The Treatment of Discontinuities in Computing the Nonlinear Energy Transfer for Finite-Depth Gravity Wave Spectra

Abstract

The calculation of nonlinear energy transfer between interacting waves is one of the most computationally demanding tasks in understanding the dynamics of the growth and transformation of wind-generated surface waves. For shallow water in particular, existing schemes for computing the full Boltzmann integral representing the nonlinear energy transfer rate converge slowly with increasing spectral resolution. This means that it is difficult to build a spectral wave model with a treatment of nonlinear interactions that is sufficiently accurate without being computationally prohibitive for practical simulations. This paper examines the behavior of terms in the Boltzmann integral with a view to identifying potential improvements in numerical methods used in its solution. Discontinuities are identified in the variation of the nonlinear interaction coefficient where quadruplets of interacting wavenumbers include pairwise matches (k1 = k3, k2 = k4) or (k1 = k4, k2 = k3), a generalization of the specific case k1 = k2 = k3 = k4 noted in earlier works. The discontinuities are not present in the deep-water case and increase in magnitude with decreasing water depth. The behavior of the interaction coefficient in their vicinity is described, and their role in limiting the accuracy of the integration procedure is considered. It is found that the discontinuities are removed when the interacting wavenumbers are constrained to satisfy the resonance conditions. Because of the asymmetrical structure of the action product term, these discontinuities should not directly cause significant errors in existing algorithms for computing nonlinear transfer rates. Indeed, the kernel of the Boltzmann integral vanishes at these points.