An asymptotic approximation of the maximum runup produced by a Tsunami wave train entering an inclined bay with parabolic cross section

Investigation of the behavior of various types of Tsunami wave trains entering bays is of practical importance for coastal hazard assessments. The linear shallow water equations admit two types of solutions inside an inclined bay with parabolic cross section: Energy transmitting modes and decaying modes. In low frequency limit there is only one mode susceptible of transmitting energy to the inland tip of the bay. The decay rates of decaying modes are controlled by the boundary conditions at the sides of the bay. Therefore a complicated eigenvalue problem needs to be solved in order to compute these decay rates. To determine the amplitude of the energy transmitting mode one should solve an integral equation, involving not just the energy transmitting mode but also decaying modes, the scattered field into the open sea, the incident wave and the reflected wave in the open sea. However, in the long wave limit, all these complications can be avoided if one applies the Dirichlet boundary conditions at the open boundary. That is to take the displacement of the free surface at the open boundary being equal to the twice of the disturbance associated with the incident wave in the open sea, just like a wall boundary condition. The runup produced by the solution obtained from this Dirichlet boundary condition, can be easily calculated using a series of images. In this model no energy is allowed to escape from the bay therefore the error arising from the simplification of the boundary conditions at the open boundary grows with time. Nevertheless the maximum runup occurs before this error becomes significant. If the characteristic wavelength of the incident wave train is equal to 5 times the width of the bay then this simple solution overestimates the first maximum of the runup only by %15 compared to the “exact” solution derived from the integral equation. This overestimation is partly due to the fact that Dirichlet boundary conditions violates the continuity of depth integrated velocities. The solution associated with Dirichlet boundary condition is perturbed in order to match fluxes inside and outside of the bay. This perturbation does not use the decaying modes inside the bay. The height of the first maximum of the runup coming from the perturbation theory is in excellent agreement with that obtained using the integral equation. This perturbation theory can also be applied to narrow bays with arbitrary cross section as long as their depth does not not change in the longitudinal direction.