The initial conditions for this simulation were obtained from a hydrostatic simulation on the same grid. Table 1 presents all descriptive details of the simulations, for which results are presented in the following sections. A leptic ratio of 1.
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The hydrostatic and m grid simulations Table 1 were used in previous studies to validate the model in hydrostatic mode [ Dorostkar et al. Figure 5 shows a temperature contour comparison between the nonhydrostatic 40 m grid results and field observations. We believe that this is because the steepening timescale associated with solitary wave evolution roughly equals the time for the surges to travel the basin length and shoal [ Boegman et al.
Cayuga Lake temperature contours at stations S2 and S3 from the a nonhydrostatic model with 40 m horizontal grid spacing, b field data.
Nonlinear Internal Waves in Lakes
Model data are interpolated to the same depths of the thermistors in the field, as shown with filled circles on the vertical axis. The field data are resampled at the model time interval s without filtering. Surges in red are due to the occurrence of internal hydraulic jumps. In the present work, the model was validated in the nonhydrostatic mode and its ability to capture NLIWs was tested by comparing the results from the finest nonhydrostatic model i.
The effects of grid refinement from 40 m Figure 6 to 22 m Figure 7 for the hydrostatic and nonhydrostatic cases are shown for surge b at S2 and S3. The time of arrival of the steepened fronts at the stations is delayed in the model; simulated waves are either generated later or propagate slower than observed. The solitary wave amplitudes are indeed slightly smaller in the model probably due to the choice of drag coefficient and horizontal grid resolution, and hence will propagate slower. Field data provided by E.
Schweitzer, Cornell University. However, the model spectra were extracted only for surge b and thus does not capture the energy peak at 3.
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Power spectra of vertically integrated potential energy signal at station S3 from 24 s a nonhydrostatic, and b hydrostatic simulated data during day Field spectra obtained from 25 s data during day — are shown in both plots with black lines. These grids have insufficient resolution to reproduce NLIWs and so dispersive effects are not evident.
As the grid was further refined to 40 m and 22 m, solitary waves evolved in the nonhydrostatic runs Figure 9 b , with little difference between grids—apart from the resolution of suspected shear instabilities trailing the surge. Spatial snapshots of simulated isotherms passing S2 indicate that the wavelength remains unchanged with grid refinement from 40 m to 22 m, demonstrating that the solitary waves are resolved on both grids and that dispersion is balancing steepening Figures 10 and Instantaneous spatial snapshots of simulated isotherms passing S2 from a—c hydrostatic 40 m grid and d—f nonhydrostatic 40 m grid.
The day of year and the location of station S2 thermistors is noted on each plot. Instantaneous spatial snapshots of simulated isotherms passing S2 from a—c hydrostatic 22 m grid and d—f nonhydrostatic 22 m grid. The time in day of year and the location of station S2 thermistors is noted on each plot. The spectral density curves from the nonhydrostatic runs closely resemble the field spectra over the shown frequency bandwidth, except where they diverge, which suggests the loss of energy to grid dissipation Figure 8 a.
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The frequency, and thus model accuracy, is increased as the grid resolution is refined. Refinement to 22 m does not change the frequency at which the spectra diverge, but rather reproduces a higher energy peak near 1. Although the generation mechanism is different, these waves are in the near buoyancy frequency bandwidth associated with shear instabilities [ Boegman et al. To check for aliasing, we compared the physical characteristics of the NLIWs to the grid resolution.
To be consistent with de la Fuente et al. Cayuga Lake bathymetric maps with 20 m contour isobaths are also shown. The time of each snapshot is shown in days. As the wave front extends across the entire width of the lake, it attains a concave curvature due to refraction; the depth is greater in the center of basin and thus the center of the front propagates faster toward the northwest slope. On day The surge then moves southeast where it interacts with a submerged ridge, again generating transverse propagating NLIW packets.
The gyres are visualized with simultaneous snapshots of the modeled currents at 0. The gyres are only captured in the upper layer where the stronger upper flow travels northwest, and they disappear as the wind calms after event W3; Figure 3 and the upper flow changes direction e. The gyres are stronger around the depth of the thermocline i. These local not basin scale gyre patterns likely develop due to sharp shoreline irregularities i.
There is a 6 m shift in easting direction between frames shown for the various depths. The time of each snapshot is given in days. Many of these applications do not have a suitable leptic ratio to fully capture nonhydrostatic effects Table 2. As the primary NLIW packets propagate along the longitudinal axis of the basin, they often propagate obliquely to the sloping bottom e. Observations of transverse shoaling may be obtained from sea surface imagery [e. Transverse shoaling also has implications for the computation of wave reflectance in nonidealized domains.
The simulated arrival time of the steepened fronts were delayed for several hours relative to the observations and the amplitudes of the simulated NLIWs were slightly underpredicted. These discrepancies could be due to the use of a uniform wind field, the coarseness of the bathymetry and grid resolution, and choice of drag coefficient and numerical errors.
They argued that insufficient bathymetric resolution causes an underprediction of barotropic currents, which leads to an underpredition of the internal wave amplitude. Conversely, de la Fuente et al. They defined the ratio of numerical to physical dispersion according to 1. The theoretical solitary wavelengths from hydrostatic physical dispersion was set to zero and nonhydrostatic KdV equations, respectively, were shown to scale with the grid lepticity as 3.
In our results, both hydrostatic and nonhydrostatic models, with large grid lepticity, show virtually identical results, as numerical dispersion dominates the solution of both equations. Our results on leptic ratio and nonhydrostatic effects are also consistent with Hodges et al. The contaminating effects of dispersion and aliasing cannot be separated, and so convergence through reduced aliasing will also occur with grid refinement section 3.
Our results e. For our observations, the minimum horizontal grid scales required by lepticity and aliasing are essentially equivalent.
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However, the required grid to resolve some NLIWs and internal solitary waves, as reported in the literature, may be up to 30 times smaller due to either an aliasing or lepticity limitation Figure Also shown are characteristic and of observed NLIWs circles in lakes, estuaries, and oceans [after Boegman et al. Grid refinement in hydrostatic models leads to considerably less dispersion as numerical dispersion decreases proportionally to the grid resolution squared.
The ratio between the computed solitary wavelengths from these grids i. Hodges et al. The current model employs 20 times the grid cells used by Zhang et al. Thus, physical dispersion in their simulations is overwhelmed by numerical dispersion by a factor of 3. Vlasenko et al. Both the present study and Lai et al. Unlike horizontal grid resolution, the vertical grid resolution was not varied in this study, and so its impact on resolving wave dynamics must be inferred from comparison of our results to those in the literature.
Similarly, Zhang et al. A search of the literature shows that, apart from recent studies, these requirements are generally not satisfied e. Grid requirements for wavelength and amplitude requirements are somewhat less restrictive. Future work will apply the model results to quantify the temporal distribution of energy and flux between different components of the internal wave field in lakes. We thank E.
Schweitzer for sharing the field observations and D. Bouffard, P. Diamessis, and S. MacIntyre for discussions on this research. Model setup files are available from the corresponding author boegmanl queensu. Volume , Issue 3.
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