Semi-Lagrangian Advection Algorithms for Ocean Circulation Models
Journal of Atmospheric and Oceanic Technology
As model resolutions increase, the Courant-Frederichs-Lewy (CFL) number based on advective motion becomes the limiting factor in setting the timestep of time-explicit circulation models. Some atmospheric models escape this limit by using an implicit or semi-implicit semi-Lagrangian formulation of advection. This formulation calculates fluid properties along parcel trajectories which follow the fluid motion and end, for each timestep, at prescribed grid-points. This work is the first application
... e first application of the semi-Lagrangian method to an operational ocean model. In this context, we solve 5 the difficulty posed by the ocean's irregular, interior boundaries by calculating parcel trajectories using a time-exponential formulation. This formulation ensures that all trajectories that are solutions to a fixed-point iteration have an origin point in the valid domain, and it does not require any prescribed extrapolation of the fluid velocities into the invalid (land) portion of the domain. We derive this method in a way that is compatible with the leapfrog timestepping scheme used in the NEMO-OPA (Nucleus for European Modelling of the Ocean, Océan Parallélisé) ocean model, and we present simulation results for a 10 simplified test-case of flow past a model island and for 10-year free runs of the global ocean on the quarter-degree ORCA025 grid. Introduction Recent work by Smith et al. (2018) has shown that over the medium term (up to seven days), a coupled forecasting system 15 involving ocean, ice, and atmospheric models can significantly improve forecasting skill over forecasts that assume persistence of initial conditions. While this is an exciting development for the future of numerical weather prediction, it creates a combined computational problem out of models and forecasting systems systems that have previously been run independently. In particular, these coupled forecasts require close integration between the atmospheric and ocean components in order to exchange information. For reasons of computational efficiency, we want each model to run with its largest admissible timestep, 20 but to support coupling we can usually only achieve this if the models have the same timestep or closely-related timesteps. While different factors cause the most stringent timestep restrictions in atmosphere and ocean circulation models, these models can adopt similar methods to alleviate the restriction. At the Canadian Meteorological Centre, our coupled numerical weather forecast uses the GEM (Geophysical Environmental Multiscale; Girard et al. (2014)) model for its atmospheric component 1 https://doi.org/10.5194/gmd-2020-9 Preprint. Discussion started: 7 April 2020 c Author(s) 2020. CC BY 4.0 License. and the NEMO-OPA (Nucleus for European Modelling of the Ocean, Océan Parallélisé; Madec (2008)) model for the ocean 25 component. The GEM model already includes several features to provide for a long model timestep, and we seek to adapt some of these to the widely-used NEMO-OPA model (version 3.1). Stretching in ocean grids While ocean currents are much slower than the fastest upper-atmosphere winds that contribute to the Courant-Frederichs-Lewy (CFL) restriction in atmospheric model 1 , global ocean models in particular suffer from grid stretching. The ORCA "tripolar" 30 grid commonly used in global NEMO-OPA model configurations (Madec and Imbard, 1996; Murray, 1996) is defined in the northern hemisphere by an elliptical coordinate system, where in the northern hemisphere circles of a latitude-like coordinate are defined by ellipses with a shared pair of foci, and hyperbolas of a longitude-like coordinate are defined by hyperbolas orthogonal to these ellipses; these coordinates match continuously at the equator to lines of latitude and longitude on a Mercator projection. By placing the foci of the ellipses on land, the grid contains no singularities in the ocean domain. 35 Unfortunately, this placement causes an abundance of small grid cells in the north polar region, especially in the Canadian Arctic Archipelago. Figure 1 depicts this situation at a nominal 1 4 • resolution: the grid point spacing of 25-30km near the equator falls to 3-4km in the archipelago. Currents in these narrow straits contribute ten times as strongly to a lateral CFL timestep restriction, compared to currents in the equatorial regions. The coordinate system is also stretched in the vertical direction. Using the z-level grid option of the NEMO-OPA model, 40 layers near the surface are spaced much more closely together than layers nearer the ocean bottom, in order to provide adequate resolution of the mixing layer. While ocean currents tend to follow nearly horizontal paths, residual upwelling or downwelling can still cause a vertical CFL restriction to be binding. Although the ocean models arrive at a binding CFL condition via a different route than for atmospheric models, semi-Lagrangian advection can alleviate this restriction in both cases. This method traces fluid parcels in a Lagrangian, fluid-45 following coordinate system, defined such that at the end of each timestep the fluid parcels arrive at the prescribed grid. The properties of the fluid parcels (in the ocean setting, temperature, salinity, and horizontal velocity) at the beginning of the time step are found by interpolating these values from their gridded locations to the origin point of the trajectory. This method provides an implicit treatment of advection, allowing timesteps with CFL numbers greater than those allowable under wholly explicit methods. 50 Existing work This technique is standard in atmospheric models (Robert, 1982) , but it is not widely applied to the ocean. In the atmosphere, especially at large scales, the effects of topography are relatively gentle, and trajectory calculations can proceed under the 1 Atmospheric models are also limited by the timestep associated with sound waves, but it is common to process at least vertically-propagating sound waves implicitly to alleviate this restriction. A comparable limit in ocean models is that arising from surface gravity waves, and here NEMO-OPA uses either an implicit or time-splitting approach for similar reasons.