Simulation of Internal Waves within an ALE ocean model: numerical challenges and modelling

Internal Solitary Waves (ISWs) are among the most important physical processes in oceanic systems. Specifically, they play a significant role in vertical mixing, energy transfer across the continental shelf, sediment resuspension, nutrient redistribution, and the regulation of thermocline structure. Their breaking and subsequent turbulent dissipation contribute significantly to the global energy cascade. Additionally, ISWs remain challenging to study: they are strongly nonlinear, inherently nonhydrostatic, and often require three-dimensional, high-resolution modelling to capture steep fronts, overturning, and mixing. Consequently, accurate numerical simulation of ISWs is vital for improving our understanding of their mechanisms and impact on ocean circulation and climate-relevant processes. 

Since the mid-20th century, numerical models have become indispensable tools for analyzing and predicting oceanic systems and processes. As such, considerable research has focused on developing discretization methods that faithfully simulate physical phenomena while minimizing numerical artifacts. Such frequent artifact is the Spurious Diapycnal Mixing (SDM), in which, due to numerical diffusion, the vertical advection scheme introduces mixing across the density layers, thus severely altering the stratification. Due to this, various methods to track and remedy SDM have been proposed [1]. 

SLS is a numerical ocean model introduced by A. Alexandris and co-authors in [2]. It uses a hybrid Finite Volume / Finite Element spatial discretization and treats the full pressure field through a Pressure Poisson equation. Thus, SLS is inherently a nonhydrostatic ocean model and can faithfully simulate dispersive phenomena, such as solitons. The main novelty of SLS is its Arbitrary Lagrangian Eulerian (ALE) scheme that suitably defines the vertical grid motion. 

Since the seminal paper, the ALE scheme of SLS was further improved through extensive numerical modelling and simulation of ISWs. To facilitate this, an optimization process was designed with the goal of reducing SDM. The optimality is expressed through a variational principle that defines the ALE grid motion through an elliptic equation. The mathematical derivation/ analysis of the scheme and its impact on SDM is organized in the preprint [3], which is submitted to Ocean Modelling and is under review. This also includes extensive simulations of ISWs including breaking and overturning on a sloping beach. 

In the present work, further experiences of simulating ISWs with SLS are presented. This includes the application of the ALE method to more challenging 3D turbulent simulations, where the ability of SLS to control SDM is further tested. Additionally, the stability of the ALE scheme is investigated, alongside analysis of some spurious behaviors that are caused by the interplay of the Lagrangian and Eulerian mesh dynamics. 

 References:

[1] Fox-Kemper, Baylor, et al. "Challenges and prospects in ocean circulation models." Frontiers in Marine Science 6 (2019): 65. 

[2] Alexandris-Galanopoulos, Andreas, George Papadakis, and Kostas Belibassakis. "A semi-Lagrangian Splitting framework for the simulation of non-hydrostatic free-surface flows." Ocean Modelling 187 (2024): 102290. 

[3] Alexandris-Galanopoulos, Andreas, and George Papadakis. "An ALE approach to reduce spurious numerical mixing through variational minimizers: application to internal waves." arXiv preprint arXiv:2511.20092 (2025)