Nesting Nonhydrostatic GCCOM within Hydrostatic ROMS ... - Cal Poly

Nesting Nonhydrostatic GCCOM within Hydrostatic ROMS for Multiscale Coastal Ocean Modeling Paul F. Choboter∗ , Mariangel Garcia† , Dany De Cecchis‡ , Mary Thomas§ , Ryan K. Walter¶ and Jos´e E. Castillo† ∗ Mathematics

Department, California Polytechnic State University, San Luis Obispo, CA 93407-0403, Email: [email protected] † Computational Science Research Center, San Diego State University, San Diego, CA 92182-1245 ‡ Centro Muldidisciplinario de Visualizaci´ on y C´omputo Cientfico (CEMVICC), Universidad de Carabobo, Venezuela § Department of Computer Science, San Diego State University, San Diego, CA 92182-1245 ¶ Physics Department, California Polytechnic State University, San Luis Obispo, CA 93407-0403 Abstract—The Regional Ocean Modeling System (ROMS) is a hydrostatic free-surface ocean model ideally suited to simulate mesoscale to basin-scale (10 km – 10000 km) ocean processes. The General Curvilinear Coastal Ocean Model (GCCOM) is a nonhydrostatic large eddy simulation (LES) model designed specifically for high-resolution (meters) simulations. In this research, a hybrid model is developed that nests a fine-grid GCCOM model within a coarse-grid ROMS. The nested GCCOM-ROMS model is tested in an idealized flow over a seamount. Keywords—numerical models, nonhydrostatic, nesting

I. I NTRODUCTION Nonlinear internal waves (NLIWs) and bores are ubiquitous features in the coastal environment. These features have important ramifications for the cross-shelf exchange of nutrients and other scalars, turbulent dissipation and diapycnal mixing, hypoxia development, and other biological/ecological processes [1]–[3]. Despite the obvious importance and a growing body of literature on the subject, many questions still remain with respect to the evolution, fate, and impact of NLIWs and bores in the coastal ocean and their connection to regional-scale processes. This study will use a new hybrid, nested model to address these issues, as well as examine other nonhydrostatic features in the coastal environment. The Regional Ocean Modeling System (ROMS, [4]) is a numerical model widely used by the oceanographic community because of its efficient numerical scheme and freely available source code. It is a large-scale model that is designed to study hydrostatic processes at kilometer- to basin-scales. Although ROMS is a popular ocean model that incorporates highorder finite-difference schemes, it is of limited applicability in regions where nonhydrostatic effects are important. This includes areas where flows over steep bathymetry generate strong vertical accelerations, shoaling of internal waves and the formation of internal bores, generation of solitary wave trains, etc., or more generally, when the vertical scales of motion are roughly equal to the horizontal scales. In general, the higher the grid resolution, the more accurate the simulation, both because the finite-difference approximations of the differential equations have lower error and because fine-scale features may only be captured with a high enough resolution. However, high resolution runs can be prohibitively

demanding of computational resources. One technique used in ocean models to obtain a higher grid resolution where it is needed, but still have a relatively efficient simulation, is to “nest” a higher resolution grid over a local region within a lower resolution grid that spans a larger area [5]. The low resolution grid simulation results are then used to supply the boundary conditions for the high-resolution grid simulations on the high resolution grid, in a technique referred to as oneway nesting [6]. In two-way nesting, the high-resolution grid also feeds information back to the low-resolution grid [7]. One of the major challenges to numerically simulating nonhydrostatic ocean phenomena is the vast range of length and time scales present. One technique to address this challenge is to use multiple grids, either through nesting [14], [15], or through adaptive refinement [11]. The General Curvilinear Coastal Ocean Model (GCCOM) is capable of simulating fully nonhydrostatic flows [8]. GCCOM employs a full 3D curvilinear coordinate system, large eddy simulation (LES) dynamics, and a rigid-lid approximation. The GCCOM model is able to accurately reproduce nonhydrostatic features such as the interaction of currents with steep bathymetric features at high resolutions. By nesting a highresolution GCCOM model within a coarser ROMS model, nonhydrostatic effects can be simulated efficiently at high resolutions, while still capturing larger-scale processes. Other numerical models have been developed that are capable of simulating nonhydrostatic ocean processes. MITgcm [9] uses a curvilinear grid in the horizontal coordinates, and z-coordinates in the vertical with a finite-volume treatment of irregular bathymetry. SUNTANS [10] employs an unstructured grid in the horizontal. SOMAR [11] uses curvilinear coordinates, and features adaptive mesh refinement. Although a nonhydrostatic version of ROMS has been developed [12], [13], it is not part of the standard ROMS ocean model available to the general community. Some of the key features of GCCOM that distinguish it from other nonhydrostatic models include the use of a curvilinear grid in all three dimensions [8], and its data assimilation capabilities [16]. The present work describes the nesting of GCCOM within ROMS. Details of the boundary conditions employed are described in Section II, and the numerical tests

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Fig. 1. Grid resolutions and bathymetry in the idealized seamount test. Nested grid shown overlayed with coarse parent grid (a, top left); bathymetry on the fine grid (b, top right); bathymetry on the fine child grid (c, bottom left) and; bathymetry on the coarse grid (d, bottom right).

are specified in Section III. Results are outlined in Section IV, and conclusions are in Section V. II. B OUNDARY CONDITIONS To perform the nesting, careful treatment of the open boundary conditions of the nested region was necessary. We use radiation conditions similar to those used by ROMS [17] for the baroclinic variables. For a dynamic variable φ, the solution is relaxed towards the externally-supplied value φext according to Δt ext (φ − φn−1 ijk ) τ where τ is a relaxation timescale specified to be a short time scale for an inward-propagating signal, and a long time scale for an outward-propagating signal. The direction of the signal is computed using discrete approximations of wave motion. The discrete approximations are computed as follows. Assuming the solution is wave-like near a boundary, φnijk = φnijk +

∂φ ∂φ ∂φ + cx + cy = 0, ∂t ∂x ∂y the d’Alembert solution of the wave equation allows the phase speeds in the x and y directions to be written in terms of time and space derivatives of the function itself, cx = 

∂φ − ∂φ ∂t ∂x 2  2 , ∂φ + ∂φ ∂x ∂y

cy = 

∂φ − ∂φ ∂t ∂y 2  2 . ∂φ + ∂φ ∂x ∂y

The discretized version of these equations, −Δt φΔx φ ,  2 2 (Δx φ)2 + Δx (Δ φ) y Δy

cx =

Δx Δt

cy =

−Δt φΔy φ Δy ,  2 Δt Δy 2 + (Δ φ)2 (Δ φ) x y Δx

are evaluated to give an approximation of direction of a signal near the boundary. III. N UMERICAL TESTS The first tests have been designed to replicate the idealized seamount tests of Abouali [8]. These simulations are over a 3600 m by 2800 m domain, in a test basin with a depth of 1000 m. Fluid of constant density flows in at x = 0. In this contribution, we focus on a flow with a constant density for simplicity, but future work will explore stratified flows. The velocity at x = 0 varies linearly with depth, from u = 0 at the bottom to u = 1 m/s at the top of the water column. Fluid flows out at x = 3600, and the dynamics are periodic in y. The bathymetry is a Gaussian shaped seamount, with an amplitude of 500 m. We have run ROMS and GCCOM in various configurations on three grid configurations: (1) a 35×24×39 grid, referred to as the coarse grid, (2) a 101×68×39 fine grid, and (3) a 61×43×39 nested grid on the same resolution as the fine grid (Figure 1).

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Fig. 3. Pointwise difference of u-component of velocity, averaged over domain for each test case, compared with ROMS fine-grid run: blue is for coarse grid ROMS, red is nested ROMS, magenta is nested GCCOM with fine boundary conditions, and green is nested GCCOM with coarse boundary conditions.

four simulations was also calculated. Spatial maps of these correlation coefficients (Figure 4) show a strong dependence of correlation on location: all five test runs agree well upwind of the seamount, but the details of the variability in the wake of the seamount are not well correlated with the fine-grid ROMS run for any of the test cases. Fig. 2. Instantaneous u velocity for the seamount test cases, at t = 10000 s. Contour interval is 0.05 cm/s, with the u = 0 contour in bold. Shown are, top to bottom, results from fine grid, coarse grid, ROMS-ROMS nest, GCCOM-ROMS with coarse boundary conditions, and GCCOM-ROMS with fine boundary conditions.

To isolate the effects of grid resolution and nesting, we have performed three ROMS simulations: one on the fine grid, a second on the coarse grid, and a third on the nested grid, with the third run’s boundary conditions supplied by the coarsegrid run. To isolate the effects of changing model physics, these runs are compared to two additional runs: one where GCCOM is nested within the fine-grid ROMS simulation, and a second where GCCOM is nested within the coarse-grid ROMS simulation. Note that these last two test cases represent successful runs of the hybrid GCCOM-ROMS model. IV. R ESULTS A qualitative comparison of the u-component of velocity at a snapshot in time for all five test cases (Figure 2) indicates that the changing of the model has a larger effect on the details of the flow than the changing of the grid resolution. To quantify these differences, we compare each run to the ROMS fine-grid run. The time evolution of the point-bypoint difference of the u-component of velocity (Figure 3) confirms that changing the model physics contributed to a more significant change in the results than changing the resolution. The correlation between time series of u at each grid point in the fine-grid ROMS simulation and the u time series at the corresponding grid point in each of the other

V. C ONCLUSION The nested GCCOM models show very distinct dynamics as compared to ROMS, and are qualitatively similar to the seamount tests of Abouali [8]. We plan to next test the nested model for stratified flow conditions. Future studies will also focus on the simulation of internal waves and bores along a sloping shelf. The authors anticipate that the coupling of GCCOM with a well-developed and efficient hydrostatic model such as ROMS will result in a useful tool for the study of nonhydrostatic processes in the coastal environment. ACKNOWLEDGMENT The authors were supported by the National Science Foundation (Grants #0753283 and #0721656), the Department of Energy (DOE # DE-GC02-02ER25516), the CSU Council on Ocean Affairs, Science and Technology (COAST) Grant Development Program, and with resources available with an NSF funded XSEDE allocation (TG-CCR110014), at the San Diego Supercomputer Center, the Texas Advanced Computing Center, and the SDSU Computational Sciences Research Center. R EFERENCES [1] R. K. Walter, C. Brock Woodson, R. S. Arthur, O. B. Fringer, and S. G. Monismith, “Nearshore internal bores and turbulent mixing in southern Monterey Bay,” J. Geophys. Res. Ocean, vol. 117. [Online]. Available: http://dx.doi.org/10.1029/2012JC008115 [2] R. K. Walter, M. E. Squibb, C. B. Woodson, J. Koseff, and S. Monismith, “Stratified turbulence in the nearshore coastal ocean: dynamics and evolution in the presence of internal bores,” J. Geophys. Res. Ocean, vol. 119, pp. 8709–8730.

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Fig. 4. Correlation between each test case and fine resolution ROMS run. Shown, from left to right respetively, are the coarse-resolution ROMS, ROMS nested in ROMS, GCCOM-ROMS model with fine grid ROMS at interface, and GCCOM-ROMS model with coarse grid ROMS at interface.

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