On the evolution and run-up of tsunamis

The first part of this work investigates the validity of the classical solitry wave paradigm for tsunamis. Our study involves the evolution of transient waves, from the initial release of a hump of water, then propagating large distances over flat bottom and finally shoaling over a mildly sloping bottom from the ocean to the beach. We monitor the development of time- and space-scales and compare with solitary wave theory. Next, we simulate the disintegration of long waves into a train of undular bores and discuss the relevance of this phenomenon for tsunami runup. We conclude that solitary wave theory is not applicable for geophysical tsunamis. In the second part of this work, we derive new analytical run-up formulas for incoming single waves and leading depression N-waves, where the time- and space-scales can be chosen freely independent of the wave height. Finally, we derive a convolution formulation for the run-up of transient waves. This is compared with a numerical simulation and the results are in good agreement.