The model equations for storm surge and tsunamis most commonly used are the shallow water equations with addition of appropriate source terms for bathymetry. Traditional approaches will need to resolve the mesh to discretize small-scale structure, which impacts the time-step size to be proportional to the size of cells. In this thesis, a novel approximate Riemann solver was developed in order to deal with the existence of barrier without restricting the time-step due to small cells. Because of the wave redistribution method and proper ghost cells setting, the novel Riemann solver maintained properties including mass and momentum conservation, the well-balancing properties and robustness at the wet-dry interface. The solver also preserves nonnegative water depth and prevents leakage. A modified h-box method is applied so the algorithm can overcome restrictions of small time-step sizes. The work has been done in the context of the GeoClaw platform with retaining the capabilities of GeoClaw solver. At the same time, the special developed Riemann solver extends the package to handle the sub-grid-scale effects of barriers. Incorporating the solver developed in this work into the GeoClaw framework has allowed to leverage GeoClaw’s ability to handle complex bathymetry and real applications.

Storm induced flooding presents a multifaceted threat to coastal communities across the world.With climate change and sea level rise this danger is expected to increase. As coastal communities become exposed to more frequent and more severe flooding, the need for protective measures will increase. To know how to optimally protect against coastal flooding requires an understanding of future flood risk, storms, and storm surge. These are challenging to estimate due to many sources of uncertainty. In this thesis I present a methodology to forecast this future flood risk. I combine multiple computational, physics and statistical models to accurately describe the fluid dynamics of flooding, the cyclones that drive surge, and how climate change will influence these different components in the future. These computational models must be fast so that they can be embedded into an optimization framework that makes many evaluations. To find an optimal protective measure I employ stochastic and derivative free optimization methods. A complete study is conducted on New York City and optimal protective strategies are found for minimizing the total cost from storm surge subject to different budget constraints.

Natural and local anthropogenic changes in estuaries (e.g., sea-level rise, navigation channel construction and loss of wetlands) interact with each other and produce non-linear effects. There is also a growing recognition that tides in estuaries are not stationary. These factors together are changing the estuarine water level regime, however the implications for extreme water levels remain largely unknown. Changes over the past century in many estuaries, such as channel deepening and streamlining for navigation have significantly altered the hydrodynamics of long waves, often resulting in amplified tides (a ~85% increase in Wilmington, NC since 1900) and storm surge in estuaries. This research focuses on establishing analytical and numerical models that simulate a wide range of systems and flow conditions that combine multiple flood sources: astronomical tide, storm surge, and high river flow. To investigate the effects of estuarine bathymetry conditions (e.g., channel depth, convergence length), hurricane conditions (e.g., pressure and wind field), river discharge, and surge characteristics (e.g., time scale and amplitude and relative phase) on tide and storm surge propagation, I develop an idealized analytical model and two numerical models using Delft-3D. The Cape Fear River Estuary, NC (CFRE), and St Johns River Estuary, FL (SJRE) are used as case studies to investigate flood dynamics. The analytical approach has been compared and verified with idealized numerical models. I use data recovery, data analysis, and idealized numerical modeling of the CFRE to investigate the effects of bathymetric changes (e.g., dredging and channel modification) on tidal and storm surge characteristics over the past 130 years. Data analysis and modeling results suggest that long-term changes in tides can be used along with the tidal analysis tools to investigate changes in storm surge. Analysis indicate that tidal range in Wilmington, NC (Rkm 47) has doubled to 1.55m since the 1880s, while a much smaller increase of 0.07m observed close to the ocean in Southport (Rkm 6) since the 1920s. Further, model results suggest that the majority of long term changes in tides of this system have been caused by deepening the system from 7m to 15.5m due to dredging, rather than by changes in the coastal tides. Numerical modeling using idealized, parametric tropical cyclones suggests that the amplitude of the worst-case, CAT-5 storm surge has increased by 40-60% since the nineteenth century. Storm surges are meteorologically forced shallow water waves with time scales that overlap those of the tidal bands. Using data, I show that the surge wave can be decomposed into two sinusoidal waves. Therefore, I analytically model surge via a 3-constituent analytical tide model, where the third constituent is the dominant semi-diurnal tide and friction is linearized via Chebyshev polynomials. A constant discharge is considered to approximate fluvial effects The analytical model is used to study how surge amplitude, surge time scale, and surge-tide relative phase affect the spatial pattern of amplitude growth and decay, and how depth changes caused by channel deepening influence the magnitude of a storm surge. I use non-dimensional numbers to investigate how channel depth, surge time scale and amplitude, surge asymmetry, and relative timing of surge to tides alter the damping or amplification of surge along the estuary. The non-dimensional numbers suggest that increasing depth has similar effects as decreasing the drag coefficient. Similarly, larger time scale has an equivalent effect on tide and surge as increasing depth due to channel deepening. Analytical model results show that the extent of the surge amplification is dependent on the geometry of the estuary (e.g., depth and convergence length) and characteristics of the surge wave. Both models show that much of the alterations of water levels in estuaries is due to channel deepening for navigation purposes and that the largest temporal change occur for surges with a high surge to D2 amplitude ratio and a short time scale. Model results farther indicate that surge amplitude decays more slowly (larger e-folding) in a deeper channel for all surge time scales (12hr-72hr). Another main finding is that, due to nonlinear friction, the location of maximum change in surge wave moves landward as the channel is deepened. Thus, changes in flood risk due to channel deepening are likely spatially variable even within a single estuary. Next, I use the verified analytical model and numerical models to investigate the effects of river flow on surge wave propagation, and spatial and temporal variability of compound flooding along an estuary. To model the historic SJRE, I digitize nautical charts of SJRE to develop a numerical model. Both the numerical and analytical models are used to investigate the contribution of tide, surge, and river flow to the peak water level for historic and modern system configurations. Numerical modeling results for hurricane Irma (2017) show that maximum flood water levels have shifted landward over time and changed the relative importance of the various contributing factors in the SJRE. Deepening the shipping channel from 5.5m to 15m has reduced the impacts of river flow on peak water level, but increased the effects of tide and surge. Sensitivity studies also show that peak water level decreases landward for all river flow scenarios as channel depth increases. Model results show that the timing of peak river flow relative to the time of maximum surge causes very large changes in the amplitude of total water level, and in river flow effects at upstream locations for modern configuration than for the historic model. Changes in surge amplitudes can be interpreted by the non-dimensional friction number, which shows that depth (h), surge time scale (T=1/w), and convergence length-scale (Le) affect the damping/amplification of both tides and surge waves. Overall, this study demonstrates that a system scale alteration in local storm surge dynamics over the past century is likely to have occurred in many systems and should be considered for system management. The results of this research give the scientists and engineer a better understanding of tide, river flow, and surge interactions, and thereby contribute to an understanding of how to predict storm surges and help mitigate their destructive impacts. Future system design studies also need to consider long-term and changes of construction and development activities on storm surge risk in a broader context than has historically been the case.