A Pressure Projection Scheme With Near‐Spectral Accuracy for Nonhydrostatic Flow in Domains With Open Boundaries

• Published in: Journal of advances in modeling earth systems 2024-06, Vol.16 (6), p.n/a
• Main Authors: Winters, K. B., Claret, Mariona, Lelong, M.‐Pascale, Ourmières, Yann
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; Approximation; Boundaries; Boundary conditions; eddies and mesoscale processes; Fluid dynamics; Fourier transforms; General circulation models; Inertial waves; internal and inertial waves; modeling; Nesting; numerical solutions; Ocean circulation; Ocean models; Oceans; Poisson equation; Poisson's equation; Simulation; Stratified flow; Upper ocean
• ISSN: 1942-2466; 1942-2466
• DOI: 10.1029/2023MS004040

We describe a pressure projection scheme for the simulation of incompressible flow in cubic domains with open boundaries based on fast Fourier transforms. The scheme is implemented in flow_solve, a numerical code designed for process studies of rotating, density‐stratified flow. The main algorithmic features of the open‐boundary code are the near‐spectral accuracy of the discrete differentiation and a dynamic two‐dimensional domain decomposition that scales efficiently to large numbers of processors. The simulated flows are not required to be periodic or to satisfy symmetry conditions at the open boundaries owing to the use of mixed series expansions combining cosine and singular Bernoulli polynomial basis functions. These expansions facilitate the imposition of inhomogeneous boundary conditions and allow the code to be used for offline, one‐way nesting within an arbitrarily embedded subdomain of a larger scale simulation. The projection scheme is designed to exploit a simple and powerful numerical engine: inversion of Poisson's equation with homogeneous Neumann boundary conditions using fast cosine transforms. Here, we describe the mathematical transformations used to accommodate the imposition of space‐ and time‐varying boundary conditions. The utility of the approach for process studies and for nesting within submesoscale‐resolving ocean models is demonstrated with simulations of wind‐driven near‐inertial waves in the upper ocean. Plain Language Summary Internal gravity waves in the upper ocean are affected by the presence of features such as fronts, eddies, and geographically constrained winds and currents. Ocean models that are able to accurately capture geographic realism often lack the capacity to resolve the intricacies of the internal waves that may be generated. This paper presents a mathematical framework that combines the realism of ocean models with high‐fidelity simulation of small‐scale dynamics. The framework is demonstrated through a series of examples of increasing complexity, including wave trapping in a wind‐perturbed anticyclonic vortex and simulation of near‐inertial waves near a density front in the Gulf of Lion. The proposed methodology has the potential to take advantage of the increasing horizontal resolution and fine‐scale realism of ocean models, while also relaxing the approximations that limit their ability to resolve the detailed dynamics of the internal wave field. Key Points A computational framework based on fast spectral