Apparent diffusion due to mountain microstructure in shallow waters

Knut Solna | Nachbin, Andre

Keywords: waves in random media | asymptotic theory | scientific computing

Wave propagation in disordered (random) media is the underlying theme. We study the effective behaviour of long coastal waves that travel over rough topographies. The topographies analyzed contain a smooth slowly varying profile together with disordered small-scale features. The mathematical model is a Conservation Law with random coefficients. The main (stochastic theory) asymptotic result is that the medium fluctuations cause the propagating pulse to broaden as it travels. The so called {\em apparent diffusion} (or pulse shaping) depends only on the traveling distance and the {\em statistics} of the random medium fluctuations. Thus, the broadening can be described in a deterministic way independently of the particular medium realization. This is confirmed numerically. Numerical experiments also show that the theory describing pulse shaping is very robust. Nonlinear shallow water simulations show that small amplitude pulse shaping is not affected by higher order terms. The robustness of the theory is observed numerically for a wide parameter regime. We vary both the microscale fluctuation level as well as the horizontal length scales of the topography. The numerical experiments produce very good results regarding the prediction for the wavefront attenuation.

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