A. I. Dyachenko

New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface

Abstract New method for numerical simulation of potential flows with a free surface of two-dimensional fluid, based on combination of the conformal mapping and Fourier Transform is proposed. The method is efficient for study of strongly nonlinear effects in gravity waves including wave breaking and formation of rogue waves.

Modulation instability of stokes wave : Freak wave

Formation of waves of large amplitude (freak waves, killer waves) at the surface of the ocean is studied numerically. We have observed that freak waves have the same ratio of the wave height to the wave length as limiting Stokes waves. When a freak wave reaches this limiting state, it breaks. The physical mechanism of freak wave formation is discussed.

Is free-surface hydrodynamics an integrable system?

Abstract A strong argument is found in support of the integrability of free-surface hydrodynamics in the one-dimensional case. It is shown that the first term in the perturbation series in powers of nonlinearity is identically equal to zero, the consequences of which are discussed as well.

Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

Abstract Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier–Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoullis equation), but also to the kinematic boundary condition. The nonlinear Schrodinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

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.

On the formation of freak waves on the surface of deep water

Numerical simulation of the fully nonlinear water equations demonstrates the existence of giant breathers on the surface of deep water. The numerical analysis shows that this breather (or soliton of envelope) does not loose energy. The existence of such a breather can explain the appearance of freak waves.

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.

Five-wave interaction on the surface of deep fluid

This article deals with the studying of the interaction of gravity waves propagating on the surface of an ideal fluid of infinite depth. The system of the corresponding equation is proven to be integrable up to the fourth order in power of steepness of the waves, but to be nonintegrable in the next, fifth, order. An exact formula for the five-wave scattering matrix element is obtained using diagram technique on the resonant surface. The stationary solutions of the five-wave kinetic equation are studied as well.

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.

Wave-vortex dynamics in drift and β-plane turbulence

Abstract For the theory of drift plasma and β-plane geophysical dynamics both large-scale vortex and small-scale wave components are important: linear excitation and dissipation occur mainly at small scales, while concentration of the energy spectrum takes place (through the inverse cascade) at large vortices. Based on the time and space separation of these scales averaged evolution equations are derived. The equation for the small scales describes the propagation of high-frequency quanta on the background of a flow produced by large-scale vortices; this equation provides the conservation of the spectral density of the potential enstrophy of small scales. The equation for the large-scale component is the Charney-Hasegawa-Mima equation with a source term having the form of the ponderomotive force and providing the inverse energy cascade from small to large scales. A new computational approach for the modeling of drift and β-plane turbulence is proposed on the basis of the equations obtained - the quantum in the cell method.