Sum-frequency triad interactions among surface waves propagating through an ice sheet

We study nonlinear resonant wave-wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence - ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.