Consistent nonlinear stochastic evolution equations for deep to shallow water wave shoaling

Abstract

Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water.Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water.

Infragravity Wave Generation by Wind Gusts

“…We assume that the wind sea is not affected by IG waves, due to their small amplitudes. Following the previous work of the authors (Vrecica & Toledo, ), the WAE for IG wave components can be defined as where the subscript p indicates frequency for all variables, a p is the wave amplitude, E p = a p a − p is the spectral component at frequency f p = pf min ( f min = f 1 is the lowest frequency under consideration), C g , p is the group velocity, and S p , nl ,3 is the nonlinear triad source term for IG wave components due to gustiness. Originally, the D p term represented a dissipation of wave energy; however, it can also describe the energy input.…”

“…As the wind speed oscillates in time and space, so does the growth rate incorporated inside the D p term. In order to account for this behavior, the model of Vrecica and Toledo () is adapted to evaluate the evolution of the wave field along characteristic lines ( s = t + C g , p x ), which follow the wave propagation in time and space. This results in a formulation of the WAE nonlinear triad source term fitting for the estimation of IG wave generation by resonant gust‐triad interactions …”

“…We assume that the wind sea is not affected by IG waves, due to their small amplitudes. Following the previous work of the authors (Vrecica & Toledo, 2016), the WAE for IG wave components can be defined as…”

“…As the wind speed oscillates in time and space, so does the growth rate incorporated inside the D p term. In order to account for this behavior, the model of Vrecica and Toledo (2016) is adapted to evaluate the evolution of the wave field along characteristic lines (s = t + C g,p x), which follow the wave propagation in time and space. This results…”

Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

“…Here, the W -term is the nonlinear interaction kernel defined in Bredmose et al [14] and Vrecica & Toledo [74] for cases without and with dissipation, respectively. D p,l can describe a linear damping or forcing term, while t = ε 2 T represents a slow time evolution, which is typically on a different scale than the spatial evolution x = εX, for T and X physical time and space variables (cf.…”