Non-hydrostatic modelling of large scale tsunamis Msc. Thesis P.B. Smit Caption : The Indian Ocean tsunami as it arrives at Koh Jum, Thailand Cover photograph by : A. Grawin (2004) Title : Msc. Thesis Delft University of technology Non-hydrostatic modelling of large scale tsunamis Author : P.B. Smit Date : Oktober 2008 Professor(s) : Prof. dr. ir. G.S.Stelling Supervisor(s) : ir. R.J. Labeur ir. J. van Thiel de Vries ir. D. Vatvani i ii Abstract The Indian Ocean Tsunami has once again revived the discussion in the tsunami modelling community if the non-linear shallow water equations are a valid model for the propagation of tsunamis. It is suggested that the mechanism of frequency dispersion which is absent in these equations might be important in the correct modelling of large scale tsunamis. In this master Thesis a non-hydrostatic numerical model based upon the scheme proposed by Stelling and Zijlema (2003) is constructed and it is investigated if it can be an effective and efficient way to include the effect of frequency dispersion in the modelling of tsunamis in their propagation and run-up. The non-hydrostatic algorithm is incorporated into the existing explicit shallow water solver of XBeach. In this way the model is extended to allow for shorter wave propagation. The main reason for doing this was to show that the employed non-hydrostatic scheme can be easily incorporated as a simple add-on. The depth averaged formulation of the XBeach model prevented an easy extension towards multiple layers but, for a single layer, the addition of the non-hydrostatic pressures was indeed straightforward. No large modifications to the existing code where required. The numerical model is based on the application of mehrstellen verfahren for the pressure gradients in the vertical. This makes it possible to exactly set the surface pressure to zero which is important for the correct modelling of surface waves. The advective terms have been included in a momentum conservative way based on Stelling and Duinmeijer (2002). This allows for the correct modelling of braking waves. The resulting 2DV model is validated with analytical solutions available for: (i) an oscillating basin (ii) the propagation of a solitary waves (iii) the run-up of long waves on a beach and (iv) the dambreak solution. Furthermore the model is verified using experimental data by Synolakis (1987) on the run-up of solitary waves on a plane beach. In all cases it is concluded that the results are satisfactory. The 2DV model is subsequently expanded into a 3D model which is validated with a 3D version of the oscillating basin and verified with the Berkhoff shoal which includes shoaling, refraction and diffraction of waves. A surprising result is that the model using only a single layer is able to satisfactorily reproduce the measurements. The numerical model is applied to two tsunami benchmark tests conducted by Briggs (1995). The first test consists of the run-up of solitary waves on a vertical wall while the second deals iii with the run-up of solitary waves on a conical island. From the first test it is concluded that the model can correctly model these types of waves using only a single layer. Furthermore, when compared to hydrostatic solutions, the model is a dramatic improvement. The over steepening, typical of the non-linear shallow water equations, does not occur. From the results of the second test it is concluded that the model can accurately predict the inundation heights. However, very fine grids where needed due to the excessive numerical diffusion introduced by the upwind approximations. It can be concluded that the non-hydrostatic model by Stelling and Zijlema can indeed be an attractive way to include frequency dispersion into large scale tsunami propagation models. It is anticipated that the non-hydrostatic terms add about fifty percent to the duration of a simulation. iv Preface and acknowledgements This thesis concludes the Master of Science program at the faculty of Civil engineering and Geosciences at Delft University of Technology, Netherlands. It was carried out over a period of roughly a year at the section for environmental fluid mechanics. Working on this master thesis has been very satisfying and, at times, very frustrating. There have been weeks in which little progress was made and weeks in which everything seemed to fall into place. Here I want to thank those who helped me through the rough times with their advise, be it totally wrong, and their support. Thank you Johan, Dirk, Mirza, Martijn and Carolina. I would also like to thank my family, who have always supported me, Sijbrand for the fruitful discussion we had, and the various student assistants and other graduates whom I had the pleasure working with. Finally I would like to thank the members of my graduating committee and the staff of fluid mechanics for all the support you provided me. I’m afraid you haven’t seen the last of me yet. Delft, November 2008 Pieter Bart Smit v vi Table of contents ABSTRACT ......................................................................................................................................... III PREFACE AND ACKNOWLEDGEMENTS..................................................................................... V TABLE OF CONTENTS ................................................................................................................... VII SECTION I: INTRODUCTION ...........................................................................................................1 1. INTRODUCTION ..............................................................................................................................1 1.1. Objective...............................................................................................................................2 1.2. Readers guide .......................................................................................................................3 2. TSUNAMIS .....................................................................................................................................4 2.1. Introduction ..........................................................................................................................4 2.2. Initially generated waves......................................................................................................7 2.3. Wave propagation............................................................................................................... 10 2.4. Leading wave...................................................................................................................... 16 3. DISPERSION OF LARGE SCALE TSUNAMIS ..................................................................................... 19 3.1. Oceanic propagation .......................................................................................................... 19 3.2. Coastal waters .................................................................................................................... 24 3.3. Discussion........................................................................................................................... 28 SECTION II: 2DV NUMERICAL MODEL...................................................................................... 31 4. MODEL DESCRIPTION ................................................................................................................... 33 4.1. Governing equations........................................................................................................... 33 4.2. Grid schematization............................................................................................................ 37 4.3. Space discretisation............................................................................................................ 41 4.4. Time discretisations............................................................................................................ 51 4.5. Determining the non-hydrostatic pressure ......................................................................... 55 4.6. Higher order spatial approximations ................................................................................. 57 4.7. Flooding and drying ........................................................................................................... 61 4.8. Boundary conditions........................................................................................................... 66 5. VALIDATION AND VERIFICATION .................................................................................................69 5.1. Linear dispersion relation .................................................................................................. 69 5.2. Solitary wave in channel..................................................................................................... 75 5.3. Long waves on a beach....................................................................................................... 80 5.4. Dam break .......................................................................................................................... 82 5.5. Run up of solitary waves..................................................................................................... 87 5.6. Discussion........................................................................................................................... 91 vii SECTION III: 3D NUMERICAL MODEL ......................................................................................