Tsunami Wave Modeling on Unstructured Grids: Tides Interaction Study

1. TSUNAMI MODELLING WITH UNSTRUCTURED GRIDS. INTERACTION BETWEEN TIDES AND TSUNAMI WAVES. Androsov A., Behrens J., Harig S., Wekerle C., Schröter J., Danilov S. Alfred Wegener Institute for Polar and Marine Research Bremerhaven, Germany. Alexey.Androsov@awi.de 24th International Tsunami Symposium, July 14-17, 2009, Novosibirsk, Russia

2. Outline • Tsunami model formulation (TsunAWI). • Model tests: • Test advection. • Onshore propagation. Wetting and drying (Benchmark 1). • Okushiri test case (Benchmark 2). • Historical (tsunami 26.12.2004) simulation. • Tide-tsunami interactions.

3. Finite element spatial discretization Linear conforming (left) and non-conforming shape functions • Advantages: • non-conforming shape functions are orthogonal => no lumping is needed • less noisy then P1 – P1 discretization • Most accurate represent wave propagation speed • Disadvantages: • increased space for velocity • very unstable advection of momentum in the original formulation by Hanert et al. 2005

4. Time discretization • Why explicit? • Large number of nodes (~4 0000 000) • „Small“ wave period (~10 min) • Wetting and and drying

5. Test Advection 1) Approach To calculate the advection term in the momentum equation we first project the velocity from the to the space in order to smooth it. Then we use it in the advection term and proceed as usual by multiplying this form with a basis functions and integrating over the domain. In contrast to the advection scheme proposed by Hanert et al (2005), it has the advantage that no boundary integral has to be computed. 2) Approach (corrected with extra boundary terms) 3) Approach In case of the combined discretization of advective transport, only divergent velocity undertakes from space .

6. Test Advection Wave in the narrow chanel Underwater sill 5 m

7. Benchmark Problems 1 TSUNAMI RUNUP ONTO A PLANE DOMAIN General approach For modelling wetting and drying we used moving boundary technique based on linear least square extrapolation through the wet-dry boundary and into the dry region. We used „dry node concept“ developed by Lynett 2002. The idea is to exclude dry nodes from the solution and then extrapolate the elevation to dry nodes from wet neighbours.

8. Benchmark Problems 1 TSUNAMI RUNUP ONTO A PLANE DOMAIN Initial condition T=0 T=160 s T=175 s T=220 s Analitycal solution Modelled SSH

9. Benchmark Problems 2 TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN bathymetry and coastal topography used in the laboratory experiment Okushiri Isl. Input wave Tsuji walley Left boundary : input wave boundary, Shut down at t = 22.5 s Right boundary : solid wall Top boundary : solid wall Bottom boundary: solid wall

10. Benchmark Problems 2 TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN 1. Numerical simulation area 5.448m X 3.402m Min depth: -0.125m (land) Max depth: 0.13535m 2. Mesh: Number of nodes – 24436 Number of elements – 48330 Max delta = 12 cm Min delta = 0.8 cm 3. Open boundary conditions: West - input wave Shut down at : 0-22.5sec 5. Model parameters: 6. Output: Wave gage position

11. Diffusion approximation Smagorinsky diffusion The adjustable coefficient c in the Smagorinsky term was set to 0.04 – 0.4. These are about the smallest values that can be used and still maintain numerical stability. The Smagorinsky diffusion dominates in regions where the flow is changing rapidly.

12. Benchmark Problems 2 TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN Wave gage Simulation with constant coefficient horizontal diffusion!

13. Benchmark Problems 2 TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN Without advection With advection

14. 26.12.2004 tsunami simulation • Numerical simulation area: Indian Ocean • 2. Mesh: • Number of nodes – 2166320 • Number of elements – 4304458 • Min delta = 500 m • 3. Model parameters: Diffusion - Smagorinsky Without inundation

15. 26.12.2004 tsunami simulation Comparison with station of observation

16. 3 5 Measurements (BPPT) 7 6 Model 8 Inundation Banda Aceh, Sumatra Jun 2004 Dec 2004 QuickBird Images

17. Tide-tsunami interactions • Numerical simulation area: South part ofJawa, Bali, Lombok, Sumbawa, Sumba. • 2. Mesh: • Number of nodes – 177132 • Number of elements – 347098 • = 160 m • = 29 km • 3. Model parameters: Diffusion - Smagorinsky With inundation

18. Tide-tsunami interactions Initialization Tsunami-Tide model Tsunami: Mw = 8.5 Tide: M2 wave. Open boundary conditions ETPXO 6.2

19. Tide-tsunami interactions Coastline stations (H=2 m) Tsunami wave propagation with tide and without tide.

20. Tide-tsunami interactions Shelf stations (H=16-101 m) Tsunami wave propagation with tide and without tide.

21. Tide-tsunami interactions Deep stations (H=1678-4113 m) Tsunami wave propagation with tide and without tide.

22. Tide-tsunami interactions Nonlinear interaction Difference in elevation between a full solution (propagation of tsunami wave on the tidal background) and composite solution (arithmetic sum of tsunami and tides computed separately). Red line – high tide; blue line – low tide.

23. Tide-tsunami interactions Nonlinear interaction. Energy. Tsunami Tide Tsunami&Tide

24. Tide-tsunami interactions Nonlinear interaction. Energy. Tide&Tsunami – (Tide + Tsunami) Tide&Tsunami

25. Tide-tsunami interactions Nonlinear-Linear interaction. Energy. Potential Energy Kinetic Energy Red line – with advection; blue line – without advection.

26. Summary • Stable methods of calculating momentum advection for nonconforming velocity elements are offered and tested. • The wetting and drying algorithm implemented in the model compares successfully with the analytical solution by Carrier et al. (2003). • Using the Smagorinsky viscosity is found necessary for providing good agreement of numerical results with observations. • The model agrees fairly well with respect to arrival time of the first crest and inundation with the available observational data.

27. Summary • There is a fairly strong nonlinear interaction between the tsunami waves and tides. • The major difference between tide and tsunami occurs in the run-up region. • The amplification of tsunami amplitude is mainly associated with strong amplification of tsunami currents. • The nonlinear interaction of the tide with tsunami is important, as it generates stronger sea level change and also stronger changes in tsunami currents. • The resulting run-up ought to be calculated for the tsunami and tide propagation together.