ASYMPTOTICALLY EXACT ZAKHAROV EQUATIONS AND THE STABILITY OF WATER WAVES WITH BIMODAL SPECTRA

摘要

The equations describing the weakly nonlinear evolution of a broad band wavenumber spectrum are derived in detail from a general set of governing equations. These equations are similar to the Zakharov equations, but are asymptotically exact. The most general broad band evolution equations require a closure assumption which must be compatible with the physics associated with the governing equations. It is shown that the Zakharov equations implicitly involve a closure assumption which is motivated neither physically nor mathematically. For narrow band spectra and unidirectional wavetrains, this closure assumption is irrelevant. However, for counterpropagating wavetrains it yields local evolution equations which are not asymptotically exact. A better closure is introduced, and it is shown that this closure leads directly to evolution equations for counterpropagating wavetrains that are asymptotic for longer times. These equations are of mean-field type. The specific case of inviscid water waves is analyzed for bimodal spectra, and the differences in the stability predictions of the two formalisms are compared for both long and finite wavelength perturbations. [References: 36]