Second-order random interfacial wave solutions for two-layer fluid with a free surface

A previous study (Song. 2004. Geophys Res Lett, 31(15):L15302) of the second-order solutions for random interfacial waves is extended in a constant depth, two-layer fluid system with a rigid lid is extended into a more general case of two-layer fluid with a top free surface. The rigid boundary condition on the upper surface is replaced by the kinematical and dynamical boundary conditions of a free surface, and the equations describing the random displacements of free surface, density-interface and the associated velocity potentials in the two-layer fluid are solved to the second order using the same expansion technology as that of Song (2004. Geophys Res Lett, 31 (15):L15302). The results show that the interface and the surface will oscillate synchronously, and the wave fields to the first-order both at the free surface and at the density-interface are made up of a linear superposition of many waves with different amplitudes,wave numbers and frequencies. The second-order solutions describe the second-order wave-wave interactions of the surface wave components, the interface wave components and among the surface and the interface wave components. The extended solutions also include special cases obtained by Thorpe for progressive interfacial waves (Thorpe. 1968a. Trans R Soc London, 263A:563～614) and standing interfacial waves (Thorpe. 1968b. J Fluid Mech, 32:489～528) for the two-layer fluid with a top free surface. Moreover, the solutions reduce to those derived for random surface waves by Sharma and Dean (1979.Ocean Engineering Rep 20) ifthe density of the upper layer is much smaller than that of the lower layer.