Abstract
The problem of formation and evolution of wind-generated wave spectra on a sea surface can be effectively solved due to the fact that the ratio of air density to water density provides a small parameter, with the consequence that characteristic wave amplitudes in typical situations are small. This justifies “a weak turbulent”approach to the problem of a statistical description, using an expansion in powers of nonlinearity.
The simplest variant of such a description is from the kinetic equation for waves, derived first by Hasselman (1962). This equation describes two fundamental processes — direct and inverse cascades of energy. The stationary equation has two remarkable exact solutions giving two energy spectra.
One spectrum describes the direct cascade — energy transport to a region of large wavenumbers (Zakharov and Filonenko, 1966). It is realized for frequencies ω ≫ ω0 where ω0 = g/V and U is a characteristic wind velocity. The other spectrum corresponds to an inverse cascade of energy in which wavelengths increase (“wave aging”) (Zakharov and Zaslavsky, 1981). It is realized at ω< ω0. The inverse cascade leads to the production of “aged waves” that are common in an open ocean (Glazman, 1989–91). Both spectra were observed experimentally by many authors. Long waves (ω < ω0) have a narrow angle distribution that can be explained (Zakharov and Shrira, 1990) by interaction between waves and a wind-generated shear flow, always existing near the sea surface.
Another fundamental phenomenon in surface wave theory is wave breaking. It is a strongly nonlinear effect that is suppressed by surface tension if the wind velocity is small enough. At larger wind speeds weak turbulence coexists with wave breaking, while at still larger wind speeds wave-breaking prevails. A simple model of wave breaking is offered, using a self-similar solution of the Euler equation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
R.E. Glazman and S.H. Pilorz, Effects of sea maturity on satellite altimeter measurements, J. Geophys. Res., 95 (1990), 2857–2870.
R.E. Glazman, Statistical problems of wind-generated gravity waves, arising in microwave remote sensing of surface winds, Trans. Geosci. Rem. Sensing, 29 (1991), 135–142.
V.G. Bondur and E.A. Sharkov, Statistical properties of whitecaps on a rough sea, Oceanology, 22 (1982), 274–279.
M.L. Banner, I.F. Jones and J.C. Trinder, Wavenumber spectra of short gravity waves, J. Fluid Mech., 198 (1989), 321–344.
K. Hasselmann, On the nonlinear energy transfer in gravity-wave spectrum, J. Fluid Mech., 12 (1962), 481–500.
V.E. Zakharov, Stability of periodic waves of finite amplitude on a surface of a deep fluid, J. Appl.Mech.Tech.Phys., (1968), No. 2, p. 190.
D.R. Crawford, H.G. Yuen and P.G. Saffman, J. Wave Motion, (1980), 1–16.
V.P. Krasitzky, On the canonical transformation of the theory of weakly nonlin-ear waves with non-decay dispersion law, Sov.Phys. JETP, (1991)
J.W. Miles, On the generation of surface waves by shear flow, J. Fluid Mech., 3 (1957), 185–204.
O.M. Phillips, Spectral and statistical properties of the equilibrium range in wind-generated gravity waves, J. Fluid Mech., (1985), 505–531.
V.E. Zakharov, Weak-turbulence in a plasma without magnetic fields, Zh. Eksp. Teor.Fiz., 51 (1966), p.688 (Sov. Phys. JETP, 24(1967), 455–459.)
V.E. Zakharov, V.S. Lvov and G. Falkovich, Kolmogorov spectra of wave turbulence, Springer-Verlag, (1992, in press).
V.E. Zakharov and N.N. Filonenko, The energy spectrum for stochastic oscillation of a fluid’s surface, Doklady Akad. Nauk, 170 (1966), 1292–1295.
Y. Toba, Local balance in the air-sea boundary processes. III. On the spectrum of wind waves, J. Oceanogr. Soc. Japan, 29 (1973), 209–220.
S.A. Kitaigorodskii, On the theory of the equilibrium range in the spectrum of wave-generated gravity waves, J. Phys.Oceanogr., 13 (1983), 816–827.
V.E. Zakharov and M.M. Zaslavskii, The kinetic equation and Kolmogorov spectra in the weak-turbulence theory of wind waves, Izv. Atm. Ocean. Phys., 18 (1982), 747–753.
V.E. Zakharov and M.M. Zaslavskii, Ranges for generation and dissipation in the kinetic equation for a low-turbulence theory of wind waves, Izv. Atm. Ocean. Phys., 18 (1982), 821–827.
V.E. Zakharov and M.M. Zaslavskii, On the problem of wind-driven waving prognosis, Doklady Akad. Nauk. SSSR, 256 (1982), 567.
V.E. Zakharov and M.M. Zaslavskii, Shape of spectrum of energy carrying components of a water surface in a weak-turbulence theory of wind waves, Izv. Atm. Ocean. Phys., 19 (1983), 207–212.
V.E. Zakharov and M.M. Zaslavskii, Dependence of wave parameters of the wind velocity, duration and its action and fetch in the weak-turbulence theory of wind waves, Izv. Atm. Ocean Phys., 19 (1983), 300–306.
V.E. Zakharov and N.N. Filonenko, Weak turbulence of capillary waves, J. Appl. Mech. Tech. Phys., 5 (1967). 62–67.
V.E. Zakharov and V.I. Shrira, On the formation of the directional spectrum of wind waves, Zh. Eksp.Teor.Fiz., 98 (1990), 1941–1958.
V.E. Zakharov and V.I. Shrira, Water wave nonlinear interaction owing to drift current: Formation of the angular spectrum of the waves, J.Fluid Mech., (1992, in press).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zakharov, V.E. (1992). Inverse and Direct Cascade in the Wind-Driven Surface Wave Turbulence and Wave-Breaking. In: Banner, M.L., Grimshaw, R.H.J. (eds) Breaking Waves. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84847-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-84847-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-84849-0
Online ISBN: 978-3-642-84847-6
eBook Packages: Springer Book Archive