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Towards Nonhydrostatic Ocean Modeling with Large-eddy Simulation Oliver B. Fringer* Ocean models are limited in the physics they can resolve by the avail- able computing resources and the available time a user has to wait for a result. At one end of the spectrum, resolution is compromised in favor of speed and efficiency in order to obtain predictive solutions faster than real time. For example, a coupled ocean-atmosphere hurricane prediction model must run much faster than real time if predictions of hurricane tracks are to inform the public on potential safety hazards. In these large scale models, grid resolution dictates the physics that can be resolved and in turn determines the physics that must be parameterized. In general, a coarser grid leads to a faster prediction, but at the expense of resolving less physics and placing more emphasis on the underlying parameteriza- tions of unresolved processes. Most ocean models require parameter- izations of unresolved physics and are known as unsteady Reynolds- averaged (URANS) simulations, whereby the higher-frequency motions are filtered out of the governing equations and parameterized. At the other end of the spectrum of ocean models are those that resolve as much of the processes as possible with grid resolutions that are dictated by available computing resources. If all of the turbulence is resolved by the grid, the simulation is termed a direct numerical simulation (DNS) and no parameterizations are required. DNS is pro- hibitively expensive because the number of grid cells required in a three- *âStanford University 81
82 OCEANOGRAPHY IN 2025 dimensional simulation is given by Re9/4, where Re = UL/Î½ is the char- acteristic Reynolds number and U and L are characteristic velocity and length scales, respectively. As an example, a DNS of a breaking internal wave for which Re = 106 (U = 0.1 m sâ1, L = 10 m) would require roughly 3 Ã 1013 grid points, or 10,000 Tb of memory in a typical simulation code, an intractable problem at least for the next two decades. Because the rate at which energy is lost to dissipation can be considered uniform over scales within roughly one order of magnitude of the smallest dissipative scale, it is possible to relax the resolution requirement and perform a so- called large-eddy simulation (LES) under the premise that a bulk of the energetics is accounted for by the large, energy-containing eddies, while the smaller, or subgrid-scale eddies, can easily be parameterized without incurring significant parameterization errors. The fundamental differ- ence between URANS and LES is that LES reverts to DNS with enough grid refinement, while the parameterization in URANS is independent of the grid resolution. While LES is much more tractable than DNS, LES of ocean processes is still computationally intensive. For example, LES of the turbulent dynamics in an upwelling front over one inertial period in a 1 km by 1 km by 500 m domain using 1 m resolution (sufficient for LES) would require a simulation with 500 million grid cells using 512 proces- sors on a supercomputer for roughly two weeks. In addition to the computational requirements associated with a large number of grid cells, LES models must also compute the nonhydrostatic pressure, which roughly doubles the computational overhead relative to a hydrostatic simulation. Strictly speaking, all LES models must be nonhydrostatic. However, all nonhydrostatic models do not necessarily require LES, since numerous problems require solution of the nonhydro- static pressure without requiring the grid resolution associated with LES (internal solitary-like waves, for example, are simulated well with non- hydrostatic URANS formulations). While some ocean models incorporate LES-style parameterizations for horizontal turbulent fluxes, there are no ocean models based strictly on the LES formulation for turbulent fluxes in all three directions, since all models possess some form of a RANS-type parameterization for the unresolved vertical turbulent fluxes, particularly at solid boundaries and at the free surface. The most optimistic projections of supercomputer performance based on Mooreâs law and recent trends in parallel computing indicate that parallel computer performance will increase by a factor of 10,000 by 2025. This implies that the highest-resolution simulations of oceanic processes will increase by a factor of 100 in each direction if the grid is refined only in the horizontal, while three-dimensional simulations could be increased by a factor of 20 in each coordinate direction. Even with these substantial increases in grid resolution, it is unlikely that LES will take center stage in
Oliver B. Fringer 83 the ocean modeling community, since an increase in the grid resolution by an order of magnitude in all three directions will still not enable LES-type simulations that fare better than URANS parameterizations in regional and larger scale ocean models, particularly at boundaries. While regional scale hydrostatic URANS models will likely not be converted to LES models, the increase in computer performance will lead to an increase in the speed of the smaller-scale nonhydrostatic URANS models, and these models could potentially be run as LES-type models. For example, at present, it takes two months for a nonhydrostatic URANS model to compute the internal wave field in Monterey Bay over a fortnight with a resolution of 20 m (using unstructured grids). In 2025, if the projec- tions are correct, it may be possible to compute this same problem with a resolution of 1 m in each direction, which would certainly come close to LES. Even with this grid resolution, however, turbulent fluxes at the free surface and bottom boundaries will still need to be parameterized using URANS-type parameterizations. Because advances in computer power by 2025 will likely not lead to a regional scale LES-type ocean modeling capability, and beyond that time it is not clear whether computing power will continue to increase at a rate that justifies implementation of LES-type ocean models, the ocean model- ing community will need to focus more on model nesting and coupling, whereby higher grid resolution is achieved via nesting of progressively finer grids within a simulation, and the grids are coupled to one another via transfer of information at the boundaries or within the individual grids. Although one-way nesting is quite common in ocean modeling, the notion of two-way coupled ocean modeling is far from mainstream and few models incorporate real two-way nesting. With increased computer power, the refined grids in two-way coupled simulations will run as LES- type models and hence will require less parameterization, and these will in turn provide the necessary subgrid-scale fluxes to the coarser grids and will effectively act as the turbulence parameterizations for those grids. Ultimately, only parts of a regional-scale domain will be resolved as an LES, while the remainder will continue to run using URANS parameter- izations. These parameterizations will in turn have significantly improved by 2025 due to findings from highly resolved LES and DNS process studies. In summary, computing power will advance enough by 2025 such that nonhydrostatic, large-eddy simulations of coastal-scale problems may be tractable. However, computers may never be fast enough to achieve large-eddy simulations of regional and larger scale oceanic domains. The alternative is to focus on two-way model coupling that will enable subgrid-scale physics as computed by LES-type models in specific regions of interest to be fed back into the larger-scale URANS models.