Non-hydrostatic modelling of large scale tsunamis

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 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.Civil Engineering and Geoscience