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Titel Simulation of wind wave growth with reference source functions
VerfasserIn Sergei I. Badulin, Vladimir E. Zakharov, Andrei N. Pushkarev
Konferenz EGU General Assembly 2013
Medientyp Artikel
Sprache Englisch
Digitales Dokument PDF
Erschienen In: GRA - Volume 15 (2013)
Datensatznummer 250076555
 
Zusammenfassung
We present results of extensive simulations of wind wave growth with the so-called reference source function in the right-hand side of the Hasselmann equation written as follows -ˆ‚Nk-+ -ˆ‡ ω -‹ -ˆ‡ N = S + S + S -ˆ‚t k x k nl in diss (1) First, we use Webb’s algorithm [8] for calculating the exact nonlinear transfer function Snl. Second, we consider a family of wind input functions in accordance with recent consideration [9] ( )s S = γ(k)N , γ(k) = γ ω ω– f (θ). in k 0 ω0 in (2) Function fin(θ) describes dependence on angle θ. Parameters in (2) are tunable and determine magnitude (parameters γ0, ω0) and wave growth rate s [9]. Exponent s plays a key role in this study being responsible for reference scenarios of wave growth: s = 4-ˆ•3 gives linear growth of wave momentum, s = 2 – linear growth of wave energy and s = 8-ˆ•3 – constant rate of wave action growth. Note, the values are close to ones of conventional parameterizations of wave growth rates (e.g. s = 1 for [7] and s = 2 for [5]). Dissipation function Sdiss is chosen as one providing the Phillips spectrum E(ω) ~ ω5 at high frequency range [3] (parameter ωdiss fixes a dissipation scale of wind waves) Sdiss = Cdissμ4wωN (k)Θ(ω - ωdiss) (3) Here frequency-dependent wave steepness μ2w = E(ω,θ)ω5-ˆ•g2 makes this function to be heavily nonlinear and provides a remarkable property of stationary solutions at high frequencies: the dissipation coefficient Cdiss should keep certain value to provide the observed power-law tails close to the Phillips spectrum E(ω) ~ ω-5. Our recent estimates [3] give Cdiss -‰ˆ 2.0. The Hasselmann equation (1) with the new functions Sin, Sdiss (2,3) has a family of self-similar solutions of the same form as previously studied models [1,3,9] and proposes a solid basis for further theoretical and numerical study of wave evolution under action of all the physical mechanisms: wind input, wave dissipation and nonlinear transfer. Simulations of duration- and fetch-limited wind wave growth have been carried out within the above model setup to check its conformity with theoretical predictions, previous simulations [2,6,9], experimental parameterizations of wave spectra [1,4] and to specify tunable parameters of terms (2,3). These simulations showed realistic spatio-temporal scales of wave evolution and spectral shaping close to conventional parameterizations [e.g. 4]. An additional important feature of the numerical solutions is a saturation of frequency-dependent wave steepness μw in short-frequency range. The work was supported by the Russian government contract No.11.934.31.0035, Russian Foundation for Basic Research grant 11-05-01114-a and ONR grant N00014-10-1-0991. References [1]   S. I. Badulin, A. V. Babanin, D. Resio, and V. Zakharov. Weakly turbulent laws of wind-wave growth. J. Fluid Mech., 591:339–378, 2007. [2]   S. I. Badulin, A. N. Pushkarev, D. Resio, and V. E. Zakharov. Self-similarity of wind-driven seas. Nonl. Proc. Geophys., 12:891–946, 2005. [3]   S. I. Badulin and V. E. Zakharov. New dissipation function for weakly turbulent wind-driven seas. ArXiv e-prints, (1212.0963), December 2012. [4]   M. A. Donelan, J. Hamilton, and W. H. Hui. Directional spectra of wind-generated waves. Phil. Trans. Roy. Soc. Lond. A, 315:509–562, 1985. [5]   M. A. Donelan and W. J. Pierson-jr. Radar scattering and equilibrium ranges in wind-generated waves with application to scatterometry. J. Geophys. Res., 92(C5):4971–5029, 1987. [6]   E. Gagnaire-Renou, M. Benoit, and S. I. Badulin. On weakly turbulent scaling of wind sea in simulations of fetch-limited growth. J. Fluid Mech., 669:178–213, 2011. [7]   R. L. Snyder, F. W. Dobson, J. A. Elliot, and R. B. Long. Array measurements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech., 102:1–59, 1981. [8]   D. J. Webb. Non-linear transfers between sea waves. Deep Sea Res., 25:279–298, 1978. [9]   V. E. Zakharov, D. Resio, and A. N. Pushkarev. New wind input term consistent with experimental, theoretical and numerical considerations. ArXiv e-prints, (1212.1069), December 2012.