Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Hamidreza Shirkhani Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the Doctorate in Philosophy degree in civil engineering Department of Civil Engineering Faculty of Engineering University of Ottawa © Hamidreza Shirkhani, Ottawa, Canada, 2016 PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Abstract The knowledge of flow hydrodynamics within the next decades is of particular importance in many practical applications. In this study, a methodological improvement has been made to the evaluation of the flow hydrodynamics under climate change. This research, indeed, proposes an approach which includes the methods that can consider the climate change impact on the flow in estuaries, gulfs, etc. It includes downscaling methods to project the required climate variables through the next decades. Here, two statistical downscaling methods, namely, Nearest Neighbouring and Quantile-Quantile techniques, are developed and implemented in order to predict the wind speed over the study area. Wind speed has an essential role in flow field and wave climatology in estuaries and gulfs. In order to make the proposed methodology computationally efficient, the flow in the estuary is simulated by a large-scale model. The finite volume triangular C-grid is analysed and shown to have advantages over the rectangular (finite difference) one. The dispersion relation analysis is performed for both gravity and Rossby waves that have crucial effects in oceanic models. In order to study the unstructured characteristic of the triangular grids, various isosceles triangles with different vertex angles are considered. Moreover, diverse well-known second-order time stepping techniques such as Leap-Frog, Adams-Bashforth and improved Euler are studied in combination with the C-grid semi discrete method. The fully discrete method is examined through several numerical experiments for both linear and non-linear cases. The results of the large-scale model provide the boundary conditions to the local coastal model. In order to model the flow over a local coastal area, a well-balanced positivity preserving central-upwind method is developed for the unstructured quadrilateral grids. The quadrilateral grid can effectively simulate complex domains and is shown to have advantages over the triangular grids. The proposed central-upwind ii PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries scheme is well-balanced and preserve the positivity. Therefore, it is capable of modelling the wetting and drying processes that may be the case in many local coastal areas. It is also confirmed that the proposed method can well resolve complex flow features. The local model incorporates the outputs of the downscaling and large-scale flow models and evaluates the flow hydrodynamics under changing climate. iii PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Acknowledgements I would like to express my deepest gratitude to my supervisors, Dr. Ousmane Seidou and Dr. Majid Mohammadian, for their invaluable guidance, advice, and continued supports. I appreciate all their contributions of time and inspiration to make my PhD experience productive and stimulating. None of this would have been possible without their dedication. I would like to acknowledge my committee members, Drs. Colin Rennie, Ioan Nistor, Tew-Fik Mahdi, and Abhijit Sarkar, for providing precious suggestions and comments on my thesis. I would like to express my sincere appreciation to Dr. Alexander Kurganov for his valuable collaboration and advice. I would like to thank my family for their everlasting encouragement and specially my parents for their unconditional love and support in all my pursuits. Most of all, I would like to give my huge thanks to my beloved wife Parna who has always been by my side with patience, endless love, and faithful support. iv PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries To Parna and My parents v PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries List of figures Fig. 2.1 - Ensemble average of raw GCMs data for different RCPs: (a) annual mean wind speed within short-term, medium-term, and long-term futures; (b) monthly mean wind speed, where shaded area represents the range of changes projected by different models; the model ensemble averages for each RCP are shown with thick lines ........................................................................... 16 Fig. 2.2 - Study area through the Qatar coast, with weather station indicated in the figure (adapted from http://d-maps.com/carte.php?num_car=56002&lang=en; http://d- maps.com/carte.php?num_car=66536&lang=en; http://d- maps.com/carte.php?num_car=3494&lang=en) ............................................................................... 17 Fig. 2.3 - Monthly distribution of observed and projected daily mean wind speed using Quantile- Quantile transformation for calibration and validation period: (a) ensemble average; (b) CM3 .... 22 Fig. 2.4 - Monthly distribution of observed and projected daily mean wind speed using nearest neighbor method for calibration and validation period (ESM2M) ................................................... 23 Fig. 2.5 - Comparison of raw and corrected (downscaled) wind speeds for ESM2M with observation within validation period ............................................................................................... 23 Fig. 2.6 - (a) Taylor diagrams for raw and corrected (downscaled) monthly mean wind speeds, comparing observations with the various models and model ensemble simulations for the historical period; (b) box plot for raw and corrected (downscaled) daily mean wind speed versus observations. .................................................................................................................................... 25 Fig. 2.7 - Time spread of monthly distribution for projected mean daily wind speed (Quantile- Quantile) for ESM2M under the RCP26 scenario ............................................................................. 26 Fig. 2.8 - GCM spread of monthly distribution for long-term projection of mean daily wind speed (Quantile-Quantile) under the RCP85 scenario ................................................................................ 27 Fig. 2.9 - Scenario spread of ensemble average for monthly distribution of short-term projection of mean daily wind speed (Quantile-Quantile) ................................................................................ 27 Fig. 2.10 - Observed and downscaled (ESM2M) wind speed for the 1981–2010 period: (a) monthly mean, Quantile-Quantile; (b) monthly mean, nearest neighbor; (c) daily mean time series, Quantile-Quantile; (d) daily mean time series, nearest neighbor .................................................... 28 Fig. 2.11 - Monthly 100-year wind speed (Quantile-Quantile) within calibration and validation periods for ensemble average .......................................................................................................... 31 vi PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Fig. 2.12 - Contour plots of (a) 100-year significant wave heights (m) from historical wind observations; (b) short-term 100-year significant wave height changes (%) for ensemble average under RCP26 ..................................................................................................................................... 33 Fig. 2.13 -Multimodel average changes of 100-year significant wave height for a range of depths and fetch lengths under (a) RCP26; (b) RCP45; (c) RCP60; (d) RCP85 emission scenarios ............... 36 Fig. 3.1 - Normal velocity and water surface elevation locations are presented by the symbols ○ and →, respectively (left); estimation of normal gradient of water surface elevation at face j (right). ... 43 Fig. 3.2 - (a) Triangular mesh grid made up of equilateral triangles of length h. (b) Gradient (left) and divergence (right) stencils of the C-grid spatial discretization. ................................................. 44 Fig. 3.3 - Discretization of spatial term in momentum equation. ..................................................... 44 Fig. 3.4 - Phase speed ratio for C-grid spatial discretization. ........................................................... 48 Fig. 3.5 - The group velocity vectors of (a) the continuous case and (b) C-grid spatial discretization. .......................................................................................................................................................... 49 Fig. 3.6 - Definition of the directions OX, OY and OD on a Cartesian coordinate. ......................... 49 Fig. 3.7 - Computational and analytical directional derivative (Gcp.d and Gan.d) in OY, OX and OD directions. ................................................................................................................................... 50 Fig. 3.8 - The phase speed ration as a surface function for CFL = 0.1. ............................................ 53 Fig. 3.9 - As for Fig. 3.8 but for CFL = 0.5. ..................................................................................... 54 Fig. 3.10 -As for Fig. 3.8 but for CFL = 0.9. .................................................................................... 55 Fig. 3.11 - Cross sections of the water surface elevation at times 2000 s and 4000 s for the Leap- Frog method ...................................................................................................................................... 57 Fig. 3.12 - Cross sections of the water surface elevation at times 2000 s and 4000 s for the Adams– Bashforth method.............................................................................................................................. 58 Fig. 3.13 -Isolines of water surface elevation of wave propagation at time 5000 s and CFL = 0.1. 59 Fig. 3.14 - Water surface elevation of the seiche wave for the domain midpoint at time 5T < t < 6T for . ............................................................................................................................... 61 Fig. 3.15 - As Fig. 3.14 but with CFL = 0.9 and k = π/2 for the Leap-Frog scheme. ....................... 62 Fig. 3.16 - Water surface elevation of the seiche (standing wave) versus time for k = π/8 with different CFL numbers. .................................................................................................................... 62 vii PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Fig. 3.17 - Comparison of numerical and exact solution of pure gravity wave at time 60 s with CFL = 0.1. ................................................................................................................................. 63 Fig. 3.18 - Water surface elevation after water reflects off the side walls of the basin at time 400 s with CFL = 0.1 using (a) Adams–Bashforth and (b) Leap-Frog schemes. ....................................... 64 Fig. 3.19 - Evolution of L2 error in Log–Log scale for pure gravity wave at t = 60 s. .................... 65 Fig. 3.20 - Organized-unstructured triangular meshes with (a) excellent, (b) good, (c) acceptable shape and (d) fully-unstructured triangular mesh. ............................................................................ 67 Fig. 3.21- Evolution of L2 error in log–log scale for pure gravity wave at t = 60 s (a) for organized-unstructured mesh with good shape (b) fully-unstructured triangular mesh. .................. 69 Fig. 3.22- Three dimensional view of water surface at t = 60 s using (a) good organized- unstructured triangular mesh (b) fully-unstructured triangular mesh. .............................................. 69 Fig. 3.23 - (a) Comparison of analytical and numerical solutions of a parabolic flood wave at t = 7 s (b) three dimensional view of the parabolic flood at t = 25 s. .......................................................... 71 Fig. 3.24 - Evolution of the L2-error for parabolic flood wave in log–log scale. ............................. 71 Fig. 3.25 - Comparison of the computed (dots) and analytical (lines) water surface elevation for parabolic flood wave at different time stage. .................................................................................... 72 Fig. 3.26 - Contour lines of still water depth over the model-lake. .................................................. 74 Fig. 3.27 - Steady-state water surface elevation and horizontal velocity field for (left) North-East wind direction att = 2 h and (right) North-West wind direction at t = 4 h. ....................................... 75 Fig. 3.28 - Horizontal velocity field (left) and water surface elevation (right) after wind direction changes at t = 3 h. ............................................................................................................................. 76 Fig. 4.1 – (a) Normal velocity and water surface elevation locations are presented by the symbols and , respectively. (b) Estimation of normal gradient of water surface elevation at face ........... 87 Fig. 4.2- Isosceles triangle cell grid with vertex angle α and side length of (right) anda sample of fully unstructured mesh grid (right) which can be generally represented by grid cell shown in (a) with various values of α. ................................................................................................................... 89 Fig. 4.3 - (a) Three possible location of normal velocity of gradient and (b) Two possible location for water surface elevation in the divergence stencils of the C-grid spatial discretization ............... 90 Fig. 4.4 - Selected directions for the triangular (left) and rectangular cell(right) ............................. 95 viii PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Fig. 4.5 - Dispersion relation results of non-dimensional frequency for (a) continuous, triangular C-grid with vertex angle of (b) , (c) , (d) . Left column ( ) is high resolution and Right column ( ) is low resolution. ............................ 96 Fig. 4.6 - Dispersion relation results of non-dimensional frequency with for triangular C-grid with vertex angle of (a) , (b) , (c) and (d) structured rectangular C-grid. ........................................................................................................... 98 Fig. 4.7 - Phase Speed Ration of various temporal schemes for ........................ 101 Fig. 4.8 - As of Fig. 4.7 but for ................................................................................... 102 Fig. 4.9 - As of Fig. 4.7 but for .................................................................................... 103 Fig. 4.10 - Damping Error associated to the Rossby mode for Improved Euler scheme with (a) , (b) and (c) . .......................................................................... 105 Fig. 4.11 - Water surface elevation around the source point for various grid resolution with vertex angle of (left), (midle) and (right). ........................................... 107 Fig. 4.12 - Vertical cross sections (1D slices along ) of the water elevation for surface gravity wave : Results obtained by analytical solution (right) and using proposed numerical scheme (left). Water surface elevation when the wave is (a) at its initial condition of Gaussian distribution, (b) moving toward the basin wall (c) moving back to the center after hitting the wall, (d) returned to the starting point for first and (e) second time. ............................................................................... 109 Fig. 4.13 - Water surface elevation for Rossby waves at : Using proposed scheme with CFL=0.1 (left) and analytical solution (right) ................................................................................ 111 Fig. 4.14 - Comparison of numerical and analytical solution of water surface elevation for Rossby wave after periods ( ) ...................................................................................................... 112 Fig. 4.15 - Evolution of error for Rossby wave after 5 periods ( in Log-Log scale .. 114 Fig. 4.16 - Computed water surface elevation for Rossby waves at using non-linear shallow water equations. ................................................................................................................. 114 Fig. 4.17 - Contour plots of the divergence field computed by unstructured C-grid scheme after using (a) low resolution and (b) high resolution mesh grids. ....................................... 117 Fig. 4.18 - As of Fig. 4.10 but after . ............................................................................. 118 Fig. 4.19 - Zoom of the bottom left corner of the basin for divergence of velocity field after (a) and (b) . T ................................................................................................... 118 ix PhD Thesis: Methodological Developments for an Improved Evaluation of Climate Change Impact on Flow Hydrodynamics in Estuaries Fig. 4.20 - Unstructured triangular mesh grids of the circular basin along with the still water depth contour lines. .................................................................................................................................. 120 Fig. 4.21 - Water surface elevation at . ........................................................................ 121 Fig. 4.22 - Velocity field component of the solution at . ............................................. 121 Fig. 4.23 - Contour plots of water surface elevation for Rossby soliton wave: (a) the initial condition (b) computed by proposed scheme at dimensionless time (Equivalant to ). ....................................................................................................................................... 123 Fig. 4.24 - 3D view of the Rossby soliton computed by proposed scheme at dimensionless time of . ........................................................................................................................................... 124 Fig. 5.1- An unstructured quadrilateral cell with its four neighbouring cells ................................. 132 Fig. 5.2 - Subdividing cell into four triangles and ........................................ 135 Fig. 5.3 - Organized unstructured quadrilateral mesh. ................................................................... 142 Fig. 5.4 - Example1: Contour plot (left) and 3D view (right) of . ............................................... 143 Fig. 5.5 - Example2: as a function of for . .......................................... 144 Fig. 5.6 - Example2: 3D view of computed using the mesh with cells for . .................................................................................................................................. 145 Fig. 5.7- Example2: for computed using the well-balanced (left column) and non- well-balanced (right column) schemes. .......................................................................................... 146 Fig. 5.8 - Example3: 1D slice of bottom topography (5.40). The plot is not to scale. .................... 147 Fig. 5.9 - Example3: computed by the proposed well-balanced (left) and non-well-balanced (right) schemes. .............................................................................................................................. 148 Fig. 5.10 - Example4: 1D slice of the bottom topography (5.42). The plot is not to scale. ............ 149 Fig. 5.11 - Example4: 3D view of for computed using quadrilaterals .................................................................................................................................. 151 Fig. 5.12 - Example4: with computed using the grids with (left column) and (right columns) cells. ...................................................................... 152 Fig. 5.13 - Example5: 3D view of the bottom topography (5.44) and initial water surface (5.45). 153 x