Session A26: Waves: Nonlinear Dynamics and Turbulence

Frequency Diffusion in Wave-Mean Flow Interaction

Several recent studies have shown that dispersive wave system, in presence of a weak (and stationary) random mean flow (of much smaller magnitude than the group velocity), may show a scattering behaviour. Starting from a set of PDEs describing dispersive waves in presence of a mean flow, we study the possibility of having a frequency shift producing statistical scattering of the waves. We identify two relevant parameters for our study : a first parameter controlling the power law of the dispersion relation; and a second parameter indicating the initial ratio between the average value of the stationary mean flow and the initial group velocity. We then investigate the corresponding phase diagram thanks to a 2D ray tracing scheme in order to identify the different asymptotic regimes. This scheme is derived from the wave-mean flow Hamiltonian system. We demonstrate that, as shown in prior studies, there is a range of dispersive systems for which the frequency stays constant and diffusion is only observed in wave number angle. Moreover, we show that, for lower values of the power law controlling the dispersion relation, the assumption that the group velocity is larger than the mean flow can break, yielding to a diffusion to larger values in frequency, as well as in wave number.

    

From dispersive wave turbulence to an acoustic-like wave regime

Wave turbulence is a statistical state in which numerous random weakly nonlinear waves interact with each other. It leads to an energy cascade from large scales down to small scales driven by resonant interactions between waves. This state is first evidenced experimentally in a one dimensional canal for dispersive gravity-capillary waves on the surface of mercury [1]. Then by using a ferrofluid and a high external magnetic field, we managed to experimentally create a transition from dispersive wave turbulence to a nondispersive regime involving coherent structures which are found to be shock waves [2]. These structures are characterized by a significant steepening with a discontinuity, and correspond to singularities in their second order derivative. Because of the discontinuities, shock waves then transfer energy at every scales instantly and become thus the main process to cascade energy over scales.

 [1] G. Ricard, and E. Falcon, Experimental quasi-1D capillary-wave turbulence, EPL, 135, 64001 (2021).

 [2] G. Ricard, and E. Falcon, From dispersive wave turbulence to an acoustic-like wave regime, planned paper.   

On the time scales of spectral evolution of nonlinear waves

Kinetic theory for nonlinear waves predicts a kinetic time scale of spectral evolution, which is O(ε^-4 ) with ε the wave steepness. However, in (Annenkov & Shrira, 2009, PRL), a “faster” evolution on the dynamic O(ε^-2 ) time scale is identified through numerical simulations of surface gravity waves, posing a challenge to the standard kinetic equation in wave turbulence. To study this problem, we consider the evolution of nonlinear wave fields via the Majda-McLaughlin-Tabak (MMT) equation, which captures wave turbulence behavior while being free of complexities associated with surface gravity waves (e.g., the canonical transformation to remove the quadratic nonlinearity terms). For the one-dimensional MMT equation, our results show that the kinetic scaling is obtained for a considerable range of nonlinearity. Below the nonlinearity range, the kinetic scaling is prohibited in the discrete wave turbulence regime. For higher nonlinearity, dynamic scaling is observed in conjunction with the emergence of coherent structures. Results from the two-dimensional MMT equation will also be presented.

    

Do surface gravity waves have a frozen turbulence state?

We study the energy transfer by exact resonances for surface gravity waves in a finite periodic spatial domain. Based on a kinematic model simulating the generation of active wave modes in a finite discrete wavenumber space S_R, we examine the possibility of direct and inverse energy cascades. More specifically, we set an initially excited region which iteratively spreads energy to wave modes in S_R through exact resonances. At each iteration, we first activate new modes from scale resonances (which generate modes with new lengths), then consider two bounding situations for angle resonances (which transfer energy at the same length scale): the lower bound where no angle resonance is included and the upper bound where all modes with the same length as any active mode are excited. Such a strategy is essential to enable the computation for a large domain S_R with the maximum wavenumber R∼10³. We show that for both direct and inverse cascades, the energy cascade to the boundaries of S_R can be established when the initially excited region is sufficiently large, otherwise a frozen turbulence state occurs, with a sharp transition between the two regimes especially for the direct cascade. Through a study on the structure of resonant quartets, the mechanism associated with the sharp transition and the role of angular energy transfer in the cascades are elucidated.   

Exploring Breather Dynamics in a Two-dimensional, Nonlinear Schroedinger Equation with Non-local Derivatives

Breather solutions to nonlinear wave equations represent an important class of coherent structure, typically featuring strong spatial localization and oscillations in time. In our talk, we present a study of the nonlinear Schroedinger equation (NLS) with non-local derivatives evaluated on a periodic, two-dimensional domain. For derivatives of certain orders, we find a novel breather solution that dominates field evolution in the regime of nonlinearity approaching zero. As nonlinearity is increased, the breathers break down, yielding to the wave-turbulence (or Rayleigh-Jeans) spectra. To better understand these dynamics, we study the phase-space trajectories associated with the breather solutions and find that they are quasi-periodic and close to trajectories of the linearized NLS. With the increase of nonlinearity, these trajectories deform before breaking down entirely, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory. We conclude by exploring this connection briefly, raising questions for future work.   

Experimental nonlinear waves along a torus of fluid

Curved interfaces such as toroidal drops are ubiquitous in nature, but are unstable, making them difficult to study. Using an original technique we create a stable torus of liquid on a superhydrophobic substrate allowing a systematic study of waves along its inner and outer border under curved and periodic conditions [1,2]. By exciting the torus, we recover the displacements of the borders and study the dispersion relation of the torus, yielding a rich spectral structure: gravity-capillary waves, sloshing modes and a center-of-mass mode [1]. We will show that nonlinear waves in form of solitons can propagate along the torus. We stress the observation of subsonic elevation solitons due to a strong influence of periodic boundary conditions in the Korteweg-de Vries equation, giving a non-trivial dependence of the soliton velocity on its amplitude. Finally, a triadic resonance instability is observed between sloshing and gravity-capillary modes, potentially allowing wave turbulence.

1. F. Novkoski, E. Falcon and C.-T. Pham, Experimental Dispersion Relation of Surface Waves along a Torus of Fluid, Phys. Rev. Lett., 127, 144504 (2021).

2. F. Novkoski, C.-T. Pham and E. Falcon, Experimental Periodic Korteweg-de Vries solitons along a torus of fluid, submitted to Europhys. Lett., (2022).   

Wave-turbulence decomposition of particle trajectories.

The ocean surface flow controls air-sea fluxes of gas, heat, and momentum, and is characterized by wind-driven waves, and turbulence, among other processes. Decomposing the relative effects of waves and turbulence on velocity signals has long been of interest for Eulerian measurements. However, Lagrangian particle trajectories in this system have received relatively little attention. The decomposition of wave and turbulence motion in the frequency range in which they overlap continues to be a challenging problem. Many approaches limit their scope to irrotational waves and assume that waves and turbulence do not interact. We aim to develop a data-driven method using Dynamic mode decomposition (DMD) that does not make any of the assumptions mentioned, and additionally provides insight into the wave-turbulence interactions. We collected data from experiments in a wind-driven wave tank. Both the laboratory and synthetically-generated data were analyzed using conventional wave-turbulence decomposition methods and the proposed DMD method. This is a promising avenue that could have broad applications in the study of Lagrangian measurements.   

Numerical evaluation of the spectral evolution of internal gravity waves due to wave-wave interactions

The kinetic energy spectra of oceanic internal gravity waves (IGWs) from recent field measurements and wave turbulence theory exhibit large variability, deviating from the standard Garrett-Munk (GM) spectrum. An improved understanding of energy cascade across scales by IGWs is needed for better parameterization of smallscale dissipation and ocean mixing. In this work, we conduct direct numerical simulations of the kinetic equation, which describes the spectral evolution of IGWs due to wave-triad interactions in the wave turbulence theory. Dominant mechanisms of energy cascade are identified, and energy fluxes in the wavenumber-frequency space for non-GM spectra are calculated. This will shed light on a new formulation of finescale parameterization incorporating varying spectral slopes of IGWs and a realistic ocean environment.