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

Zhiling Liao,Shaowu Li,Qingping Zou,Ye Liu

doi:10.1175/jpo-d-20-0164.1

Zhiling Liao, Shaowu Li + Show 2 more

Open Access

https://doi.org/10.1175/jpo-d-20-0164.1

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Abstract

AbstractThe theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for nonbreaking conditions. In the present study, analytical solutions of the group-forced IG waves at O(β1) (β1 = hx/(Δkh), hx = bottom slope, Δk = group wavenumber, h = depth) in intermediate water and at in shallow water are derived separately. In case of off-resonance [β1μ−1 = O(β1), where is the resonant departure parameter, cg = group speed] in intermediate water, additional IG waves in quadrature with the wave group forcing (hereinafter, the nonequilibrium response or component) are induced at O(β1) 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 nonequilibrium response is mainly attributed to the spatial variation of the equilibrium bound IG wave amplitude instead of group-forcing. In case of near-resonance [β1μ−1 = O(1)] in shallow water; however, both the equilibrium and nonequilibrium 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 generated 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.

Highlights

Infragravity (IG) waves are ocean surface waves of low frequencies, typically between 0.004 and 0.04 Hz, that distinct from the wind waves or swells of frequencies between 0.04 and 1 Hz (Bertin et al 2018)
Where FI, FII, and FIII are the additional IG response functions induced by spatial gradients of the group-forcing, bound IG wave amplitude described by LHS62 solution, and amplitude of the IG wave described by FH, i.e., 8 < : FI FII FIII
The spatial evolution of infragravity (IG) waves induced by wave group over topography in a wide range of resonance intensity and water depth under nonbreaking conditions has been investigated with a newly defined complex response function

Summary

Introduction

Infragravity (IG) waves are ocean surface waves of low frequencies, typically between 0.004 and 0.04 Hz, that distinct from the wind waves or swells of frequencies between 0.04 and 1 Hz (Bertin et al 2018). The off-resonant solution provides new insight for the relative importance of different generation mechanisms of the nonequilibrium bound IG waves in intermediate depths, while the nearly-resonant solution predicts the shallow water shoaling rate of bound IG waves on a plane sloping beach to be ;h21 for the first time. It indicates that in shallow water where strong resonance occurs, the group-forced IG wave is not locally determined but relies on its spatial evolution history.

Governing equations

Methodology

Boundary condition

Discussions on group-forced infragravity waves over a symmetric topography

Conclusions