A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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116 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation The driving force for this problem is the variation in water surface elevation ζ at the seaward boundary that is caused by a simple harmonic tide 2πt ζ() t = Asin⎛------- ⎞, (5.1) ⎝ t ⎠ p where the time t is expressed in hours from the start of the simulation. The tidal amplitude is A = 3 ft (0.9 m) and the tidal period is t p = 12.4 hours. A period of 12.4 hours corresponds with the principal lunar constituent of the astronomical tide known as the M 2 tide. The dimensions of the channel length and depth were chosen so that the wavelength 49 of the M 2 tide is close to four times the length L of the tidal channel. Under these conditions, the traveling wave entering the channel interferes constructively (adds together) with the wave reflected from the wall boundary, thereby setting up a resonant oscillation in the channel. The result is a tidal range of water surface elevation that is greater at the closed boundary than at the open boundary. This effect is illustrated in figure 5.14 by using a solution to the problem showing water surface elevation at six locations along the channel against time. 49The wavelength of the tide can be estimated as Lw = tp ⋅ gH0 with tp expressed in seconds. WATER SURFACE ELEVATION, IN FEET 5 4 3 2 1 0 −1 −2 −3 −4 −5 0 1 2 TIME, IN DAYS EXPLANATION 0 feet 60,000 feet 120,000 feet 180,000 feet 240,000 feet 300,000 feet Figure 5.14. Water surface elevations at six locations along the tidal channel. The distances are measured landward from the seaward boundary.
5. Numerical Experiments 117 Each test simulation for the tidal problem is run for a total of six days (144 hours). This allows sufficient time for any solution instabilities to appear. The results of each simulation are presented at the location x = 180,000 ft (55 km) from the seaward boundary of the tidal channel. A solution for water surface elevation and velocity is shown in figure 5.15. After simulating ten cycles of the M 2 tide, results from the 11th cycle are examined in detail and used in the computation of error measures. 5.2.2.1 Two-Level Scheme Figure 5.16 shows three solutions of the tidal problem using the two-level scheme; one is the base solution included in both graphs. The solid lines present the base solution obtained by using the two-level scheme with a very small grid size of Δt = 0.5 minutes, Δx = 400 ft (122m) and with five iterations (niter = 5). The base solution is assumed to be free of numerical error for the purposes of computing error measures for the solutions with coarser grid sizes. The two solutions indicated by the dashed curves in figure 5.16 are computed using large time steps of 60 minutes (fig. 5.16A) and 30 minutes (fig. 5.16B.); both solutions use a space step (Δx) of 6,000 ft (1,828 m) and two iterations (niter = 2). Figure 5.17 shows a plot of the volumetric transport (U = uh) from the solution using Δt = 60 minutes compared with the volumetric transport of the base solution. The root mean square error (RMS e) of the solution is computed relative to the base solution using the volumetric transport variable defined in table 5.5. The RMS e is 8.7 percent for the 60-minute solution and 2.3 percent for the 30-minute solution. The other error measures defined in are computed as A e = 5.2 percent, P e = 0.35 percent, and M e = 4.7 percent for the solution using Δt = 60 minutes, and A e = – 0.8 percent, P e = 0.16 percent, and M e = 1.0 percent for the solution using Δt = 30 minutes. WATER SURFACE ELEVATION, IN FEET 5 4 3 2 1 0 −1 −2 −3 −4 −5 Water surface elevation Velocity 0 1 2 3 4 5 6 TIME, IN DAYS 11th cycle Figure 5.15. Solution of the tidal problem for water surface elevation and velocity at the location x = 180,000 feet. The results of simulations for the 11th cycle of the M 2 tide are examined in detail in figure 5.16. VELOCITY, IN FEET PER SECOND
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In cooperation with the Interagency
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U.S. Department of the Interior Dir
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vi 4.2 Semi-Implicit One-Dimensiona
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viii Figure 5.6. Graph showing solu
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x Tables Table 2.1. Dimensionless n
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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