Alexander O. Korotkevich

Mesoscopic wave turbulence.

We report results of simulation of wave turbulence. Both inverse and direct cascades are observed. The definition of “mesoscopic turbulence” is given. This is a regime when the number of modes in a system involved in turbulence is high enough to qualitatively simulate most of the processes but significantly smaller than the threshold, which gives us quantitative agreement with the statistical description, such as the kinetic equation. Such a regime takes place in numerical simulation, in essentially finite systems, etc.

Weak turbulent Kolmogorov spectrum for surface gravity waves.

We study the long-time evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in moderately small wave numbers. We numerically implemented the primitive Euler equations for the potential flow of an ideal fluid with free surface written in Hamiltonian canonical variables, using the expansion of the Hamiltonian in powers of nonlinearity of terms up to fourth order. We show that because of nonlinear interaction processes a stationary Fourier spectrum of a surface elevation close to <|eta(k)|(2)> approximately k(-7/2) is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.

Weak turbulence of gravity waves

For the first time weak turbulent theory was demonstrated for surface gravity waves. Direct numerical simulation of the dynamical equations shows Kolmogorov turbulent spectra as predicted by analytical analysis [1] from kinetic equation.

Coexistence of weak and strong wave turbulence in a swell propagation

By performing two parallel numerical experiments-solving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equation-we examined the applicability of the theory of weak turbulence to the description of the time evolution of an ensemble of free surface waves (a swell) on deep water. We observed qualitative coincidence of the results. To achieve quantitative coincidence, we augmented the kinetic equation by an empirical dissipation term modeling the strongly nonlinear process of white capping. Fitting the two experiments, we determined the dissipation function due to wave breaking and found that it depends very sharply on the parameter of nonlinearity (the surface steepness). The onset of white capping can be compared to a second-order phase transition. The results corroborate the experimental observations of Banner, Babanin, and Young.

Decay of the Monochromatic Capillary Wave

It was demonstrated by direct numerical simulation that, in the case of weakly nonlinear capillary waves, one can get resonant waves interaction on the discrete grid when resonant conditions are never fulfilled exactly. The waves’s decay pattern was obtained. The influence of the mismatch of resonant condition was studied as well.

Communication through plasma sheaths

We wish to transmit messages to and from a hypersonic vehicle around which a plasma sheath has formed. For long distance transmission, the signal carrying these messages must be necessarily low frequency, typically 2 GHz, to which the plasma sheath is opaque. The idea is to use the plasma properties to make the plasma sheath appear transparent.

Numerical verification of the weak turbulent model for swell evolution

Abstract The purpose of this article is to numerically verify the theory of weak turbulence. We have performed numerical simulations of an ensemble of nonlinearly interacting free gravity waves (a swell) by two different methods: by solving the primordial dynamical equations describing the potential flow of an ideal fluid with a free surface, and by solving the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. In both cases we have observed effects predicted by this theory: frequency downshift, angular spreading and formation of a Zakharov–Filonenko spectrum I ω ∼ ω −4 . To achieve quantitative coincidence of the results obtained by different methods, we have to augment the Hasselmann kinetic equation by an empirical dissipation term S diss modeling the coherent effects of white-capping. Using the standard dissipation terms from the operational wave predicting model (WAM) leads to a significant improvement on short times, but does not resolve the discrepancy completely, leaving the question about the optimal choice of S diss open. In the long run, WAM dissipative terms essentially overestimate dissipation.

Simultaneous numerical simulation of direct and inverse cascades in wave turbulence.

The results of the direct numerical simulation of isotropic turbulence of surface gravity waves in the framework of Hamiltonian equations are presented. For the first time, the simultaneous formation of both direct and inverse cascades has been observed in the framework of the primordial dynamical equations. At the same time, a strong long wave background has been developed. It has been shown that the Kolmogorov spectra obtained are very sensitive to the presence of this condensate. Such a situation has to be typical for experimental wave tanks, flumes, and small lakes.

Complex Singularity of a Stokes Wave

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower half-plane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase in the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value Hmax/λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here, H is the wave height and λ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance νc from the lowest singularity in the upper half-plane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the square-root branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for νc → 0 we suggest a new power law scaling νc ∝ (Hmax − H)3/2 as well as new estimate Hmax/λ ≃ 0.1410633.

On dissipation function of ocean waves due to whitecapping

The Hasselmann kinetic equation [1] provides a statistical description of waves ensemble. Several catastrophic events are beyond statistical model. In the case of gravity waves on the surface of the deep fluid may be the most frequent and important events of such kind are whitecapping and wave breaking. It was shown earlier that such effects leads to additional dissipation in the energy contaning region around waves spectral peak, which can be simulated by means of empiric dissipative term in kinetic equation. In order to find dependence of this term with respect to nonlinearity in the system (steepness of the surface) we preformed two numerical experiments: weakly nonlinear one in the framework of 3D hydrodynamics and fully nonlinear one for 2D hydrodynamic. In spite of significantly different models and initial conditions, both these experiments yielded close results. Obtained data can be used to define analytical formula for dependence of the dissipative term of dissipation coefficient with respect to ...