On the runup of long waves on a plane beach

Using the records of free surface fluctuations at several locations during the 2011 Japan Tohoku tsunami, we first show that the leading tsunami waves in both near-field and far-field regions are small amplitude long waves. These leading waves are very different from solitary waves. We then focus on investigating the evolution and runup of non-breaking long waves on a plane beach, which is connected to a constant depth region. For this purpose, we develop a Lagrangian numerical model to solve the nonlinear shallow water equations. The Lagrangian approach tracks the moving shoreline directly without invoking any additional approximation. We also adopt and extend the analytical solutions by Synolakis (1987) and Madsen and Schffer (2010) for runup and rundown of cnoidal waves and a train of multiple solitary waves. The analytical solutions for cnoidal waves compare well with the existing experimental data and the direct numerical results when wave amplitudes are small. However, large discrepancies appear when the incident amplitudes are finite. We also examine the relationship between the maximum runup height and the leading wave form. It is concluded that for a single wave the accelerating phase of the incident wave controls the maximum runup height. Finally, using the analytical solutions for the approximated wave forms of the leading tsunamis recorded at Iwate South station from the 2011 Tohoku Japan tsunami, we estimate the runup height.