A Tsunami in Lisbon Engenharia Civil

A Tsunami in Lisbon Where to run?

Daniel André Silva Conde

Dissertação para obtenção do Grau de Mestre em Engenharia Civil

Júri

Presidente: Prof. Doutor António Jorge Silva Guerreiro Monteiro Orientador: Prof. Doutor Rui Miguel Lage Ferreira Co-Orientador: Prof. Doutor Carlos Alberto Ferreira de Sousa Oliveira Vogal: Prof. Doutora Maria Ana Viana Baptista Vogal: Engenheira Maria João Martins Telhado

Outubro 2012

To my parents, Armando e Guilhermina. Thank you. Aos meus pais, Armando e Guilhermina. Obrigado.

Acknowledgments

This dissertation was developed at Instituto Superior Técnico - Technical University of Lisbon, Portugal, under the guidance of Professors Rui M. L. Ferrira and C. Sousa Oliveira. During this time, part of the work was supported by CEHIDRO - Centro de Estudos de Hidrossistemas - with two research initiation scholarships, also under the guidance of Professor Rui M. L. Ferreira.

For the great opportunities that were given to me with no hesitation and for all the support, knowledge and motivation transmitted throughout this year, my sincere gratefulness to Professor Rui M. L. Ferreira. And I really do mean sincere.

My many thanks to Ricardo for all the support and valuable help with STAV-2D. And of course to Edgar, for keeping the morale high when odds weren’t.

Also, to Professora Maria Ana Baptista, my sincere thanks for all the kindness and availability. Without her contribution this work wouldn’t be possible. To Professor C. Sousa Oliveira for the thoughtful insights about various topics on seismicity and tsunamis.

To my dear colleagues, specially André Marques and Pedro Pinotes, for making those hardworking hours as enjoyable as they could have been. Also to my fellow colleagues and friends at the Residência Engo Duarte Pacheco for providing the most hilarious 5 years that one could have had.

My non-translatable-to-words gratitude to my parents. To my mother, Mira, and my father, Armando, for the constant support, guidance and love throughout my entire existence.

To my grandfather, Joaquim, for the sensitive human being that he is and inspires to be. To my grandmother, Celeste, for being the craziest grandmother on this earth. And I mean this as the truest compliment. To my grandmother, Alcínia, for all the support and welcoming aﬀections every single time she laid eyes on me.

To my godparents, Fátima and Sérgio, for all the support, aﬀection and enjoyable moments that we have had together so far. Also, my aﬀectionate gratitude to my older "godfather", Sérgio, whose memory will last forever.

A very special place for the younger family members. To Zi and Lino for having the most inspiring relation that can be seen between two brothers. Everyone feels better if you’re around. Also, to the rising star in the game (football!), Miguel for the sincere feelings and aﬀections. To Catarina for the special

i connection we share. No matter how long or how far I am away from all of you, I will never cease to have four brothers.

And that’s about it. Oh wait... (ahah!) my loving gratitude to my girlfriend, Dora, for making the last 5 years probably the best ones that I have had so far. Thank you for fulﬁlling a long lasting void.

It would take forever to thank everyone I owe some part of myself or my life. To all those I have not mentioned, my apologies and sincere thanks.

ii Resumo

Uma revisão recente do histórico de tsunamis em Portugal mostrou que o estuário do Tejo foi afetado por vários tsunamis catastróﬁcos ao longo dos últimos dois milénios. Dada esta herança histórica, e a crescente consciencialização devida a fenómenos recentes com mediatização mundial, o presente trabalho procura fornecer informação relevante quanto à exposição das zonas ribeirinhas do estuário do Tejo a um tsunami semelhante ao ocorrido em 1 de Novembro de 1755.

Novas abordagens, teóricas e metodológicas, foram usadas para modelar um cenário de Tsunami, com especial enfase para a propagação em terra. Um dos aspetos mais relevantes desta fase de um tsunami é a sua capacidade de incorporar detritos, sejam eles de origem natural, como sedimentos ou vegetação, ou artiﬁcial, como ruínas de infraestrutura construída ou outros equipamentos do quotidiano humano.

Um modelo conceptual específico para escoamentos deste tipo - debris flows - foi revisto e ampliado com um flexível esquema de condições de fronteira. O modelo numérico contempla as mais recentes técnicas de discretização matemática e uma implementação computacional eficiente. Os seus fundamentos teóricos são apresentados a um nível introdutório enquanto que as novas funcionalidades são exaustivamente explicadas, demonstradas e discutidas.

Um esforço muito considerável foi direcionado para a discretização da cidade de Lisboa. O ambiente con- struído deixa de ser um simples pano de fundo para visualizar resultados e passa a deﬁnir, explicitamente, a superfície do terreno. Todas as estruturas relevantes foram adequadamente modeladas com modelos digitais de terreno e malhas computacionais muitíssimo ﬁnas para a escala do problema em mão.

Os resultados são apresentados em vários níveis diferentes, desde perspetivas abrangentes a análises detalhadas ao nível de cada arruamento. Nestas últimas, é fornecida informação pormenorizada quanto aos tempos de chegada do tsunami, as suas propriedades hidrodinâmicas e os padrões de deposição de detritos subsequentes à inundação.

Este estudo demonstrou que vários locais na frente ribeirinha do estuário do Tejo são especialmente preocupantes no que a um cenário de Tsunami diz respeito. Este documento fornece informação relevante para apoiar o planeamento das respostas de socorro e eventual evacuação da cidade, constituindo mais um passo na promoção da segurança dos cidadãos de Lisboa.

Palavras-Chave: Tsunamis, modelação matemática, estuário do Tejo, Lisboa

iii iv Abstract

A recent revision of the catalogue of tsunamis in Portugal has shown that the Tagus estuary has been aﬀected by catastrophic tsunamis numerous times over the past two millennia. Provided this historical heritage, and the growing awareness due to recent events worldwide, the present work aims at providing relevant data on the exposure of the Tagus estuary waterfront to a tsunami similar to the one occurred on the 1st November of 1755.

New theoretical and methodological approaches to tsunami modeling were employed, with special emphasis given to propagation over dry land. One of the most relevant features of this stage of the tsunami is its ability to incorporate debris, either natural sediment incorporated from the bottom boundary or remains of human built environment. The increased mass and momentum of the run-up can inﬂict tremendous damage and, regrettably, severe human losses.

A conceptual model specifically suited for describing debris flows is reviewed and extended with flexible boundary conditions. The employed numerical tool features the most up to date mathematical discretization techniques and an efficient computational performance. The theoretical fundamentals of the numerical model are presented at an introductory level while the new features implemented are given a thorough explanation and discussion.

A very considerable effort was directed to Lisbon’s topology discretization. The built environment is no longer seen as a simple background layer for result visualization, it actually explicitly defines the terrain surface. All relevant structures are adequately modeled with very fine elevation models and computational meshes, provided the scale of the problem at hand.

The results are provided at diﬀerent levels, ranging from generic overviews to street-level analysis on designated areas. At these locations detailed data on wave arrival times, hydrodynamic features of the advancing inundation front and urban debris scattering patterns are provided.

This study has shown that several locations on the Tagus estuary waterfront are especially worrisome in what respects to a tsunami impact scenario. The present document provides relevant information to support the design of emergency response plans, comprising one further step in promoting the safety of Lisbon’s citizens.

Keywords: Tsunamis, mathematical modelling, Tagus estuary, Lisbon.

v vi Notation

A Cell area [L2] C Sediment concentration [−]

CL Homogenous, depth-averaged sediment concentration in layer L [−]

Cf Friction coeﬃcient [−] c Shallow water wave velocity [ms−1] −1 c˜ik Approximate c in k edge [ms ]

ds Reference sediment diameter [m] (n) −1 e˜ik n eigenvector [ms ] E Flux vector

fsij Stress tensor [P a] F~ Generic force [N] F Flux vector in x G Flux vector in y g Gravitic acceleration [ms−2] H Source terms vector h Fluid height [m] H Average depth [m]

hL Thickness of layer L [m]

hL Fluid height on the left side of a shock [m]

hR Fluid height on the right side of a shock [m] ~n Unit normal to a plane [m] 1/3 −1 Ks Manning-Strickler coeﬃcient [m s ] L Wavelength (KdV notation) [−]

Ll Lower interface of a given layer [−]

Lu Upper interface of a given layer [−] p Bed porosity [−] p Hydrostatic Pressure [P a]

PL Depth-averaged hydrostatic pressure in Layer L [P a] 3 −1 qs Solid discharge [m s ] ∗ 3 −1 qs Solid discharge capacity [m s ] R Friction source term vector s Speciﬁc sediment gravity [−] S Shock speed [ms−1]

tik Unit tangent to k edge, in natural rotation from nik [m] T Bottom slope source terms numerical ﬂux matrix

Tij Depth integrated turbulent tensions tensor [P a] −1 uφ Velocity associated with the vertical mass ﬂux [ms ] −1 uI Interface velocity [ms ]

uL Velocity on the left side of a shock [m]

uR Velocity on the right side of a shock [m] −1 u∗ Friction velocity [ms ] −1 UiL Depth averaged velocity in layer L, in the xi direction [ms ] −1 UL Depth averaged velocity in layer L, in the x direction [ms ] U Independent variables vector

vii V Primitive variables vector −1 VL Depth averaged velocity in layer L, in the y direction [ms ]

wi Weight factor for cell i [m] −1 ws Sediment settling velocity [ms ]

Zb Bed elevation [m] (n) αik Wave strengths in k edge [m] (n) βik Bottom source ﬂux coeﬃcient in k edge [m]

δij Kronecker delta [−] Depth-averaged turbulent kinetic energy rate of dissipation [m2s−3] −1 ΦLu Net vertical flux across the lower interface [ms ] −1 ΦLu Net vertical flux across the upper interface [ms ] −1 Φa,b Net vertical flux from layer a to layer b [ms ] φ Generic function [m3s−1] η Free-surface elevation (KdV notation) [−] θ Shields parameter [−] κ Depth-averaged turbulent kinetic energy [m2s−2] ˜(n) −1 λik n eigenvalue of [ms ]

λt Wavelength [m]

λL Characteristics speed on the left side of a shock [m]

λR Characteristics speed on the right side of a shock [m] Λ Adaptation length [m] µ Dynamic viscosity of the ﬂuid [P a.s] ν Poisson coeﬃcient [−]

νT Turbulent viscosity [P a.s] ρ Mixture density [kgm−3] −3 ρL Depth-averaged density of the mixture on layer L [kgm ] ρ(w) Clean water density [kgm−3]

sigmaij Stress tensor [P a]

τij Turbulent stress tensor [P a]

τb Bed shear stress [P a]

τy Yield stress [P a]

τv Viscous stress [P a]

τt Turbulent stress [P a]

viii Acronyms

BC Boundary Conditions BKBC Bottom Kinematic Boundary Condition CFD Computational Fluid Dynamics CFL Courant-Friedrichs-Lewy CPU Central Processing Unit DEM Digital Elevation Model FEM Finite Element Method FDM Finite Diﬀerence Method FSDBC Free Surface Dynamic Boundary Condition FSKBC Free Surface Kinematic Boundary Condition FVM Finite Volume Method GIS Geographical Information Systems IC Initial Conditions KdV Kortweg-de Vries (equations) PDE Partial Diﬀerential Equation RH Rankine-Hugoniot (conditons) RP Riemann Problem STAV-2D Strong Transients in Alluvial Valleys 2D

ix x Contents

Acknowledgments ...... i Resumo ...... iii Abstract ...... v Notation ...... vii Acronyms...... ix List of Figures ...... xv

1 Introduction 1 1.1 Motivation and framework ...... 1 1.2 Proposed objectives ...... 2 1.3 Structure ...... 2

2 Tsunamis 5 2.1 Introduction ...... 5 2.2 What is a tsunami? ...... 5 2.3 Generation mechanisms ...... 7 2.4 Tsunami dynamics ...... 10 2.5 Tsunami consequences ...... 11 2.5.1 The 1755 Lisbon tsunami ...... 11 2.5.2 Recent tsunamis in the Indian Ocean and Japan ...... 13

3 Conceptual Model 17 3.1 Basic equations of fluid and sediment dynamics ...... 17 3.1.1 Material derivative and transport theorem ...... 17 3.1.2 Conservation of mass ...... 18 3.1.3 Conservation of momentum ...... 19 3.2 Governing equations for stratified flows ...... 21 3.2.1 Conservation of mass ...... 24 3.2.2 Conservation of momentum ...... 27 3.3 Closure equations ...... 29

4 Numerical Model 33 4.1 Discretization scheme ...... 33

xi 4.1.1 Introduction ...... 33 4.1.2 Finite volume schemes: The case of the 2D shallow-water equations ...... 34 4.1.3 Stability region ...... 37 4.1.4 Wetting-drying algorithm ...... 38 4.1.5 Entropy correction ...... 38 4.2 Boundary conditions ...... 39 4.2.1 Introduction ...... 39 4.2.2 Rankine-Hugoniot conditions ...... 39 4.2.3 Shocks ...... 40 4.2.4 Expansion waves ...... 41 4.2.5 Analytical solutions ...... 42 4.2.6 Physical feasibility of the mathematical solutions ...... 43 4.2.7 Results ...... 44

5 Integration with Geographic Information Systems 49 5.1 Introduction ...... 49 5.2 Pre-Processing utilities ...... 49 5.2.1 Generating boundaries ...... 49 5.2.2 Reﬁning meshes ...... 50 5.3 Post-processing utilities ...... 51 5.3.1 Resampler and raster converter ...... 51

6 A 1755 tsunami in today’s Tagus estuary topography 55 6.1 Pre-Processing ...... 55 6.1.1 Lisbon and Tagus estuary topography ...... 55 6.1.2 Initial Conditions ...... 58 6.1.3 Boundary Conditions ...... 58 6.2 Brief framework for result presentation and discussion ...... 61 6.3 Overview of a tsunami scenario in Lisbon ...... 62 6.3.1 General description of tsunami propagation ...... 62 6.3.2 Impact at Carcavelos ...... 63 6.3.3 Impact at Caxias and Cruz Quebrada ...... 64 6.3.4 Impact at Belém and Algés ...... 65 6.3.5 Impact at Caparica and Trafaria ...... 66 6.4 Street-level results for inland propagation ...... 67 6.4.1 Results for Alcântara ...... 67 6.4.2 Results for Downtown ...... 73 6.4.3 Results for Almada ...... 80 6.5 Morphological impacts ...... 87 6.5.1 The relevance of ﬂow-morphology interaction ...... 87 6.5.2 Vale do Zebro Royal complex ...... 87 xii 6.5.3 Urban debris washed at Downtown and Alcântara ...... 89 6.5.4 Morphological impacts on other locations ...... 93

7 Conclusions and recommendations 95 7.1 Main conclusions ...... 95 7.2 Directions for future work ...... 96

Bibliography 97

Appendices 101

xiii xiv List of Figures

2.1 Oscillatory and translatory waves ...... 6 2.2 Three main stages of tsunami propagation ...... 7 2.3 Schematics of a seismic generation mechanism ...... 9 2.4 A soliton or solitary wave ...... 10 2.5 A regular wave evolving into a bore ...... 11 2.6 1755 copper engraving depicting the 1755 Lisbon earthquake and tsunami (City Museum) 12 2.7 Tide gauge measurements of the 2004 tsunami at several locations (Rabinovich & Thomson, 2007) ...... 13 2.8 Morphological impact of the 2004 Indian Ocean at Lhoknga, northwest coast of Sumatra, and Phuhket, Thailand ...... 14 2.9 Pressure gage readings for the 2011 Tohoku tsunami (Maeda et al., 2011) ...... 15 2.10 Morphological impact of the 2011 tsunami at Natori and Fukushima, south of Tohoku (ABC News, 2012) ...... 15

3.1 Layer set featured in the conceptual model ...... 22 3.2 A non-material surface ...... 23 3.3 Layer set featured in the conceptual model ...... 27

4.1 A shock (compressive) at the boundary, ∂Ω ...... 41 4.2 Physical admissibility with respect to the entropy condition ...... 44 4.3 Comparison results for the implemented boundary conditions ...... 45 4.4 Steady conditions at time t = 100 [s] and t = 2000 [s] ...... 47 4.5 Steady conditions at time t = 100 s (still the same at t = 2000 s) ...... 47

5.1 Pre-processing samples: example of ArcMap isoline defined boundary (left) and conversion to Gmsh (right) ...... 50 5.2 Example of mesh refinement procedure ...... 51 5.3 Point-in-polygon (PIP) problem examples ...... 52 5.4 Examples of VTK file resampling results ...... 53

6.1 DEM Raster with 5x5 [m] resolution ...... 56 6.2 Bathimetry with 10x10 [m] resolution ...... 56

xv 6.3 Example of DEM built environment enhancement ...... 57 6.5 Mean annual flow duration curve (Ómnias - Sacavém hydrometric station) ...... 58 6.4 Warm-up setup and results ...... 59 6.6 Supplied data for a 1755-like tsunami ...... 60 6.7 Supplied data for a 1755-like tsunami (2) ...... 61 6.8 Relevant locations for result presentation and discussion ...... 61 6.9 Tsunami propagation overview at the Tagus estuary ...... 62 6.10 Tsunami run-up at Carcavelos ...... 63 6.11 Tsunami run-up at Caxias and Cruz Quebrada ...... 64 6.12 Tsunami run-up at Belém and Algés ...... 65 6.13 Tsunami run-up at Caparica and Trafaria ...... 66 6.14 One-way traffic zones (top) and tsunami approach to Alcântara (bottom; t=10’20”) . . . . 67 6.15 Detailed results for Alcântara, t = 12 minutes 20 seconds ...... 69 6.16 Detailed results for Alcântara, t = 14 minutes 20 seconds ...... 70 6.17 Detailed results for Alcântara, t = 16 minutes 20 seconds ...... 71 6.18 Detailed results for Alcântara, t = 20 minutes 20 seconds ...... 72 6.19 One-way traffic zones (top) and tsunami approach to Downtown (bottom; t=13’20”) . . . 73 6.20 Detailed results for Downtown, t = 15 minutes ...... 75 6.21 Detailed results for Downtown, t = 15 minutes 40 seconds ...... 76 6.22 Detailed results for Downtown, t = 16 minutes 20 seconds ...... 77 6.23 Detailed results for Downtown, t = 17 minutes 20 seconds ...... 78 6.24 Detailed results for Downtown, t = 20 minutes 20 seconds ...... 79 6.25 Detailed results for Cacilhas, t = 14 minutes 30 seconds ...... 81 6.26 Detailed results for Cacilhas, t = 16 minutes 10 seconds ...... 82 6.27 Detailed results for Cacilhas, t = 17 minutes 20 seconds ...... 83 6.28 Detailed results for Cacilhas, t = 22 minutes 50 seconds ...... 84 6.29 Detailed results for Cacilhas, t = 24 minutes 20 seconds ...... 85 6.30 Detailed results for Cacilhas, t = 27 minutes 40 seconds ...... 86 6.31 Tsunami approaching the Royal Complex at Vale do Zebro, t=25’ ...... 87 6.32 Tsunami run-up at Coina river (Vale do Zebro) ...... 88 6.33 Tsunami impact at the Royal complex ...... 89 6.34 Idealized setup for urban debris, mainly composed of vehicle-like solid materials ...... 90 6.35 Mobile material distribution along the riverfront ...... 90 6.36 Debris scattering patterns in Alcântara at the aftermath of the tsunami ...... 91 6.37 Debris scattering patterns in Downtoen at the aftermath of the tsunami ...... 91 6.38 Comparison between debris and clearwater formulations ...... 92 6.39 Morphological impacts on remaining locations ...... 93

xvi Chapter 1

Introduction

1.1 Motivation and framework

Often considered as rare events when compared to other natural hazards tsunamis have an enormous devastating potential. The frequency concept associated to these phenomena is to be understood within consistent timescales, namely the geological timescale. At this level, tsunamis and earthquakes, as well as many other geophysical occurrences such as volcanic eruptions or major landslides, become relatively common events whose predictability, at ﬁner timescales, becomes practically impossible. Other diﬃculties arise when studying tsunamis since one of the main uncertainties lies in the potential of an oceanic earthquake to generate tsunamis. Not all oceanic earthquakes occur in such a way that the seabed gets vertically displaced and vertical momentum is transferred to the water column.

All these variables highlight a much required awareness of the stochastic nature of tsunamis and other geophysical phenomena, in general. Furthermore, seismic risk mitigation is presently a standard in most structural design regulations and public safety procedures. In coastal countries like Portugal, where tectonic activity is likely to concentrate in oceanic faults, accounting for seismic risks while negletcing potential tsunami impacts seems, at least, inconsistent and incautious.

The large ratio between seafront extension and continental territory made Portugal hugely dependant on its maritime Economic Exclusive Zone. Whether it is due to turism, fishing activities or transoceanic transport, the economic contribution of the seafront to the global wealth of the country has been very significant and drove most of Portuguese towards the coast. Over centuries, this generalized demographic behaviour ultimately converged into a country with two noticeably distinct development levels: a relatively unpopulated countryside on interior regions contrasting with densely populated and commercially active coastal cities. Although the last decades had a significant contribution in fading such dissimilari- ties, drawn by major communication and technological progresses, the heritage left by recent centuries is unavoidable: most of the population, infrastructure and valuable historical assets are located in coastal areas. This emphasizes the relevance of seismic risk mitigation procedures that involve structural damage as well as inundation extents.

1 1. Introduction

Recent revisions of the Portuguese catalog of tsunamis (Baptista & Miranda, 2009) have also shown that the seafront has been struck by numerous events over the past two millenia.

Presently, numerous shallow water models are used within the scientific community for tsunami propagation (Imamura et al., 2006). While these include all the relevant components for offshore propagation, their ability to account for very specific features in waterfront and overland propagation is limited. One of the most relevant features of tsunami propagation over solid boundaries is its ability to incorporate debris, either natural sediment incorporated from the bottom or remains of human built environment. Despite acknowledged, this feature is commonly neglected in most tsunami inland propagation studies, probably by virtue of a relatively incomplete understanding of the involved phenomena when sediment transport imply significant changes in the rheological behaviour of the flow.

1.2 Proposed objectives

A computational ﬂuid dynamics (CFD) model based on the ﬁnite volume method (FVM) (Canelas, 2010) - STAV2D (Strong Transients over Alluvial Valleys 2D) - featuring enhanced sediment transport capabilities (Canelas, 2010) and up to date numerical discretization schemes (Murillo, 2006) was employed to analyze a tsunami scenario in the Tagus estuary. The tsunami parameters resemble the widely known event that struck Lisbon on the 1st November of 1755.

Not only is the present work based on improved conceptual and numerical models, but also, the physical environment discretization was taken to a level not known in other studies. The human built environment was detailed at a very reﬁned scale, featuring buildings, streets and even urban mobile debris.

The main objective of this work lies in providing detailed data on the exposure of designated areas along the Tagus estuary riverfront. Lisbon’s Downtown, Alcântara and Cacilhas were thoroughly discretized in order to perform a street level description, a signiﬁcant improvement over the present knowledge of a tsunami impact on these locations. Morphological changes and debris scattering were also analyzed by taking full advantage of the solid transport capabilities of the numerical model.

A set of secondary objectives was also delineated, consisting of computational work on the CFD code: a new scheme of boundary conditions was implemented and pre/post-processing toolkits were developed to promote functional integration with commercial Geographical Infromation Systems (GIS).

1.3 Structure

The ﬁrst chapter is devoted to a brief bibliographical review on the subject of Tsunamis. Particular emphasis is given to the hydrodynamic concepts and numerical modeling trends. Historical context is also provided on a very relevant event for the present work: The 1755 Great Lisbon Earthquake and Tsunami. Recent events that were given great interest in worldwide media and scientiﬁc communities

2 1.3. Structure are also revised.

Theoretical background on the conceptual model that supports the present work and a revision of the employed closure models are provided in Chapter 2, with special care being given to particular mathematical formulations inherent to multiphase ﬂows.

A brief description of the discretization scheme is presented in the following chapter. The implemented boundary condition scheme is thoroughly explained and its performance assessed with a set of functional tests.

The fourth chapter introduces the developed toolkit for pre and post-processing tasks. It is particularly useful for future reference and also highlights the expected ease of use of these tools.

The last chapter features a comprehensive description of the simulated scenario. All details concerning the setup and outsourced inputs for these simulations are provided, including boundary and initial conditions and the employed topographic datasets. The results are displayed at diﬀerent levels: a tsunami scenario overview on the whole Tagus estuary, a detailed street-by-street analysis on designated areas and morphological impact assessment.

3 1. Introduction

4 Chapter 2

Tsunamis

2.1 Introduction

Human communities living near the ocean have always experienced the devastating eﬀects of a tsunami. These waves have surely claimed hundreds of thousand of lives and have caused incalculable material losses. However, since they are not frequent in industrialized western countries, they have not been given relevant media attention. The morning of the December 24th changed this forever when an earthquake of magnitude 9.1 stroke the coast of Sumatra in Indonesia. The tremendous energy released in this oceanic earthquake generated a vertical displacement of the seabed, rising the sea surface to form a devastating tsunami that killed 280,000 people and aﬀected the lives of several million others (Helal, 2008).

While people living along the coastline of Sumatra had little time and no warning on the approaching tsunami, those living along the further coasts of Thailand, Sri Lanka, India and East Africa had plenty of time to escape to higher grounds. Unaware, they stayed at lower ground, as no tsunami warning system was implemented in the Indian Ocean at that time. Unfortunately, it has taken a disaster of such magnitude to awake the scientiﬁc communities to this silent but fearful natural hazard. The present chapter aims to describing some of the most common physical features and modeling trends on tsunamis.

2.2 What is a tsunami?

The term tsunami originates from the Japanese tsu and nami meaning wave and harbor, respectively. It may have been originally used by ﬁshermen, who would sail out, encountering no visually unusual waves while out at sea, and later returned to ﬁnd an overwhelmed hometown.

Tsunamis are waves with large wavelengths that are originated by some kind of sudden disturbance in standing 1 water and are characterized as shallow-water waves. These are diﬀerent waves from the ones most commonly observed on a beach, which are caused by the wind blowing across the ocean’s surface.

1 By standing water one means a water mass whose√ velocity is much lower than the velocity of a propagating tsunami, which is close to that of shallow-water waves celerity ( gH).

5 2. Tsunamis

Wind-generated waves usually have periods of ﬁve to thirty seconds and a wavelength of 100 to 200 [m] whereas tsunamis can have periods and wavelengths that are much higher (Helal, 2008).

A major diﬀerence between these waves and tsunamis is the mass transport carried by the wave. While wind-generated waves induce nearly-closed trajectories 2 to the particles a tsunami displaces them several meters from their initial positions. The volume of water that gets transported and the speeds reached by tsunami waves allows them to carry enough energy to wipe out entire towns and cities (Helal, 2008). Wind-generated waves are called oscillatory waves whereas tsunamis are translatory waves (Figure 2.1).

Oscillatory Wave Translatory Wave

Stokes Drift

Figure 2.1: Oscillatory and translatory waves

Tsunami propagation comprises three main stages: oceanic propagation, near shore shoaling and inland propagation (Figure 2.2 on page 7). At the ﬁrst stage, within deep ocean waters, the period of a tsunami can range from 5 minutes to 2 hours and the wavelength, λt, can be as high as 500 [km].

Waves are governed by shallow-water equations if the ratio between the wavelength and the depth of the ocean is over 20. Given that the maximum mean depth of the sea-ﬂoor, H, is about 5 km, the ratio of the

ﬂow depth to the wavelength is small: H/λt ≈ 0.001, a common value, for instance, in river ﬂows. Hence, tsunamis will always behave as shallow-water waves no matter where on the globe they are being modeled √ or observed. The velocity of a shallow-water wave is also equal to gH, where g is the acceleration of gravity (9.8 [m/s2]) and H is the depth of the water column.

Thus, in very deep water, a tsunami will propagate at high speeds: in the Paciﬁc Ocean, for instance, the average depth is approximately 4000 [km], which means that the velocity of a tsunami in this water mass is nearly 200 [m/s]. Considering that the distance across this ocean, for instance from Seattle to Hong Kong, is roughly 8000 [km] this would yield a total of 11 hours for travel time.

2Trajectories are not exactly closed because the ﬁnal position of the particles is not coincident with the initial position. This is called the Stokes Drift (Stokes, 1847). Furhter details on surface wave dynamics can be found in the Appendix 1. Nonetheless, the mass transport in this drift is negligible when compared to that of a tsunami.

6 2.3. Generation mechanisms

Oﬀshore tsunami amplitudes are likely below 1 [m] and wavelengths are certainly a few hundred kilometers long. Given that the order of magnitude of the hydraulic gradient can be given by the wave amplitude to wavelength ratio, this leads to practically null energy dissipation: the hydraulic gradient is 10−5 for a rather small wavelength of 100 [km]. This yields basically no head loss for a tsunami propagating over deep waters, reaching distant shores with high energy levels.

As a tsunami approaches shallower waters near shore the depth of the water decreases, meaning that the velocity of the tsunami will also decrease. However, its energy and mass are maintained, suggesting that the height of the tsunami must grow enormously. It is at this stage that the tsunami becomes visible, resembling the atrocious footage spread out on the media in 2004 (Indian Ocean) and 2011 (Japan).

The third stage of a tsunami deals with breaking as it approaches shore and depends greatly on the bathymetry. Overland tsunami propagation may form bores: abrupt wave forms mathematically represented as discontinuities. These discontinuous and highly turbulent flows are likely to incorporate debris, whether natural sediment or human built environment remains. The dynamics of this final stage of a tsunami is somewhat similar to that of flood waves caused by dam breaking, dyke breaking or overtopping. Very high concentrations of debris are carried on the water column and may change the hydrodynamic properties and rheological behaviour of the fluid (Dias, 2006).

Translatory wave low amplitude mass is conserved bores may form high wavelength amplitude increases shore mean sea level

continental shelf velocity

sea bed

Figure 2.2: Three main stages of tsunami propagation

2.3 Generation mechanisms

Tsunamis can are mainly generated by seismic activity (seismicity), landslides or volcanic activity. The seismic generation mechanism is responsible for most tsunami occurrences and, as a major source, must be discussed with further detail. Tsunamis can be generated by seismicity when the sea ﬂoor abruptly rises and vertically displaces the overlying seawater. Tectonic earthquakes are a particular kind of earthquake that are associated with the continental drift. When these earthquakes occur in the ocean, the water above the deformed area is displaced from its initial position (Dias, 2006). Namely, a tsunami can be generated when thrust faults, associated with convergent or destructive plate boundaries, suddenly slip and displace large volumes of water due to the vertical component of momentum involved (see Figures 2.3a to 2.3d).

7 2. Tsunamis

Mathematical modeling of this mechanism requires a displacement ﬁeld to be parametrized at the sea bottom. Volterra (1907) originally solved this problem in the case of an entire elastic space. Hence, and for simplicity’s sake, that will be the approach herein presented.

Considering the thrust mechanism as a fracture, the discontinuities in the displaced components across the fractured surface can be described using the theory of elasticity, namely surfaces across which the displacement ﬁeld is discontinuous (Steketee, 1958). For simplicity’s sake some assumptions are introduced.

The curvature of the earth, its gravity, temperature, magnetism and non-homogeneity are neglected. A semi-infinite, homogeneous and isotropic medium is considered instead. Further details on how the earth’s curvature, topography, homogeneity and isotropy influence the displacement fields can be found on the works published by McGinley (1969), Ben-Mehanem et al. (1970a,b) and Masterlark (2003). Furthermore, one assumes that the the linear elasticity theories are valid within the fractured region.

Let O be the origin of a Cartesian coordinate system, xi the Cartesian coordinates with i = 1, 2, 3, and ~ei ~ k a unit vector in the positive xi direction. A force F = F~ek at O generates a displacement ﬁeld ui (P,O) at point some point P , given by,

2 2 k F ∂ r ∂ r ui (P,O) = δik 2 − α (2.1) 8πµ ∂xn ∂xi∂xk

where δik is the Kronecker delta, r is the distance from P to O and α is a coeﬃcient deﬁned as,

λ + µ 1 α = = (1 − ν) (2.2) λ + 2µ 2 where λ and µ are the Lamé’s constans and ν is the Poisson’s ratio 3 . Hooke’s Law easily yields the stresses due to the displacement ﬁelds in equation (2.2)

k ∂uk ∂ui ∂uj σij = λδij + µ + (2.3) ∂xk ∂xj ∂xi

+ − The dislocation is defined as a surface Γ across which one finds a discontinuity ∆ui = ui − ui in the displacement fields. This yields the following formula (Volterra, 1907) describing the displacement field for a particular point (x1, x2, x3)

1 ZZ u (x , x , x ) = ∆u σk · n dS (2.4) k 1 2 3 F i ij j Γ

. In general, the displacement ﬁeld represented by equation (2.4) is reproduced in the ocean free surface (Figure 2.3c on page 9), posing the initial conditions for tsunami propagation (Imamura et al., 2006).

Tsunamis often have a small amplitude oﬀshore which is why they generally pass unnoticed at sea, forming only a slight swell usually less than half a meter above the normal sea surface. A tsunami can occur in

3Transverse to axial strain ratio when a material is compressed.

8 2.3. Generation mechanisms any tidal state and even at low tide can still severely inundate coastal areas. The 1964 Alaska earthquake (Magnitude, Mw, 9.2), the 2004 Indian Ocean earthquake (Mw 9.2) and the 2011 Tohoku earthquake (Mw 9.0) are recent examples of powerful thrust earthquakes that generated powerful tsunamis, capable of crossing entire oceans and still inﬂict severe devastation in coastal areas. Other potential generation mechanisms may occur, although with fewer recorded occurrences. These include landslides, explosive volcanic eruptions, large atmospheric depressions and even meteor impacts.

Tsunamis generated by landslides are called sciorrucks. These phenomena rapidly displace large water volumes, as energy from falling debris or expansion is transferred to the water at a rate faster than it can absorb. Their existence was conﬁrmed in 1958, when a giant landslide in Lituya Bay, Alaska, caused the highest wave ever recorded, which had a height of 524 [m]. The wave didn’t travel far, as it struck land almost immediately. Two people ﬁshing in the bay were killed, but another boat amazingly managed to ride the wave (Fine et al., 2006). Scientists named these waves megatsunami and discovered that extremely large landslides from volcanic island collapses can also generate these kind of waves.

Meteotsunamis are generated by deep depressions that cause tropical cyclones, generating a storm surge often called a meteotsunami that can raise tides several meters above normal levels. The displacement comes from low atmospheric pressure within the centre of the depression.

SWL - Still Water Level SWL - Still Water Level

Overriding Plate Overriding Plate Subducting Plate Subducting Plate

(a) Converging plates before earthquake (b) Overriding plate bends under strain

Overriding Plate Overriding Plate Subducting Plate Subducting Plate

Figure 2.3: Schematics of a seismic generation mechanism

9 2. Tsunamis

2.4 Tsunami dynamics

A simple mathematical formulation to demonstrate the physics behind a tsunami wave is herein presented Helal (2008). In this approach a tsunami is conceptualized as a non-linear bi-dimensional wave propagating over a horizontal ground.

The governing law for this solution is the Korteweg-de Vries equation (KdV). It is a nonlinear, dispersive partial diﬀerential equation for a function of two real variables, space, x, and time, t. For some function φ the Korteweg-de Vries equation is stated as,

3 ∂tφ + ∂xφ + 6φ∂xφ = 0 (2.5) where the constant 6 in front of the last term is merely conventional: multiplying t, x, and φ by constants can be used to vary the constants preceding each of the featured terms.

For hydrodynamic applications this equation is often presented in the form,

∂η ∂η 3 ∂η ξ ∂3η + + η + = 0 (2.6) ∂t ∂x 2 ∂x 6 ∂x3 and featuring the following non-dimensional variables,

δ − h h h2 η = ; = ; ξ = (2.7) H H L2 where is the wave amplitude corresponding to the depth h; L is the wavelength; η(x, t) is the relative movement of the free surface relating to the maximum height and ξ is the squared depth to wavelength ratio. This form of the KdV equation is an numerical simulation standard when it comes to oﬀshore propagation and its solutions are provided in the form of solitons (Figure 2.4) or other dispersive waves.

H mean sea level

h δ

sea bed

Figure 2.4: A soliton or solitary wave

The dispersive term (fourth in equation (2.6)) counterbalances the quasi-linear terms (second an third terms in equation (2.6)) allowing for the self-similar propagation of the wave-form shown in Figure 2.4. As the wave approaches shore non-linear terms will become more important and the wave will cease to be decribed by the KdV equation.

10 2.5. Tsunami consequences

When a tsunami reaches the seashore and propagates overland ( last stage in Figure 2.2) it may develop into an abrupt wave form, or bore, which is mathematically considered as a discontinuity. Bores are non-linear wave forms withour relevant dispersive terms, especially if debris are incorporated into the water column. t +2dt t +dt 0 t 0 0

Figure 2.5: A regular wave evolving into a bore

In this case, the dynamics of the tsunami is governed by shallow-water equations. In fact, dispersive eﬀects are often disregarded and shallow-water equations are frequently used to describe the propagation of tsunamis in open waters (Imamura & Shuto, 1990).

The shallow-water equations will not be described in this section. They constitute the core of the mathematical simulation tools employed in this dissertation and will be carefully derived in Chapter 3, p. 17 to 21.

2.5 Tsunami consequences

2.5.1 The 1755 Lisbon tsunami

Around 9:40 AM on the 1st November of 1755, the All Saints’ Day Catholic holiday, Lisbon was struck 4 by a violent earthquake with a magnitude ranging from 8.5 Mw to 9 Mw , (Abe, 1989) with an epicenter in the Atlantic Ocean 5 that became known as the Great 1755 Lisbon Earthquake. This earthquake was followed by several ﬁres and most importantly, for the purpose of this dissertation, by a major tsunami.

At that time Lisbon was inhabited not only by Portuguese people but also by a considerable foreign population, mainly French, Spanish, English and German. This contributed to the wide diversity of publications, in both Portuguese and English, regarding the Great 1755 Lisbon Earthquake and made it the most well documented historic tsunami event (Baptista et al., 1998b).

The following are amongst the most relevant historic reports (Baptista et al., 1998a, 2006),

"There was another great shock after this that pretty much aﬀected the river, but, I think not so violent as the preceding, though several people afterwards assured that as they were riding on horseback, in the great road to Belem, one side of which lays upon the river, the waters rushed

4 Moment Magnitude Scale, Mw 5Located south of the west coast of Algarve and northeast of the Archipelago of Madeira(Baptista et al., 1998b)

11 2. Tsunamis

in so fast that they were forced to gallop as fast as possible to the upper grounds for fear of being carried away" (British Accounts, 1755)

"A large quay, piled up with goods near the Custom House sunk the ﬁrst shock, with about 600 persons upon it, who all perished" (British Accounts, 1755)

"(...) a lot of people run to the river bank trying to escape from the ruins. Suddenly the sea came in through the bar and ﬂooded the river banks..." (Moreira de Mendonça, 1758).

"(...) waters with their ebb and ﬂow ﬂooded the Customs, the Square and the Vedoria".

Figure 2.6: 1755 copper engraving depicting the 1755 Lisbon earthquake and tsunami (City Museum)

To avoid interpretation and subsequent distortions Baptista et al. (1998a) carried a direct analysis of historical data between 1755 and 1759 where all the documents were original from the 18th century and were analyzed in their original language. This study pointed out very useful aspects about the earthquake and tsunami that destroyed Lisbon on that morning of the November 1st.

The earthquake shook the city at about 9h30 a.m. and lasted about 9min. When most people were heading to the riverfront in a desperate eﬀort to avoid the ﬁres and ruining buildings in the inner city and also out of curiosity about the unusual low tide, which was in fact a drawback and the only warning of an incoming tsunami, a huge water wall hit the riverfront. The tsunami traveled for about 90 minutes in the open sea and should have hit the city at about 11h00 a.m. climbing the shores up to a mean run-up level of 5 [m] and penetrating the city as deep as 250 [m]. The death toll only due to the tsunami is estimated in around 900 people whereas the earthquake itself should be held responsible for a number lives ranging from 10.000 up to 100.000 (Álvaro S. Pereira, 2006).

About eighty-five percent of the city’s built environment collapsed, including famous palaces and libraries constituting some of the most brilliant examples of the unique Manueline architectural style. Even buildings that had suffered little damage from the earthquake were destroyed in the subsequent fires. The Royal Ribeira Palace, which was located near the riverfront in the modern square of Terreiro do Paço, was destroyed by the earthquake and tsunami. Inside, the 70.000 volume royal library, also filled

12 2.5. Tsunami consequences with hundreds of works of art by notorious painters and sculptors of that time were forever lost. The scientiﬁc and historian communities were also deprived of priceless intelligence on the early campaigns led by the famous navigators Vasco da Gama, Fernão de Magalhães and Pedro Alvares Cabral.

All the major churches in Lisbon were ruined by the earthquake, namely the Lisbon Cathedral, the Basilicas of São Paulo, Santa Catarina, São Vicente de Fora, and the Misericórdia Church. The Royal Hospital of All Saints in the Rossio Square was consumed by ﬁre and many patients burned to death. Visitors to Lisbon may still walk the ruins of the Carmo Convent, which were preserved to remind Lisbonners of that day’s destruction.

2.5.2 Recent tsunamis in the Indian Ocean and Japan

Two recent tsunami events in the last decade have surely awaken people and governments to the reality and unpredictability of these natural hazards, namely the 2004 Indian Ocean and the 2011 Tohoku tsunamis. The 2004 Indian Ocean earthquake was an undersea megathrust earthquake 6 that occurred on Sunday, 26 December 2004, with an epicenter oﬀ the west coast of Sumatra, Indonesia. The resulting tsunami is given various names, including 2004 Indian Ocean tsunami or Boxing Day tsunami.

The earthquake was caused by subduction and triggered a series of devastating tsunamis along the coasts of most landmasses bordering the Indian Ocean, killing over 230.000 people in fourteen countries, and inundating coastal communities.

Asia Hillarys 20˚N Salalah Cocos Africa Hanimaadhoo Male Colombo 200 cm 0˚ Lamu Port Louis Gan M w =9.3 Zanzibar Colombo Pointe Diego Garcia La Rue Cocos 3 hr 20˚S Port Louis Pointe La Rue Australia 5 hr Hanimaadhoo Hillarys 7 hr St. Paul Sea level 40˚ 9 hr Salalah 11 hr Kerguelen Male Indian Ocean 13 hr 60˚ Lamu Gan

Mawson Zanzibar Diego Garcia Casey Dumont d'Urville SyowaSyowa Zhong Shan Davis Antarctica 0˚ 20˚ 40˚ 60˚ 80˚ 100˚ 120˚ 140˚E 26 27 28 26 27 28

Figure 2.7: Tide gauge measurements of the 2004 tsunami at several locations (Rabinovich & Thomson, 2007)

Figure 2.7 displays sea level measurements from several buoys along the Indian Ocean. As expected, oﬀshore amplitudes were relatively small. For instance, the tide gauge located at the Cocos Islands, 1700 [km] from the epicentre, revealed a small 30 [cm] high wave followed by a long sequence of oscillations,

6A megathurst earthquake is an earthquake generated by seismicity at convergent plate boundaries, where the subducting plate shallowly dips under the overriding plate. All recorded major earthquakes, with Mw over 9.0, are of the megathurst type.

13 2. Tsunamis

Higher readings were provided by the gauges located at Gan and Diego Garcia Islands where amplitudes were just below 1 [m]. As the tsunami approached continental platforms the wave height grew considerably. Gauge data yields amplitudes of nearly 3 [m] (Titov et al., 2005) at Pointe La Rue, Salalah and Colombo. At Banda Aceh, Indonesia, water was reported to have risen by 3 [m] in under 1.5 minutes (Chanson, 2005).

Important morphological changes were observed in Sri Lanka and Indonesia with some bed variations reaching over 5 [m] (Chanson, 2005; Goto et al., 2011) at relevant harbors, for instance, the Lhoknga harbour depicted in Figure 2.8, and at most seafront mobile reaches. This highlights the solid transport capacity of the overland propagation stage, clearly visible on the aftermath pictures in Figure 2.8, and the importance of eﬀectively accounting for this feature in numerical simulations. Waste and debris are mostly deposited inland but can also be carried oﬀshore, when draw-down occurs, dispersing along the shore.

(a) Lhokonga before (b) Lhokonga after the (c) Phuket before the (d) Phuket after the the tsunami impact tsunami impact tsunami impact tsunami impact (Chanson, 2005) (Chanson, 2005) (NASA) (NASA)

Figure 2.8: Morphological impact of the 2004 Indian Ocean at Lhoknga, northwest coast of Sumatra, and Phuhket, Thailand

This tsunami was one of the deadliest natural disasters in recorded history. Indonesia was the hardest- hit country, followed by Sri Lanka, India, and Thailand. With a magnitude of Mw 9.1 – 9.3, it is the third largest earthquake ever recorded on a seismograph. The plight of the aﬀected people and countries prompted a worldwide humanitarian response.

The 2011 earthquake off the Pacific coast of Tohoku and also known as the the Great East Japan Earthquake or the 3/11 Earthquake, was a magnitude 9.0 (Mw) undersea megathrust earthquake off the coast of Japan that occurred on Friday, 11 March 2011, with the epicenter approximately 70 kilometres east of the Oshika Peninsula of Tohoku and the hypocenter at an underwater depth of 32 [km]. It was the most powerful known earthquake ever to have hit Japan, and one of the five most powerful earthquakes in the world since modern record-keeping began in 1900. The earthquake triggered powerful tsunami waves that travelled as far as 10 [km] inland with up to 4 [m] high.

14 2.5. Tsunami consequences

Tohoku

Fukushima

Figure 2.9: Pressure gage readings for the 2011 Tohoku tsunami (Maeda et al., 2011)

Pressure gauge readings for this earthquake, such as the ones shown in Figure 2.9, show oﬀshore heights up to 1.2[m]. As previously referenced these small waves are prone to shoaling and the water height that hits the shore is expected to be much higher. Several amateur footage was spread out on the media showing ﬂow depths ranging from 2 to 4 [m] high (Biggs & Sheldrick, 2011).

Significant morphological impacts occurred throughout all Japanese coast. Mobile reaches suffered severe bed configuration changes as depicted in Figure 2.10. Built areas were completely wiped by the advancing front of a turbulent, debris laden, flow.

(a) Natori before the tsunami impact (b) Natori after the tsunami impact

Figure 2.10: Morphological impact of the 2011 tsunami at Natori and Fukushima, south of Tohoku (ABC News, 2012)

15 2. Tsunamis

The Japanese National Police Agency conﬁrmed 15,867 deaths. The tsunami also caused a number of nuclear accidents, primarily the level 7 meltdowns at three reactors in the Fukushima Daiichi Nuclear Power Plant. Many electrical generators were taken down, and at least three nuclear reactors suﬀered explosions due to hydrogen gas that had built up after the cooling system failed.

Early estimates placed insured losses from the earthquake alone at US$14.5 to $34.6 billion.[31] The Bank of Japan had to oﬀer $15 trillion (US$183 billion) to the banking system on 14 March in an eﬀort to normalize market conditions. The World Bank’s estimated economic cost was US$235 billion, making it the most expensive natural disaster in world history.

16 Chapter 3

Conceptual Model

3.1 Basic equations of ﬂuid and sediment dynamics

The purpose of the present section is to introduce the fundamental equations of fluid dynamics within the continuum hypothesis (Atkin & Crane, 1976). Under this hypothesis individual sediment grains are ignored and the mass of sediment is uniformly distributed over the volume of the system. Furthermore, this assumption is only valid at scales larger than the grain diameter and assumes homogeneity within the sediment-fluid mixture. Along these lines, variables such as density, velocity or suspended solids concentration are to be described as continuous time-dependent scalar or vector fields on R3.

The physical phenomena that govern flow dynamics are to be mathematically represented by a set of three laws, namely conservation of mass, momentum and energy. The derivation of these equations can follow one of two distinct paths: a space-averaged estimation of flow effects over a finite region, usually referred to as the integral form, or a more refined point-to-point description of the flow, the differential form. The conceptual model, described in section 3, is significantly more consistent with the differential form and so this will be the undertaken approach to derive the referred conservation equations.

3.1.1 Material derivative and transport theorem

The time-derivative of a given property φ(xi, t) of a fluid element within a velocity field ui(xi, t) can be obtained by tracing this same element along its trajectory. In such case there will be two distinct contributions for the total variation of φ: the first is due to the time dependence of the φ field, named the local term, and the other is induced by the motion of the fluid element through spatial gradients of φ, the advective term. The space coordinates xi are x1 = x, x2 = y and x3 = z, where x, y are the horizontal coordinates and z is a vertical coordinate aligned with the gravity field. The time coordinate is t.

The material derivative, denoted by Dφ/Dt, is thus given by,

Dφ ∂ = φ(t, x (t)) (3.1) Dt ∂t i

17 3. Conceptual Model which in turn can be expanded, by means of the chain rule, to,

Dφ ∂φ ∂φ dxi ∂φ ∂φ = + = + ui (3.2) Dt ∂t ∂xi dt ∂t ∂xi

where t stands for time and ui are the velocity vector components. This form clearly distinguishes the two contributory terms for the total variation of φ: the local term, ∂tφ, and the advective term, ui ∂xi φ.

Another rather important concept in fluid mechanics is the Reynold’s Transport Theorem which states that for any material volume V (t) and differentiable scalar field ϕ it is valid that,

d Z Z ∂ϕ ∂ ϕ dV = + (ϕui) dV (3.3) dt ∂t ∂xi V (t) V (t) which provides the necessary framework to formulate the conservation laws for mass, momentum and energy (Wesseling, 2009).

3.1.2 Conservation of mass

This conservation law states that the rate of change of mass must equal the rate of mass production in an arbitrary material volume V (t). Mathematically this is expressed by,

d Z Z ρ dV = σ dV (3.4) dt V (t) V (t) where ρ is the apparent density of an homogeneous granular-ﬂuid mixture and σ is the rate of mass production per unit volume. The deﬁnition of ρ is as follows,

δM ρ = lim (3.5) δ∀→0 δ∀ where δ an arbitrary diﬀerence operator, δM is the mass of the ﬂuid-granular mixture and δ∀ is the volume occupied by that mass. The mass of the mixture is M = Mw + Ms where M is the total mass and Ms is the sediment mass and hence equation (3.5) becomes,

δM δ∀ δM δ∀ ρ = lim w w + s s (3.6) δ∀→0 δ∀ δ∀w δ∀ δ∀s

where ∀w and ∀w are the volumes of water and sediment, respectively. Note that ∀w = ∀ − ∀s and (3.6) converges to ρ = ρw (1 − C) + ρsC. Defining the specific gravity as s = ρs/ρw on obtains the final form of the mixture density,

ρ = ρw [1 + (s − 1)C] (3.7)

18 3.1. Basic equations of ﬂuid and sediment dynamics

By using the transport theorem, equation (3.3), one can expand equation (3.4) to,

Z ∂ρ ∂ Z + (ρui) dV = σ dV (3.8) ∂t ∂xi V (t) V (t)

where ui is the velocity vector ﬁeld and t stands for time. The mass production rate per unit volume, σ, accounts for singular sinks or sources inside V (t). In this case there are no sinks or sources since all inputs and outputs to the control volume can be expressed in terms of incoming or outgoing ﬂuxes, respectively.

Provided that equation (3.8) holds for every arbitrary V (t), the diﬀerential form of the mass conservation law can be obtained by extracting the left-hand integrand,

∂ρ ∂ + (ρui) = 0 (3.9) ∂t ∂xi which is commonly known as the continuity equation as it requires no further assumptions other than that both density and velocity are continuum functions (White, 2002).

The conceptual model is aimed at river flows which are greatly influenced by the transported sediment in what concerns hydrodynamics. Since both water and sediment are incompressible so must be their mixed flow but this alone does not imply a constant density value. Following the material derivative definition, equation (3.2), one further obtains,

Dρ ∂u + ρ i = 0 (3.10) Dt ∂xi

3.1.3 Conservation of momentum

The momentum conservation equation is derived from Newton’s 2nd Law, the fundamental physical prin- ciple governing dynamics. Employing this law to balance the rate of change in linear momentum with all the applied external forces on a ﬂuid element will yield,

d Z Z Z ρu dV = ρg dV + f n dS (3.11) dt i i Sij j V (t) V (t) ∂V (t)

where ∂V (t) is the boundary surface of V (t), gi is the gravitational force per unit mass, nj is the outward unit vector and fSij are the surface forces per unit area. Expanding the left term according to the Reynold’s Transport Theorem will further demonstrate the implications of this conservation equation,

Z Z Z ∂ρui ∂ρui + uj dV = ρgi dV + fSij nj dS (3.12) ∂t ∂xj V (t) V (t) ∂V (t)

where ui stands for the velocity component being conserved.

19 3. Conceptual Model

The source terms on the right hand side are gravity (body force), pressure and viscous stresses (surface forces). It can be shown that the surface forces are deﬁned by the following 2nd order tensor, (Aris, 1989)

fSij = −p δij + τij (3.13)

where p is the hydrostatic pressure, δij is the Kronecker delta and τij is the deviatoric stress tensor (viscous stress tensor, in the case of pure water). By applying the Divergence Theorem to the surface integral in (3.12) one obtains,

Z Z ∂fSij fSij · njdS = dV (3.14) ∂xj ∂V (t) V (t) which, in further combination with equation (3.12), must give,

Z Z ∂ρui ∂ ∂ + ui (ρuj) dV = ρgi + fSij dV (3.15) ∂t ∂xj ∂xj V (t) V (t) which, being valid for any material volume V (t), can be condensed into a diﬀerential form by extracting the integrands. Expanding the surface forces according to equation (3.13) will further yield,

∂ρui ∂ρuiuj ∂p ∂τij + = − + ρgi + (3.16) ∂t ∂xj ∂xi ∂xj which completes the momentum conservation law. For the special case of incompressible ﬂows, as the null divergence hypothesis still holds, (3.16) can be further simpliﬁed to,

∂ρui ∂ρui ∂p ∂τij + uj = − + ρgi + (3.17) ∂t ∂xj ∂xi ∂xj

The Navier-Stokes equations

In order to complete the system of equations it is still necessary to relate the deviatoric stress tensor

τij with the velocity ﬁeld by means of a constitutive relation. For clear water, the simplest of these constitutive relations establishes that the viscous stresses are proportional to the strain rates and to the dynamic viscosity, µ, (newtonian ﬂuid) and is then given by, (Batchelor, 1967)

∂ui ∂uj 2 τij = µ + − ∇ · ~uδij (3.18) ∂xj ∂xi 3 which, for incompressible ﬂows, simpliﬁes to,

∂ui ∂uj τij = µ + (3.19) ∂xj ∂xi

20 3.2. Governing equations for stratiﬁed ﬂows

Combining the momentum conservation equation for incompressible ﬂows, equation (3.17), with the previous consititutive relation (3.19) will yield the widely known Navier-Stokes equations.

∂ρui ∂ρui ∂p ∂ ∂ui ∂uj + uj = − + ρgi + µ + (3.20) ∂t ∂xj ∂xj ∂xj ∂xj ∂xi which describe the ﬂow in case of absence of suspended sediments.

3.2 Governing equations for stratiﬁed ﬂows

The conceptual model underlying the present dissertation idealizes river flow as a stratified planar flow of both water and sediment. Holding the hypothesis that the layers composing this stratified flow are continua, the conservation equations suitable for describing such media are also applicable to each layer and ultimately to the whole layer set. Each of these layers is a mixture of sediment and water characterized by its respective sediment concentration and ensuing density.

As the model is designed in 2D further work is required on the equations presented in Chapter 3.1. Speciﬁcally these equations must be brought from their generic 3D arrangement down to a 2D nearly equivalent form. A consistent path to performing such transition is depth-averaging those conservation equations. For a given function, for instance f(x, y, z, t), the depth-averaging is deﬁned as

1 Z y0+hL FL(x, y, t) = f(x, y, z, t) dz (3.21) hL y0

where FL(x, y, t) is the depth-averaged f(x, y, z, t) function within a layer L with depth hL.

It can be shown that integrating any function f(x, y, z, t) over a ﬁnite domain of the dependant variable z will result in some function of the form Fz(x, y, t) which may not depend on z itself. Hence,

∂F L = −1 (3.22) ∂z

Neglecting the vertical velocity (2D) in the conservation equations can be acceptable if the studied phenomena have an horizontal scale that happens to be much greater than the vertical scale, which is consistent with the hydrodynamics of a tsunami wave. Although the propagation of this type of wave occurs within the deep waters of the oceans, its wave length is great enough so that the tsunami will always interact with the seabed, thus generating friction, leading to a shallow-water wave behaviour.

The applicability of this conceptual model is restricted to flows where there is a clear distinction between suspended sediment, transported within the water column, and bedload, composed by coarser sediment transported near the riverbed Ferreira (2005). The layers featured in this conceptual model are those depicted in Figure 3.1. The first layer, from herein onwards Layer [1] or clear water layer, is the upper water column in which fine suspended sediments are transported. Underneath lies the contact load layer,

21 3. Conceptual Model

Layer [2] or bedload layer, which has a high concentration of coarse and suspended sediment. One must note that the thickness of Layer [2] is expected to be much smaller than the one of Layer [1], and so the set-up presented in Figure 3.1 only serves illustrative purposes. The bed, or Layer [b], was modeled as a passive, non-moving repository of sediment that is allowed to undergo either deposition or scour.

Clearwater Layer Layer [1]

Bedload Layer Layer [2] Bed Layer [b]

Figure 3.1: Layer set featured in the conceptual model

As previously introduced in section 3.1 the governing equations are a set of partial differential equations (PDEs) that describe fluid dynamics. The featured conservation equations consist of two mass conservation equations, for either total and solid phases, two momentum conservation equations, one for each direction of the x−y plane and a set of closure equations. Since two distinct phases are featured, the bedload layer and the clear water layer, the 2D conservation equations must be manipulated with special care, so that all fluxes are accounted for and all quantities remain conserved.

Each layer is bounded by two interfaces. These are deﬁned as continuum surface functions of the form,

SL(x, y, z, t) = IL(x, y, t) − z = 0 (3.23)

These surfaces can be defined as material or non-material. Material surfaces are not prone to fluxes which implies that the fluid velocity vector at every point of these surfaces must equal the velocity vector of the surface itself. On the contrary, non-material surfaces can be crossed and so no equality is imposed between the velocity vectors of the fluid and surface (see Figure 3.2).

At some instant, t, the interface is defined by the surface z = St and the fluid particles lying on it are 0 defined by the surface z = St. After an infinitesimal time step, dt, both of these surfaces will change their 0 configuration into St+dt and St+dt, respectively. The particles initially lying on the interface may now be completely detached from it, meaning that the surface and these particles will have different velocities, respectively ~uI and ~u. However these can be related to one another by an equivalent flux velocity, ~uφ, according to ~u = ~uI + ~uφ. The particular case where ~uφ = 0 defines a material surface, as it is the case of the free-surface.

Whether a surface is deﬁned as material or non-material it should always be a continuous surface and by such deﬁnition one can use equation (3.2) and obtain,

DS S (x, y, z, t) = 0 =⇒ L = 0 (3.24) L Dt

22 3.2. Governing equations for stratiﬁed ﬂows

u n uS S uF u S‘ F,t = t+dt uF,n F

St+dt uS

St=S’t

Figure 3.2: A non-material surface

that leads to the corresponding interface continuity equation,

DS ∂I ∂I ∂I L = L + u L + v L − w = 0 (3.25) Dt ∂t I ∂x I ∂y I

If ﬂuid velocity is the vector sum of the surface velocity and the equivalent ﬂux velocity then ~uI = ~u − ~uφ and one can rewrite equation (3.25) as,

∂I ∂I ∂I L + (u − u ) L + (v − v ) L − (w − w ) = 0 (3.26) ∂t φ ∂x φ ∂y φ which will simply yield,

∂I ∂I ∂I ∂I ∂I L + u L + v L − w + u L + v L − w = 0 (3.27) ∂t ∂x ∂y φ ∂x φ ∂y φ

The flux across the interface, ΦIL , is defined as the projection of the equivalent flux velocity ~uφ on the interface normal since the tangential component of ~uφ represents a displacement along the surface and not through it, thus carrying no flux, as displayed in Figure 3.2. The surface normal is herein defined as a unit vector collinear with the surface gradient.

∂I ∂I ∇~ I L + L − 1 Φ = ~u · ~n = ~u · L = ~u · ∂x ∂y (3.28) IL φ IL φ φ r k∇~ I k 2 2 L ∂IL ∂IL ∂x + ∂y +1

For the sake of simplicity an additional hypothesis is proposed: the interfaces are surfaces who happen 2 ∂IL to be smooth enough so that ∂x ≈ 0 and so equation (3.28) can be written as,

∂I ∂I Φ = ~u · ∇~ I = u L + v L − w (3.29) IL φ L φ ∂x φ ∂y φ

Merging equations (3.29) and (3.27) and multiplying all the resulting terms by the ﬂuid density at the

23 3. Conceptual Model interface will yield

∂IL ∂IL ρ|z=IL + [ρui]|z=IL − [ρw]|z=IL = ΦIL ρ|z=IL (3.30) ∂t ∂xi

3.2.1 Conservation of mass

Subscript notation can be used to write a more compact form of the mass conservation equation for incompressible ﬂows derived in Chapter 3.1, (equation (3.10))

∂ρ ∂ρu + i = 0 (3.31) ∂t ∂xi

To integrate the terms of this equation over the depth of each layer it is necessary to make systematic use of the Leibniz’ Integral Rule, which states that,

Z b(β) ∂ ∂ Z b(β) ∂a(β) ∂b(β) f(x, β) dx = f(x, β) dx + f(a(β), β) − f(b(β), β) (3.32) a(β) ∂β ∂β a(β) ∂β ∂β and where all variables are generic. For presentation and readability purposes the depth-averaging will be carried out separately for each term. The integration sequence will also follow the order in which these terms appear in the mass conservation equation.

Recalling the depth-averaged function deﬁnition and the Leibniz Integral Rule, equations (3.21) and (3.32), respectively, the depth-averaging of the local term yields,

Z Lu Z Lu ∂ ∂ ∂Ll ∂Lu ρ dz = ρ dz + ρ|z=Ll − ρ|z=Lu Ll ∂t ∂t Ll ∂t ∂t ∂ ∂L ∂L = [ρ h ] + l ρ| − u ρ| (3.33) ∂t L L ∂t z=Ll ∂t z=Lu

where t is time, xi are the spatial coordinates (x, y, z) assuming that z is a vertical coordinate, ~ez = ~g/|~g|,

ρL is the depth-averaged density within a layer L enclosed by lower and upper interfaces, denoted by Ll and Lu respectively. This procedure can be also used to integrate the convective term, thus obtaining,

Z Lu Z Lu ∂ ∂ ∂Ll ∂Lu ρui dz = ρui dz + [ρui]|z=Ll − [ρui]|z=Lu Ll ∂xi ∂xi Ll ∂t ∂t ∂ ∂Ll ∂Lu = [ρLUiL hL] + [ρui]|z=Ll − [ρui]|z=Lu (3.34) ∂xi ∂xi ∂xi which is accurate for both xx and yy axis. The zz axis is a particular case of this integration,

Z Lu Z Lu ∂ ∂ ∂Ll ∂Lu ρw dz = ρw dz + [ρw]|z=Ll − [ρw]|z=Lu (3.35) Ll ∂z ∂z Ll ∂z ∂t because the ﬁrst term on the right hand side of equation (3.35) is obviously null. Following equation

24 3.2. Governing equations for stratiﬁed ﬂows

(3.22) and provided that the interfaces are deﬁned as surfaces of the form FL(x, y, z, t) = IL(x, y, t) − z, equation (3.35) simpliﬁes to,

Z Lu ∂ ρw dz = [ρw]|z=Lu − [ρw]|z=Ll (3.36) Ll ∂z concluding the depth-averaging of the continuity equation. Finally, summing equations (3.33), (3.32) and (3.36) will yield,

Z Lu ∂ρ ∂ρu ∂ρv ∂ρw + + + dz = 0 Ll ∂t ∂x ∂y ∂z

Equation (3.37) must still undergo further manipulation to clearly reveal the vertical ﬂuxes through the interfaces of overlying layers. Most of the interfaces featured in this model are non-material, being the free-surface, the topmost surface, the only exception. Finally, by recalling equation (3.37)

∂ ∂ ∂Ll ∂Ll [ρLhL] + [ρLUiL hL] + ρ|z=Ll + [ρui]|z=Ll − [ρw]|z=Ll ∂t ∂xi ∂t ∂xi ∂Lu ∂Lu − ρ|z=Lu + [ρui]|z=Lu − [ρw]|z=Lu = 0 ∂t ∂xi one can notice that the terms inside the brackets are in fact the vertical fluxes, as defined in (3.30). Performing such substitution will yield the mass conservation equation for the conceptual model in its final differential form,

∂ ∂ ∂ [ρ h ] + [ρ U h ] + [ρ V h ] = Φ ρ| − Φ ρ| (3.38) ∂t L L ∂x L L L ∂y L L L Lu z=Lu Ll z=Ll

where ΦLu ρ|z=Lu and ΦLu ρ|z=Ll are the net mass ﬂuxes between Layer L and the layer above, through interface Lu, and also with the layer underneath, through interface Ll.

Another particular feature of this formulation is that every layer that happens to have an appreciable concentration of suspended sediment justiﬁes two distint mass conservation equation: one for the solid phase - suspended sediment - and another for the liquid phase - clean water. Recalling the deﬁnition of mixture density, introduced in (3.5),

ρL = ρw (1 + CL(s − 1)) (3.39)

25 3. Conceptual Model and bearing in mind that it evidences two distinct phases,

ρL = ρw(1 − CL) + ρsCL (3.40) substituting (3.40) in (3.38) will result in a single equation that implicitly contains both solid and water mass conservation equations. For the water mass conservation equation one has,

∂ ∂ ∂ [(1 − C )ρ h ] + [(1 − C )ρ U h ] + [(1 − C )ρ V h ] = ∂t L w L ∂x L w L L ∂y L w L L

= (1 − CL)ΦLu ρ|z=Lu − (1 − CL)ΦLl ρ|z=Ll (3.41) while the solid mass conservation equation is simply written as the supplementary part,

∂ ∂ ∂ [C ρ h ] + [C ρ U h ] + [C ρ V h ] = C Φ ρ| − C Φ ρ| (3.42) ∂t L s L ∂x L s L L ∂y L s L L L Lu z=Lu L Ll z=Ll

Applying equations (3.41) and (3.42) to the layers featured in the conceptual model will complete the ﬁnal set of mass conservation equations.

For the suspended sediment layer, Layer [1] in Figure 3.1, one can assume that the sediment concentration is negligible and therefore the depth-averaged layer density is roughly the same as clean water density and so only equation (3.41) is relevant. If one considers the water density to be constant in both time and space then it can be removed from the conservation equation,

∂ ∂ ∂ h + [U h ] + [V h ] = − Φ (3.43) ∂t 1 ∂x 1 1 ∂y 1 1 1,2

where U and V are the depth-averaged velocities in the ~ex and ~ey directions, respecively, and I1,2 is the interface between Layer [1] (clear water) and Layer [2] (bedload). High concentration of sediment is expected in Layer [2] which leads to using both (3.38) and (3.42),

∂ ∂ ∂ [h ] + [U h ] + [V h ] = Φ − Φ (3.44) ∂t 2 ∂x 2 2 ∂y 2 2 1,2 b

∂ ∂ ∂ [C h ] + [C U h ] + [C V h ] = C Φ − (1 − p)Φ (3.45) ∂t 2 2 ∂x 2 2 2 ∂y 2 2 2 2 1,2 b

where b is the bed, p is its respective porosity, Φ1,2 is the mass net ﬂux between layers [1] and [2], Ib is the bed interface and Φb is the net ﬂux between the bed and Layer [2]. Figure 3.3 can further clarify the physical meaning of the previous variables.

Since the bed is assumed to be saturated and to have no velocity at all there is only need for a sediment

26 3.2. Governing equations for stratiﬁed ﬂows mass conservation equation in this layer. Hence the only relevant equation is (3.42)

∂ (1 − p) Z = (1 − p)Φ = D − E (3.46) ∂t b b

where p is the bed porosity, Zb is the bed elevation and D and E are equivalent deposition and erosion rates, respectively.

U1 Clearwater Layer C1 ≈ 0 h1 Layer [1] Φ 1,2 I I1,2 1,2 U Bedload Layer h C Φ 2 Layer [2] 2 2 b I I Z b b b Bed Layer [b]

Figure 3.3: Layer set featured in the conceptual model

3.2.2 Conservation of momentum

In this section the momentum conservation equation derived in Chapter 3.1, equation (3.16), will undergo the same depth-averaging technique as the mass conservation equation, in the previous section. Recalling the momentum conservation equation,

∂ρui ∂ρui ∂p ∂τij + uj = − + ρgi + ∂t ∂xj ∂xi ∂xj one can use it to justify the hydrostatic pressure ﬁeld. If the vertical velocity is negligible so must be the vertical acceleration which leads to that both left-hand side of equation (3.16) and stresses τzj are null. This will simply resume (3.16) to,

∂p Z h ∂p Z h = ρg ⇔ dz = ρg dz ⇔ p(h) = ρgh (3.47) ∂z 0 ∂z 0 while showing why the pressure ﬁeld is assumed to be hydrostatic.

Again, to improve the readability and understanding of the following steps, all integrations will be performed separately for each hand of the momentum conservation equation, and also a term at a time whenever deemed necessary. One should also note that the horizontal components of gi are null. Applying the Leiniz’ Rule to the integration of the left-hand member over the depth of a given layer L will yield,

27 3. Conceptual Model

Z Lu ∂ρui ∂ ∂Ll ∂Lu dz = [ρLUiL hL] + [ρui] |z=Ll + [ρui] |z=Lu (3.48) Ll ∂t ∂t ∂t ∂t Z Lu ∂ρui ∂ ∂Ll ∂Lu dz = [ρLUiL hL] + [ρui] |z=Ll + [ρui] |z=Lu (3.49) Ll ∂xj ∂xj ∂xj ∂xj where the i index now stands for the 2D horizontal space directions - i = [1, 2]. One should note the existing similarities between these two equations and equations (3.33) and (3.34). The steps used to derive the vertical mass ﬂuxes can also be applied to full extent on equations (3.48) and (3.49). After those manipulations one obtains,

which represents the vertical net ﬂux of momenta, in the ~ei direction, through the interfaces of some layer L. The depth-averaged left-hand member of (3.16) can be therefore written as

∂ ∂ [ρLUiL hL] + ρLUiL ULj hL + ΦLl [ρui] |z=Ll − ΦLu [ρui] |z=Lu (3.52) ∂t ∂xj

For the right-hand member of (3.16), the applied external forces, the same logic must be followed in order to achieve a consistent set of depth-averaged quantities.

The ﬁrst term to be integrated is the pressure ﬁeld, which was shown to be hydrostatic,

Z Lu Z Lu ∂p ∂ ∂Ll ∂Lu dz = p(z) dz + p|z=Ll − p|z=Lu (3.53) Ll ∂xi ∂xi Ll ∂xi ∂xi

The depth-averaged pressure is obtained by using the variable substitution ξ = z−Ll while also accounting for for the weight of all the nL layers overlying layer L,

n Z Lu Z hL 1 XL p(z) dz = p(ξ) dξ = γ h2 + γ h h = P (3.54) 2 L L Ll i L L Ll 0 i=L+1

and γL is the same as ρLg. The depth-averaging of the viscous stresses is performed by integrating the deviatoric stress tensor over the depth of a layer, τij, yielding,

Z Lu Z Lu ∂τij ∂ ∂Ll ∂Lu dz = τij dz + τij|z=Ll − τij|z=Lu Ll ∂xj ∂xj Ll ∂xj ∂xj ∂ ∂Ll ∂Lu = TLij hL + τij|z=Ll − τij|z=Lu (3.55) ∂xj ∂xj ∂xj

Adding equations (3.52) to (3.55) will yield a 2D depth-averaged equation of the momentum conservation

28 3.3. Closure equations equation that, even while apparently more complicated, is in fact simpler to solve as it involves fewer unknowns, due to the removal of the vertical velocity, and thus reduces the governing system to three conservation laws. So, the 2D version of equation (3.16) is given by,

Applying equation (3.56) to Layer [1] will yield the following result,

∂ ∂ [ρ1U1i h1] + ρ1U1i U1j h1 + ΦI1,2 [ρui] |z=I1,2 = ∂t ∂xj ∂ ∂I1,2 ∂ ∂I1,2 = − P1 − p|z=I1,2 + T1ij hL + τij|z=I1,2 (3.57) ∂xi ∂xi ∂xj ∂xj and similarly for Layer [2] one should obtain,

∂ ∂ ∂ ∂Zb [ρ2U2i h2] + ρ2U2i U2j h2 − Φ1,2 [ρui] |z=I1,2 = − P2 − p|z=Zb ∂t ∂xj ∂xi ∂xi ∂I1,2 ∂ ∂Zb ∂I1,2 + p|z=I1,2 + T2ij h2 + τb,j − τij|z=I1,2 (3.58) ∂xi ∂xj ∂xj ∂xj

A total momentum conservation equation, accounting for both layers, is obtained if one sums equations (3.57) and (3.58). Considering the total values of momentum, momentum ﬂux and applied external forces for both layers as a whole one obtains,

∂ ∂ ∂ ∂Z ∂ ∂Z [ρU h] + [ρU U h] = − P − b gh + T h + b τ (3.59) i T i j T T T ijT T b,j ∂t ∂xj ∂xi ∂xi ∂xj ∂xj where all the variables with index T represent the sum of their equivalents over both layers 1 and P2 2: generically ϕT = i=1 ϕi, where ϕ can be ρUih, ρUiUjh, P , h or Tijhi. Due to the hydrostatic pressure ﬁeld it is also equivalent to write ∂xi Zb ghT instead of ∂xi Zb p|z=Zb . Provided that there are no discontinuities in the velocity ﬁeld across the interface that bounds both layers 1 and 2, Ferreira et al.

(2009) showed that the vertical ﬂuxes of momentum Φ1,2 [ρui] |z=I1,2 and the terms ∂xi I1,2 p|z=I1,2 are cancelled when (3.57) and (3.58) are summed.

The governing system of equations has thus been formally derived and its complete deﬁnition depends only on adequate closures or closure equations, which will be derived in the next section.

3.3 Closure equations

The proposed system of governing equations, namely (3.43), (3.44), (3.45), (3.57) and (3.58), requires closures for the dynamics of the bedload layer, h2 and U2, for the evolution of bed morphology, Φb or

D − E, and ﬁnally for ﬂow resistance, τb,j and TLij (Canelas, 2010; Ferreira et al., 2009).

29 3. Conceptual Model

The closures for the dynamics of the bedload layer were derived under the premise that greater kinetic energy ﬂuxes will lead to an increase of the bedload layer thickness, thus allowing for a complete dissipation of the kinetic energy ﬂuctuating components (Ferreira, 2005).

The proposed closure for the bedload layer thickness, h2, is given by, (Ferreira, 2005)

h2 = m1 + m2θ (3.60) ds

where ds is the significant particle diameter, m1 and m2 are parameters that are dependent upon the mechanical properties of the particles and fluid viscosity and lastly θ is the Shield’s parameter, calculated 2 as θ = Cf k~uk / (gds(s − 1)) where Cf is the friction coefficient. The bedload layer depth-averaged velocity,

U2, is given by a power law derived by Ferreira (2005),

h 1/6 U = U 2 (3.61) 2 h

Bed morphology closures relate the existing imbalance between solid discharge and transport capacity with bed elevation changes. The solid discharge, qs, is given by qs = CρU while several formulas can be ∗ used for the transport capacity, qs , namely Meyer-Peter & Muller (1947), Bagnold (1966) and a specific formula for stratified flows and debris flows by Ferreira (2005). The closure equation for ∂tZb is thus given by,

∂Z q − q∗ b = s s (1 − p)−1 (3.62) ∂t Λ where p is the bed porosity and Λ is an adaptation length deﬁned as a two horizontal asymptotic curve

Λ(Λmin, Λmax, θref , θ) whose arguments Λmin, Λmax and θref are parameters used to ensure that the value of Λ will never grow too large nor will it decrease to nearly zero.

Closures for ﬂow resistance feature bottom friction, viscous stresses and turbulent stresses. The bedload layer is expected to carry a high concentration of sediment leading to the same rheological behavior of an hyperconcentrated ﬂow.

In such conditions the total shear stress is the sum of three main components: the yield stress, τy, the viscous stresses, τν , and the turbulent stresses, τt. The cohesive nature of very ﬁne particles is taken into account through constant values of the yield stress, since most of the presently available formulae are only supported by very singular experiments.

A useful property of a Newtonian ﬂuid is that the viscous stresses can be expressed in terms of both dynamic viscosity, µ, and vertical velocity gradients, ∂zui. As for the turbulent stresses induced by bottom interaction they can be associated to the second power of the vertical velocity gradient as these

30 3.3. Closure equations are related to the kinetic energy of the ﬂow (Takahashi, 2007). Total shear stress is thus given by

∂u ∂u τ = τ + τ + τ = τ + µ i + ρ d2 c i (3.63) i y ν t y ∂z s f ∂z

The bed shear stress is given by,

τbi = Cf ρk~ukui (3.64)

where the friction coeﬁcient Cf can be given by the Manning-Strickler formula,

1 gh 3 Cf = 2 (3.65) Ks or by a specific formula concerning stratified flows or debris flows (Ferreira et al., 2009),

k~ukds Cf = (3.66) hωs

where ωs is the settling velocity of the sediment particles (Jimenez & Madsen, 2003) and Ks is the Manning coeﬃcient.

For the turbulent stresses a simple κ − model is applied (Pope, 2000). This model ignores small-scale eddies in the motion and calculates a large-scale motion with an eddy viscosity that carries and dissipates the energy of the smaller-scale ﬂow.

Under these assumptions one can deﬁne the depth-averaged stress tensor Tij as,

∂ui ∂uj Tij = ρνT + (3.67) ∂xj ∂xi

where νT is the eddy viscosity deﬁned in terms of the depth-averaged kinetic energy, κ, and turbulent kinetic energy dissipation rate, ,

2 22 κ 4.375u∗ νT = Cµ = 0.09 3 (3.68) 5u∗/h

√ where u∗ is the shear velocity or friction velocity, calculated as u∗ = τb/ρ.

The system of equations is now completely deﬁned featuring governing equations (3.43), (3.44), (3.45), (3.57), (3.58) and closure equations (3.63) to (3.68).

31 3. Conceptual Model

32 Chapter 4

Numerical Model

4.1 Discretization scheme

4.1.1 Introduction

Eulerian mesh-based methods featuring finite difference, finite element or finite volume schemes are amongst the most widely used tools in computational fluid dynamics (CFD) (Hircsh, 1988). The employed numerical model (Canelas, 2010; Murillo & García-Navarro, 2010) stands on a fully conservative finite volume method (FVM), that has now became a standard within the CFD world. Its important property of providing a proper discrete representation of the conservation laws when discontinuities are present in the domain (Hircsh, 1988; LeVeque, 1990; Toro, 2009) is a valuable advantage over other methods.

Only one flow layer is currently implemented in the computational model and hence the governing equations derived in Chapter 3 can be simplified. The absence of adjacent flow layers vanishes all the vertical flux terms obtained when depth-averaging the governing equations with Leibniz’s Integral Rule, as demon- strated for the total momentum conservation equation (3.59) in chapter 3. The only exceptions are the total and sediment mass fluxes between the flow layer and the bed. The governing system features total mass conservation, equation (4.1), total momentum conservation for the x and y directions, equations (4.2) and (4.3), respectively, and sediment mass conservation, (4.4).

∂th + ∂x (hu) + ∂y (hv) = −∂tZb (4.1)

2 1 2 1 1 τb,x ∂t (uh) + ∂x u h + 2 gh + ∂y (uvh) = −gh∂xZb − ∂xhTxx − ∂yhTxy − + Sx (4.2) ρm ρm ρm 2 1 2 1 1 τb,y ∂t (vh) + ∂x (vuh) + ∂y v h + 2 gh = −gh∂yZb − ∂xhTyx − ∂yhTyy − + Sy (4.3) ρm ρm ρm

∂t (Cmh) + ∂x (Cmhu) + ∂y (Cmhv) = −(1 − p)∂tZb (4.4) where x, y are the space coordinates, t is time, h is the depth, u and v are the depth-averaged velocities in the x and y directions, respectively, Zb is the bed elevation, ρm and Cm represent the depth-averaged

33 4. Numerical Model

density and concentration of the mixture, respectively, Tij are the depth-averaged turbulent stresses and τb represents the friction exerted by the bed on the ﬂuid. Sx and Sy are source terms induced nd 1 by 2 order perturbations due to stratiﬁcation (Ferreira, 2005). The bed variation ∂tZb in given by ∗ ∗ (1−p)∂tZb = (qs −qs )/Λ where qs is the sediment discharge, qs is its capacity value and Λ is an adaptation length. Sediment inertia and added pressure are neglected in the momentum equations.

Bed slope and friction source terms are of special importance for well-balanced CFD models. The employed formulation (Murillo & García-Navarro, 2010) represents an eﬀort to correctly introduce these terms while ensuring compatibility with steady solutions 2, a feature lacked by several other models.

4.1.2 Finite volume schemes: The case of the 2D shallow-water equations

FVM’s discretize the physical domain into control (finite) volumes where the conservation laws are applied using an Eulerian perspective of the flow: fluxes and accumulated quantities must be balanced for each finite volume according to the system of equations derived in Chapter 3. One should note that the numerical model herein presented uses a different notation from the Conceptual Model presented in chapter 3: vectors are represented by bold characters, for instance the conservative variables vector, U, the i index refers to cells and the k index refers to cell edges.

Considering some quantity U : R × R → Rm, where m is the system dimension, defined in an arbitrary volume Ω, fixed in both space and time, bounded by a closed set of surfaces Γ then the local value of U is dependent on incoming and outgoing fluxes, denoted by E ∈ Rm×m, expressing the contributions of the neighboring volumes. Singular sources also contribute and are herein denoted by H : R × R → Rm. The fluxes are functions of the quantity in question, which is to say E = E(U). A conservation law for the quantity U can be integrated over the the domain Ω yielding,

Z I Z ∂t UdΩ + ∇ · E = HdΩ (4.5) Ω Γ Ω

Expanding this equation for a ﬁxed control volume, Ωi, a discrete form of the conservation law is obtained,

N X Ai∂tU i + (E · n)Lk = AiHi (4.6) k=1

where U i is the dependant variable vector, Hi is the source term vector, both deﬁned as cell-averages, Ai is the area of cell Ωi which is bounded by N permeable boundaries. The system composed by equations (4.1) to (4.4) can be written in a compact form as,

∂tU(V ) + ∇ · E(U) = H(U) ⇔ ∂tU(V ) + ∂xF (U) + ∂yG(U) = H(U) (4.7) where the dependant non-conservative variables, V , dependant conservative variables U and ﬂuxes F

1For further details refer to Appendix 3, page 107 2Such as the case of still water in hydrostatic equilibrium over steep bottom topography.

34 4.1. Discretization scheme and G are deﬁned as,

        h h uh vh      2 1 2    u   uh  u h + gh   vuh  V =   ; U =   ; F =  2  ; G =   (4.8)  v   vh   uvh  v2h + 1 gh2        2  Cm Cmh Cmhu Cmhv and the source term vector H is H = R+T+S, where R is the parcel of source terms not susceptible to be treated as non-conservative fluxes, T are non-physical fluxes and S are stratification induced source terms. These terms are treated as fluxes in order to obtain a well-balanced numerical scheme (Vázquez-Cedón, 1999) and are defined as,

 (D−E)    − 1−p 0  τb,x     − (w)  −gh∂xZb R =  ρ  ; T =   . (4.9)  − τb,y  −gh∂ Z   ρ(w)   y b −(D − E)) 0

To obtain the FVM discretization, the system (4.7) is integrated over a cell i and the Gauss theorem is applied to the resulting divergence term. This procedure leads to,

Z I Z ∂t U(V )dS + E (U) · ndl = H(U)dS (4.10)

Ωi Γi Ωi

T where E · n = F nx + Gny with n = (nx, ny) being the outward normal to cell Ωi. Denoting the cell area by Ai and given that all quantities are piecewise constant ﬁelds, one obtains,

I ∂tAi hU ii + E (U) · ndl = Ai hHii (4.11)

Γi where the h i represents the spatial average in the cell, given by

1 Z hξii = ξ (x, y, t) dS. (4.12) Ai Ωi

Performing the surface integrals on the N boundaries of cell i, equation (4.11) becomes

N X ∂tAi hU ii + LkhE · niik = Ai hHii (4.13) k=1

where Lk is the length of the k edge. The local discretization of the system becomes,

ni ∆ hU ii X A + L ∆ h(E − T ) · ni = A hR i (4.14) i ∆t k ik i i k=1

35 4. Numerical Model

th] where the flux variation across the k edge of cell i, ∆ik hE − T i, is expressed as a function of the local variation of the dependent conservative variables, using Roe’s approximate Riemann solvers (Roe, 1981). Note that the flux vector of a system of shallow-flow conservation laws is not homogeneous and, hence, it is not possible to perform an exact flux vector splitting. Assuming a local linearization of the conservative flux vector orthogonal to the edge in question, one obtains (Canelas, 2010; Toro, 2001),

NW X ˜(n) (n) (n) ∆ik hE · ni = (hEji − hEii) · nik = λik αik e˜ik (4.15) n=1

˜(n) (n) where NW is the dimension of the eigenspace, λik and e˜ik are the approximate eigenvalues and eigenvectors, respectively, and are deﬁned as,

˜(1) ˜(2) ˜(3) ˜(4) λik = (u˜ · n − c˜)ik; λik = (u˜ · n)ik; λik = (u˜ · n +c ˜)ik; λik = (u˜ · n)ik (4.16)

        1 0 1 0         (1) u˜ − cn˜ x (2) −cn˜ y (3) u˜ +cn ˜ x (4) 0 e˜ =   ; e˜ =   ; e˜ =   ; e˜ =   (4.17) ik v˜ − cn˜  ik  cn˜  ik v˜ +cn ˜  ik 0  y   x   y    C˜m 0 C˜m 1 ik ik ik ik

(n) ˜ where coeﬃcients αik are the wave-strengths for each of the eigenvectors and u˜, v˜, c˜ and Cm are the approximate values for the two velocity components, (u, v), free-surface disturbance celerity, c, and (n) sediment concentration, Cm, respectively. The eigenvector wave-strengths αik are given by,

(1) ∆ikhhi 1 α = − (∆ik huhi − u˜ik∆ik hhi) · nik ik 2 2˜cik (2) 1 αik = (∆ik huhi − u˜ik∆ik hhi) · tik c˜ik (4.18) (3) ∆ikhhi 1 α = + (∆ik huhi − u˜ik∆ik hhi) · nik ik 2 2˜cik (4) ˜ αik = ∆ik hhCmi − Cm∆ik hhi

and the approximate values of u, v, c and Cm are deﬁned as, (Toro, 2001)

√ p √ p r √ p ui hi + uj hj vi hi + vj hj h + h Cm hi + Cm hj u˜ = √ ;v ˜ = √ ;c ˜ = g i j ; C˜ = i √ j ik p ik p ik mik p hi + hj hi + hj 2 hi + hj (4.19) Murillo & García-Navarro (2010) expressed the variation of the non-conservative ﬂux vector as a linear expansion in the basis formed by the approximate eigenvalues,

NW X ˜(n) (n) (n) ∆ik hT · ni = λik βik e˜ik (4.20) n=1

(n) where βik are wave-strengths corresponding to the eigenvalues and pb is the pressure exerted by the bottom boundary on the ﬂow over a bottom step. Expressions for pb (Canelas, 2010) that guarantee a well-balanced scheme and energy dissipation at jumps have been derived by Murillo & García-Navarro

36 4.1. Discretization scheme

(2010). The wave-strengths associated with the eigenvalues are given by,

(1) 1 pb (2) (3) (1) (4) βik = − ; βik = 0; βik = −βik ; βik = 0 (4.21) 2˜cik ρw

n+1 The remaining source term, T , is discretized in a point-wise formulation. T i is computed in an intermediate step with the updated variables of the homogeneous part of equation (4.13). The shear-rate in the deﬁnition of the depth-averaged turbulent stresses is calculated from directional derivatives. The discretized directional derivative ∇abF , of a generic diﬀerentiable variable F , in the direction of the vector rab that connects the barycentres of cells a and b is,

F (xb, yb) − F (xa, ya) DabF = (4.22) |ri,j|

At each cell, three directional derivatives can be deﬁned, for each of the neighboring cells. In a Cartesian referential the directional derivatives are written as ∇abF = ∇F erab , with ∇ = ∂x + partialy and erab being the directional unit vector from the barycentre of cell a to the barycentre of cell b. With the unknowns as the gradients in the Cartesian referential, a pair of directional derivatives can be used to calculate ∂xF and ∂yF . In general

! ! ! D cos(η ) sin(η ) ∂ F ab = ab ab x (4.23) Dac cos(ηac) sin(ηac) ∂yF

where ηab and ηac are the angles that the segments linking the barycentres of cells a and b and a and c, respectively, make with the x direction. Inverting the system allows for the calculation of ∂xF and ∂yF .

The eigenspace dimension for this scheme is NW = 4 and since the cells are of triangular shape then N = 3. Introducing equations (4.15) and (4.20) into (4.14) and following the Godunov’s method (LeVeque, 2002), the volume integral in the cell at time tn+1 is given by 3

3 4 − n+1 n ∆t X X ˜(n) (n) (n) (n) n+1 U i = U i − Lk λ α − β e˜ik + ∆t Ti (4.24) Ai ik k=1 n=1 which comprises the desired ﬂux-based ﬁnite volume scheme.

4.1.3 Stability region

When cell-averaging the solution, the time step is chosen small enough to guarantee that there is no interaction between waves obtained as the solution of the Riemann Problem (RP) at adjacent cells. The stability region considering the homogeneous part of the system becomes

3Further details on source term treatment and ﬂux-vector splitting techniques can be found in (Canelas, 2010)

37 4. Numerical Model

˜ ˜ min (χ , χ ) ∆t ≤ CFL ∆tλ ∆tλ = i j (4.25) ˜(n) max |λ |n=1,2,3 where the CFL value is always lower than 1 and χ is an equivalent 1D distance deﬁned as,

Ai χi = (4.26) max (Lk)i,k=1,2,3

Details on time step restrictions due to source terms can be found in Murillo & García-Navarro (2010)

4.1.4 Wetting-drying algorithm

In wet/dry interfaces with discontinuous bed level, the time step can become practically null since negative ﬂuid depths can be expected enforcing positivity requires a very small time step. In order to ensure positivity and conservation in the solution for all cases, the ﬂuxes for the update of the conserved variables are computed as (Murillo & García-Navarro, 2010),

If hn = 0 and h∗∗∗ < 0 then (∆E − T )− = (∆E − T ) and (∆E − T )− = 0 j j ik k jk (4.27) n ∗ − − If hi = 0 and hi < 0 then (∆E − T )jk = (∆E − T )k and (∆E − T )ik = 0 otherwise, the update is performed by (4.24). These restrictions efficiently prevent the appearance of negative fluid depths: the flux is zero at cells where an intermediate step predicts negative flow depths. In areas were h approaches zero, the computation of the primitive variables u = uh=h leads to potentially large round-off errors. Instead of defining a threshold for h above which the cell is allowed momentum, a threshold for the flow depth (hmin) is set to compute velocities using Kurganov & Petrova (2007)

√ uh 2h u = (4.28) ph4 + max(h4, )

√ where = hmin max(1, 2A), hmin is a mesh dependant parameter and A is the cell area.

4.1.5 Entropy correction

Non-physical solutions generated by Roe’s approximate Jacobian can occur, since there is no guarantee that the entropy condition 4. is satisﬁed. To enforce a physically admissible solution an entropy correction must be employed. In cases were the ﬂow is subcritical in one side and supercritical on the other side of an edge, entropy violating stationary shocks are computed by the described FVM (Canelas, 2010).

After the local linearization of the system the RP’s left and right rarefaction waves are identiﬁed by (1) (1) (3) (3) λik < 0 < λjk and λik < 0 < λjk , respectively, with the wave speeds being the eigenvalues computed 4Detailed in (LeVeque, 2002)

38 4.2. Boundary conditions

with the independent variables of the cells sharing edge k: λik = λ(U i) and λjk = λ(U j) as opposed to the approximated λ˜(n) used in the update.

A version of the Harten-Hyman entropy ﬁx (LeVeque, 2002) is implemented, were the objective is to forcefully promote the propagation of information in any expansion wave by diﬀusing the shock. Murillo ˜(1) (3) & García-Navarro (2010) decomposed the initial jump associated to λik and λik in two jumps.

(n) The same concept can be applied to the bottom source terms coefficients, βik , but only if a conservative ˜(n) splitting is verified. A proportional splitting to the new waves associated to λik results in bad solutions (Murillo & García-Navarro, 2010) due to numerical instabilities, and the time step suffers drastic reductions 5.

4.2 Boundary conditions

4.2.1 Introduction

Ensuring the existence of unique solutions requires suitable conditions at the boundary ∂Ω of any domain Ω. To accurately model physical phenomena the solution should depend continuously on the variable values inside Ω and those imposed on ∂Ω (Wesseling, 2009). This leads to the concept of well-posed problems, according to which, mathematical models of physical phenomena should provide unique solutions, whose behavior is smooth when small disturbances are introduced in either initial or boundary conditions.

The boundary condition scheme ﬁrstly implemented in the numerical model relies on a Riemann’s Invari- ant solution. Whilst this is quite advantageous from the computational performance and mathematical well-posedness perspectives it also carries a major drawback: if ﬂow direction at an inlet is opposite to the inner normal of ∂Ω then a unique solution does not exist.

In the interest of overcoming this limitation, new boundary conditions were developed and implemented. The derived and implemented scheme is based on the Rankine-Hugoniot shock conditions.

4.2.2 Rankine-Hugoniot conditions

When the characteristics of partial diﬀerential equations (PDE) intersect, a shock is formed. These are very thin surfaces propagating rapid transitions in physical variables, such as velocity or pressure, and may be mathematically represented by discontinuities. The Rankine - Hugoniot relations or jump conditions (Rankine, 1869) suitably describe the relationship between the states on both sides of the shock wave. Considering the following hyperbolic conservation law for a quantity ϕ,

∂ϕ ∂ + f(ϕ) = 0 (4.29) ∂t ∂x

5Further details can be found in Murillo & García-Navarro (2010)

39 4. Numerical Model

and should the solution exhibit a discontinuity at x = xs(t), integrating over the domain x ∈ [xR, xL], where xR < xs(t) < xL, will yield,

d Z xs(t) d Z xR f(ϕ(xL, t)) − f(ϕ(xR, t)) = ϕ(x, t) dx + ϕ(x, t) dx (4.30) dt xL dt xs(t) where the subscripts L and R denote the sides immediately to the left (upstream) and right (downstream) of the shock, respectively. Using Leibniz’ Rule, equation (3.32), on the integral derivatives of (4.30) will further obtain, (LeVeque, 1990; Toro, 2009)

Z xR d xs d xs ∂ϕ f(ϕ(xL, t)) − f(ϕ(xR, t)) = ϕ(xsL (t), t) − ϕ(xsR (t), t) + dx (4.31) dt dt xL dt

where ϕ(xsL (t), t) and ϕ(xsR (t), t) are the limits of ϕ(xs(t), t) as x converges to xs(t) from left or right, respectively.

By forcing the x domain limits, xL and xR, to converge on the shock location xs(t) one obtains the following result,

f(ϕ(xL, t)) − f(ϕ(xR, t)) = [ϕ(xsL (t), t) − ϕ(xsR (t), t)] S (4.32)

where S is the speed of the shock, dxs/dt. Equation (4.32) is commonly referred to as the Rankine- Hugoniot shock conditions and is often represented by a more compact form,

∆f = ∆u S (4.33)

where the jumps are deﬁned as ∆f = f(ϕ(xL, t)) − f(ϕ(xR, t)) and ∆ϕ = ϕ(xL, t) − ϕ(xR, t).

4.2.3 Shocks

Despite being discontinuities, shocks are still required to obey the conservation equations derived in Chapters 3.1 and 3, namely the mass conservation equation, (3.10), and the momentum conservation equation, (3.17). The conservation of mass across a shock is thus given by,

uLhL − uRhR = S (hL − hR) (4.34)

where uk and hk with k = L, R are the velocities and water depths immediately to the left and right of the shock wave while S is the shock speed.

The bottom elevation is assumed to have a zero-gradient type of boundary condition and thus, at the boundary edges, the gravitational force is neglectable. Hence, the momentum conservation equation

40 4.2. Boundary conditions across a shock at ∂Ω is,

1 1 u2 h + h2 − u2 h + h2 = S (u h − u h ) (4.35) L L 2 L R R 2 R L L R R

Expressing the shock speed, S, in terms of unit discharge and water depth jumps will yield,

u h − u h S = L L R R (4.36) hL − hR and, additionally, replacing (4.36) in (4.35) further obtains,

2 1 2 2 1 2 uLhL − uRhR uLhL + hL − uRhR + hR = (uLhL − uRhR) (4.37) 2 2 hL − hR which is a signiﬁcantly more useful expression since all variables are velocities or water depths. Figure 4.1 illustrates a compressive shock at an edge of the boundary, ∂Ω.

hR UR qL

Figure 4.1: A shock (compressive) at the boundary, ∂Ω

4.2.4 Expansion waves

While the Rankine-Hugoniot conditions are suitable to describe compressive shocks, the provided solutions are not physically feasible for expansive shocks (due to entropy condition violation). To handle such shocks, for instance when the mass ﬂux entering Ω is lower that the mass ﬂux on the inner side of ∂Ω, an alternative formulation, consisting of expansion waves, is proposed.

The propagation speed of these expansion waves is assumed to have the same value of the characteristic

λR. The expression itself is very close to the one of the Rankine-Hugoniot condition but, since the speed of propagation is known, only one conservation equation is required. Since the momentum conservation law features more information (velocity, water depth and pressure) it was the chosen equation to deﬁne the expansion wave conditions,

1 1 u2 h + h2 − u2 h + h2 = λ (u h − u h ) (4.38) L L 2 L R R 2 R R L L R R

41 4. Numerical Model

√ and, since λR = uR + ghR, one can further obtain,

1 1 h i u2 h + h2 − u2 h + h2 = u + pgh (u h − u h ) (4.39) L L 2 L R R 2 R R R L L R R which deﬁnes the expansion wave condition and where all terms have the same meaning of those in equation (4.37).

4.2.5 Analytical solutions

6 Equations (4.37) and (4.39) must be expressed in terms of the primitive variables inside Ω, uR and hR, and the hydrographs imposed at the permeable boundaries. These hydrographs deﬁne the discharges (Q) or the water depths (h) as time-dependent functions, Q(t) at the inlets and h(t) at the outlets.

Multiplying (4.37) by hL will deﬁne the compressible shocks at the inlets as,

1 1 u2 h2 + h3 − u2 h h + h2 h (h − h ) = (u h − u h )2 h (4.40) L L 2 L R R L 2 R L R L L L R R L where L and R denote variables upstream and downstream of ∂Ω, respectively. Expanding all terms will yield,

1 1 1 1 q2 h + h4 − h3 h − u2 h h2 + h h2 − h3 h = −2q u h h (4.41) L R 2 L 2 L R R R L 2 R L 2 R L L R R L

th which after further rearrangement will evidence a 4 order polynomial in order to hL,

1 1 1 1 − gh h4 + gh2 h3 + q2 + gh3 h2 + gh4 − 2q u h h + q2 h2 = 0 (4.42) 2 R L 2 R L R 2 R L 2 R L R R L L R

As for the expansion waves, equation (4.39), at the inlets, a similar procedure will obtain,

1 1 − g h3 + (u h − q ) u + pgh − u2 h − gh2 h + q3 = 0 (4.43) 2 L R R L R R R R 2 R L L

where qL is the value of the discharge hydrograph, Q(t). Equations (4.42) and (4.43) will provide the 7 required solutions at inlet boundaries, hL and uL , weather using compressive shocks or expansion waves.

For the outlet boundaries these polynomials must be expressed in terms of uL, as hL is imposed by the h(t) hydrograph, and so the compressive shocks are given by the following alternative arrangement of

6 ~ uR is deﬁned as the normal component of the velocity vector at ∂Ω, uR = U · ~n∂Ω where ~n∂Ω is the inner normal of ∂Ω. The tangential component of U~ does not inﬂuence compressive shocks (nor expansion waves). 7 uL is simply given by uL = qL/hL

42 4.2. Boundary conditions equation (4.37),

1 [−h h ] u2 + [2u h h ] u + g h3 + h3 − h2 h + h2 h − u2 h h = 0 (4.44) L R L R R L L 2 R L L R R L R R L and also for the expansion waves at the outlets one has,

h i 1 [−h ] u2 + u + pgh h u + g h2 − h2 − u + pgh u h + u2 h = 0 (4.45) L L R R L L 2 R L R R R R R R

In summary, the new boundary condition scheme is built upon equations (4.37) and (4.39) which describe shocks or expansion waves, respectively. These equations must be solved with respect to hL or uL, whether the solution applies to an inlet or an outlet, respectively, yielding equations (4.42), (4.43), (4.44) and (4.45).

Equation (4.42) is a 4th order polynomial that can be solved analytically by the Cardano-Ferrari’s method rd (Jao, Version 5) providing four mathematical solutions (hL1 , hL2 , hL3 , hL4 ) ∈ C. For (4.43), a 3 order polynomial, the applied solving method was the Cardano’s formula (Jao, Version 8) yielding three possible nd solutions (hL1 , hL2 , hL3 ) ∈ C. Both (4.44) and (4.45) are 2 order polynomials that are ordinarily solved with the quadratic formula, rendering two real solution candidates (hL1 , hL2 ) ∈ R.

4.2.6 Physical feasibility of the mathematical solutions

A wide range of solutions can be as cumbersome as no solution at all. Well-posed boundary conditions should provide unique and smooth values for the PDE’s being solved inside the domain, which arises the problem of selecting the only physically correct value within the mathematical array of possible solutions, a particularly sensitive matter when dealing with 4th or 3rd order polynomials. The purpose of the present section is to derive adequate conditions that can be employed as validation criteria for analytic solutions.

As time advances, a shock should have characteristics intersecting on its x−t path (Figure 4.2a). If these characteristics are emanating from the shock (Figure 4.2b) then its propagation is unstable even with small perturbations, thus endangering the well-posedness of the solution. This condition is called the entropy condition (LeVeque, 1990; Toro, 2009) and is given by,

λL ≥ S ≥ λR (4.46)

where λL and λR are the characteristics on the left and right sides of the shock, respectively, and S is the shock propagation speed. Considering the Rankine-Hugoniot conditions, equation (4.32), this condition can be satisﬁed only if (Figure 4.2a), (LeVeque, 1990; Toro, 2009)

ϕ(xsL (t), t) ≥ ϕ(xsR (t), t) (4.47)

43 4. Numerical Model

where ϕ(xsL (t), t) and ϕ(xsR (t), t) are the states on the left and right sides of the shock. This behavior was observed during the debug tests of these boundary conditions as the Rankine Hugoniot jump conditions consistently failed to fulﬁll the entropy condition when hL ≤ hR (Figure 4.2b).

t λ S λ t λ S λ t L > > R R> > L

S λ S S L λ L λ L

λ λ R R λ R x x x (a) Entropy-satisfying shock - (b) Entropy-violating shock - (c) Expansion wave - physically physically admissible physically non-admissible admissible

Figure 4.2: Physical admissibility with respect to the entropy condition

This led to the conclusion that the Rankine-Hugoniot solutions are only physically possible when hL ≥ hR.

When hL ≤ hR an expansion wave occurs and its corresponding entropy condition is given by,

ϕ(x (t), t) − ϕ(x (t), t) sL sR < (4.48) xsL (t) − xsR (t) t where t is time and is some constant > 0. Figure 4.2c depicts and expansion wave with a fan of characteristics emanating from the discontinuity’s initial position (x, t) = (x0, t0). This is consistent with the entropy condition for an expansion wave, which is given by λR > λL.

4.2.7 Results

The results herein presented were obtained using an horizontal fixed bed with 10 [m] x 25 [m] and a constant discharge of 5 [m3/s] where flow resistance was modeled using Manning-Strickler’s Law with a 1/3 −1 Ks coefficient of 33 [m s ]. An unstructured triangulated mesh was used containing approximately 10.000 elements with an average characteristic length of 0.25 [m]. The initial conditions were similar to a dam-break scenario.

Comparison with the Riemann’s invariant solution

Overall, the comparison with the previously existent boundary conditions yielded very consistent results. The Rankine-Hugoniot boundary conditions exhibited a slightly faster wave front (Figure 4.3c) but the general behavior of the solutions was, in practice, nearly coincidental.

Figure 4.3 shows the evolution of a dam-break scenario with an upstream inﬂow. The shocks at the inlet, located at x = 0, held a solution that was very close to that of a Riemann’s invariant in most of the time steps. The outlet was forcing the ﬂow to a minimum energy level.

44 4.2. Boundary conditions

0.30 0.30

0.25 Riemann’s Invariant Solution 0.25 Riemann’s Invariant Solution Rankine-Hugoniot Solution Rankine-Hugoniot Solution 0.20 0.20

0.15 0.15 h (m) h (m)

0.10 0.10

0.05 0.05

0.00 0.00 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m)

(a) Time t = 0 s (b) Time t = 5 s

0.30 0.30

Riemann’s Invariant Solution 0.25 0.25 Rankine-Hugoniot Solution

0.20 0.20

0.15 0.15 h (m) h (m)

0.10 0.10 Riemann’s Invariant Solution

0.05 0.05 Rankine-Hugoniot Solution

0.00 0.00 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (c) Time t = 10 s (d) Time t = 15 s

Figure 4.3: Comparison results for the implemented boundary conditions

Handling ﬂow reversal

The last test was performed to ensure that the boundary conditions could successfully handle ﬂow reversal, which is to say, velocity vectors that point outward at inlet cells (or inward at outlet cells). In this test a steady condition was attained and a wave was then forced at the outlet (Figure 4.4a).

The wave traveled upstream (Figure 4.4c), propagating negative velocities (Figure 4.4d), and eventually collided with the inlet (Figures 4.4g to 4.4j). At this stage the wave was reﬂected and was now propagating downstream (Figures 4.4k and 4.4l) until steady conditions were recovered once again (Figures 4.4m and 4.4n).

0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (a) Water Level at t = 8 s (b) Velocity at t = 8 s

45 4. Numerical Model

0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (c) Water Level at t = 11 s (d) Velocity at t = 11 s

0.5 0.8 0.6 0.4 0.4 0.2 0.3 0.0 v (m/s) h (m) 0.2 -0.2 -0.4 0.1 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (f) Velocity at t = 14 s (e) Water Level at t = 14 s

0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (g) Water Level at t = 15 s (h) Velocity at t = 15 s 0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (i) Water Level at t = 17 s (j) Velocity at t = 17 s

46 4.2. Boundary conditions

0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (k) Water Level at t = 20 s (l) Velocity at t = 20 s 0.5 0.8 0.6 0.4 0.4 0.3 0.2 0.0 h (m) 0.2 v (m/s) -0.2 0.1 -0.4 -0.6 0.0 0 5 10 15 20 25 0 5 10 15 20 25 x (m) x (m) (m) Water Level at t = 60 s (n) Velocity at t = 60 s

Figure 4.4: Steady conditions at time t = 100 [s] and t = 2000 [s]

Attaining steady conditions

The implemented boundary conditions had suitably attained a steady ﬂow after a simulation time of roughly 100 seconds. At the end of the simulation, 2000 seconds, the free-surface proﬁle was exactly the same, ensuring the well-posedness of this new feature.

0.5

0.4

0.3 h (m) 0.2

Riemann’s Invariant Solution 0.1 Rankine-Hugoniot Solution

0.0 0 5 10 15 20 25 x (m)

Figure 4.5: Steady conditions at time t = 100 s (still the same at t = 2000 s)

47 4. Numerical Model

48 Chapter 5

Integration with Geographic Information Systems

5.1 Introduction

In the interest of providing a better (graphical) user interaction and also promoting interoperability between STAV-2D and geographic information systems (GIS) some auxiliary sub-programs (Add-Ons) were developed, allowing the user to deﬁne all the necessary geometric entities for mesh generation and reﬁnement within a GIS environment.

STAV2D current Add-Ons only feature ArcMap, a geospatial processing program and a main component of the broadly known ESRI’s ArcGIS suite. Some of the most powerful attributes of ArcMap include viewing, plotting, editing, creating and analyzing geospatial data in either matrix or vector formats. These features highlight the advantages of having prompt information exchanges between STAV-2D, GIS and a mesh generator which was, in this case, the open-source code Gmsh (Geuzaine & Remacle, 2009).

5.2 Pre-Processing utilities

5.2.1 Generating boundaries

Deﬁning complex domain geometries, for instance isoline delimited boundaries, can become rather cumbersome if carried on a point-by-point basis. The purpose of this Add-On was to quickly convert boundary polygons drawn in ArcMap to a format compatible with Gmsh. This enables the user to directly deﬁne the boundaries over the altimetric dataset, avoiding the manual transfer of (x, y) coordinates into Gmsh.

The usage of this Add-On is quite simple: (the following steps only apply to ArcMap)

1. Draw the boundary polygon in ArcMap using a shapeﬁle (.shp) format.

2. Apply the Feature Vertices to Points tool to the .shp layer

49 5. Integration with Geographic Information Systems

3. Apply the Add XY Coordinates to the .shp layer

4. Export the Attribute Table to ./Input/Boundary.txt

5. Start GIS2Gmsh.exe and follow on-screen instructions

6. Use the generated ﬁle (boundary.geo) in ./Output as a geometry ﬁle in Gmsh

7. Use Gmsh to mesh the domain inside the boundary delimited in 1.)

(a) Domain boundary polygon drawn in ArcMap (b) Domain boundary polygon converted to Gmsh’s using .shp format (yellow line) .geo format

Figure 5.1: Pre-processing samples: example of ArcMap isoline deﬁned boundary (left) and conversion to Gmsh (right)

5.2.2 Reﬁning meshes

Mesh detail levels can greatly influence the performance of STAV-2D. In the case of greater detail re- quirements at singular locations one should be able to locally refine the mesh without having to do so to the whole domain. The mesh refinement add-on is able to import several refinement polygons from ArcMap and convert these into a Post-Processing Gmsh file (.POS) that can be applied as a background refinement mesh. This enables the user to visually define refinement levels, hence promoting productivity, since there is no need to explicitly parametrize these regions with coding in Gmsh’s geometry files (.geo).

The usage of this add-on can be summarized as:

1. Draw the reﬁnement polygons in ArcMap using shapeﬁles (.shp).

2. Name the shapeﬁles sequentially: 1.shp, 2.shp (...)

3. Apply the Feature Vertices to Points tool to all .shp layers.

4. Apply the Add XY Coordinates to all .shp layer

5. Export the Attribute Table of all .shp ﬁles to ./Input/X.txt where X is the shapeﬁle number.

6. Modify all X.txt ﬁles according to the sample ﬁles in "./Input/Templates".

7. Start GIS2Gmsh.exe and follow on-screen instructions

8. Start Gmsh and open the geometry (.geo) ﬁle previously generated for the boundary

9. Apply the Post-Processing (Background_mesh.pos) ﬁle in ./Output as a background mesh

10. Use Gmsh to mesh the domain using the prescribed cell lengths in 1.)

50 5.3. Post-processing utilities

The following ﬁgures only serve illustrative purposes and may not reﬂect the actual geometries of the simulations presented in Chapter 6.

(a) Reﬁnement polygons (red and white) drawn in (b) Target element size throughout the domain ArcMap (Gmsh, (Geuzaine & Remacle, 2009)))

(e) Coarser cells at the inlet (f) Coarser cells at the outlet

Figure 5.2: Example of mesh reﬁnement procedure

5.3 Post-processing utilities

5.3.1 Resampler and raster converter

The VTK resampler is a tool that interpolates flow variables, such as velocity or water height, between two different meshes. Its most valuable use lies in warming-up roughly meshed problems until steady or other desired conditions are attained. The obtained solution can then be interpolated to a more refined mesh and used as an initial condition for further simulations.

51 5. Integration with Geographic Information Systems

The raster converter allows the user to import STAV-2D output into GIS environments so that this data can be referenced and displayed alongside with national geographic information datasets.

Geometry processing and interpolation algorithms

In these post-processing utilities two distinct interpolation methods were used: a cell-to-cell assignment or a space-averaged neighborhood assignment. The cell-to-cell assignment method relies on a Ray-Casting algorithm (Foley et al., 1996) to determine if a given cell of the target mesh is (or not) inside a cell of the original mesh.

This is solved as a point-in-polygon (PIP) problem (Foley et al., 1996) where the point is the barycenter of the target mesh cell and the polygon is the cell of the original mesh.

To solve the problem one must draw a ray towards the barycenter of the target mesh cell (a line that connects the barycenter of that cell to a point at the "inﬁnity") and intersect this ray with the edges of the original mesh cell. If the number of intersections is odd then the barycenter of the target mesh cell is inside the original mesh cell, otherwise it is outside.

PIP In ∞ PIP Out ∞

Target Mesh Cell Original Mesh Cell

Original Mesh Cell

Target Mesh Cell

(a) "Inside" PIP condition: odd intersection count (b) "Outside" PIP condition: even intersection count

Figure 5.3: Point-in-polygon (PIP) problem examples

In the second method, the neighborhood space-average, an Inverse Distance Weighing (IDW) algorithm is used. Denoting by ~x the target cell barycenter and by ~xi the barycenter of its N nearest neighbors in the original mesh, the IDW algorithm is expressed as,

N X wi(~x,~xi)A(~xi) A(~x) = (5.1) PN i=1 i=1 wi(~x,~xi) where A(~x) is the value of the interpolated quantity A assigned to the target mesh cell and A(~x) is the value of this same quantity inside the nearest original mesh cells. The weights wi are deﬁned as,

1 wi(~x,~xi) = p (5.2) D(~x,~xi)

52 5.3. Post-processing utilities

where D(x , x i) is the distance between the barycenters x and x i and p is a power parameter. If N and p, which are user inputs, are deﬁned as one and zero, respectively, the IDW algorithm gets reduced to the also widely used Nearest Neighbor (NN) method.

The usage of this Add-On can be resumed to:

1. Place the VTK ﬁles to be converted or resampled in ./Input

2. Name these ﬁles sequentially: Input_1.vtk, Input_2.vtk (...)

3. When resampling also include a VTK ﬁle with the reﬁned mesh in ./Input and name it In- put_New.vtk

4. Start GIS2Gmsh.exe and follow on-screen instructions

5. Raster and resampled .vtk ﬁles are saved in ./Output

(a) Original VTK ﬁle (Simple testbed with 5.000 (b) Resampled VTK ﬁle (Same testbed upsampled cells) to 37.000 cells)

(c) Original VTK ﬁle (Ponte 25 de Abril close up (d) Resampled VTK ﬁle (Same location on the on the warm-up simulation using 750.000 cells) main simulation upsampled to 2.400.000 cells

Figure 5.4: Examples of VTK ﬁle resampling results

53 5. Integration with Geographic Information Systems

54 Chapter 6

A 1755 tsunami in today’s Tagus estuary topography

6.1 Pre-Processing

6.1.1 Lisbon and Tagus estuary topography

The present section focuses on the geographical information datasets employed in this work and their importance in obtaining accurate results since detailed geometries are required to comprehensively describe wave run-up and inundated areas.

Two altimetric datasets were employed. The most detailed was a 5x5 [m] resolution grid with approximately ninety million cells featuring poorly defined and diffuse geometries but, nonetheless, satisfactory elevation accuracy for the built environment (see Figures 6.3a and 6.3e for a sample view). A smoother 10x10 [m] resolution grid with twenty-five million cells was also used despite having no apparent built environment geometries (Luis, 2007) (Figures 6.1, 6.2, 6.3b and 6.3f). The Tagus estuary bathimetry was also provided as a 10x10 [m] resolution grid (Luis, 2007).

To properly define the built environment geometries these two Digital Elevation Models (DEM) had to be used alongside with a third information layer. This auxiliary layer was a vector dataset that featured all the buildings in Lisbon as polygonal entities 1 (see Figure 6.3c and 6.3g). A simple zonal statistic procedure was then followed to improve the definition of the existing DEMs: each polygon (or "building") was converted to a single valued raster using the highest elevation of the 5x5 [m] grid inside that same polygon whereas outside the polyogns (the "streets") the smoother 10x10 [m] grid was used. That is to say the buildings defined by the combination of vector and 5x5 [m] grid information were overlaid on the 10x10 [m] grid 2. The obtained results are displayed in Figures 6.3d and 6.3h.

1This vector dataset was supplied by the Department of Geographic Information and Cadastre (DIGC) of Lisbon Mu- nicipality and featured up-to-date building plan geometries. 2All grids were previously upsampled (using surface interpolation) to 2.0[m] cells to ensure adequate consistency with very ﬁne meshes (CL ≈ 2.5[m]). The designation of 5x5 or 10x10 [m] resolution only concerns the data measurement grid and not the actual cell size used for STAV-2D inputs.

55 6. A 1755 tsunami in today’s Tagus estuary topography

The tower foundations of Ponte 25 de Abril were also carefully modeled since these structures may also inﬂuence an incoming tsunami. As for the south bank of the Tagus, the existing harbors were also given appropriate detail since these areas are expected to have a major impact on the tsunami run-up.

Elevation (m) 0 50 100 150 250 300

Figure 6.1: DEM Raster with 5x5 [m] resolution

Bathymetry (m) -50 -40 -30 -20 -10 0

Figure 6.2: Bathimetry with 10x10 [m] resolution

56 6.1. Pre-Processing

(a) Lisbon downtown in the 5x5 [m] DEM (b) Lisbon downtown in the 10x10 [m] DEM

(e) Alcântara in the 5x5 [m] DEM (f) Alcântara in the 10x10 [m] DEM

(g) Alcântara in the vector layer (h) Alcântara in the ﬁnal DEM

Figure 6.3: Example of DEM built environment enhancement

57 6. A 1755 tsunami in today’s Tagus estuary topography

6.1.2 Initial Conditions

In this work two simulations were computed: a warm-up simulation and a main simulation. The warm- up simulation purpose is to build a steady ﬂow consistent with the mean annual discharge in the Tagus river. This setup is aimed at building initial conditions for the main simulation. To do so, nearly constant velocity and water level readings were required at 4 virtual gauges scattered throughout the estuary (see Figures 6.4e to 6.4h on page 59). The resulting output (see Figures 6.4a and 6.4b) was then resampled with the Add-On introduced in Chapter 5 and used as the initial condition for the main simulation.

The initial condition for this warm-up simulation was a still water mass at high tide level (+2 [m]). The discharge hydrograph at the inlet was not strictly constant (Figure 6.4d on page 59) because the shocks produced by a sudden discharge of 450 [m3/s] would likely introduce small waves that could propagate inland, corrupting the correct wet area and possibly compromising data resampling on waterfront cells. The outlet condition was a constant high tide (+2 [m]) water level hydrograph.

6.1.3 Boundary Conditions

Upstream

A simple hydrological analysis was performed to obtain the modular flow to be imposed at the inlet (or upstream) boundary. A mean annual flow duration curve was derived from a 25 year flow time-series registered at the Ómnias - Sacavém hydrometric station (see Figure 6.5).

5000

4000

3000 3

Q (m /s) 2000

1000 3 Q mod = 450 m /s

0 0 2000 4000 6000 8000 10000 t (d)

Figure 6.5: Mean annual ﬂow duration curve (Ómnias - Sacavém hydrometric station)

The corresponding modular ﬂow is given by,

R tf Qd(t) dt t0 3 Qmod = ≈ 450 [m /s] (6.1) tf − t0

where Qd(t) is the mean annual ﬂow duration curve, t is time (ranging from t0 to tf ) and Qmod is the modular ﬂow. The obtained value corresponds to an event with a 1.5 year return period.

58 6.1. Pre-Processing

(a) Water level at the end of warm-up (b) Velocity magnitude at the end of warm-up

500

400

300 3

Q (m /s) 200

100

0 0 2000 4000 6000 8000 10000 t (s) (c) Virtual gauges location (d) Warm-up inlet discharge hydrograph

2.5 0. 5 2.5 0. 5

2.0 0. 4 2.0 0. 4

1.5 0.3 1.5 Velocity 0.3 h (m) h (m) V (m/s) 1.0 0. 2 V (m/s) 1.0 Water Level 0. 2

0.5 0. 1 0.5 0. 1 Water Level Velocity 0.0 0.0 0.0 0.0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t (d) t (d) (e) Virtual gauge 1 readings (f) Virtual gauge 2 readings

2.5 0. 5 2.5 0. 5

2.0 0. 4 2.0 0. 4

1.5 0.3 1.5 Velocity 0.3 h (m) h (m) V (m/s) 1.0 0. 2 1.0 Water Level 0. 2 V (m/s) Water Level 0.5 0. 1 0.5 0. 1 Velocity 0.0 0.0 0.0 0.0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t (d) t (d) (g) Virtual gauge 3 readings (h) Virtual gauge 4 readings

Figure 6.4: Warm-up setup and results

59 6. A 1755 tsunami in today’s Tagus estuary topography

Downstream

The downstream boundary condition was obtained from oﬀshore propagation simulations performed at the University of Lisbon facilities. The model employed was a modiﬁed version of the COMCOT code by Wang & Liu (2006). The provided data describes the tsunami at the atlantic outlet arc. This data was supplied pointwise and was implemented in the perspective of a worst case scenario: the most severe wave series was imposed at all outlet cells. Several combinations were attempted to ensure that the wave propagating inside the domain represented the worst case scenario within the supplied data.

Figures 6.6a to 6.7b display the supplied data as a function h(t). In order to save computation time the ﬁrst 2000 seconds were ignored since there are no relevant waves within this time interval, which is to say, the hydrograph shown in Figure 6.6a was shifted by 2000 seconds towards the origin t0 = 0[s]. Simulation time was 10400 seconds (3 hours) and allowed for a complete propagation and draw back of all the waves from the hydrograph depicted in Figure 6.6a. If there was no data available, t ≥ 7200[s], a still high tide level was imposed (+2[m]).

h (m) 4

0 0 2000 4000 6000 8000 10 000 t (s)

(a) Tsunami level imposed at the atlantic boundary

10 10

8 8

6 6 h (m) h (m) 4 4

2 2

0 0 2000 2200 2400 2600 2800 3000 4700 4800 4900 5000 5100 5200 t (s) t (s) (b) First and main wave of the tsunami event (c) Second wave

Figure 6.6: Supplied data for a 1755-like tsunami

60 6.2. Brief framework for result presentation and discussion

10 10

8 8

6 6 h (m) h (m) 4 4

2 2

0 0 5550 5600 5650 5700 5750 5800 6950 7000 7050 7100 7150 7200 t (s) t (s) (a) Third wave (b) Fourth and ﬁfth waves

Figure 6.7: Supplied data for a 1755-like tsunami (2)

6.2 Brief framework for result presentation and discussion

The previous chapters introduced the mathematical and theoretical concepts upon which the STAV-2D model is built and all the information layers that compose the framework of a potential tsunami scenario in Lisbon. The results herein presented are mainly focused on the eﬀects of the ﬁrst wave of a 1755-like tsunami (Figure 6.6b), by far the most destructive and the earliest of the whole event.

Display and discussion of the obtained results is carried at two distinct levels: an overview of the main morphological impacts and run-up extents at the seafronts of Carcavelos, Oeiras, Belém, Caparica and Barreiro and a more detailed street-level analysis at Alcântara, Lisbon Downtown and Cacilhas. This last set of results ads great value to the tsunami modelling eﬀorts carried by the Portuguese scientiﬁc community since no previous studies were known to explicitly discretize the built environment features.

Another important contribution arises from the inclusion of solid transport capabilities in STAV-2D, allowing debris flow simulations to be performed. Recent events showed that the hydrodynamics of tsunami inland propagation is much closer to that of a debris flow than to a clear water flow. The last section in this chapter is devoted to debris scattering patterns and main morphological impacts.

Figure 6.8: Relevant locations for result presentation and discussion

61 6. A 1755 tsunami in today’s Tagus estuary topography

6.3 Overview of a tsunami scenario in Lisbon

6.3.1 General description of tsunami propagation

Figure 6.9 depicts the propagation of the first tsunami wave for 27 minutes. The description of the first wave at the Tagus mouth is well documented (Baptista et al., 2011) in a magazine dated 19 November 1755 (Urban, 1755): “[...] Castle of Bugio was so far overcome by the water, that the garrison fired several guns as signals for help [...]. By my best judgment, the water rose in five minutes about 16 feet [...]”. From this report the wave height at Bugio Tower should be approximately 5 [m]. After hitting Bugio the tsunami propagates towards the seafront of Oeiras causing massive run-ups, at very high speeds. The wave then propagates inward the estuary through the Tagus channel, accelerating due to the higher depths while flooding Lisbon and Cacilhas, a particularly hazardous stage of this event.

(a) Water level as the tsunami hits Bugio Tower (b) Water level, t = 7 minutes

(e) Water level, t = 20 minutes (f) Water level, t = 27 minutes Water level (m) 2.0 3.0 4.0 5.0 6.0

Figure 6.9: Tsunami propagation overview at the Tagus estuary Tsunami impact at Bugio Castle used as relative time reference

62 6.3. Overview of a tsunami scenario in Lisbon

When the tsunami reaches the estuary it propagates radially and the wave height drastically reduces as a consequence of the widening cross section. Nonetheless, it still carries immense energy with signiﬁcant run-ups occurring at Barreiro, on the south bank of the estuary.

6.3.2 Impact at Carcavelos

Carcavelos is one of the busiest beaches in Lisbon. The prominent shape of the seafront at this location makes it one of the most hazardous areas in the case of a tsunami. The wave does not undergo any kind of deﬂection or attenuation prior to impact, which is nearly head-on with the beach. This causes an extensive and destructively fast inundation. Maximum run-up is as high as 20 [m] above mean-sea level (MSL) and may extend to 600 [m] away from shore, with velocities just above 5 [m/s] (Figure 6.10).

(a) Water level, t = -1 minutes (b) Velocity magnitude, t = -1 minutes

(e) Water level, t = 3 minutes (f) Velocity magnitude, t = 3 minutes

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.10: Tsunami run-up at Carcavelos Tsunami impact at Bugio Castle used as relative time reference

63 6. A 1755 tsunami in today’s Tagus estuary topography

6.3.3 Impact at Caxias and Cruz Quebrada

The head-on impact of the tsunami at this location (Figure 6.9 on page 62) inﬂicts massive wave heights and velocities when the tsunami propagates overland. Run-up is expected to propagate inland until the ground rises to 10 [m]. The very mild topography of the reaches at Caxias and Cruz Quebrada implies that a 10 [m] run-up can lead to an inundated area as far as 1500 [m] from the seafront. A mild terrain inﬂicts much less resistance to the incoming wave leading to a maximum velocity of 6 [m/s]. Algés is also a vulnerable area with a inundation of 300 [m] inland, up to 8 [m] high.

Inland penetration reaching further than 1 [km] is only observed at Caxias. Poor built environment deﬁnition (and low density) can also favor these extensive inundations since macro ﬂow resistance is practically ignored.

(a) Water level, t = 6 minutes (b) Velocity magnitude, t = 6 minute

(e) Water level, t = 8 minutes (f) Velocity magnitude, t = 8 minutes

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.11: Tsunami run-up at Caxias and Cruz Quebrada Tsunami impact at Bugio Castle used as relative time reference

64 6.3. Overview of a tsunami scenario in Lisbon

6.3.4 Impact at Belém and Algés

Belém is recognized for its concentration of national monuments and public spaces, composing a mixture of historical monuments and modern symbols of the Portuguese culture. This combination of iconic landmarks makes it one of the most attractive locations in the whole city, bringing the discussion of a tsunami impact at this location to a very sensitive level. Mosteiro dos Jerónimos, Torre de Belém, Centro Cultural de Belém and Padrão dos Descobrimentos are all at jeopardy due to their riverfront location, facing inundation depths of 3.5 [m] and velocities higher than 5 [m/s]. Run-up extent is limited to 550 [m] from the riverfront, up to 5 [m] high (Mosteiro dos Jerónimos, Figure 6.12).

(a) Water level, t = 10 minutes (b) Velocity magnitude, t = 10 minute

(e) Water level, t = 13 minutes (f) Velocity magnitude, t = 13 minutes

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.12: Tsunami run-up at Belém and Algés Tsunami impact at Bugio Castle used as relative time reference

65 6. A 1755 tsunami in today’s Tagus estuary topography

6.3.5 Impact at Caparica and Trafaria

Another rather hazardous area is the Atlantic seafront between Costa da Caparica and Trafaria. At south of Costa da Caparica the wave will propagate inland as further as 800 [m]. Smaller inundation lengths occur at Costa da Caparica and at Almada’s northern seafront due to dense built environment and forest areas. Another concerning detail of this seafront is the presence of many camping sites bordering beach areas. These heavily populated sites, namely during bathing seasons, are worryingly prone to catastrophic life losses due to inherent evacuation and access diﬃculties.

Important wave shoaling is expected due to the mild slope of the seabed. Preceding further inland propagation this seafront will be devastatingly swept by a 10 [m] wave at a speed of roughly 7 [m/s]. A later discussion on this chapter is reserved to the severe morphological changes that are to occur.

(a) Water level, t = 1 minutes (b) Velocity magnitude, t = 1 minutes

(e) Water level, t = 7 minutes (f) Velocity magnitude, t = 7 minutes

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.13: Tsunami run-up at Caparica and Trafaria Tsunami impact at Bugio Castle used as relative time reference

66 6.4. Street-level results for inland propagation

6.4 Street-level results for inland propagation

6.4.1 Results for Alcântara

As a result of the 1755 earthquake and tsunami, the Portuguese Royal Family and the government moved to Alto da Ajuda. At that time, and following the catastrophe, it was the best place to hold the national power entities: shallow bedrocks allowed better seismic structural behaviour as foundations could be embedded, the elevation was enough to surpass any incoming tsunami and airy surroundings protected the Royal family from any developing plague due to the many fatalities in the lower city. Nobility, workers and municipal services followed this relocation. The increasingd traﬃc between Alto da Ajuda and the old Lisbon city allowed this otherwise suburban area to expand at a fast pace and develop into what is nowadays one of the most central spots in Lisbon.

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.14: One-way traﬃc zones (top) and tsunami approach to Alcântara (bottom; t=10’20”) Tsunami impact at Bugio Castle used as relative time reference

Known to have suffered severe damage from the 1755 event this area was refined with a 2.5 [m] mesh and the built environment was modeled with actual city planimetry. The urban mesh is quite dense and organically adapted to the underlying topography. This poses some difficulties to traffic management and, to avoid typical rush-hour bottlenecks, many loops are implemented with restrictive one-way traffic streets. These are especially worrisome as they can drive traffic towards the riverfront. Figure 6.14 shows 5 concerning zones in what respects to restrictive circulation.

67 6. A 1755 tsunami in today’s Tagus estuary topography

Roughly 10 minutes after the tsunami hits Bugio Tower, the water level has risen to the point of riverfront overtopping (Figure 6.14 on 67) and a minor 0.5 [m] high inundation front takes place at Avenida da Índia. At this stage velocities range from 0.3 to 2.5 [m/s]. Impact at Cordoaria Nacional is imminent.

Two minutes later (Figure 6.15 on page 69), at 12 minutes and 20 seconds, most of the riverfront between Cordoaria Nacional and Lisbon harbour is already flooded. At this stage the inundation front already approaches Hospital Egas Moniz as Rua da Junqueira and Congress Center are swept by a devastating wave front with nearly 1.6 [m] at a speed of almost 4 [m/s]. Vila Galé Ópera Hotel and Alcântara/Santo Amaro Docks are also severely hit. At the hotel water depths are comparable to a minor flooding, about 0.6 [m], but with velocities over 4 [m/s] heavy debris can be carried inside the hotel halls, causing many casualties. At Alcântara Docks the marina dampens the wave and slows down the flow at the cost of higher depths, which can reach 1.5 [m]. The harbour is a concerning location as high flow depths and great velocities can carry a large amount of containers, and even ships, towards the streets.

Water is already reaching the inner grounds of Hospital Egas Moniz by the time of 14’20” (Figure 6.16 on page 70). Patient evacuation becomes severely constrained since three of the four surrounding streets are one-way minor arteries and lead towards the inundated areas, namely Calçada da Boa Hora, Travessa Conde Ribeira (Zone 3) and Rua da Junqueira (Zone 1 - completely flooded at this stage). Only one street can be used to effectively reach higher grounds: Rua Filipe Vaz towards Travessa Giestal (Zone 1). Alcântara-Mar Train Station is inundated with flow depths reaching more than 1 [m] and velocities around 3 [m/s]. According to traffic circulation allowances Zones 1, 2, 3 and 4 are now completely isolated with possible access to Zones 1, 3 and 4 only possible if such rules are neglected. Zone 2 is completely surrounded by water. Avenida 24 de Julho gets inundated by a fast advancing front as the flow breaks through the maintenance facilities on the east area of the harbour and heads towards the National Museum on Ancient Art. A significant draw-down starts shaping at Cordoria Nacional with flow velocity towards the river reaching 2.5 [m/s] and flow depths of approximately 1.7 [m].

Two minutes later, 16’20” (Figure 6.17 on page 71), the tsunami crest has already left the harbour area and now heads towards Downtown. Nonetheless, the tsunami hazard is still far from stabilized at Alcântara. Draw-downs can be as destructive as run-ups and all the area is now subjected to a generalized draw- down. Velocities higher than 3 [m/s] are observed at the Alcântara Docks and Congress Center. Through all inundated area the flow patterns are clearly affected by the buildings, acting as obstacles, as seen in the velocity map. Inundation extent is at its maximum and nearly coincides with the 5 [m] elevation isoline and Avenida 24 de Julho. Water has now reached Santos, approximately half-way between Alcântara and Downtown, with water depths around 1.7[m] and velocities of 2.4 [m/s]. Unusual flow patterns occur inside the harbour as two noticeably large eddies begin to form (Figure 6.17b).

After another four minutes the draw-down is still intense at the harbour and riverfront areas with outgoing flow reaching 3 [m/s] (Figure 6.17, page 71). Depths are generally below 1 [m]. The flow pattern displayed in the velocity map clearly shows the main drawing-down flow trajectories. The buildings act as bottlenecks inducing high velocities mainly at the harbour area, including the Alcântara Docks.

68 6.4. Street-level results for inland propagation

Tsunami impact at Alcântara waterfront. 12’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 12 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 12 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.15: Detailed results for Alcântara, t = 12 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

69 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Alcântara waterfront. 14’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 14 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 14 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.16: Detailed results for Alcântara, t = 14 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

70 6.4. Street-level results for inland propagation

Tsunami impact at Alcântara waterfront. 16’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 16 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 16 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.17: Detailed results for Alcântara, t = 16 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

71 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Alcântara waterfront. 20’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 20 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 20 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.18: Detailed results for Alcântara, t = 20 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

72 6.4. Street-level results for inland propagation

6.4.2 Results for Downtown

The Downtown area comprises the Pombaline Town, Cais do Sodré and the Alfama district beneath the Lisbon Castle, and extends northwards towards Rossio and Figueira squares and Avenida da Liberdade, a well-known boulevard noted for the shops and cafes. The Pombaline Lower Town is a distinct district built after the 1755 Lisbon earthquake and tsunami. The tsunami impact is well documented and is known to have inundated, for instance, Largo de São Paulo, Terreiro do Paço and Cais do Sodré. Further details on the historical details of the 1755 event can be found in Baptista et al. (2006).

This area is one of the first examples of earthquake-resistant construction, chareacterised by buildings featuring a technique known as gaiola pombalina Bento et al. (2004). Its modern layout features a narrow but regular grid where traffic circulation is, presently, arranged in several looping cells that can become worrisome when traffic is forced to head towards the riverfront. Figure 6.19 shows several streets that can pose difficulties in safely reaching higher grounds.

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.19: One-way traﬃc zones (top) and tsunami approach to Downtown (bottom; t=13’20”) Tsunami impact at Bugio Castle used as relative time reference

73 6. A 1755 tsunami in today’s Tagus estuary topography

Figure 6.19 shows the tsunami approaching Cais do Sodré. At 13’20” water level is beginning to rise to the point of overtopping Avenida 24 de Julho. The inundation front rapidly advances and less than two minutes later, 15’ after the tsunami hit Bugio Tower, Cais do Sodré has been almost completely overwhelmed by the tsunami. Rua do Instituto Industrial is already flooded half-way despite being almost 250 [m] away from the riverfront. The vulnerable Avenida Ribeira das Naus is also overtopped, blocking one of the main exit routes from Downtown to Northern riverfront areas. The inundation front is advancing toward Terreiro do Paço and a nearby fluvial interface. Arsenal da Marinha is also severely exposed. The traffic in Zone 1 and Zone 2 is particularly critical since some streets force traffic towards the incoming wave, namely Boqueirão Ferreiros, Boqueirão do Duro, Rua da Moeda and Rua do Arsenal. Zone 3 is a converging point for the traffic from Zones 1, 2, 4 and 5 which can lead to a complete clogging as Zone 3 is one of the first areas to become inundated. Traffic would be completely paralyzed in all five zones.

Fourty seconds later the extent of the inundation area is quite disturbing (Figure 6.20 on page 75). Terreiro do Paço is midway submerged with water depths reaching 1.5 [m] and velocities as great as 3.0 [m/s]. Arsenal da Marinha is also severely hit. Run up has penetrated inland as deep as 380 [m] at Rua do Instituto Industrial and Boqueirão Ferreiros. Cais do Sodré is now completely flooded with water depth reaching almost 2 [m] and velocities can be higher than 3 [m/s]. Zones 1, 2 and 3 are now completely isolated. The main concern is the traffic that is driven towards the riverfront and Cais do Sodré due to a few one-way streets, namely Rua do Arsenal, Rua do Comércio, Rua do Ouro and Rua da Prata. Many administrative facilities are at jeopardy of imminent tsunami impact, including the Ministry of Agriculture, the Ministry of Justice, the Ministry of Finance, Lisbon’s City Hall and several courts. The urban grid next to Terreiro do Paço is also a very important financial area with several major national and international banks holding their headquarters at Rua do Comércio and Rua S. Julião.

By 16’20” Terreiro do Paço has now been completely inundated as the wave front heads north, towards the Pombaline Lower Town grid (Figure 6.21 on page 76). Most administrative facilities are, at this stage, completely overwhelmed by the incoming tsunami. Inundation fronts are now taking over east downtown with valuable churches and museums being consumed by the incoming water. All the streets are two-way and should promote an eﬀective evacuation. The tsunami crest is now ashore of Cais do Sodré, meaning that the inundation is still being fed at this location. The water propagates deeper inside the urban mesh at Cais do Sodré reaching Rua da Boavista. A moderate draw-dawn at Santos is also evident.

The Tsunami crest is shorewards of Alfama by the 17th minute, leaving extensive inundation and strong draw-downs at Cais do Sodré and Terreiro do Paço (Figure 6.22 on page 77). Largo de São Paulo was known to have been flooded by the 1755 tsunami and the obtained results are consistent with this report. Rua do Comércio, Rua São Julião and Rua Augusta are at this stage suffering minor flooding, with water depths of 0.6 [m] and velocities of approximately 0.7 [m/s]. Impact at Santa Apolónia Train Station is considerably strong, with flow depths of nearly 1 [m] and a velocity around 1.9 [m/s]. Inundated extent is, again, nearly coincident with the 5 [m] elevation isoline. At the 20th minute (Figure 6.24 on page 79) the tsunami is heading towards the estuary interior. Generalized draw-down occurs throughout the

74 6.4. Street-level results for inland propagation riverfront, especially in narrow sections between buildings, where velocities can reach almost 4 [m/s].

Tsunami impact at Downtown waterfront. 15’ after hitting Bugio Tower

(a) Water level at Alcântara, t = 15 minutes

(b) Velocity magnitude at Alcântara, t = 15 minutes

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.20: Detailed results for Downtown, t = 15 minutes Tsunami impact at Bugio Castle used as relative time reference

75 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Downtown waterfront. 15’40” after hitting Bugio Tower

(a) Water level at Downtown, t = 15 minutes 40 seconds

(b) Velocity magnitude at Downtown, t = 15 minutes 40 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.21: Detailed results for Downtown, t = 15 minutes 40 seconds Tsunami impact at Bugio Castle used as relative time reference

76 6.4. Street-level results for inland propagation

Tsunami impact at Downtown waterfront. 16’20 after hitting Bugio Tower

(a) Water level at Downtown, t = 16 minutes 20 seconds

(b) Velocity magnitude at Downtown, t = 16 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.22: Detailed results for Downtown, t = 16 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

77 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Downtown waterfront. 17’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 17 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 17 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.23: Detailed results for Downtown, t = 17 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

78 6.4. Street-level results for inland propagation

Tsunami impact at Downtown waterfront. 20’20” after hitting Bugio Tower

(a) Water level at Alcântara, t = 20 minutes 20 seconds

(b) Velocity magnitude at Alcântara, t = 20 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.24: Detailed results for Downtown, t = 20 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

79 6. A 1755 tsunami in today’s Tagus estuary topography

6.4.3 Results for Almada

Almada has a strong tie with the Tagus river and fluvial activities, mainly due to several shipyards and transportation services across the Tagus. The wave starts to approach Cacilhas 14’30” after the impact at Bugio (Figure 6.25 on page 81). A very fast inundation front will hit Rua do Ginjal, where velocities will reach 2.85 [m/s] and flow depths of about 1.6 [m]. The inland propagation towards Largo Alfredo Dinis is very fast as flow velocity can reach 3.68 [m/s] and flow depths will range from 0.7[m] to 1.12 [m]. At this stage the inundation reaches Avenida Aliança Povo MFA with a water column 0.81 [m] high and a velocity of 2.14 [m/s]. This advancing front is rather concerning as it will prevent access to northern higher grounds that would otherwise be easily accessible by taking the Avenida Aliança Povo MFA towards Rua Cândido dos Reis. The only possible route to escape the inundation is Avenida Alinaça Povo MFA towards Avenida António José Gomes.

Ninety seconds later, 16’10” (Figure 6.26 on page 82) after Bugio is hit, the inundation front has already clashed with the Fireman Headquarters at Cacilhas, where speeds can be as high as 2.61 [m/s] and flow depths will be around 1.54 [m], the highest depths so far observed at Cacilhas. This is a critical loss for the response mechanisms since Fireman will not be able to help this population even after a complete draw-down. The advancing front rapidly propagates through Avenida Aliança Povo MFA, with a maximum velocity of 3.61 [m/s] and flow depths as high as 1.14 [m]. At LISNAVE (naval warehouses) the inundation front has a flow depth of 0.73 [m] and a velocity of 3.33 [m/s]

Figure 6.27 (page 83) shows the inundation at 17’40”. The advancing front is already claiming the Professional School of Almada with ﬂow depths of 1.08 [m] and velocities reaching 2.81 [m/s] at Avenida Aliança Povo MFA. The maximum ﬂow depth and velocity at this stage occur at the LISNAVE shipyards, reaching 1.73 [m] and 4.21 [m/s], respectively. Lisbon Naval is struck by a water column with 0.83 [m] propagating at 2.71 [m/s].

Avenida Aliança Povo MFA should be completely ﬂooded by 22’50” (Figure 6.28 on page 84). The available alternatives are restricted to Rua Manuel Febrero e Avenida António José Gomes. where the 0.99 [m] ﬂood is still advancing at a speed of 2.64 [m/s]. Lisbon Naval Base is now completely overwhelmed the incoming water that are now heading towards Rua Manuel José Gomes.

The wave-front heads towards Largo 5 de Outubro, leaving Rua Tenente Valadim and Rua Francisco Ferrer completely ﬂooded in a matter of just two minutes (Figure 6.28 on page 84). Flow depths of 0.69, 0.79 and 1.12 [m] are registered at Rua José Alves de Almeida, Rua António José Gomes and Lisbon Naval Base, respectively. Also, velocities can range from 0.45 to 2.16 [m/s] at these locations. Signiﬁcant draw-downs occur at the northern areas, namely the LISNAVE shipyards.

The tsunami crest leaves Cacilhas shoreline when 27’40” have passed since the impact on Bugio Tower. The progress of the inundation is almost unnoticeable as there is only one new location to add to the ﬂood extent, Largo 5 de Outubro. Flow depths read 0.51 [m] and velocities range from 0.4 to 1.45 [m/s]. Figure 6.28 (on page 84) shows the maximum run-up extent.

80 6.4. Street-level results for inland propagation

Tsunami impact at Cacilhas waterfront. 14’30” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 14 minutes 30 seconds

(b) Velocity magnitude at Cacilhas, t = 14 minutes 60 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.25: Detailed results for Cacilhas, t = 14 minutes 30 seconds Tsunami impact at Bugio Castle used as relative time reference

81 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Cacilhas waterfront. 16’10” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 16 minutes 10 seconds

(b) Velocity magnitude at Cacilhas, t = 16 minutes 10 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.26: Detailed results for Cacilhas, t = 16 minutes 10 seconds Tsunami impact at Bugio Castle used as relative time reference

82 6.4. Street-level results for inland propagation

Tsunami impact at Cacilhas waterfront. 17’40” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 17 minutes 20 seconds

(b) Velocity magnitude at Cacilhas, t = 17 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.27: Detailed results for Cacilhas, t = 17 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

83 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Cacilhas waterfront. 22’50” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 22 minutes 50 seconds

(b) Velocity magnitude at Cacilhas, t = 22 minutes 50 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.28: Detailed results for Cacilhas, t = 22 minutes 50 seconds Tsunami impact at Bugio Castle used as relative time reference

84 6.4. Street-level results for inland propagation

Tsunami impact at Cacilhas waterfront. 24’20” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 24 minutes 20 seconds

(b) Velocity magnitude at Cacilhas, t = 24 minutes 20 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.29: Detailed results for Cacilhas, t = 24 minutes 20 seconds Tsunami impact at Bugio Castle used as relative time reference

85 6. A 1755 tsunami in today’s Tagus estuary topography

Tsunami impact at Cacilhas waterfront. 27’40” after hitting Bugio Tower

(a) Water level at Cacilhas, t = 27 minutes 24 seconds

(b) Velocity magnitude at Cacilhas, t = 27 minutes 40 seconds

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.30: Detailed results for Cacilhas, t = 27 minutes 40 seconds Tsunami impact at Bugio Castle used as relative time reference

86 6.5. Morphological impacts

6.5 Morphological impacts

6.5.1 The relevance of ﬂow-morphology interaction

Tsunami overland propagation is most of the times modelled with clear water equations where bed interaction is restricted to flow resistance. Solid transport features implemented in STAV-2D attempt to simulate a noticeable feature of this stage of a tsunami, as seen in most footage of recent events, which is the inclusion of debris in the flow. Erodible beds are expected to suffer major morphological changes if inland propagation attains significant speeds.

This section provides insights on the aftermath of a 1755-like tsunami, focusing on the morphological changes that were left behind. One should note that these morphological changes might not refer only to bed and riverfront conﬁgurations but also to built environment appliances and vehicles, which are often carried further inland by the incoming ﬂow.

6.5.2 Vale do Zebro Royal complex

This wheat processing facility borders the Coina river, near Barreiro, on the south of the Tagus estuary. It was an important asset during the Portugese Discovoeris era, being owned and directly managed by the Portuguese Crown, comprising over 27 ovens for bread and biscuit production. Its location was strategically chosen: nearby woods and cereal ﬁelds would provide fuel and wheat for the production process and the Coina river would serve as a convenient ﬂuvial runway. This complex is known to have been destroyed by the combination of earthquake and tsunami that occurred on the November 1st of 1755. In 1961 it was converted into a military academy and museum for the Portuguese Marines.

Figure 6.31: Tsunami approaching the Royal Complex at Vale do Zebro, t=25’

Important morphological changes that occurred due to the tsunami are of particular interest to calibrate the solid transport parameters required for STAV-2D. The aim of this procedure was to obtain a run-up consistent with relevant morphological changes in the river bed along with signiﬁcant momentum in the ﬂooding of the complex.

87 6. A 1755 tsunami in today’s Tagus estuary topography

Prior to reaching Barreiro, the tsunami had already suffered major diffraction on the Tagus mouth and channel. Furthermore, the wider cross section of the estuary also contributed to lower energy fluxes, decreasing the wave amplitude and speed. The propagation of the tsunami along the Coina river is depicted in Figure 6.32 (image sequence is subsequent to Figure 6.9).

(a) Water level, t = 29’ (b) Velocity magnitude, t = 29’

(e) Water level, t = 49’ (f) Velocity magnitude, t = 49’

Water level (m) Velocity (ms-1) 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0

Figure 6.32: Tsunami run-up at Coina river (Vale do Zebro) Tsunami impact at Bugio Castle used as relative time reference

Figure 6.33 (page 89) displays the time-evolving bed elevation and water level curves at a cross section that intersects the Royal complex. The building itself is represented by a noticeable prominence on the left bank (closeup figures on the left column). The advancing front has a severe impact on the ground due to aggressive scouring (over 0.5 [m]) around the factory (Figure 6.33, t=37’). Water depths are around 0.5 [m] at this stage and velocities nearly reach 2 [m/s]. Roughly ten minutes after, the factory is almost completely destroyed (Figure 6.33, t=47’). The draw-down is also an important stage of this event as it might induce great velocities. At the complex the draw-down does significant damage to the remnants of the factory while inflicting further erosion on the riverbank (Figure 6.33, t=37’).

88 6.5. Morphological impacts

10 5 8 4 6 4 3 m  m   

2 Z Z 2 0 -2 1 -4 0 -1000 -500 0 500 1000 0 100 200 300 400 500 X m X m (a) Bed (−) and water (−−) levels, t = 37’ (b) Bed (−) and water (−−) levels (closeup)

    10 5 8 4 6 4 3 m  m   

2 Z Z 2 0 -2 1 -4 0 -1000 -500 0 500 1000 0 100 200 300 400 500 X m X m (c) Bed (−) and water (−−) levels, t = 47’ (d) Bed (−) and water (−−) levels (closeup)

    10 5 8 4 6 4 3 m  m   

2 Z Z 2 0 -2 1 -4 0 -1000 -500 0 500 1000 0 100 200 300 400 500 X m X m (e) Bed (−) and water (−−) levels, t = 57’ (f) Bed (−) and water (−−) levels (closeup)

Figure  6.33: Tsunami impact at the Royal complex  Tsunami impact at Bugio Castle used as relative time reference

6.5.3 Urban debris washed at Downtown and Alcântara

A simplified setup was used to simulate debris transport at Alcântara and Downtown. A layer of mobile material was added to the base DEM, which was held fixed during this simulation. This layer was delimited by the lanes of the main riverfront avenues, namely Avenida da Índa, Avenida 24 de Julho and Avenida Ribeira das Naus (Figure 6.34). The mobile material was set with the same height and width as those of an average vehicle, roughly 1.5 [m] and 1.75 [m], respectively, and a density of 1.5 [kg/m3], corresponding to the density of an average car filled with water (up to 75 percent of its volume).

89 6. A 1755 tsunami in today’s Tagus estuary topography

Real Real Modeled Built Env. 1.75 1.5 1.5 1.75 H = 1.50

Overview 4.5 (m) W = 1.75 W = 1.75

Modeled Built Env. Cross SectionCross Continuous

1.5 (m) H = 1.50

W = 1.75 W = 1.75

Bedrock layer Mobile material Mobile material

Figure 6.34: Idealized setup for urban debris, mainly composed of vehicle-like solid materials

The container storage area at the harbour was also modeled as mobile material, using the same vehicle- like density. The height of these elements was set to that of two overlying containers, roughly 4.8 [m]. Increasing the height further would be pointless since the maximum depth expected at this location was about 3 to 4 [m]. Figure 6.35 displays the locations where mobile material was overlaid to the base DEM.

A debris speciﬁc formulation (Ferreira et al., 2009) was employed when modeling solid transport. The main purpose of this simulation was to assess deposition patterns, a relevant element to deﬁne preferable rescue routes. The outcome can be seen in Figures 6.36 and 6.37 (pages 91 and 91, respectively). The blue coloring stands for erosion areas, which must be consistent with Figure 6.35, and red indicates debris deposition. Debris scattering patterns are within the expectations for this kind of phenomena: large quantities are carried inland and are deposited in streets and squares while smaller quantities are also deposited close to the riverfront as water returns to the river during the draw-down stage.

Figure 6.35: Mobile material distribution along the riverfront

The aftermath of this scenario showed relevant debris deposition at both designated areas. At Alcântara, signiﬁcant deposits are expected near the Centro de Congressos. Particularly, Travessa Conde Ribeira, Travessa Guarda and Travessa Galé are specially prone to deposition. Avenida 24 de Julho may be blocked

90 6.5. Morphological impacts due to debris scattering near Docas de Alcântara and at Santos area. At the harbour, major scattering occurs and due to the large size of the containers, signiﬁcant damage and accessibility diﬃculties are expected.

Figure 6.36: Debris scattering patterns in Alcântara at the aftermath of the tsunami

Figure 6.37: Debris scattering patterns in Downtoen at the aftermath of the tsunami

The downtown area is plentiful in what concerns squares and open spaces. These locations are specially prone to debris accumulation and may prevent rescue teams from deploying ﬁeld hospitals and other victim support services. Terreiro do Paço, Largo da Casa dos Bicos, Arsenal da Marinha, Praça Dom Luís I and Jardim de Santos are some of the squares where signiﬁcant deposition occurs.

A brief comparison between the clearwater and debris formulations is herein presented to assess if the solid material carried in the water column has any signiﬁcant inﬂuence on the wave front depths and speeds. The results for three monitoring gauges of water levels and velocities are displayed in Figure 6.38 on page 92.

91 6. A 1755 tsunami in today’s Tagus estuary topography

6.5 5 6.0 4 5.5

s  3 5.0 m  m    h

4.5 V 2 4.0 1 3.5 3.0 0 15 20 25 30 15 20 25 30 T min T min (a) Water level at Jardim de Santos: (-) Debris ﬂow (b) Velocity magnitude at Jardim de Santos: (-) (- -) clear water Debris ﬂow (- -) clear water    

6.5 5 6.0 4 5.5

s  3 5.0 m  m    h

4.5 V 2 4.0 1 3.5 3.0 0 15 20 25 30 15 20 25 30 T min T min (c) Water level at Cais do Sodré: (-) Debris ﬂow (- (d) Velocity magnitude at Cais do Sodré: (-) Debris -) clear water ﬂow (- -) clear water    

6.5 5 6.0 4 5.5

s  3 5.0 m  m    h

4.5 V 2 4.0 1 3.5 3.0 0 15 20 25 30 15 20 25 30 T min T min (e) Water level at Terreiro do Paço: (-) Debris ﬂow (f) Velocity magnitude at Terreiro do Paços: (-) (- -) clear water Debris ﬂow (- -) clear water     Figure 6.38: Comparison between debris and clearwater formulations

At Terreiro Paço significant differences are observable in velocity measurements. Such differences are due to a barrier effect on the incoming flow since the gauge was relatively close to Avenida Riberida das Naus, an avenue were mobile material was added. The remaining readings reflect what was expected. In the case of a debris flow, the wave propagating inland is slower and higher, which leads to later arrival times for the debris flow advancing front. Nonetheless, its increased mass and momentum are by far more devastating. The steady water level in the presented charts represents the inundation that remains after the wavefront passage. In the case of a debris flow, this inundation level is higher since the deposited debris have a ground rising effect.

92 6.5. Morphological impacts

6.5.4 Morphological impacts on other locations

Early simulations performed with a mobile bed condition throughout the whole domain showed massive morphological changes on the seafronts of Caparica, Caravelos and Oeiras. However these can be overestimated since the employed mesh was coarser, 25 [m] cells on average, and no built environment features were at the time included in the DEM. Boundary conditions were the ones presented earlier on this chapter. The water level that follows the early overland propagation is higher in the debris ﬂow case, as debris deposition is, virtually, an increasing ground level.

A clear pattern of deposition-erosion-deposition is noticeable on Figures 6.39a and 6.39c. As the run-up progressed to higher grounds its momentum and transport capacity were decreasing and the material, ﬁrstly dragged from the riverfront, settled much further from there. When the water recedes back to the ocean its velocity increases and it incorporates debris again, dragging them towards the waterfront. During the last stages of draw-down, when velocities are lower and transport capacity decreases, new deposits of debris occur by the waterfront.

(a) Morphological changes at Caparica - Trafaria seafront

(b) Morphological changes at Carcavelos (c) Morphological changes at Oeiras

Bed variation (m) -1.0 -0.5 0.0 0.5 1.0

Figure 6.39: Morphological impacts on remaining locations

93 6. A 1755 tsunami in today’s Tagus estuary topography

94 Chapter 7

Conclusions and recommendations

7.1 Main conclusions

The main objective of the dissertation was to generate numerical data on the inundation and run-up of designated areas in the the Tagus estuary.

To achieve this goal three main tasks were undertaken. In general terms:

1. The conceptual model of STAV2D was reviewed and complemented with appropriate boundary conditions. In particular, a new boundary condition scheme had to be implemented. The Rankine- Hugoniot algorithm, which was developed as a secondary objective of the present dissertation, added significant flexibility to the CFD model. The performed functional tests attained very satisfactory results, with particular emphasis to flow reversal handling, as this feature is a significant enhancement in the applicability range of the STAV2D model.

2. The geometrical description of the estuary waterfront was considerably enhanced. A very considerable effort was employed to achieve a reality-like discretization of the city’s buildings. These were carefully modelled to ensure that the streets would behave as small canals for the overland propagating flow. Considerable effort was also placed in articulating STAV2D with geographical information tools. The Geographical Information Systems toolkit provides useful tools for pre and post processing operations. It provides a much required compatibility platform, linking simulation outputs and geographical information with few user steps. The overall effort allowed for numerical simulations with a spatial discretization of 2.5 [m] in an exact digital model of the built environment.

3. Numerical simulations were performed to generate a database of water depth and velocity at several locations in the Tagus estuary. Initial and boundary conditions were built from bibliography (ocean boundary) and hydrologic analysis (Tagus boundary). Special emphasis was placed in the charac- terization of the inundation and run-up on the Alcântara and Downtown waterfronts, in Lisbon, Almada waterfront (Cacilhas) and Vale do Zebro (Barreiro). At these locations, the database allows for detailed estimates of wave arrival, its hydrodynamic properties and debris deposition patterns.

95 7. Conclusions and recommendations

The conceptual model that provides the theoretical basis for this dissertation adds significant value to tsunami modelling efforts, in the sense that it allows for a more quantitative assessment of the impact of debris transport by the wave. In fact, from the simulations, it is clear that the impacts of the tsunami propagation are different depending on whether it is considered a debris flow or a clear-water wave front.

The results obtained and discussed in chapter 6 for maximum run-up levels and extents are consistent with previous works, namely thos published in Baptista et al. (2006), with an average run-up height of around 5 [m] at the waterfronts of Alcântara and Downtown. Nonetheless, the governing equations feature higher detail concerning debris transport, rheological behaviour changes and ﬂow resistance laws. When compared to clear water results the debris ﬂow had higher destructive capacity, which highlights the purpose of the present work.

When using meshes with such detail, numerical stability issues often arise. Even so,the numerical model successfully performed throughout all scenario attempts, with an average performance of 1:30, which is to say, CPU time was 30 times the simulation time.

The outcome of these simulations is believed to be particularly useful for the Civil Protection services. Detailed data is provided in what concerns arrival times, run-up depths and velocities and debis scattering that may lead to unusable areas on the aftermath of the tsunami. This information can be useful in the design of evacuation plans, victim support and rescue operations and even simulacra training.

7.2 Directions for future work

The present work provides the primary basis to successfully converge academic research with public service in what concerns to this kind of natural hazard. In fact, it may even extend to other phenomena within the ﬂuid dynamics ﬁeld. A solid interaction between public safety services and research teams can greatly contribute to better response mechanisms and, ultimately, to human life protection.

Emphasizing on tsunami modelling, lagrangian methods, such as Smoothed Parcticle Hydrodynamics, are especially ﬁtted for describing the overland propagation stage of tsunamis (Dias, 2006). Their ability to discretely distinguish between solid and liquid phases make this technique specially appealing for this kind of numerical study.

Lastly, further improvements can be made on the built environment discretization. Recent methods for ultrasonic scanning of city streets would provide a very accurate topographic basis for this type of numerical work. Fine scale detail that is otherwise lost could improve result reliability in particularly complex locations if such methods were available

This work provided a new contribution for the forecast of the impacts of a large tsunami in the Tagus estuary, by employing ideas and methods not known to have been previously explored. It is believed that this research line can be further explored, in conjunction with Civil Protection competent authorities, in order to promote the safety of Lisbon’s citizens if the worst - a tsunami event - is to happen.

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100 Appendices

Appendix 1: Wave dynamics

A brief description of the classic mathematical description of waves is herein presented. The most basic assumptions are that water is incompressible, and has a constant density value, ρ, and that dissipation due to bottom friction can be neglected for this type of theoretical analysis as it is often included as a source term in most models (Dias, 2006).

The irrotational ﬂow of an incompressible ﬂuid is governed by the conservation of mass,

∂w ∇ ~u + = 0 (1) 2D ∂z and the conservation of momentum,

D~u Dw ∂p ρ = −∇ ~p ; ρ = − − ρ g (2) w Dt 2D w Dt ∂z w where ~u is the horizontal velocity (u, v), w is the vertical velocity, and g is the gravitational acceleration. A more comprehensive version of the momentum equations will be subjected to an in-depth analysis in Chapters 3.1 and 3. From the deﬁnition of From the deﬁnition of irrotacionality,

∇ × (u, v, w) = 0 (3) and thus there exists a potential function φ(x, y, z, t) such that,

∂φ ∇ φ = ~u ; = w (4) 2D ∂z

Use of the potential function is particularly convenient when dealing with a free surface boundary condition (Mellor, 1998). The continuity equation now becomes the Laplace’s equation,

∂2φ ∇2 φ + = 0 (5) 2D ∂z2

101 and the momentum equation can be written in terms of φ if integrated over z, yielding Bernoulli’s equation,

2 ∂φ 1 2 1 ∂φ p + |∇2Dφ| + + gz + = 0 (6) ∂t 2 2 ∂z ρw which are directly used in solving ﬂows bounded with solid surfaces. Since the velocity normal to a surface is null, the value of w = ∂φ/∂z = 0 must be null at the bottom.

The surface boundary condition, which is moving, is far more complicated since two boundary conditions are required. The free surface is represented by F (x, y, z, t) = η(x, y, t) − z = 0 and the bottom is given by z = −h(x, y). The necessary boundary conditions can be derived from either a kinematic or dynamic point of view. The free-surface kinematic boundary condition (FSKBC) is given by,

DF ∂η ∂φ = + ∇ φ · ∇ η − = 0 (7) Dt ∂t 2D 2D ∂z and the dynamic boundary condition at the free-surface (FSDBC) and at the bottom (BDBC) are given by Bernoulli’s equation at z = η(x, y, t) and z = −h(x, y), respectively. The FSDBC is given by,

∂φ 1 1 ∂φ2 + |∇ φ|2 + + gη = 0 (8) ∂t 2 2D 2 ∂z and the BDBC is expressed as,

∂φ ∇ φ · ∇ h + = 0 (9) 2D 2D ∂z

The goal is to solve the Laplace’s equation algonside with the FSKBC, FSDBC and BDBC equations. The conservation of momentum equation is not required in the solution procedure and it is used a posteriori to ﬁnd the pressure p once η and φ have been found (Dias, 2006).

102 Appendix 2: Conservation of energy

The energy conservation equation is herein derived with the single purpose of completing the set of conservation laws in ﬂuid dynamics. This equation is not implemented in the numerical model as it becomes redundant when used alongside with both mass and momentum conservation equations. Furthermore this derivation applies only to the mechanical energy as no thermodynamic features are presently implemented in the numerical model.

The balance between local inertia and applied external forces expressed in the momentum conservation law, equation (3.17), may be converted into an energy conservation law by taking the corresponding work of those forces. One might do so by working with the rate of work, deﬁned as,

d dx W = F i = F u (10) dt k ki dt ki i

where Wk is the work of force F~k along the ~x direction. This suggests that multiplying equation (3.17) by ui, and summing over i, will yield the energy conservation equation (Kundu et al., 1990). Thus, after some algebraic manipulations, one should obtain the following result,

D 1 2 2 Dρ ∂p ∂τij ρ ui + ui = −ui + ui ρgi + ui (11) Dt 2 Dt ∂xj ∂xj which is the energy conservation equation, although only suitable to incompressible ﬂows.

103 104 Appendix 3: Implemented boundary condition algorithm

In the interest of future reference this subsection summarizes the implemented subroutines. The complete Fortran90 code was replaced by an equivalent pseudo-code.

NOTE: During debug tests it was noticed that for very small ∆h, for instance ∆h < 0.1[mm], the characteristics λR and λL had virtually the same values and therefore a consistent entropy condition could not be veriﬁed for either shocks or expansion waves. In order to overcome this drawback the following threshold was implemented: if ∆h < 0.1[mm] then uL = uR and hL = hR, that is to say, the boundary replicates the values of the inner cells from time ti to ti+1.

Implemented subroutine for INLETS

Acquire ql from q(t) hydrograph Acquire hr and ur from the inner cells

if ( abs(hr - hl) < 1E-4) then !Avoid numerical residue hr = hl

else if (hl > hr) then !Shock Condition Solve equation (3.14), 4th order (Ferrari-Cardano’s Method) Obtain 4 possible solutions for hl Reject complex solutions (Im > 1E6) Reject negative real solutions (hl <= 0) Test the entropy condition for the remaining solution/s Use the only validated solution

else if (hl < hr) then !Expansion Wave Condition Solve equation (3.15), 3rd order (Cardano’s Formula) Obtain 3 possible solutions for hl Reject complex solutions (Im > 1E6) Reject negative real solutions (hl <= 0) Test the entropy condition for the remaining solution/s Use the only validated solution

end if

Return hl Compute ul = ql/hl Assign the (u,v) velocity vector accordingly with the edge normal

105 Implemented subroutine for OUTLETS

Acquire hl from h(t) hydrograph Acquire hr and ur from the inner cells

if ( abs(hr - hl) < 1E-4) then !Avoid numerical residue hr = hl

else if (hl > hr) then !Shock Condition Solve equation (3.16), 2nd order (Quadratic Formula) Obtain 2 possible solutions for ul Reject complex solutions (Im > 1E6) Test the entropy condition for the remaining solution/s Use the only validated solution

else if (hl < hr) then !Expansion Wave Condition Solve equation (3.17), 3rd order (Cardano’s Formula) Obtain 2 possible solutions for ul Reject complex solutions (Im > 1E6) Test the entropy condition for the remaining solution/s Use the only validated solution

end if

Return ul Assign the (u,v) velocity vector accordingly with the edge normal

106 Appendix 3: Stratiﬁcation Source Terms

The 2nd order perturbations in the shallow-water equations induced by stratiﬁcation are parametrized in a source term represented by the symbol S(U), deﬁned as,

T S(U) = [ 0 Sx Sy 0 ] ; (12) where,

1 2 Sx = − uh∂tρm + u h∂xρm + uvh∂yρm ρm 1 − ∂t (−ρmuh + ρ2u2h2 + ρ1u1h1) ρm 1 2 2 2 1 2 1 2 1 2 − ∂x −ρmu h + ρ2u2h2 + ρ1u1h1 − gρmh + gρch2 + gρ1h1 + gρ1h1h2 (13) ρm 2 2 2 1 − ∂y (−ρmuvh + ρ2u2v2h2 + ρ1u1v1h1) ρm 1 − (−ghρm∂xZb + (ρ2h2 + ρ1h1) g∂x(Zb)) ρm

1 2 Sy = − vh∂tρm + uvh∂xρm + v h∂yρm ρm 1 − ∂t (−ρmvh + ρ2v2h2 + ρ1v1h1) ρm 1 − ∂x (−ρmuvh + ρ2u2v2h2 + ρ1u1v1h1) (14) ρm 1 2 2 2 1 2 1 2 1 2 − ∂y −ρmv h + ρ2v2h2 + ρ1v1h1 − gρmh + gρch2 + gρ1h1 + gρ1h1h2 ρm 2 2 2 1 − (−ghρm∂yZb + (ρch2 + ρ1h1) g∂y(Zb)) ρm

where ρm, u and v are the depth-averaged density and velocity components of the mixture, ρ1 and ρ2 are depth averaged densities of layers 1 and 2, respectively, h1 and h2 are the depths of layers 1 and 2, respectively, u1 and u2 are the depth-averaged velocities in the x direction of layers 1 and 2, respectively and v1 and v2 are the depth-averaged velocities in the x direction of layers 1 and 2, respectively

107