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

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  • 1 State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, P. R. China.
  • | 2 The Lyell Centre, Institute for Infrastructure and Environment, Heriot-Watt University, Edinburg, UK.
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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 O(β1) (β1=hx/Δkh, hx =bottom slope, Δk =group wavenumber, h =depth) in intermediate water and at O(β11) in shallow water are derived separately. In case of off-resonance (β1μ1=O(β1), where μ=1cg2/gh 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 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 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 (β1μ1=O(1)) in shallow water, however, both the equilibrium and non-equilibrium components are O(β11) 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.

Corresponding authors: Ye Liu, liuye2009@tju.edu.cn; Qingping Zou, q.zou@hw.ac.uk.

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 O(β1) (β1=hx/Δkh, hx =bottom slope, Δk =group wavenumber, h =depth) in intermediate water and at O(β11) in shallow water are derived separately. In case of off-resonance (β1μ1=O(β1), where μ=1cg2/gh 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 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 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 (β1μ1=O(1)) in shallow water, however, both the equilibrium and non-equilibrium components are O(β11) 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.

Corresponding authors: Ye Liu, liuye2009@tju.edu.cn; Qingping Zou, q.zou@hw.ac.uk.
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