Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations

Abstract

International audienceExcept for vertical diffusion (and possibly the external mode and bottom drag), oceanic models usually rely on explicit time-stepping algorithms subject to Courant-Friedrichs-Lewy (CFL) stability criteria. Implicit methods could be unconditionally stable, but an algebraic system must be solved at each time step and other considerations such as accuracy and efficiency are less straightforward to achieve. Depending on the target application, the process limiting the maximum allowed time-step is generally different. In this paper, we introduce offline diagnostics to predict stability limits associated with internal gravity waves, advection, diffusion, and rotation. This suite of diagnostics is applied to a set of global, regional and coastal numerical simulations with several horizontal/vertical resolutions and different numerical models. We show that, for resolutions finer that 1/2◦, models with an eulerian vertical coordinate are generally constrained by vertical advection in a few hot spots and that numerics must be extremely robust to changes in Courant number. Based on those results, we review the stability and accuracy of existing numerical kernels in vogue in primitive equations oceanic models with a focus on advective processes and the dynamics of internal waves. We emphasize the additional value of studying the numerical kernel of oceanic models in the light of coupled space-time approaches instead of studying the time schemes independently from spatial discretizations. From this study, we suggest some guidelines for the development of temporal schemes in future generation multi-purpose oceanic models

The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives

“…Fig. 3 in Lemarié et al, 2015). Same stability constraint is obtained if the order of integration between momentum and tracer equations is reversed in (7.12) and (7.13).…”

“…if q is initialised with a constant, it remains so). An overview of the time and space discretisations of (7.1) used in large and mesoscales oceanic models can be found in or Lemarié et al (2015).…”

Mixed layer formation and restratification in presence of mesoscale and submesoscale turbulence

“…Following Boccaletti et al (2007), this net ζ increase at small scale could be consistent with the development of MLIs, which release submesoscale eddy kinetic energy (EKE) extracted from available potential energy (APE) by slumping of the isopycnals. Part of the submesoscale energy would be then transferred to mesoscale through an inverse cascade and part of it would be dissipated by the diffusive advection scheme (Mohammadi-Aragh et al, 2015) and the temporal Assellin filter (Soufflet et al, 2015;Lemarie et al, 2015) used in NEMO, resulting in the mesoscale eddy field observed at day 90. Indeed, Figure 7, which represents the kinetic energy spectra, computed along the zonal direction and averaged meridionaly over the middle box for each panel of Figure 5 shows an increase of kinetic energy at all scales for days 35 and 45 which could be due to the APE-EKE conversion and an associated inverse cascade.…”

The Finite-volumE Sea ice–Ocean Model (FESOM2)

“…As demonstrated in Lemarié et al (2015), in practice, the strongest Courant number limitation comes from vertical advection in isolated patches adjacent to the coast. The code numerical efficiency can be augmented if some measures are taken to stabilize it with respect to vertical advection.…”