Asymptotic behavior of frequency and wave number spectra of nearshore shoaling and breaking waves

Abstract

Wave number spectra from shoaling and breaking waves in five laboratory tests are compared to a recently published parameterization describing the evolutionary characteristics of surf zone wave spectra. This parameterization proposes two regions in which different wave number spectral shapes are present; the Zakharov range (2.5 kp ≤ k ≤ 1/h) has a k−4/3 dependence, while the Toba range (k > 1/h) has a k−5/2 shape. Comparison to laboratory data reveals that the spectral slopes from the Zakharov range of the data trend toward increased agreement with the parameterization in the range of incipient breaking. In contrast, the spectral slopes from the Toba range evolve slowly, likely owing to nonlinear interactions with lower, more energetic wave numbers. However, in the inner surf zone the spectral shape for both parameterized ranges tend toward k−2 (or equivalently, f−2 in shallow water). The dissipation coefficient α(f) is extracted from time series of free surface elevations; it is found that α(f) has a reciprocal frequency dependence with that of the frequency spectra S(f) of the data, and that this correspondence increases as the waves enter the surf zone. Additionally α(f) ∼ 1/S(f) ∼ f2 in the inner surf zone. It is concluded that the wave number spectrum parameterization, particularly in the Zakharov range, reasonably describes the spectral shape while the surf zone is still only partially saturated. However, continued breaking moves the spectral shape away from the parameterized slope toward a f−2 (or k−2) spectral shape, representative of the sawtooth‐like shape of surf zone waves. This spectral shape is a clear inner surf zone asymptote.