Water wave scattering from a mass loading ice floe of random length using generalised polynomial chaos

We consider the scattering of water waves in a two dimensional domain from a floating sea ice floe of random length. The length is treated as a random variable governed by a prescribed probability distribution. To keep the focus on the random length aspect we choose a simple mass loading model to characterise the ice floe. We compute the expectation and variance of the reflection and transmission coefficients using two different methods derived from the framework of generalised polynomial chaos (gPC), as part of which unknown quantities of the problem are expanded in a basis of orthogonal polynomials of the random variable. The polynomials are chosen optimally for the particular probability distribution of the random variable to minimise an approximation error. We devise a stochastic collocation method, which involves computing the reflection and transmission coefficients deterministically for a number of carefully sampled lengths and fitting polynomial expansions to them. The second approach is based on the stochastic Galerkin method, for which the governing equations are transformed to accommodate the random length parameter. We also use a standard Monte Carlo (MC) approach for comparison. The gPC methods are shown to be numerically efficient and exhibit desirable exponential convergence properties, as opposed to the slow inverse square root convergence of the MC approach. Finally, we use the statistic collocation method to demonstrate that the floe size distribution can have a significant impact on the expected transmission coefficient.