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# Understanding discrete capillary-wave turbulence using a quasi-resonant kinetic equation

Abstract

Experimental and numerical studies have shown that, with sufficient nonlinearity, the theoretical capillary-wave power-law spectrum derived from the kinetic equation (KE) of weak turbulence theory can be realized. This is despite the fact that the KE is derived assuming an infinite domain with continuous wavenumber, while experiments and numerical simulations are conducted in realistic finite domains with discrete wavenumbers for which the KE theoretically allows no energy transfer. To understand this, we first analyse results from direct simulations of the primitive Euler equations to elucidate the role of nonlinear resonance broadening (NRB) in discrete turbulence. We define a quantitative measure of the NRB, explaining its dependence on the nonlinearity level and its effect on the properties of the obtained stationary power-law spectra. This inspires us to develop a new quasi-resonant kinetic equation (QKE) for discrete turbulence, which incorporates the mechanism of NRB, governed by a single parameter $\unicode[STIX]{x1D705}$ expressing the ratio of NRB and wavenumber discreteness. At $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}\approx 0.02$ , the QKE recovers simultaneously the spectral slope $\unicode[STIX]{x1D6FC}_{0}=-17/4$ and the Kolmogorov constant $C_{0}=6.97$ (corrected from the original derivation) of the theoretical continuous spectrum, which physically represents the upper bound of energy cascade capacity for the discrete turbulence. For $\unicode[STIX]{x1D705}<\unicode[STIX]{x1D705}_{0}$ , the obtained spectra represent those corresponding to a finite domain with insufficient nonlinearity, resulting in a steeper spectral slope $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{0}$ and reduced capacity of energy cascade $C>C_{0}$ . The physical insights from the QKE are corroborated by direct simulation results of the Euler equations.

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Annenkov S. Y. & Shrira V. I. 2001 Numerical modelling of water-wave evolution based on the Zakharov equation. J. Fluid Mech. 449, 341371.
Brazhnikov M. Y., Kolmakov G. V., Levchenko A. A. & Mezhov-Deglin L. P. 2002 Observation of capillary turbulence on the water surface in a wide range of frequencies. Europhys. Lett. 58 (4), 510.
Connaughton C., Nazarenko S. & Pushkarev A. 2001 Discreteness and quasiresonances in weak turbulence of capillary waves. Phys. Rev. E 63 (4), 046306.
Deike L., Bacri J.-C. & Falcon E. 2013 Nonlinear waves on the surface of a fluid covered by an elastic sheet. J. Fluid Mech. 733, 394413.
Deike L., Berhanu M. & Falcon E. 2014a Energy flux measurement from the dissipated energy in capillary wave turbulence. Phys. Rev. E 89 (6), 023003.
Deike L., Daniel F., Berhanu M. & Falcon E. 2014b Direct numerical simulations of capillary wave turbulence. Phys. Rev. Lett. 112 (1), 234501.
Denissenko P., Lukaschuk S. & Nazarenko S. 2007 Gravity wave turbulence in a laboratory flume. Phys. Rev. Lett. 99 (1), 014501.
Dyachenko S., Newell A. C., Pushkarev A. & Zakharov V. E. 1992 Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57 (1), 96160.
Falcon E., Laroche C. & Fauve S. 2007 Observation of gravity–capillary wave turbulence. Phys. Rev. Lett. 98 (9), 94503.
Galtier S., Nazarenko S. V., Newell A. C. & Pouquet A. 2002 Anisotropic turbulence of shear-Alfvén waves. Astrophys. J. Lett. 564 (1), L49.
Kartashova E. A. 1990 Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Physica D 46 (1), 4356.
Lighthill M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Lvov V. S. & Nazarenko S. 2010 Discrete and mesoscopic regimes of finite-size wave turbulence. Phys. Rev. E 82 (5), 056322.
Lvov Y. V., Polzin K. L. & Tabak E. G. 2004 Energy spectra of the ocean’s internal wave field: theory and observations. Phys. Rev. Lett. 92 (12), 128501.
Nazarenko S. 2006 Sandpile behaviour in discrete water-wave turbulence. J. Stat. Mech. 2006 (02), L02002.
Nazarenko S. 2011 Wave Turbulence, vol. 825. Springer Science and Business Media.
Newell A. C. & Rumpf B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43, 5978.
Pan Y.2016 Understanding of weak turbulence of capillary waves. PhD thesis, Massachusetts Institute of Technology.
Pan Y. & Yue D. K. P. 2014 Direct numerical investigation of turbulence of capillary waves. Phys. Rev. Lett. 113 (9), 094501.
Pan Y. & Yue D. K. P. 2015 Decaying capillary wave turbulence under broad-scale dissipation. J. Fluid Mech. 780, R1.
Piscopia R., Polnikov V., DeGirolamo P. & Magnaldi S. 2003 Validation of the three-wave quasi-kinetic approximation for the spectral evolution in shallow water. Ocean Engng 30 (5), 579599.
Polnikov V. G. & Manenti S. 2009 Study of relative roles of nonlinearity and depth refraction in wave spectrum evolution in shallow water. Engng Appl. Comput. Fluid Mech. 3 (1), 4255.
Pushkarev A., Resio D. & Zakharov V. 2003 Weak turbulent approach to the wind-generated gravity sea waves. Physica D 184 (1), 2963.
Pushkarev A. N. & Zakharov V. E. 1996 Turbulence of capillary waves – theory and numerical simulation. Phys. Rev. Lett. 76 (18), 33203323.
Pushkarev A. N & Zakharov V. E. 2000 Physica D 135 (1), 98116.
Wright W. B., Budakian R. & Putterman S. J. 1996 Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76 (24), 45284531.
Xia H., Shats M. & Punzmann H. 2010 Modulation instability and capillary wave turbulence. Europhys. Lett. 91 (1), 14002.
Zakharov V. E. 2010 Energy balance in a wind-driven sea. Phys. Scr. 2010 (T142), 014052.
Zakharov V. E. & Filonenko N. N. 1966 The energy spectrum for stochastic oscillations of a fluid surface. Dokl. Akad. Nauk SSSR 170, 12921295.
Zakharov V. E. & Filonenko N. N. 1967 Weak turbulence of capillary waves. J. Appl. Mech. Tech. Phys. 8, 3740.
Zakharov V. E., L’vov V. S. & Falkovich G. 1992 Kolmogorov Spectra of Turbulence 1. Wave Turbulence, vol. 1, p. 275. Springer, ISBN: 3-540-54533-6.
Zaslavskii M. M. & Polnikov V. G 1998 Three-wave quasi-kinetic approximation in the problem of the evolution of a spectrum of nonlinear gravity waves at small depths. Izv. Atmos. Ocean. Phys. 34, 609616.
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Journal of Fluid Mechanics
• ISSN: 0022-1120
• EISSN: 1469-7645
• URL: /core/journals/journal-of-fluid-mechanics
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