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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
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Medientyp |
Artikel
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Sprache |
Englisch
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Digitales Dokument |
PDF |
Erschienen |
In: GRA - Volume 15 (2013) |
Datensatznummer |
250076555
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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
-Nk-+ - Ï -
- N = S + S + S
-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 Ï Ï– f (θ).
in k 0 Ï0 in
(2)
Function fin(θ) describes dependence on angle θ. Parameters in (2) are tunable and
determine magnitude (parameters γ0, Ï0) and wave growth rate s [9]. Exponent s plays a key
role in this study being responsible for reference scenarios of wave growth: s = 4-3
gives linear growth of wave momentum, s = 2 – linear growth of wave energy and
s = 8-3 – 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(Ï) ~ Ï5 at high frequency range [3] (parameter Ïdiss fixes a dissipation scale of wind
waves)
Sdiss = Cdissμ4wÏN (k)Î(Ï - Ïdiss)
(3)
Here frequency-dependent wave steepness
μ2w = E(Ï,θ)Ï5-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(Ï) ~ Ï-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
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. |
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