Reciprocal elevation and flow Green's functions and the simulation of nearshore tsunami inundation

Abstract

In addition to the elevation Green's function which is known as the numerical Green's function in previous tsunami studies, a new type of Green's function for shallow water equations is proposed. This new Green's function is the flow response due to impulsive initial sea surface increase and is dubbed the flow Green's function. Based on the momentum equation, its reciprocity to sea surface response for impulsive initial flow is suggested and is shown numerically. That is, the x- and y-volume fluxes at the vulnerable city (x, y) caused by an impulsive increase of the unit sea surface η at the source point (xs, ys) are exactly the η response at (xs, ys) caused by an impulsive unit x- and y-volume fluxes at (x, y) multiplied by the negative gravitational acceleration and the water depth at (x, y). The later will be called a reciprocal flow Green's function. As reciprocal flow Green's functions include information on wave direction, by combing reciprocal elevation Green's functions (known as reciprocal Green's functions in previous tsunami studies), they can be used as an accurate boundary condition for the simulation of nearshore inundation. This new approach is proposed and compared with direct tsunami simulation for both normally (Case 1) and obliquely (Case 2) incident tsunamis. The accuracy of the new approach is similar to a direct simulation, and is a significant improvement compared with a simplified bell-shaped wave approach. The computer time is significantly reduced compared with direct simulations.