Abstract
Abstract. This paper presents MQGeometry, a multi-layer quasi-geostrophic (QG) equation solver for non-rectangular geometries. We advect the potential vorticity (PV) with finite volumes to ensure global PV conservation using a staggered discretization of the PV and stream function (SF). Thanks to this staggering, the PV is defined inside the domain, removing the need to define the PV on the domain boundary. We compute PV fluxes with upwind-biased interpolations whose implicit dissipation replaces the usual explicit (hyper-)viscous dissipation. The discretization presented here does not require tuning of any additional parameter, e.g., additional eddy viscosity. We solve the QG elliptic equation with a fast discrete sine transform spectral solver on rectangular geometry. We extend this fast solver to non-rectangular geometries using the capacitance matrix method. Subsequently, we validate our solver on a vortex-shear instability test case in a circular domain, on a vortex–wall interaction test case, and on an idealized wind-driven double-gyre configuration in an octagonal domain at an eddy-permitting resolution. Finally, we release a concise, efficient, and auto-differentiable PyTorch implementation of our method to facilitate future developments on this new discretization, e.g., machine-learning parameterization or data-assimilation techniques.
Funder
H2020 European Research Council
Reference32 articles.
1. Arakawa, A. and Lamb, V. R.: A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. Weather Rev., 109, 18–36, 1981. a, b
2. Blayo, E. and LeProvost, C.: Performance of the Capacitance Matrix Method for Solving Helmhotz-Type Equations in Ocean Modelling, J. Comput. Phys., 104, 347–360, 1993. a
3. Borges, R., Carmona, M., Costa, B., and Don, W. S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 3191–3211, 2008. a, b, c, d
4. Boris, J. P., Grinstein, F. F., Oran, E. S., and Kolbe, R. L.: New insights into large eddy simulation, Fluid Dynam. Res., 10, 199, https://doi.org/10.1016/0169-5983(92)90023-P, 1992. a
5. Brown, N.: A comparison of techniques for solving the Poisson equation in CFD, arXiv [preprint], https://doi.org/10.48550/arXiv.2010.14132, 2020. a