10:45
Wave Turbulence
10:45
15 mins
#188
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Turbulence of capillary waves forced by steep gravity waves
Michael Berhanu, Eric Falcon, Luc Deike
Abstract: In a recent article, we have studied experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2-0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.
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11:00
15 mins
#169
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Experience of internal wave turbulence in the Coriolis facility
Clément Savaro, Antoine Campagne, Nicolas Mordant
Abstract: We designed experimentations dedicated to establish stratified 3D turbulence forced by internal waves in the Coriolis facility.
The resulting flow fields is measured using large scale spatially and temporally resolved PIV technique.
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11:15
15 mins
#249
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Early stage of integrable turbulence in 1D NLS equation: the semi-classical approach to statistics
Giacomo Roberti, Gennady El, Stéphane Randoux, Pierre Suret
Abstract: The concept of integrable turbulence, introduced by Zakharov [1, 2, 3], has been recently recognised as a novel theoretical paradigm of major importance for a broad range of physical applications from photonics to oceanography. In the context of nonlinear fibre optics, modern laser and amplifier systems are known to suffer from spontaneous random fluctuations now identified as optical turbulence. This phenomenon can be mathematically described in the framework of the integrable one-dimensional nonlinear Schrödinger equation (1D-NLSE).
We consider the evolution of an initial partially coherent wave field with Gaussian statistics in the framework of the 1D-NLSE, and we analyse the normalised fourth order moment of the field’s amplitude, which characterises the “tailedness” of the probability density function (PDF). The relation between this statistical quantity and the spectral width of the field has been recently provided in Onorato et al. [4], however, it requires knowledge of the spectral width at each step in time. In our work, thanks to the combination of tools from the wave turbulence theory [5] and the semi-classical theory of 1D-NLSE, we derive for the first time an analytical formula for the short time evolution of the fourth order moment as a function of the statistical characteristics of the initial condition. This formula provides a quantitative description of the appearance of the "heavy" ("low") tail of the PDF in the focusing (defocusing) regime of the 1D-NLS at the initial stage of the development of integrable turbulence, and our theoretical predictions exhibit a good agreement with the numerical simulations.
References
[1] V. E. Zakharov. Turbulence in integrable systems. Stud. Appl. Math., 122(3):219–234, 2009.
[2] V. E. Zakharov and A. A. Gelash. Nonlinear stage of modulation instability. Phys. Rev. Lett., 111:054101, Jul 2013.
[3] D. S. Agafontsev and V. E. Zakharov. Integrable turbulence and formation of rogue waves. Nonlinearity, 28(8):2791, 2015.
[4] M. Onorato, D. Proment, G. El, S. Randoux, and P. Suret. On the origin of heavy-tail statistics in equations of the nonlinear Schrodinger type. Physics Letters A, 380(39):173–3177, 2016.
[5] S. Nazarenko. Wave Turbulence. 10.1007/978-3-642-15942-8. Lecture Notes in Physics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.5
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11:30
15 mins
#296
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Coexistence of solitons and extreme events in deep water surface waves
Annette Cazaubiel, Guillaume Michel, Simon Lepot, Benoit Semin, Sébastien Aumaître, Michael Berhanu, Félicien Bonnefoy, Eric Falcon
Abstract: We study experimentally, in a large-scale basin, the propagation of unidirectional
deep water gravity waves stochastically modulated in phase. We observe the emergence
of nonlinear localized structures that evolve on a stochastic wave background. Such a
coexistence is expected by the integrable turbulence theory for the nonlinear Schrödinger
equation (NLSE), and we report the first experimental observation in the context of
hydrodynamic waves. We characterize the formation, the properties, and the dynamics
of these nonlinear coherent structures (solitons and extreme events) within the incoherent
wave background. The extreme events result from the strong steepening of wave train
fronts, and their emergence occurs after roughly one nonlinear length scale of propagation
(estimated from the NLSE). Solitons arise when nonlinearity and dispersion are weak, and
of the same order of magnitude as expected from the NLSE.We characterize the statistical
properties of this state. The number of solitons and extreme events is found to increase all
along the propagation, the wave-field distribution has a heavy tail, and the surface elevation
spectrum is found to scale as a frequency power law with an exponent −4.5 ± 0.5. Most
of these observations are compatible with the integrable turbulence theory for the NLSE
although some deviations (e.g., power-law spectrum, asymmetrical extreme events) result
from effects proper to hydrodynamic waves.
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11:45
15 mins
#578
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On the Convergence of the Normal Form Transformation in discrete wave Turbulence Theory for the Charney-Hasegawa-Mima (CHM) Equation
Shane Walsh, Miguel Bustamante
Abstract: A crucial problem in discrete wave turbulence theory concerns extending the validity of the normal form transformation beyond the weakly nonlinear limit. One of the main issues occurs when the linear and nonlinear timescales are no longer separated, and some of the assumptions of wave turbulence theory no longer hold. This allows finite amplitude interactions to occur through terms which are usually eliminated through normal form transformations. The specific finite amplitude phenomenon which we wish to study is precession resonance. We investigate this for the CHM equation, Galerkin-truncated to 4 Fourier modes. We first study this reduced 4-mode system from a dynamical systems point of view to understand the manifold structure of the resonance in phase space. By calculating the normal form transformation up to 7th order (keeping all resonances up to 8-wave resonant interactions), we then numerically calculate the rate of convergence of the transformation as a function of a scaling parameter of our initial conditions. Our findings show that the scaling amplitude at which the normal form transformation diverges is of the same order as the amplitude at which precession resonance occurs. This implies that precession resonance cannot be described in the classical theory of wave turbulence through normal form transformations, so a more general theory for intermediate nonlinearity is required.
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12:00
15 mins
#348
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MEAN FLOW INSTABILITY OF SURFACE GRAVITY WAVES PROPAGATING IN A ROTATING FRAME
Kannabiran Seshasayanan, Basile Gallet
Abstract: We study the stability properties of the Eulerian mean flow generated by monochromatic surface-gravity waves propagating in a rotating frame, see illustration in figure \ref{fig1a}. The wave averaged equations, also known as the Craik-Leibovich equations \cite{craik1976rational}, govern the evolution of the mean flow. For propagating waves in a rotating frame these equations admit a steady depth-dependent base flow sometimes called the Ekman-Stokes spiral \cite{huang1979surface,gnanadesikan1995structure,mcwilliams2012wavy,polton2005role}, because of its resemblance to the standard Ekman spiral. This base flow profile is controlled by two non-dimensional numbers, the Ekman number $Ek = \frac{\nu}{f \lambda^2}$ and the Rossby number $Ro = \frac{U_s}{f \lambda}$. Here $\lambda$ is the wavelength of the surface waves, $f$ is twice the rotation rate, $U_s$ is the Stokes drift velocity associated with the surface waves and $\nu$ is the kinematic viscosity.
We show that this steady laminar velocity profile is linearly unstable above a critical Rossby number $Ro_c (Ek)$. We determine the threshold Rossby number as a function of $Ek$ using a numerical eigenvalue solver, before confirming the numerical results through asymptotic expansions in the large/low $Ek$ limit. We show the instability threshold in figure \ref{fig1b} (data points) along with the asymptotic results (dashed lines). These were also confirmed by nonlinear simulations of the Craik-Leibovich equations. When the system is well above the linear instability threshold, $Ro \gg Ro_c$, the resulting flow fluctuates chaotically.
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12:15
15 mins
#225
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Anomalous scaling in gravitational wave turbulence
Sébastien Galtier, Sergey Nazarenko, Éric Buchlin, Simon Thalabard
Abstract: The non-linear nature of the Einstein equations of general relativity suggests that space-time can be turbulent. Such a turbulence is expected during the primordial universe (first second) when gravitational waves (GW) have been excited through eg. the merger of primordial black holes. The analytical theory of weak GW turbulence was published in 2017: it is based on a diagonal space-time metric reduced to the variables t, x and y. The theory predicts the existence of a dual cascade driven by 4–wave interactions with a direct cascade of energy and an inverse cascade of wave action. In the latter case the isotropic Kolmogorov-Zakharov spectrum N(k) has the power law index -2/3 implying an explosive phenomenon. In that context, we developed a fourth-order and a second-order nonlinear diffusion models in spectral space to describe GW turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations. We show numerically by computing the second-order diffusion model that in the non-stationary regime the isotropic wave action spectrum N(k) exhibits an anomalous scaling which is understood as a self-similar solution of the second kind. The regime of weak GW turbulence is in fact limited to a narrow window in wavenumbers and turbulence is expected to become strong at the largest scales. Then, the phenomenology of critical balance can be used. The formation of a condensate may happen, and its rapid growth can be interpreted as an accelerated expansion of the universe which could be at the origin of the cosmic inflation. We can show that the fossil spectrum obtained after the turbulent inflation is compatible with the latest data obtained with the Planck satellite.
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12:30
15 mins
#337
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Integrable turbulence: experimental realization of a soliton gas
Nicolas Mordant, Ivan Redor, Eric Barthélemy, Hervé Michallet, Miguel Onorato
Abstract: We report an experimental realization of a soliton gas in shallow water. We generate a bidirectional soliton gas by taking advantage of reflections at both ends of our 34m-long wave flume. We perform a time-space resolved measurement of the wave elevation to investigate the statistical properties of the gas. We observe that although dissipation is unavoidable in experiments and the statistical properties are consistent with numerical simulation of a truly integrable KdV soliton gas.
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