Limited fetch revisited: Comparison of wind input terms, in surface wave modeling

Abstract

Results pertaining to numerical solutions of the Hasselmann kinetic equation (HE), for wind driven sea spectra, in the fetch limited geometry, are presented. Five versions of source functions, including the recently introduced ZRP model [1], have been studied, for the exact expression of S nl and high-frequency implicit dissipation, due to wave-breaking. Four of the five experiments were done in the absence of spectral peak dissipation for various S in terms. They demonstrated the dominance of quadruplet wave-wave interaction, in the energy balance, and the formation of self-similar regimes, of unlimited wave energy growth, along the fetch. Between them was the ZRP model, which strongly agreed with dozens of field observations performed in the seas and lakes, since 1947. The fifth, the W AM 3 wind input term experiment, used additional spectral peak dissipation and reproduced the results of a previous, similar, numerical simulation described in [2], but only supported the field experiments for moderate fetches, demonstrating a total energy saturation at half of that of the Pierson-Moscowits limit. The alternative framework for HE numerical simulation is proposed, along with a set of tests, allowing one to select physically-justified source terms.

Weak‐Turbulent Theory of Wind‐Driven Sea

“…As an alternative to multiple wind source terms, noncompliant with the set of nonlinear tests (see Pushkarev & Zakharov, ), the new ZRP wind‐input source term, constructed through the self‐similarity analysis of Hasselmann equation, has been proposed (Zakharov et al, ): where ρ a and ρ w are the air and water density correspondingly. Frequencies f min and f d depend on the wind speed and should be found empirically: The wind speed at 10‐m height was taken as U 10 = 10 m/s and U 10 = 5 m/s, f min = 0.1 Hz.…”

“…The dissipation was chosen in the “implicit” form: the dynamical part of the spectrum was continued from f d = 1.1 Hz (Resio et al, ) by Phillips () spectrum ε ( f , θ )∼ ω −5 , decaying faster than the equilibrium spectrum ε ( f , θ )∼ ω −4 and providing therefore high‐frequency dissipation (Pushkarev & Zakharov, ).…”

Ocean swell within the kinetic equation for water waves

“…An alternative setup of fetch-limited evolution (∂/∂t ≡ 0, ∇ r = 0) introduces dispersion of wave harmonics as a competing mechanism that can change the swell evolution dramatically. Recent advances in wave modeling (Pushkarev and Zakharov, 2016) make the problem of spatial-temporal swell evolution feasible and specify the perspectives of our first step study. The theoretical background for the classic fetch-limited setup when solutions depend on the only spatial coordinate (i.e., ∂/∂x = 0, ∂/∂y ≡ 0) is sketched in Sect.…”

“…2) dictates "magic relations" (in the words of Zakharov, 2015, 2016) between exponents p τ , q τ and p χ , q χ…”

“…Duration-limited evolution of the swell has been simulated with the Pushkarev et al (2003) version of the code based on the WRT algorithm (Webb, 1978;Tracy and Resio, 1982). Features of the code and numerical setups have been described in previous papers (Badulin et al, 2002(Badulin et al, , 2004(Badulin et al, , 2005a(Badulin et al, , b, 2007Zakharov et al, 2007;Badulin et al, 2008aBadulin et al, , 2013Zakharov, 2015, 2016). The frequency resolution for log-spaced grid has been set to (ω n+1 − ω n )/ω n = 1.03128266.…”