Wind-Wave Nonlinearity Observed at the Sea Floor. Part II: Wavenumbers and Third-Order Statistics

Abstract

This is Part 2 of a study of nonlinear effects on natural wind-generated surface gravity waves in 13-m depth, 30 km offshore of Virginia. At the sea floor in this depth, free surface gravity waves are only weakly attenuated at sea and swell frequencies (0.05?0.30 Hz) but are very strongly attenuated at frequencies higher than about 0.35 Hz. Hence, above 0.35 Hz, relatively long wavelength forced waves, excited by nonlinear interactions between directionally opposing free wind waves, are exposed at the sea floor. An array of pressure transducers at middepth was used to estimate the frequency-directional spectrum of (free) primary sea and swell waves, and the associated (forced) secondary pressure fluctuations were measured with an array on the sea floor. In Part 1, it was shown that forced-wave energy levels at the sea floor increase sharply in response to directionally opposing wind waves, in agreement with weakly nonlinear theory. In Part 2, wavelengths, propagation directions, and non-Gaussian phase coupling between free and forced waves are examined on three occasions with relatively high forced-wave energy levels. A root-mean-square wavenumber magnitude and a vector-averaged mean wave propagation direction (both functions of frequency) can be expressed accurately in terms of the pressure array cross-spectra. The wavenumber estimates at the sea floor show the theoretically expected sharp transition between a 0.05?0.30 HZ frequency range dominated by free sea and swell waves and a 0.35?0.60 Hz range dominated by forced waves with wavelengths that are long relative to free waves of the same frequency. In the ?free-wave frequency range,? wavenumber estimates are usually well within 10% of the linear dispersion relation and wave direction estimates are in excellent agreement with the directional spectra extracted from the middepth array. In the ?forced-wave frequency range,? wavenumber and direction estimates agree with nonlinear theory predictions, confirming that the observed forced waves have the sum vector wavenumber of the interacting directionally opposing wind waves. Phase coupling between free and forced waves is examined through third-order statistics of the sea floor pressure data. Consistent with theory, the normalized bispectrum has small imaginary parts scattered approximately randomly about zero and relatively large negative real parts at frequencies that correspond to directionally opposing seas and swell. Estimates of the bispectrum integrated for constant sum frequency confirm that nearly all the energy at double sea frequencies is nonlinearly coupled to directionally opposing wind waves. In qualitative agreement with nonlinear theory predictions, bispectral levels are sometimes significantly reduced by directional spreading of the interacting free waves.