Mathematical modelling of tsunami wave generation
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1 Mathematical modelling of tsunami wave generation DENYS DUTYKH 1 Chargé de Recherche CNRS 1 CNRS-LAMA, Université de Savoie Campus Scientifique Le Bourget-du-Lac, France Institut Jean le Rond d Alembert DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 1 / 53
2 Acknowledgements Collaborators: Frédéric Dias: Professor, University College of Dublin Youen Kervella: PhD student at IFREMER, Brest Special thanks to: Costas Synolakis: Professor, Univeristy of Southern California Emile Okal: Professor, Northwestern University DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 2 / 53
3 Outline 1 Introduction 2 Tsunami generation 3 Dislocation dynamics 4 Sediments effect DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 3 / 53
4 Outline 1 Introduction 2 Tsunami generation 3 Dislocation dynamics 4 Sediments effect DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 4 / 53
5 Why study tsunami generation? Main motivation for this research Objectives: 1 Understand seismology/ hydrodynamics coupling 2 What are main physical effects to take into account? 3 Provide an initial condition for various tsunami propagation codes Figure: Global seismic wave propagation model Difficulties: Almost no information about sea-bed deformations DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 5 / 53
6 Okada s solution Seismic deformation computation Physical assumptions: Isotropic homogeneous half-space Linear elasticity Static solution Fault is a rectangular dislocation Figure: U 1 strike-slip, U 2 dip-slip, U 3 tensile fault Reference Y. Okada. Surface deformation due to shear and tensile faults in a half-space. Bull. Seism. Soc. Am., 1985, 75, DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 6 / 53
7 Okada s solution Three elementary solutions parameter value δ 13 d 40 km L 150 km W 80 km ν 0.27 E Pa U 15 m Figure: Dip-slip deformation Reference: F. Dias, D. Dutykh. Dynamics of tsunami waves. In book: Extreme Man-Made and Natural Hazards in Dynamics of Structures, 2006 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 7 / 53
8 Okada s solution Three elementary solutions parameter value δ 13 d 40 km L 150 km W 80 km ν 0.27 E Pa U 15 m Figure: Strike-slip deformation Reference: F. Dias, D. Dutykh. Dynamics of tsunami waves. In book: Extreme Man-Made and Natural Hazards in Dynamics of Structures, 2006 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 7 / 53
9 Okada s solution Three elementary solutions parameter value δ 13 d 40 km L 150 km W 80 km ν 0.27 E Pa U 15 m Figure: Tensile deformation Reference: F. Dias, D. Dutykh. Dynamics of tsunami waves. In book: Extreme Man-Made and Natural Hazards in Dynamics of Structures, 2006 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 7 / 53
10 Main ingredients of Okada s solution - I Volterra s theory of dislocations Annales scientifiques de l E.N.S., 24, 1907 Displacement field in an elastic medium: u k = 1 b i τ k 8πµ ijν j dσ Σ DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 8 / 53
11 Main ingredients of Okada s solution - II Fundamental solution for an isotropic homogeneous half-space Entire space: Green-Somigliana s tensor Half-space: Mindlin s solution Combination of nuclei of strain in entire space Physics, 7, , 1936 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 9 / 53
12 Finite fault inversion algorithm Data for December 2004 event: courtesy of Chen Ji, Caltech Figure: Surface projection of the slip distribution Figure: Comparison of data (black) and synthetic seismograms (red) DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 10 / 53
13 Application to the Indian Ocean Tsunami 2004 Initial condition construction (Ref: Grilli et al. (2007)) Figure: Dislocation fault segments Figure: Initial surface elevation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 11 / 53
14 Indian Ocean Tsunami 2004 Global simulation by V. Titov (NOAA) DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 12 / 53
15 Outline 1 Introduction 2 Tsunami generation 3 Dislocation dynamics 4 Sediments effect DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 13 / 53
16 Traditional approach to tsunami generation - I Passive generation Put coseismic displacements directly on the free surface and let it propagate: DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 14 / 53
17 Traditional approach to tsunami generation - I Passive generation Put coseismic displacements directly on the free surface and let it propagate: DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 14 / 53
18 Tsunami generation process Video illustration by courtesy of J. Sharpe DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 15 / 53
19 Traditional approach to tsunami generation - II A few drawbacks Some obvious shortcomings: Interaction between moving bottom and ocean is neglected Time characteristics of source are not taken into account Initial velocity field is neglected u t=0 = 0 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 16 / 53
20 Comparison between linear and nonlinear models First minutes of tsunami propagation Main objective: To check the importance of nonlinear effects Frequency dispersion 1 Complete water wave problem 2 Cauchy-Poisson problem 3 Nonlinear Shallow-Water Equations (NSWE) Y. Kervella, D. Dutykh, F. Dias. Comparison between three-dimensional linear and nonlinear tsunami generation models, Theoretical and Computational Fluid Dynamics, 2007 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 17 / 53
21 Complete water wave problem Free-surface potential flow Governing equations: φ = 0, η t + η φ = φ z, z = η, φ t φ 2 + gη = 0, z = η, φ z + h t + φ h = 0, z = h. Solved by BIEM (PhD thesis of P. Guyenne) Accelerated by Fast Multipole Algorithm Reference: C. Fochesato, F. Dias. (2006), A fast method for nonlinear three-dimensional free-surface waves, Proceedings of the Royal Society of London A 462, DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 18 / 53
22 Cauchy-Poisson problem Linearized water-wave problem Governing equations: φ = 0, η t = φ z, z = η, φ t + gη = 0, z = η, φ z + h t = 0, z = h. Fourier-Laplace transform An analytical solution for separable geometries Reference: D. Dutykh, F. Dias. Water waves generated by a moving bottom. In book Tsunami and nonlinear waves, Kundu Anjan (Editor), 2007 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 19 / 53
23 Nonlinear Shallow Water Equations (NSWE) Saint-Venant equations Govering equations: η t + ((h+η) u ) = h t u t u 2 + gη = 0 Finite volume scheme FVCF Reference: J.-M. Ghidaglia, A. Kumbaro, G. Le Coq. On the numerical solution to two fluid models via a cell centered finite volume method. Eur. J. Mech. B-Fluids, 20, 2001 D. Dutykh, F. Dias. Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting. Accepted to Mathematics and computers in simulation, 2008 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 20 / 53
24 Comparison results - I Weakly Dispersive and weakly nonlinear waves ε := a 0 h 0 = , µ 2 := ( h 0 l ) 2 = 10 4, S := ε µ 2 = 5. Figure: Linearized solution, Fully nonlinear, NSWE DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 21 / 53
25 Comparison results - II Dispersive and weakly nonlinear waves ε := a 0 h 0 = , µ 2 := ( h 0 l ) 2 = 0.02, S := ε µ 2 = Tide gauge 1 Tide gauge z, m 0 z, m t, s t, s 0.04 Tide gauge 3 Tide gauge z, m 0 z, m t, s t, s Tide gauge 6 Tide gauge z, m 0 z, m t, s t, s Figure: Linearized solution, Fully nonlinear, NSWE DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 22 / 53
26 Cauchy-Poisson problem Analytical solution for flat bottom General solution by Fourier-Laplace transform: η( x, t) = 1 (2π) 2 R 2 1 cosh( k h) 2πi e i k x µ+i µ i s 2 ζ( k, s) s 2 +ω 2 est ds d k ω 2 = g k tanh( k h) ζ( x, t): is the unknown dynamic sea-bed displacement There is no available analytical solution for fault dynamics Issue: Use static solution and make assumptions about the dynamics Reference: D. Dutykh, F. Dias. Water waves generated by a moving bottom. In book Tsunami and nonlinear waves, Kundu Anjan (Editor), 2007 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 23 / 53
27 Dynamic sea-bed deformation Main ingredients Variables separation: ζ( x, t) = D( x)t(t) Celebrated Okada s solution provides D( x) Assumptions about time evolution T(t) (dynamic scenarios): Instantaneous Exponential Trigonometric Linear T i (t) = H(t) T e (t) = (1 e αt )H(t) T c (t) = H(t t 0 )+ 1 2 [1 cos(πt t 0 )]H(t 0 t) T l (t) = ( H(t t 0 )+ t t 0 H(t 0 t) ) H(t) Application to linear waves: η( x, t) = 1 (2π) 2 R 2 D( k)e i k x 1 cosh( k h) 2πi µ+i µ i s 2ˆT(s) s 2 +ω 2 est ds d k DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 24 / 53
28 Application to tsunami generation problems - I How large is error in translating sea-bed deformation onto free surface? Passive: deformation translated on free surface η( x, t) = 1 D( (2π) i 2 k)e k x cosωt d k Active: instantaneous scenario η i ( x, t) = 1 D( k)e i k x (2π) 2 cosh( k h) cosωt d k Reference: D. Dutykh, F. Dias, Y. Kervella. Linear theory of wave generation by a moving bottom, CRAS, 343, 2006 R 2 R 2 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 25 / 53
29 Application to tsunami generation problems - II Numerical results and conclusions z, m Active generation Passive generation time (s) z, m time (s) η active η passive / η active time (s) η active η passive / η active time (s) z, m time (s) z, m time (s) η active η passive / η active time (s) η active η passive / η active time (s) Figure: Tide gauge records Figure: Relative difference between two solutions New drawbacks: wave amplitude is always slightly exceeded water has effect of low-pass filter DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 26 / 53
30 Outline 1 Introduction 2 Tsunami generation 3 Dislocation dynamics 4 Sediments effect DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 27 / 53
31 Towards more realistic dynamic source model Fault dynamics modelling Earth crust is a linear viscoelastic material (Kelvin-Voigt model) Isotropic homogeneous or heterogeneous medium Fault modeled as a Volterra dislocation Displacement field is increased by the amount of the Burgers vector along any loop enclosing the dislocation d u = b C Simplified situation with respect to fracture mechanics: location and displacement jump are known Reference: D. Dutykh, F. Dias. Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting. Accepted to Mathematics and computers in simulation, 2008 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 28 / 53
32 Haskell s model (1969) Rupture propagation and rise time b( x, t) = 0 t ζ/v < 0 ( b 0 /T)(t ζ/v) 0 t ζ/v T b0 t ζ/v > T z x T is the rise time, V the rupture velocity O y ζ is a coordinate along the fault Front propagates unilaterally along the y axis W L 2 d L 2 δ DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 29 / 53
33 Coupled computation Seismology/hydrodynamics coupling Viscoelastodynamics Space derivatives are discretized by P2 FEM Second-order implicit time stepping Hydrodynamics Governing equations are NSWE: Solved by FVCF scheme η t + ((h+η) v ) = t h, v t v 2 + g η = 0. Coupling with FEM computation is done through the bathymetry h = h( x, t) DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 30 / 53
34 Comparison of two approaches Remarks about simulation Active generation: We simulate only first 10 s of the earthquake and couple it with hydrodynamic solver For t > 10 s we assume that bottom remains in its latest configuration Passive generation: Translate static dislocation solution onto free surface as initial condition Multiscale nature: Two different scales: elastic waves and water gravity waves DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 31 / 53
35 Results of numerical computation Comparison between passive and active generation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 32 / 53
36 Discrepancy with tide gauges records Chilean 1960 event Reference: J.C. Borrero, B. Uslu, V. Titov, C.E. Synolakis (2006). Modeling tsunamis for California ports and harbors. Proceedings of the thirtieth International Conference on Coastal Engineering, ASCE DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 33 / 53
37 Outline 1 Introduction 2 Tsunami generation 3 Dislocation dynamics 4 Sediments effect DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 34 / 53
38 Sediments Definition and formation Definition: Sediment is any particulate matter that can be transported by fluid flow and which eventually is deposited as a layer of solid particles on the bottom of water. Formation mechanisms: Material washed into the sea from the land Remains of tiny forms of sea life Layers of sand settle on the ocean floor DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 35 / 53
39 DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 36 / 53
40 Global Seismic Hazard Map Figure: The map shows the peak ground acceleration (PGA) that a site can expect during the next 50 years with 10 percent probability DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 37 / 53
41 Previous studies: plate movement modelling Ref: C.W. Fuller, S.D. Willett and M.T. Brandon. Geology (2006) It appears that the most severe subduction zone earthquakes occur in areas where sediment-filled basins are found Method: Thermo-mechanical modelling of plate movement Frictional plastic rheology and thermally-activated viscous rheology Numerics: 2D FEM DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 38 / 53
42 Previous studies: simulation results Ref: Fuller et al. Geology (2006) Figure: No sedimentation Figure: Sedimentation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 39 / 53
43 Previous studies: simulation results Ref: Fuller et al. Geology (2006) Figure: No sedimentation Figure: Sedimentation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 39 / 53
44 Previous studies: simulation results Ref: Fuller et al. Geology (2006) Figure: No sedimentation Figure: Sedimentation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 39 / 53
45 Previous studies: conclusions Reference: C.W. Fuller, S.D. Willett and M.T. Brandon. Formation of forearc basins and their influence on subduction zone earthquakes. Geology, 34, no. 2, pp (2006) Conclusion: Our numerical simulations demonstrate that sedimentation stabilizes the underlying wedge, preventing internal deformation beneath the basin [... ] The lack of deformation in these stable regions increases the likelihood of thermal pressurization of the subduction thrust, allows the fault to load faster, and allows greater healing of the fault between rupture events. DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 40 / 53
46 Numerical scheme Solving elastodynamic governing equations Variational form of governing equations: Ω ρ 2 u 2 t v dω+ Ω σ( u) : v dω = Ω ρ g v dω+ f v ds Γ N Numerics: Unstructured triangular mesh (can be adapted) Discretization in space by a P2 FEM Fully implicit discretization scheme in time Programming: FreeFem++ : program takes 200 lines with comments! DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 41 / 53
47 Two test-cases description Parameters used in computations and geometry of the domain. parameter d δ L b G g G s value 4000 m m 10 m Pa Pa ν g 0.27 ν s 0.3 (c s ) g (c s ) s 3600 m s 330 m s Table: From Mei and Foda (1981) Figure: Test-case with homogeneous medium y O d L x DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 42 / 53
48 Two test-cases description Parameters used in computations and geometry of the domain. parameter d δ L b G g G s value 4000 m m 10 m Pa Pa ν g 0.27 ν s 0.3 (c s ) g (c s ) s 3600 m s 330 m s Table: From Mei and Foda (1981) h s y O d Figure: Test-case with sediment layer L x DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 42 / 53
49 Static solution Comparison with/without sediment layer computations Static sea bed deformation Homogeneous half space With sand layer y, m x, km Figure: Volterra dislocation source: h s = 600 m DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 43 / 53
50 Dynamic computation - I Sand layer thickness h s = 600 m DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 44 / 53
51 Dynamic computation - II Sand layer thickness h s = 100 m DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 45 / 53
52 Sediment amplification factor - I How to quantify the influence of the sediment layer on maximal sea-bed amplitude? Definition: S a = max v s(x, t) (x,t) max v 0(x, t) 1 (x,t) v 0 (x, t): Vertical displacement at the sea-bed in homogeneous case v s (x, t): The same quantity in presence of the sediment layer Question: How does it depend on sediment layer relative thickness h s /d? DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 46 / 53
53 Sediment amplification factor - II Dependence on sediment layer relative thickness 0.8 Sediment Impact factor Author: Denys Dutykh, CMLA, ENS de Cachan S impact h /d s DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 47 / 53
54 Application to the Tsunami Boxing Day 2004 Seismic parameters and sediments thickness estimation Figure: Sediment thickness map DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 48 / 53
55 Application to the Tsunami Boxing Day 2004 Seismic parameters and sediments thickness estimation Figure: Seismic parameters DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 48 / 53
56 Application to the Tsunami Boxing Day 2004 Seismic parameters and sediments thickness estimation 0.8 Sediment Impact factor Author: Denys Dutykh, CMLA, ENS de Cachan S impact h s /d Figure: Sediment impact factor DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 48 / 53
57 New Scientist Phone conversation with Stephen Battersby... D. Dutykh, F. Dias. Influence of sedimentary layering on tsunami generation. Computer Methods in Applied Mechanics and Engineering. In Press DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 49 / 53
58 Further study of sediments effects In collaboration with E. Okal (Northwestern University) Normal modes framework : Numerous cases were considered with real-world parameters Sediments effect is computed in the near field and in the far field Some examples are shown below Conclusion: Results sometimes show amplification, but not in the most conditions! DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 50 / 53
59 Conclusions Conclusions: Sediment layer has no effect on static deformation, but... It makes big difference on dynamics: Slows down elastic and Rayleigh wave propagation Has strong effect on the amplitude: amplification factor up to 70% We should (maybe) revise initial condition for tsunami wave modelling in some subduction zones according to Sediment Thickness Map No information about sediments in Mediterranean region Perspectives: Perform these computations in 3D Take into account porosity in the mud layer DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 51 / 53
60 Other research directions Tsunami wave modelling (with F. Dias and R. Poncet) Tsunami energy estimation Visco-potential flows (with F. Dias) Spectral methods for water wave problem (with D. Clamond, LJAD) Two-phase flows (with LAMA, CMLA, UCD) Powder snow avalanches (in collaboration with D. Bresch & C. Acary-Robert) Modelling (Kazhikov-Smagulov, Brenner-NS,... ) Numerical simulations Protecting structures efficiency estimation DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 52 / 53
61 Thank you for your attention! dutykh DENYS DUTYKH (CNRS LAMA) Tsunami wave generation Institut d Alembert 53 / 53
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