“Simulating Tsunamis in the Indian Ocean with real bathymetry by using a high-order triangular Discontinuous Galerkin oceanic shallow

water model”

By: D. Alevras, F.X. Giraldo, T. Radko

(Copyright Anders Grawin, 2006)

Purpose

♦ This study, as being part of a research regarding the development of a triangular discontinuous Galerkin oceanic shallow water model, focuses on formatting real bathymetry data of the Indian Ocean in order to simulate the propagation stage of the Indian Ocean Tsunami of 26 December 2004 by using this DG model. ♦ In order to validate this simulation the study uses real measurements. The model results are compared to tide gauge data from several stations around the Indian Ocean, to satellite altimetry and field measurements. Notices: - No inundation algorithms are included in this simulation - Shorelines are being treated as fixed wall boundaries

Professor Francis X. Giraldo Associate Professor and Associate Chair for Research

Department of Applied Mathematics Naval Postgraduate School

Monterey, California

Professor Timour Radko Associate Professor

Department of Oceanography Naval Postgraduate School

Monterey, California

Co - Advisors

♦ I gratefully acknowledge the Alfred-Wegener Institute for Polar and Marine Research (tsunami modeling group) for providing the real bathymetry data of the Indian Ocean region. ♦ I acknowledge the NOAA for providing satellite altimetry data used to compare model results. The University of Hawaii Sea Level Center, Honolulu, Hawaii, the Survey of India, Delhi, and the National Institute of Oceanography, Goa, India for providing the sea-level records of the 2004 Indian Ocean.

Acknowledgments

♦ Background

♦ Simulation of the Indian Ocean Tsunami

- Model description

- Real bathymetry data

- Tsunami source model

- Real measurement data

♦ Simulations results

- Comparison to tide gauge records

- Comparison to satellite altimetry

- Comparison to field measurements

♦ Conclusions

♦ Questions

Outline

Background

♦ Shallow water equations in conservation form: where are the conservation variables, is the flux tensor, and is the source function, where: - the nabla operator is defined as: - is the wind stress, - denotes the tensor product operator, - is the bottom friction - is the geopotential height, - is the identity matrix - is the bathymetry, (rank-2) - is the velocity vector, - is the Coriolis parameter, - is the unit normal vector of the x-y plane, (continued)

( ) ( )St

∂ + ⋅ =∂q F q q∇

( ),TTφ φq = u

( ) ( )22

12

φ

φ φ ν φ

⎛ ⎞⎜ ⎟= ⎜ ⎟⊗ + −⎝ ⎠

uF q

u u I u∇

( ) ( )

0

bSf τφ φ φ γφ

ρ

⎛ ⎞⎜ ⎟= −⎜ ⎟× + ∇ − +⎜ ⎟⎝ ⎠

qk u u

( ),T

x y∂ ∂∇ =⊗ghφ =

( ), T= u vu

bφ

( )0 mf f y yβ= + −( )0,0,1 T=k

τ

2Iγ

♦ Primitive Equations: ♦ Approximate the solution as:

–  Interpolation O(N) ♦ Write Primitive Equations as: ♦ Weak Problem Statement: Find such that Integration O(2N)

Background

(continued)

( )St

∂ +∇⋅ =∂

Fq q

1

NM

N i ii

q qψ=

=∑ ( )N N=F F q ( )N NS S= q

( ) NN N NR S

tε∂≡ +∇⋅ − =

∂qq F

( )Nq ψ∈Σ Ω ∀ ∈Σ1

eN

ee=

Ω = ΩU

{ }2( ) : ( )N e eL Pψ ψΣ = ∈ Ω ∈ Ω ∀Ω

( )/

0e

NR q dψΩ Ω

Ω =∫

♦ Time Integrator. By writing the semi-discrete (in space) equations as the strongly stability preserving (SSP) Runge-Kutta third order RK3 temporal discretization of this vector equation is where and the coefficients and are given from the following table:

Background

( )St

∂ =∂q q

( )1 10 1

k k n k k k ktSα α β− −= + + Δq q q q0 3 1,n n+= =q q q q α β

k=1 k=2 k=3 α0 1 3/4 1/3 α1 0 1/4 2/3

β 1 1/4 2/3

Model description

♦ Boundary conditions: - No inundation algorithms are included - Shorelines are being treated as fixed wall boundaries - Points with depth less than 11 m are treated as 11 m ♦ Time step was Δt=1.5 sec with 10 hrs total simulation. ♦ Total number of used elements was 130445 created by 66715 grid points ♦ Tidal , wind forcing and Coriolis effect were not included.

Real bathymetry data

♦ Real bathymetry data were providing by the Alfred-Wegener Institute. ♦ The unstructured mesh was produced with the mesh generator TRIANGLE by Jonathan Shewchuk [Shewchuk, 1996]. ♦ Resolution of mesh is: 14 km in the deep ocean, 500 m in the coastal areas and less than 100 m in the Northern tip of Sumatra. (continued)

Real bathymetry data

♦ Conversion of the real bathymetry data from latitude / longitude coordinates into cartesian coordinates (x,y) was made by using the MATLAB function m_ll2xy as described in http://www.eos.ubc.ca/rich/map.html ♦ Properties of the used Mercator projection:

Min Longitude Max Longitude Min Latitude Max Latitude 0300 E 1100 E 350 S 350 N

Tsunami source model

♦ Kowalik, Knight, Logan and Whitmore (2005), Numerical Modeling of the Global Tsunami: Indonesian tsunami of 26 December 2004, Science of Tsunami Hazards, 2005; 23: 40-56 pp. ♦ Based on Okada (1985), Surface deformation due to shear and tenside faults in a half space, Bulletin of the Seismological Society of America, 1985; 75: 1135-1154 pp. ♦ Max Uplift: 5.07 m Max Subsidence: -4.75 m

Real measurements data - Tide gauge stations

Black color: National Institute of Oceanography (NIO), Goa, India http://www.nio.org/datainfo/tidegauge.html Red color : University of Hawaii's Sea Level Center database (Honolulu) http://ilikai.soest.hawaii.edu/uhslc/iot1d/index.html (continued)

Real measurements data - Tide gauge stations

♦ R. Pawlowicz, B. Beardsley, and S. Lentz, Classical tidal harmonic analysis including error estimates in MATLAB using t_tide, Computers and Geosciences, 2002; 28: 929-937 pp. ♦ R. Pawlowicz, t_tide Harmonic Analysis Toolbox, (http://www.eos.ubc.ca/rich/t_tide)

(continued)

Real measurements data - Tide gauge stations

Real measurements data - Satellite altimetry

♦ NOAA, Tsunami event - December 26, 2004 Indonesia (Sumatra), http://nctr.pmel.noaa.gov/info1204.html

Real measurements data - Field measurements

♦ During the Indian Ocean tsunami a Belgian yacht, the "Mercator", was an- chored about 1.6 km of the Phuket coast (Thailand). ♦ The yacht's depth gauge was operating and measured changing wave heights during the tsunami [Siffer, 2005].

Simulations results – Tide gauges

Group A: Received tsunami waves directly from the source area Group B: Received tsunami waves emanating from the source and to the influence of tsunami waves reflected from the southwestern coasts of the Indonesian Islands.

Simulations results – Tide gauges Group A

A

B C

D

(continued)

Simulations results – Tide gauges Group A

A

B C

D

(continued)

Simulations results – Tide gauges Group A

A

B

(continued)

Simulations results – Tide gauges Group A

Simulations results – Tide gauges Group B

A

C

B

D

(continued)

Simulations results – Tide gauges Group B

A B

C D

(continued)

Simulations results – Tide gauges Group B

Simulations results – Satellite Altimetry (Jason-1)

♦ The model estimates very accurately the leading wave crest at about 50 S and the double peak structure between this and the equator (underestimates max amplitudes). ♦ Between 50 N to 120 N the model's crest is in contrast to the observation's trough. ♦ Between 120 N to 200 N the model and satellite data are in total agreement and specifically, the model estimates very accurately the leading wave crest at about 200 N.

A B

Simulations results – Field measurements

♦ Based on the observations: - the 3 main tsunami waves had trough-to-crest wave heights of 6.6, 2.2 and 5.5 m. - the first wave (trough) struck the yacht's location at 02:38 UTC (1 hr 39 min).

Simulations results – Maximum Amplitudes

♦ The model results depict the west south-west propagation of most of the energy due to the orientation of the earthquake rupture. (The tsunami wave amplitude, squared, is proportional to the potential energy).

♦ The mid-ocean ridges played a major role as wave guides that transferred the tsunami energy to far-field regions outside the source area in the Indian Ocean.

♦ Areas that the tsunami focused most of its energy.

A

Simulations results: Time-evolution of the free surface

♦ The triangular DG Oceanic Shallow Water Model is run with real bathymetry for the December 26, 2004 Tsunami in the Indian Ocean.

♦ The simulation is run for 8 hours with output every 5 minutes

Notices:

-  no wetting and drying algorithms are included in this simulation.

-  no-flux boundary conditions

Simulations results – Mass Conservation

Conclusions

♦ The model predicts accurately the positive (wave crest) first wave at all tide gauge stations positions due to the uplift of the western side of the initialization area, and the negative (wave trough) at "Mercator" 's location due to the subsidence on the eastern side. ♦ The model simulates very accurately the arrival times and the amplitude's of the first tsunami waves for the Tropical's Indian Ocean, Africa and southwest Indian tide gauge stations (Group A). All these stations received tsunami waves directly from the source area. ♦ The differences in arrival times or in maximum amplitudes between simulation results and records from the Indian tide gauge stations and at Hanimaadhoo (Maldives) are due to the absence of inundation algorithms. ♦ The model underestimates the amplitude of the arrived tsunami waves at "Mercator" location due to misrepresentation of the water depth by the used mesh grid.

(continued)

Conclusions

♦ The differences between simulation results and satellite altimetry are due to the approximation of the tsunami phenomenon by the shallow water equations. ♦ The model conserves mass up to machine precision. ♦ We are encouraged with the results and next plan to add inundation algorithms .

Questions