Simulation of wind wave growth with reference source functions
Abstract
We present results of extensive simulations of wind wave growth with the socalled reference source function in the righthand side of the Hasselmann equation written as follows 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 = ?(k)N , ?(k) = ? ? ? f (?). in k 0 ?0 in (2) Function fin(?) describes dependence on angle ?. Parameters in (2) are tunable and determine magnitude (parameters ?0, ?0) and wave growth rate s [9]. Exponent s plays a key role in this study being responsible for reference scenarios of wave growth: s = 43 gives linear growth of wave momentum, s = 2  linear growth of wave energy and s = 83  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(?) ~ ?5 at high frequency range [3] (parameter ?diss fixes a dissipation scale of wind waves) Sdiss = Cdissμ4w?N (k)θ(?  ?diss) (3) Here frequencydependent wave steepness μ2w = E(?,?)?5g2 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 powerlaw tails close to the Phillips spectrum E(?) ~ ?5. Our recent estimates [3] give Cdiss ? 2.0. The Hasselmann equation (1) with the new functions Sin, Sdiss (2,3) has a family of selfsimilar 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 fetchlimited 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 spatiotemporal 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 frequencydependent wave steepness μw in shortfrequency range. The work was supported by the Russian government contract No.11.934.31.0035, Russian Foundation for Basic Research grant 110501114a and ONR grant N000141010991. References [1] S. I. Badulin, A. V. Babanin, D. Resio, and V. Zakharov. Weakly turbulent laws of windwave growth. J. Fluid Mech., 591:339378, 2007. [2] S. I. Badulin, A. N. Pushkarev, D. Resio, and V. E. Zakharov. Selfsimilarity of winddriven seas. Nonl. Proc. Geophys., 12:891946, 2005. [3] S. I. Badulin and V. E. Zakharov. New dissipation function for weakly turbulent winddriven seas. ArXiv eprints, (1212.0963), December 2012. [4] M. A. Donelan, J. Hamilton, and W. H. Hui. Directional spectra of windgenerated waves. Phil. Trans. Roy. Soc. Lond. A, 315:509562, 1985. [5] M. A. Donelan and W. J. Piersonjr. Radar scattering and equilibrium ranges in windgenerated waves with application to scatterometry. J. Geophys. Res., 92(C5):49715029, 1987. [6] E. GagnaireRenou, M. Benoit, and S. I. Badulin. On weakly turbulent scaling of wind sea in simulations of fetchlimited growth. J. Fluid Mech., 669:178213, 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., 102:159, 1981. [8] D. J. Webb. Nonlinear transfers between sea waves. Deep Sea Res., 25:279298, 1978. [9] V. E. Zakharov, D. Resio, and A. N. Pushkarev. New wind input term consistent with experimental, theoretical and numerical considerations. ArXiv eprints, (1212.1069), December 2012.
 Publication:

EGU General Assembly Conference Abstracts
 Pub Date:
 April 2013
 Bibcode:
 2013EGUGA..15.5284B