Capillary Wave Turbulence Research Articles

Experimental evidence for the weak turbulence on the surface of liquid hydrogen

We present new results of experimental investigations of turbulence of capillary waves on the charged surface of liquid hydrogen. The high-frequency boundary of the inertial interval has been observed for the first time. The boundary frequency ω b is shifted towards higher frequencies with increasing the wave amplitude η p at a pumping frequency ω p in accordance with the theory of weak wave turbulence, ω b∼ η p 4/3 ω p 23/9.

The turbulence of capillary waves on the surface of liquid hydrogen

It is experimentally demonstrated that the surface excitation of liquid hydrogen at a low frequency results in the turbulent mode in a system of capillary waves. The experimental results are in good agreement with the theory of weak wave turbulence. The pair correlation function of the surface deviations is described by the exponential function ωm. The exponent m decreases in magnitude from m=−3.7±0.3 to −2.8±0.2 when the pumping at a single resonant frequency changes to broadband noise excitation. Measurements are made of the dependence of the boundary frequency ωb of the upper edge of the inertial range in which the Kolmogorov spectrum is formed on the wave amplitude ηp at the pumping frequency. It is demonstrated that the obtained data are well described by a function of the form ωb∝η p 4/3 ω p 23/9 .

Measurement of the boundary frequency of the inertial interval of capillary wave turbulence at the surface of liquid hydrogen

The boundary frequency was experimentally measured for the upper edge of an inertial interval corresponding to the Kolmogorov spectrum for energy distribution over the oscillation frequencies at the surface of liquid hydrogen. It is shown that the dependence of boundary frequency ωb on the wave amplitude ηp at the pump frequency ωp is well described by the power law ωb∼η p 4/3 ω p 23/9 .

Discreteness and quasiresonances in weak turbulence of capillary waves.

A numerical study is presented which deals with the kinematics of quasiresonant energy transfer in a system of capillary waves with a discrete wave number space in a periodic box. For a given set of initially excited modes and a given level of resonance broadening, the modes of the system are partitioned into two classes, one active, the other forbidden. For very weak nonlinearity the active modes are very sparse. It is possible that this sparsity explains discrepancies between the values of the Kolmogorov constant measured in numerical simulations of weakly turbulent cascades and the theoretical values obtained from the continuum theory. There is a critical level of nonlinearity below which the set of active modes has finite radius in wave number space. In this regime, an energy cascade to dissipative scales may not be possible and the usual Kolmogorov spectrum predicted by the continuum theory not realized.

Turbulence of capillary waves — theory and numerical simulation

An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics I k = C( P 1/2/ σ 3/4) k −19/4. The Kolmogorov constant C is found analytically together with the interval of locality in K → -space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost istropic Kolmogorov spectrum in the large k → region. Besides, at small amplitudes of the pumping, an esentially new phenomenon is found: “frozen” turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k → . This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.

Turbulence of capillary waves.

A numerical model for direct simulation of the surface of ideal fluid based on the expansion of theHamiltonian of the surface up to terms of fourth order is developed. For the case of capillary wavewe observe the formation of powerlike spectrum of spatial elevations close to one predicted by weak-turbulent theory

Large-scale properties of wave turbulence.

Wave turbulence for systems with only direct (small-scale) turbulent cascades is analyzed at scales much larger than the scale of the pumping. At such scales, the turbulence spectrum is shown to turn into an equilibrium Rayleigh-Jeans distribution with the temperature determined by the pumping scale and energy dissipation rate (the turbulent flux). The behavior of the damping of the waves changes drastically at a scale determined by the mean free path of turbulent waves. Two particular examples of acoustic and capillary-wave turbulence are considered. We also carried out numerics which confirm the theoretical predictions.

Two stages of decay turbulence of capillary waves

Evolution of decay turbulence of capillary waves in deep water is considered in the framework of the isotropic kinetic equation. It is shown that the evolution comprises of two stages. During the first stage an arbitrary localized large-scale wave distribution explosively evolves into a small-scale Kolmogorov spectrum. The second stage starts at the moment the Kolmogorov spectrum reaches dissipative scales. For systems with non-linear damping, the characteristic time of this stage is much longer (up to thousand times) than the first stage. The energy distribution is close to the Kolmogorov spectrum and decaying follows a self-similar law.

Experiments on capillary wave turbulence

Measurements of the response of a fluid surface to parametric excitation at a frequency that excites ripples, displays a number of qualitatively distinct regimes. As the amplitude is increased, standing waves (with higher harmonics) turn into a stationary, square ‘‘crystal,’’ pattern. At higher amplitudes, ‘‘dislocations’’ migrate through the pattern. While at the highest amplitudes, the spectrum becomes broadband with no discernable spatial pattern. Measurements are being made with a thin wire probe as well as the shadowgraph technique. The wire indicates that the broadband spectrum follows a power law: and the surface FFT indicates that many modes participate in the motion. Improvements in this absolute calibration are required before one can determine whether wave turbulence has been observed. [Work supported by the US DOE Division of Engineering and Geophysics and NASA−Microgravity.]

Weak turbulence of capillary waves

In recent years the theory of weak turbulence, i.e. the stochastic theory of nonlinear waves [I, 9], has been intensively developed. In the theory of weak turbulence nonlinearity of waves is assumed to be small; this enables us, using the hypothesis of the random nature of the phases of individual waves, to obtain the kinetic equation for the mean squares of the wave aplitudes.