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

OutlineTsunami 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.

Governing EquationswhereBoundary conditions:Initial conditions:

Finite element spatial discretization Linear conforming (left) and non-conforming shape functionsAdvantages:non-conforming shape functions are orthogonal => no lumping is neededless noisy then P1 P1 discretizationMost accurate represent wave propagation speedDisadvantages:increased space for velocityvery unstable advection of momentum in the original formulation by Hanert et al. 2005

Time discretizationWhy explicit?Large number of nodes (~4 0000 000)Small wave period (~10 min)Wetting and and drying

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.1) Approach2) Approach (corrected with extra boundary terms)3) Approach In case of the combined discretization of advective transport, only divergent velocity undertakes from space .Test Advection

Test AdvectionUnderwater sill 5 mWave in the narrow chanel

General approachBenchmark Problems 1TSUNAMI RUNUP ONTO A PLANE DOMAIN 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.

T=0T=160 sT=175 sT=220 sInitial conditionModelled SSHAnalitycal solutionTSUNAMI RUNUP ONTO A PLANE DOMAIN Benchmark Problems 1

Benchmark Problems 2Left boundary : input wave boundary, Shut down at t = 22.5 sRight boundary : solid wallTop boundary : solid wallBottom boundary: solid wallTSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN Input waveOkushiri Isl.Tsuji walleybathymetry and coastal topography used in the laboratory experiment

Benchmark Problems 2TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN 1. Numerical simulation area 5.448m X 3.402m Min depth: -0.125m (land) Max depth: 0.13535m2. Mesh: Number of nodes 24436 Number of elements 48330 Max delta = 12 cm Min delta = 0.8 cm3. Open boundary conditions: West - input wave Shut down at : 0-22.5sec5. Model parameters:

6. Output: Wave gage position

Smagorinsky diffusionThe 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.Diffusion approximation

Benchmark Problems 2Wave gageSimulation with constant coefficient horizontal diffusion!TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN

Benchmark Problems 2TSUNAMI RUNUP ONTO A COMPLEX THREEE-DIMENSIONAL DOMAIN Without advectionWith advection

26.12.2004 tsunami simulation Numerical simulation area: Indian Ocean 2. Mesh: Number of nodes 2166320 Number of elements 4304458 Min delta = 500 m3. Model parameters:

Without inundationDiffusion - Smagorinsky

26.12.2004 tsunami simulation Comparison with station of observation

InundationJun 2004Dec 2004Banda Aceh, SumatraQuickBird Images

Tide-tsunami interactionsNumerical simulation area: South part of Jawa, Bali, Lombok, Sumbawa, Sumba. 2. Mesh: Number of nodes 177132 Number of elements 347098 = 160 m = 29 km

3. Model parameters:

Diffusion - SmagorinskyWith inundation

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

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

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

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

Tide-tsunami interactionsNonlinear interactionDifference 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.

Tide-tsunami interactionsNonlinear interaction. Energy.TsunamiTideTsunami&Tide

Tide-tsunami interactionsNonlinear interaction. Energy.Tide&TsunamiTide&Tsunami (Tide + Tsunami)

Tide-tsunami interactionsRed line with advection; blue line without advection.Nonlinear-Linear interaction. Energy.Potential EnergyKinetic Energy

SummaryStable 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.

SummaryThere 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.