Numerical model for multi-layered tsunami waves

In order to develop a generalized numerical model for multi-layered tsunami wave system, a three-layer system was considered. Six governing equations, two for each layer were derived from Euler equations of motion and continuity for three layers, assuming long wave approximation, negligible friction and interfacial mixing. From derived equations, it is found that only top layer equations are independent of number of intermediate layers; equations for all other layers are dependent on number, extent and density of intermediate layer(s). Momentum and continuity equations for the top layer are exactly same as in the case of earlier developed governing equations for two-layered system. Continuity equation for the bottom layer is also exactly same as in the case of two-layered system. Momentum equation for the bottom layer is dependent on extent and density of top layer as well as all intermediate layers. Continuity equation for intermediate layer is affected by levels of immediate bottom layer. Momentum equation for the intermediate layer is affected by extent and density of upper layer(s). Developed governing equations were converted to a numerical model using staggered Leap-Frog scheme for the computations of water level and discharge in each layer in one-dimensional propagation. Developed numerical model results were compared with an earlier developed model for two layers, which was rigorously verified by analytical solution. It was found that this three-layer model produces same results when it is converted to two-layer through mathematical manipulation (i.e. by assuming a negligible/zero depth or similar density of adjacent layer for any layer). The details properties of three-layer model were discussed through numerical simulations for different scenarios. The developed model can be easily converted to a multi-layer (any number) model and can be applied confidently to simulate the basic features of different practical tsunami problems similar to that investigated in this study.