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

Figures
Figure 1.1. Sketch of density-driven (gravitational) circulation in an estuary
caused by the longitudinal density gradient.................................................................... 4
Figure 2.1. Sketch showing orientation of Cartesian coordinate system with
corresponding velocity components................................................................................. 18
Figure 2.2. Sketch defining the free-surface elevation ............................................................... 31
Figure 2.3. Sketch showing velocities for mass flux balance on a surface control
volume in three dimensions................................................................................................ 34
Figure 2.4. Sketch showing viscous stresses on a surface layer of a water body
affected by a wind stress with components τxs and τys................................................ 35
Figure 2.5. Sketch showing flow over the bottom boundary of an estuary with
roughness elements of varying sizes................................................................................ 37
Figure 2.6. Sketch showing velocities for mass flux balance on a bottom control volume
in three dimensions.............................................................................................................. 38
Figure 2.7. Sketch showing viscous stresses on a bottom layer of a water body with
boundary stress components τxb and τyb ........................................................................ 39
Figure 2.8. Grids showing horizontal stretching of coordinates using two regions in both
the x and y directions............................................................................................................. 47
Figure 2.9. Plan view of an estuary shown in (A) physical space and (B) computational space
using the transformation of Boericke and Hall (1974) .................................................... 48
Figure 2.10. Orthogonal curvilinear grid used by Blumberg and Herring (1987) to model a
portion of the California coast............................................................................................ 49
Figure 2.11. Nonorthogonal curvilinear grid used by Johnson and others (1991, 1993) to
model Chesapeake Bay....................................................................................................... 50
Figure 2.12. Sketch of a multilayered system........................................................................................ 52
Figure 2.13. Three-dimensional model layering schemes using 10 layers to resolve the
deep water channel ............................................................................................................. 54
Figure 3.1. Multilevel model computational grid showing the stacked layers and the
location of variables............................................................................................................. 56
Figure 3.2. Diagram of a 3-dimensional horizontal velocity profile approximated by layeraveraged
values ................................................................................................................... 58
Figure 4.1. Definition sketch for 1-dimensional channel ................................................................... 70
Figure 4.2. Two-time-level staggered grid for the 1-dimensional model........................................ 71
Figure 4.3. Three-time-level staggered grid for the 1-dimensional model ..................................... 75
Figure 4.4. Computational stencils for (A) explicit leapfrog differencing method and
(B) implicit (Crank-Nicolson type) leapfrog differencing method used in the
three-time-level 1-dimensional scheme........................................................................... 76
Figure 4.5. The space-staggered horizontal grid used in the 3-dimensional model showing
the location of variables...................................................................................................... 80
Figure 4.6. Multilevel staggered grid for the 3-dimensional model. ................................................ 81
Figure 4.7. Representation of five-diagonal coefficient matrix ........................................................ 89
Figure 4.8. Illustration of the "natural" ordering scheme used in forming the matrix system
of equations for the free-surface elevation........................................................................ 89
Figure 5.1. Channel characteristics and initial flow conditions for the 1-dimensional
hydrograph test problem..................................................................................................... 95
Figure 5.2. Hydrographs computed with the two-level, semi-implicit scheme using a
large spatial step size of 5,000 feet and time step size of 5 minutes............................ 96
Figure 5.3. Base solution of the hydrograph problem shown at 125-minute intervals during
the simulation........................................................................................................................ 98
Figure 5.4. Graphs showing error measures using the two-level semi-implicit scheme on the
hydrograph test problem.....................................................................................................100
Figure 5.5. Graph showing two-level scheme solutions for the hydrograph problem using
three different values for space steps and a time step of 5 minutes...........................102
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