Nonlinear filtering stabilizations for the quasi-geostrophic equations
Abstract
Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).
Related
- Non-decaying weak solutions to the 2D quasi-geostrophic equations
- Nonlinear profiles on solutions to the surface quasi-geostrophic equation
- Uniqueness and stability of nonlinear filtering equations with unbounded random coefficients
- Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation
- Global solutions to the Boussinesq equations and the planetary geostrophic approximation
BibTeX
@misc{besabe2026nonlinear,
title = {{Nonlinear filtering stabilizations for the quasi-geostrophic equations}},
author = {Lander Besabe and Sachin Kumar and Annalisa Quaini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16983},
eprint = {2609.16983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16983},
}
RIS
TY - UNPB
TI - Nonlinear filtering stabilizations for the quasi-geostrophic equations
AU - Lander Besabe
AU - Sachin Kumar
AU - Annalisa Quaini
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16983
UR - https://arxiv.org/abs/2609.16983
AB - Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).
ER -