An Analytical Spectral Model for Infragravity Waves over Topography in Intermediate and Shallow Water under Nonbreaking Conditions

Abstract

The theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for non-breaking conditions. In the present study, analytical solutions of the group-forced IG waves at (, hx =bottom slope, Δk =group wavenumber, h =depth) in intermediate water and at in shallow water are derived separately. In case of off-resonance (, where is the resonant departure parameter, cg = group speed) in intermediate water, additional IG waves in quadrature with the wave group forcing (hereinafter as the non-equilibrium response or component) are induced at relative to the equilibrium bound IG wave solution of Longuet-Higgins and Stewart (1962) in phase with the wave group. The present theory indicates that the non-equilibrium response is mainly attributed to the spatial variation of the equilibrium bound IG wave amplitude instead of group-forcing. In case of near-resonance () in shallow water, however, both the equilibrium and non-equilibrium components are at the leading order. Based on the nearly-resonant solution, the shallow water limit of the local shoaling rate of bound IG waves over a plane sloping beach is derived to be ~ h−1 for the first time. The theoretical predictions compare favorably with the laboratory experiment by Van Noorloos (2003) and the present numerical model results using SWASH. Based on the proposed solution, the group-forced IG waves over a symmetric shoal are investigated. In case of off-resonance, the solution predicts a roughly symmetric reversible spatial evolution of the IG wave amplitude, while in cases of near- to full- resonance the IG wave is significantly amplified over the shoal with asymmetric irreversible spatial evolution.

A nonlinear, non-dispersive energy balance for surfzone waves: infragravity wave dynamics on a sloping beach

“…List 1992;Janssen et al 2003;Battjes et al 2004), excitation of free IG waves over a sloping bed (e.g. Mei & Benmoussa 1984;Nielsen & Baldock 2010;Contardo et al 2021;Liao et al 2021) and breakpoint generation (Symonds et al 1982). However, within the spectral framework all these mechanisms are represented as either contributions to nonlinear flux gradients or nonlinear interactions, so that identifying the dominant mechanism that drives the interaction is in general not possible.…”

“…2021; Liao et al. 2021, and many others). Such theoretical background combined with numerical modelling (e.g.…”

“…2015; Liao et al. 2021). A subset of nonlinear contributions to the energy flux can be included by using a fully nonlinear low-frequency energy balance (e.g.…”

“…Many studies assumed small nonlinear contributions, and estimated the IG flux from linear theory (e.g. Thomson et al 2006;Van Dongeren et al 2007;Torres-Freyermuth, Lara & Losada 2010;de Bakker et al 2015;Liao et al 2021). A subset of nonlinear contributions to the energy flux can be included by using a fully nonlinear low-frequency energy balance (e.g.…”

Unified analytical solution for group-induced infragravity waves based on Green's function

“…As the group propagating velocity approaches the free long-wave propagating velocity in shallow water, resonance occurs between group forcing and subharmonics propagating in the same direction as wave groups, causing the solutions based on the perturbation method to diverge. In this case, implicit solutions in integral form were derived by Symonds et al (1982), Van Leeuwen (1992) and Schäffer (1993) for a plane beach, and by Liao et al (2021) for arbitrary topography with a mildly sloping bottom. The near-resonant solution of Liao et al (2021) indicates that, with diminishing depth on a plane beach, the group-induced subharmonic asymptotically leads the group forcing by π/2 at leading order, and its amplitude increases as ∝ h −1 (h = depth), a shoaling rate lower than the shallow-water limit of the LHS62 solution (∝ h −2.5 ) but higher than the free infragravity wave growth rate (∝ h −0.25 , known as Green's law; Green 1838).…”