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Dispersion and Tidal Dynamics of Channel-Shoal Estuaries By Dispersion and Tidal Dynamics of Channel-Shoal Estuaries by Christopher Dean Holleman A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering { Civil and Environmental Engineering in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Mark T. Stacey, Chair Professor Fotini K. Chow Professor Philip S. Marcus Fall 2013 Dispersion and Tidal Dynamics of Channel-Shoal Estuaries Copyright 2013 by Christopher Dean Holleman 1 Abstract Dispersion and Tidal Dynamics of Channel-Shoal Estuaries by Christopher Dean Holleman Doctor of Philosophy in Engineering { Civil and Environmental Engineering University of California, Berkeley Professor Mark T. Stacey, Chair Estuaries, and the varied ecosystems they support, are affected by human action in many ways. One of the fundamental environmental questions pertaining to estuaries is how material is mixed into ambient waters and transported within and beyond the estuary. Char- acterizing transport and mixing is essential for tackling local environmental questions, such as gauging the water-quality impacts of a wastewater outfall, or how dredging may alter es- tuarine circulation. In many estuaries the dominant physical dispersion mechanism is shear dispersion. A fundamental and idealized approach to the scalar transport problem is under- taken, in which canonical shear dispersion regimes are augmented with a new, intermediate regime. This mode of shear dispersion, in which the longitudinal variance of the plume in- crease with the square of time, is likely to occur in channel-shoal basins where bathymetric transitions can create sheared velocity profiles. A key tool in the engineer's arsenal for pur- suing studies of scalar transport is the numerical model, but care must be taken to separate physical mixing effects from numerical artifacts. Unstructured models are particularly well- suited to estuarine problems as the grid can be adapted to complex local topography, but the numerical errors of these models can be difficult to characterize with standard methods. An in-depth error analysis reveals a strong dependence of numerical diffusion on the orientation of the grid relative to the flow. Flexible grid generation methods allow for optimizing the grid in light of this dependence, and can decrease across-flow numerical diffusion by a factor of two. The most far-reaching impact of our actions, though, must certainly be global climate change. A period of rising sea level is being ushered in by climate change, and the final research-oriented chapter seeks to further our understanding of how rising sea levels will affect basin-scale tidal dynamics. Numerical and analytic approaches show that as basins get deeper tidal amplification becomes more effective and tidal range increases. The crux of the analysis, though, is the inclusion of inundation within these scenarios. Inundated areas dissipate incident tidal energy, countering the added amplification due to basins becoming deeper. The net effect, in the case of San Francisco Bay, is that tidal amplification under 2 sea level rise actually decreases, such that a particular rise in mean higher high water in the coastal ocean is predicted to raise mean higher high water within the bay by a lesser amount. i To my parents, my brothers, my friends. ii Contents Contents ii List of Figures iv List of Tables viii 1 Introduction 1 2 Transient Shear Dispersion Regimes 6 2.1 Introduction . 6 2.2 Analytical Derivation . 9 2.3 Estimation of Regimes and Parameters . 14 2.4 Tidally-forced Channel-Shoal Domain . 19 2.5 Discussion . 24 2.6 Conclusions . 25 3 Numerical Diffusion on Unstructured, Flow-aligned Meshes 26 3.1 Introduction . 26 3.2 Analytical Grid Diffusion . 29 3.3 Idealized Simulations . 35 3.4 Automated Grid Alignment in an Idealized Flow . 37 3.5 Application to a Physical System . 44 3.6 Discussion . 54 3.7 Conclusions . 56 4 Coupling of Sea Level Rise, Tidal Amplification and Inundation 57 4.1 Introduction . 58 4.2 Physical Domain . 62 4.3 Numerical Model . 64 4.4 Energy Flux and Tidal Phase Analysis . 67 4.5 Tidal Amplification and Damping . 76 4.6 Overtides . 82 4.7 Discussion . 84 iii 4.8 Conclusions . 86 5 Conclusion 89 A Calculation of Diffusion Coefficients 99 B Connectivity-based Model Bathymetry 102 C Model Calibration and Validation 106 D Tools for Estuarine Modeling 109 D.1 Changes to the Numerical Kernel . 109 D.2 Grid Generation . 113 D.3 Boundary Forcing . 120 D.4 Visualization . 122 iv List of Figures 2.1 Schematic of the three dispersion regimes . 11 2.2 Schematic of plume interaction with bathymetry variability . 13 2.3 Accumulation of strain by a plume in the finite shear, infinite domain case . 15 2 2.4 Relationship between σx and the time origins for covariance and lateral variance in the case of infinite shear. 17 2.5 Asymptotic behavior and lack of accumulated state for linear shear dispersion. The upper panel shows a sample tracer distribution, with corresponding time series of covariance and longitudinal variance in the lower panels. 18 2.6 Bathymetry profiles idealized domains, (a) narrow channel and (b) wide channel. Horizontal, dashed lines bracketing z = 0 denote the range of the tidal boundary condition. Vertical, dotted lines denote particle release locations. 19 2.7 Plan view of the central portion of the wide channel idealized domain. Locations of particle plume releases are shown by empty circles, and the Lagrangian residual field is shown by arrows. Greyscale depth breaks shown at depths of 11, 15 and 19m. ......................................... 21 2.8 Instantaneous plume distributions after 0 (gray), 7 (blue), 20 (green), 50 (orange) and 100 (red) tidal periods. Greyscale depth breaks at 11, 15, and 19 m, with the lower portion of the plot falling within the channel region and the upper half in the shoal region. Ellipses are drawn according to the 2-D covariance matrix. 21 2 2.9 Cumulative growth of σx for the same particle cloud as depicted in figure 2.8. 2 The time series of σx has been partitioned into the cumulative contributions of ∗ the three shear dispersion regimes, based on the observed values for αi , plus the constant contribution of the imposed K0. ..................... 22 2.10 Correlation between predicted and observed growth coefficients. Diagonals show the 1:1 lines and grey perpendiculars intersect at (0; 0). Panels show correlation for (a) the infinite shear regime, (b) the finite shear / unbounded regime, and (c) the linear, Fickian limit of bounded shear dispersion. Correlation coefficients are ∗ ∗ ∗ R = 0:890, 0.867, and 0.565 for α2, α1 and α0, respectively. 23 2.11 Flux view of quadratic dispersion in an estuarine reach which is not laterally well- mixed. Significant salt flux occurs even though gradients in the cross-sectionally averaged salinity hsi are negligible. 25 v 3.1 Schematic for derivation of recurrence relation . 29 3.2 Schematic showing vanishing of lateral diffusion for θ = 0. Solid lines denote zero-flux faces of the cells, and dashed edges have nonzero flux. The arrows along the center strip show the vector (∆x; ∆y) for which the Taylor series expansion is evaluated for each cell's upwind neighbor. The nonzero y component of the vectors leads to an overpredicted Ky unless the two types of cells are treated simultaneously. 32 3.3 Idealized domain with initial condition for passive scalar. l = 500m, θ = 15◦. 36 3.4 Measured versus predicted numerical diffusion, using the average of the indepen- dent modified equations. (a) Longitudinal diffusion (b) Lateral diffusion. (c) Cross diffusion. Grid alignment in (a)-(c) is shown by the orientation of the triangular markers, relative to a left-to-right flow as illustrated in (d). 38 3.5 Measured versus predicted numerical diffusion using a recurrence relation to unify the two types of cells. (a) Longitudinal diffusion (b) Lateral diffusion (c) Cross diffusion. Grid alignment in (a)-(c) is shown by the orientation of the triangular markers, relative to a left-to-right flow as illustrated in (d). 39 3.6 Unaligned grid for rigid-body rotation test case, upper-right quadrant . 40 3.7 Principal streamlines extracted from the unaligned simulation . 43 3.8 Upper-right quadrant of the aligned grid for the rigid-body rotation test case . 43 3.9 Initial scalar field on the unaligned grid. Grayscale graduations indicate 0.1 contours, with a maximum concentration of 1.0 . 44 3.10 Computational domain for the San Francisco Bay model runs. Nominal grid edge-length is 5000m in the coastal ocean, 500m in North San Francisco Bay, and 250m in South San Francisco Bay. For computational efficiency and boundary condition simplicity the Sacramento-San Joaquin Delta has been replaced by a pair of long dissipative channels. The dashed line represents the tidal ocean boundary condition. 45 3.11 South San Francisco Bay bathymetry used in the physically realistic model runs. Contours at -15, -10, -5 (shaded) and -2.5m (NAVD88). Transect sampling loca- tions are shown as black dots, with the start of the transect (0km) at the southern end. 47 3.12 Principal velocity ellipses in the vicinity of the study area, derived from model output from the unaligned grid. Ellipses are calculated for all cells, but are shown interpolated onto a coarse regular grid and normalized by major axis magnitude for clarity. 48 3.13 Flow-aligned contours, traced from the principal velocities, ready for input to the re-gridding process. The separation between adjacent contours scales with the grid resolution, leading to sparser contours in the north and west. 49 3.14 (a) Distribution of cell-center spacing for the unaligned grid. (b) Distribution of skewness for the unaligned grid. (c) Distribution of cell-center spacing for the aligned grid.
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