Abstract
Refraction of narrow-band surface waves in coastal areas can result in wave-focal zones where due to interference, wave statistics vary rapidly and on similar length scales as those of individual waves. However such interference patterns, or wave coherence, are not accounted for in conventional stochastic wave models that are based on the energy balance equation or radiative transfer equation. In this work we present a quasi-coherent theory, which is an extension of the radiative transfer equation and quasi-homogeneous theory. We show that this new stochastic modelling approach can resolve rapid variations in wave statistics that occur in the vicinity of a wave caustic. The results compare favourably to those obtained from ensemble averages calculated with a deterministic phase resolving model (SWASH) and, in a focal zone, constitute a significant improvement over those obtained with a conventional stochastic wave model based on an energy balance equation (SWAN).
Highlights
The evolution of ocean waves in coastal areas is strongly affected by variations in depth and currents, which has important implications for wave-driven coastal circulation and nearshore transport processes (e.g. Hoefel and Elgar, 2003)
For many coastal applications such wave statistics are obtained from stochastic wave models (e.g. The WAMDI Group, 1988; Tolman 1991; Booij et al 1999), that are based on some form of the wave action balance equation
Wave statistics can exhibit fast-scale variations associated with cross-correlations between non-collinear wave components, induced by refraction of a coherent incident wave field; such cross-correlations are not considered by the radiative transfer equation (e.g. O’Reilly and Guza, 1991)
Summary
The evolution of ocean waves in coastal areas is strongly affected by variations in depth and currents, which has important implications for wave-driven coastal circulation and nearshore transport processes (e.g. Hoefel and Elgar, 2003). In the vicinity of the caustics and in the focal region enclosed by the caustics, the inclusion of cross-variance contributions, as present in the QC model, is essential to accurately resolve the rapid variations in wave statistics This is likely to be important in natural focal zones in coastal areas where waves are refractively focused for instance over tidal currents in inlets, or coastal bathymetry. These cross-variance contributions appear in the CM spectrum as additional peaks in the spectrum, located midway (spatially and spectrally) between the auto-variance contributions of the correlated wave components (Hlawatsch & Flandrin, 1997), and oscillate on the scale of the interference pattern. The cross-variance contributions change from positive when two components interfere constructively, to negative when two components interfere destructively and their values oscillate on the scale of the interference pattern, resulting in the observed rapid changes in the CM spectrum within the region enclosed by the caustics
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