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Book ChapterDOI

Assessment of Nonlinear Quadruplet Interactions for Measured Spectra in Deep Waters on the East Coast of India Through Gauss–Legendre Quadrature Method

TL;DR: In this paper, the Gauss-Legendre Quadrature Method (GLQM) was used to estimate the nonlinear energy transfer rate between the higher and lower frequencies of a measured spectrum of a moored buoy in deep waters.
Abstract: Existing methods, such as Discrete Integration Algorithm (DIA) or Multiple DIA (MDIA) for evaluating Boltzmann integral to assess the nonlinear energy transfer within a given energy spectrum at a given location, do not account for all the contributing wave resonating quadruplets (QPs) for want of computational ease in the wave models such as WAM and WWIII. By virtue of employing the state-of-the-art Gauss–Legendre Quadrature Method (GLQM), the transfer integral becomes free of singularities, and fast estimation of all the contributing QPs is possible; and hence, this method provides both accuracy and efficiency. It also works for different frequency and angular resolutions of the input spectral grid. In this paper, GLQM is validated with EXACT-NL and WRT methods for a theoretical spectrum. It is then applied to a measured spectrum of a moored buoy of National Institute of Ocean Technology, off Machilipattinam (DS5) in deep waters for evaluating the nonlinear QP interactions based on one-month data during July 2005. A characteristic monthly averaged 1D frequency spectrum has been chosen which represented a double-peaked sea-dominated spectrum. It is then fitted with a theoretical JONSWAP spectrum with 99.5% confidence. The nonlinear energy transfer rate (Snl) between the higher and lower frequencies of this fitted spectrum has been evaluated using GLQM and are quantified. The nonlinear coupling between the sea and swell parts is found to be absent as the ratio of the swell and sea frequencies is less than 0.6 (Masson in J Phys Oceanogr 23:1249–1258, 1993 [1]). Few hypothetical cases have been studied to understand Snl behaviour further.
References
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Journal ArticleDOI
TL;DR: In this article, the energy flux in a finite-depth gravity-wave spectrum resulting from weak non-linear couplings between the spectral components is evaluated by means of a perturbation method.
Abstract: The energy flux in a finite-depth gravity-wave spectrum resulting from weak non-linear couplings between the spectral components is evaluated by means of a perturbation method. The fifth-order analysis yields a fourth-order effect comparable in magnitude to the generating and dissipating processes in wind-generated seas. The energy flux favours equidistribution of energy and vanishes in the limiting case of a white, isotropic spectrum. The influence on the equilibrium structure of fully developed wave spectra and on other phenomena in random seas is discussed briefly.

1,220 citations

Journal ArticleDOI
TL;DR: In this article, a more efficient method of computing the nonlinear transfer in a surface wave spectrum is developed which is symmetrical with respect to all wavenumbers of the resonant interaction quadruplets.
Abstract: A more efficient method of computing the nonlinear transfer in a surface wave spectrum is developed which is symmetrical with respect to all wavenumbers of the resonant interaction quadruplets. This enables a large number of computations to be carried out, as required for investigations of the spectral energy balance or the development of parameterizations. New results are presented for finite-depth surface waves. By filtering out regions in interaction phase space, the assumptions involved in the narrow-peak and local-interaction approx-imations are investigated. Both approximations are found to be useful but are generally not sufficiently accurate to replace exact computations or provide adequate parameterizations for wave models.

364 citations

Journal ArticleDOI
TL;DR: In this article, Hasselmann's equation was studied numerically for a Pierson-Moskowitz spectrum and it was shown that the apparent creation of order, due to the non-linear enhancement of the peak of the spectrum, occurs as the by-product of a large amount of disorder, created at high wavenumbers.

155 citations

Journal ArticleDOI
TL;DR: In this article, the authors present an economical method to evaluate the complete Boltzmann integral, which uses selected scaling properties and symmetries of the nonlinear energy transfer integrals to construct the integration grid.
Abstract: Nonlinear transfer due to wave-wave interactions was first described by the Boltzmann integrals of Hasselmann (1961) and has been the subject of modelling ever since. We present an economical method to evaluate the complete integral, which uses selected scaling properties and symmetries of the nonlinear energy transfer integrals to construct the integration grid. An important aspect of this integration is the inherent smoothness and stability of the computed nonlinear energy transfer. Energy fluxes associated with the nonlinear energy transfers and their behaviour within the equilibrium range are investigated with respect to high-frequency power law, peak frequency, peakedness, spectral sharpness and angular spreading. We also compute the time evolution of the spectral energy and the nonlinear energy transfers in the absence of energy input by wind or dissipated by wave breaking. The response of nonlinear iterations to perturbations is given and a formulation of relaxation time in the equilibrium range is suggested in terms of total equilibrium range energy and the nonlinear energy fluxes within the equilibrium range.

135 citations

Journal ArticleDOI
TL;DR: It is shown that Webb's method produces an attractive set of integrable equations, and the Jacobian term arising from the integration over the frequency delta-function in the Boltzmann integral has a singularity well outside the energy containing part of the wave spectrum.

114 citations