Benjamin Miquel

Nonstationary wave turbulence in an elastic plate.

We report experimental results on the decay of wave turbulence in an elastic plate obtained by stopping the forcing from a stationary turbulent state. In the stationary case, the forcing is seen to induce some anisotropy and a spectrum in disagreement with the weak turbulence theory. After stopping the forcing, almost perfect isotropy is restored. The decay of energy is self-similar and the observed decaying spectrum is in better agreement with the prediction of the weak turbulence theory. The dissipative part of the spectrum is partially consistent with the theoretical prediction based on previous work by Kolmakov. This suggests that the nonagreement with the weak turbulence theory is mostly due to a spurious effect of the forcing related to the finite size of the system.

The role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov-Zakharov spectrum

The weak turbulence theory has been applied to waves in thin elastic plates obeying the Föppl-Von Kármán dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the weak turbulence theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Föppl-Von Kármán equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves γ_{k}=a+bk{2} is used. This confirms that the Föppl-Von Kármán equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus, it is shown that dissipation has a major influence on the scaling properties of stationary solutions of weakly nonlinear wave turbulence.

Transition from Wave Turbulence to Dynamical Crumpling in Vibrated Elastic Plates

We study the dynamical regime of wave turbulence of a vibrated thin elastic plate based on experimental and numerical observations. We focus our study on the strongly nonlinear regime described in a previous Letter by Yokoyama and Takaoka. At small forcing, a weakly nonlinear regime is compatible with the weak turbulence theory when the dissipation is localized at high wave number. When the forcing intensity is increased, a strongly nonlinear regime emerges: singular structures dominate the dynamics at large scales whereas at small scales the weak turbulence is still present. A turbulence of singular structures with folds and D cones develops that alters significantly the energy spectra and causes the emergence of intermittency.

Wave turbulence buildup in a vibrating plate

We report experimental and numerical results on the buildup of the energy spectrum in wave turbulence of a vibrating thin elastic plate. Three steps are observed: first a short linear stage, then the turbulent spectrum is constructed by the propagation of a front in wave number space and finally a long time saturation due to the action of dissipation. The propagation of a front at the second step is compatible with scaling predictions from the Weak Turbulence Theory.

A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit

A simple model of nonlinear salt-finger convection in two dimensions is derived and studied. The model is valid in the limit of a small solute to heat diffusivity ratio and a large density ratio, which is relevant to both oceanographic and astrophysical applications. Two limits distinguished by the magnitude of the Schmidt number are found. For order one Schmidt numbers, appropriate for astrophysical applications, a modified Rayleigh–Benard system with large-scale damping due to a stabilizing temperature is obtained. For large Schmidt numbers, appropriate for the oceanic setting, the model combines a prognostic equation for the solute field and a diagnostic equation for inertia-free momentum dynamics. Two distinct saturation regimes are identified for the second model: the weakly driven regime is characterized by a large-scale flow associated with a balance between advection and linear instability, while the strongly-driven regime produces multiscale structures, resulting in a balance between energy input through linear instability and energy transfer between scales. For both regimes, we analytically predict and numerically confirm the dependence of the kinetic energy and salinity fluxes on the ratio between solutal and thermal Rayleigh numbers. The spectra and probability density functions are also computed.

Hybrid Chebyshev function bases for sparse spectral methods in parity-mixed PDEs on an infinite domain

Abstract We present a numerical spectral method to solve systems of differential equations on an infinite interval y ∈ ( − ∞ , ∞ ) in presence of linear differential operators of the form Q ( y ) ( ∂ / ∂ y ) b (where Q ( y ) is a rational fraction and b a positive integer). Even when these operators are not parity-preserving, we demonstrate how a mixed expansion in interleaved Chebyshev rational functions T B n ( y ) and S B n ( y ) preserves the sparsity of their discretization. This paves the way for fast O ( N ln ⁡ N ) and spectrally accurate mixed implicit-explicit time-marching of sets of linear and nonlinear equations in unbounded geometries.

Intermittency and emergence of coherent structures in wave turbulence of a vibrating plate

We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long-time numerical simulations makes this system extremely valuable for wave turbulence studies. The purely 2D character of dynamics of the elastic plate makes it much simpler to handle compared to much more complex 3D physical systems that are typical of geo- and astrophysical issues (ocean surface or internal waves, magnetized plasmas or strongly rotating and/or stratified flows). When the forcing is small the observed wave turbulence is consistent with the predictions of the weak turbulent theory. Here we focus on the case of stronger forcing for which coherent structures can be observed. These structures look similar to the folds and D-cones that are commonly observed for strongly deformed static thin elastic sheets (crumpled paper) except that they evolve dynamically in our forced system. We describe their evolution and show that their emergence is associated with statistical intermittency (lack of self similarity) of strongly nonlinear wave turbulence. This behavior is reminiscent of intermittency in Navier-Stokes turbulence. Experimental data show hints of the weak to strong turbulence transition. However, due to technical limitations and dissipation, the strong nonlinear regime remains out of reach of experiments and therefore has been explored numerically.

Wavelet analysis of the slow non-linear dynamics of wave turbulence

In wave turbulence, the derivation of solutions in the frame of the Weak Turbulence Theory relies on the existence of a double time-scale separation: first, between the period of the waves and caracteristic nonlinear time tNL corresponding to energy exchange among waves; and secondly, between tNL and the characteristic dissipation time td. Due to the lack of space and time resolved measurement, this hypothesis have remained unverified so far. We study the turbulence of flexion waves in thin elastic plates. td is measured using the decline stage of the turbulence whereas a wavelet analysis is performed to measure the characteristic non-linear time tNL.