Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations

Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section channel is inclined linearly to the horizon, and that the wave flow is uniform in the cross-section. As a result, 1-D nonlinear shallow-water equations are reduced to a linear, semi-axis variable-coefficient 1-D wave equation by using the generalized Carrier-Greenspan transformation [Carrier and Greenspan (J Fluid Mech 1:97-109, 1958)] recently developed for arbitrary cross-section channels [Rybkin et al. (Ocean Model 43-44:36-51, 2014)], and all characteristics of the wave field can be expressed by implicit formulas. For detailed computations of the long wave runup process, a robust and effective finite difference scheme is applied. The numerical method is verified on a known analytical solution for wave runup on the coasts of an inclined parabolic bay. The predictions of the 1-D model are compared with results of direct numerical simulations of inundations caused by tsunamis in narrow bays with real bathymetries.