinate framework, this method is validated on internal solitary waves, including shoaling, wedge interactions, and breaking, demonstrating reduced spurious diapycnal mixing while preserving physical fidelity. The approach is general to any GCL-based vertical coordinate model and provides concrete guidelines for selecting coefficients and monitor functions to balance smoothness, Lagrangian bias, and gradient-focused refinement.

Abstract

Spurious numerical mixing is a frequent phenomenon in ocean models. In the present paper, we present an efficient and robust methodology that defines the vertical grid motion so that this mixing is reduced. This motion is defined through the solution of an optimization problem that -- using the ideas of the calculus of variations -- results in an elliptic equation. This framework is generally applicable to any ocean model that uses an 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] and its applicability in fully nonlinear internal waves is investigated for the demanding test cases of wave breaking and overturning. These numerical benchmarks show the ability of the method to reduce spurious mixing, while attaining the physical relevancy of the results.

Figures (14)

• Figure 1: The free-surface Euler system
• Figure 2: The Finite Volume scheme with the reconstructed non-conforming mesh.
• Figure 3: Sketch of the Finite Volume fluxes, alongside the reconstructed variables on the cell's faces: $\vec{U}^{L/R}$ and $\vec{U}^{+ / -}$.
• Figure 4: Diagrams of the Lagrangian-Smoothing ratio of eq.(\ref{['eq:ratio']}) based on the wavenumber for different values of $a_\vartheta/a_{x\xi}$ at $\omega_*=2\pi$. The y axis is in logarithmic scale. Lagrangian dominated areas are colored with red, while Smoothing dominated ones with blue.
• Figure 5: Snapshot of the soliton of §\ref{['sec:djl']} (see Tab.\ref{['tab:isw']}) at $t=10s$ on a $1000 \times 100$ mesh. Contours of the normalized density, horizontal and vertical velocity are presented on a bounding box with the y axis being magnified with a factor of 10.