Hamiltonian formalism in wave turbulence problems

University Of Delaware, Newark DE

Investigators

Abstract

This project contributes to a better understanding of nonlinear phenomena related to ocean waves. Of special interest are two problems that involve complex interactions between surface waves and underlying currents or between surface waves and floating structures. These coupled processes are still not well understood and raise challenging questions in oceanography and engineering. Wave-current interactions play a key role in many circumstances like the generation of rogue waves or the transport of contaminants in the open ocean, as well as the mechanism of sediment transport which drives beach erosion in coastal areas. All these have far-reaching implications for a broad range of human activities related to the shipping, tourism, fishing or oil industries, and coastal infrastructure. An important application of wave-structure interactions is to wave power extraction by floating wave energy converters. Ocean waves have great potential as a source of renewable energy but entail many scientific challenges. Wave farms where arrays of wave energy converters are placed in a geometric configuration over extended maritime areas have been considered as a serious option. Determination of the optimal configuration under various wave conditions is crucial for maximizing power absorption in such a system. This research develops new mathematical models for these coupled phenomena that have so far been poorly represented in operational wave forecasting yet are of great relevance in the context of climate change and energy crisis. This project also provides opportunities for the participation and training of graduate students. Under consideration are situations where nonlinear wave interactions occur over a wide range of length and time scales in a complex environment, which poses serious difficulties for their asymptotic analysis and numerical simulation. Examples include ocean waves interacting with a vortical current and ocean waves interacting with an array of floating wave energy converters. In both cases, a Hamiltonian formulation can be established to describe the problem and therefore Hamiltonian techniques are ideal to properly analyze it. Such techniques however are still not sufficiently advanced in the context of nonlinear partial differential equations. The investigator constructs building blocks for this Hamiltonian formalism where the presence of multiple scales can be naturally accommodated in the asymptotic analysis while producing approximations that preserve important structural properties such as energy conservation. This research contributes to the development of the theory of weak wave turbulence in complex media. Both deterministic and statistical viewpoints are adopted to obtain reduced nonlinear models for the long-time evolution of the wave amplitude and wave spectrum. Exact equilibrium solutions of these model equations associated with invariants of motion are derived and numerical simulations for more general nonlinear cases are performed to complement the theoretical predictions. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.