Andrei N. Pushkarev

Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schro¨dinger equation

The nonlinear Schrodinger (NLS) equation iΨt + ∇2Ψ + α⋎Ψ⋎sΨ = 0 is a canonical and universal equation which is of major importance in continuum mechanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension and s is the order of nonlinearity, can be understood in terms of a combination of weak turbulence theory and condensate and collapse formation. The results are derived in the broad context of a class of Hamiltonian systems of which NLS is a member, so that the reader can gain a perspective on the ingredients important for the realization of the various equilibrium spectra, thermodynamic, pure Kolmogorov and combinations thereof. We also present time-dependent, self-similar solutions which describe the relaxation of the system towards these equilibrium states. We show that the number of particles lost in an individual collapse event is virtually independent of damping. Our numerical simulation of the full governing equations is the first to show the validity of the weak turbulence approximation. We also present a mechanism for intermittency which should have widespread application. It is caused by strongly nonlinear collapse events which are nucleated by a flow of particles towards the origin in wavenumber space. These highly organized events result in a cascade of particle number towards high wavenumbers and give rise to an intermittency and a behavior which violates many of the usual Kolmogorov assumptions about the loss of statistical information and the statistical independence of large and small scales. We discuss the relevance of these ideas to hydrodynamic turbulence in the conclusion.

Turbulence of capillary waves: theory and numerical simulation

Abstract 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 Ik=C(P1/2/σ3/4)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.

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 approach to the wind-generated gravity sea waves

Abstract We performed numerical simulation of the kinetic equation describing behavior of an ensemble of random-phase, spatially homogeneous gravity waves on the surface of the infinitely deep ocean. Results of simulation support the theory of weak turbulence not only in its basic points, but also in many details. The weak turbulent theory predicts that the main physical processes taking place in the wave ensemble are down-shift of spectral peak and “leakage” of energy and momentum to the region of very small scales where they are lost due to local dissipative processes. Also, the spectrum of energy right behind the spectral peak should be close to the weak turbulent Kolmogorov spectrum which is the exact solution of the stationary kinetic (Hasselmann) equation. In a general case, this solution is anisotropic and is defined by two parameters—fluxes of energy and momentum to high wave numbers. Even in the anisotropic case the solution in the high wave number region is almost proportional to the universal form ω−4. This result should be robust with respect to change of the parameters of forcing and damping. In all our numerical experiments, the ω−4 Kolmogorov spectrum appears in very early stages and persists in both stationary and non-stationary stages of spectral development. A very important aspect of the simulations conducted here was the development of a quasi-stationary wave spectrum under wind forcing, in absence of any dissipation mechanism in the spectral peak region. This equilibrium is achieved in the spectral range behind the spectral peak due to compensation of wind forcing and leakage of energy and momentum to high wave numbers due to nonlinear four-wave interaction. Numerical simulation demonstrates slowing down of the shift of the spectral peak and formation of the bimodal angular distribution of energy in the agreement with field and laboratory experimental data. A more detailed comparison with the experiment can be done after developing of an upgraded code making possible to model a spatially inhomogeneous ocean.

On the kolmogorov and frozen turbulence in numerical simulation of capillary waves

Abstract Numerical simulation of dynamical equations for capillary waves excited by long-scale forcing shows the presence of both Kolmogorov spectrum at high wavenumbers (with the index predicted by weak-turbulent theory) and non-monotonic spectrum at low wavenumbers. The value of the Kolmogorov constant measured in numerical experiments happens to be different from the theoretical one. We explain the difference by the coexistence of Kolmogorov and “frozen” turbulence with the help of maps of quasi-resonances. Observed results are believed to be generic for different physical dispersive systems and are confirmed by laboratory experiments.

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.

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.

Computer simulation of Langmuir collapse

Abstract The problem of Langmuir wave collapse in 2D and 3D plasma is considered. A new approach for computer simulation of this phenomenon is proposed, which includes two different theoretical models: averaged dynamical equations and Vlasovs set of equations. It allows to take into account all essential effects during the whole process of collapse and, hence, to get a reliable picture of the collapse in detail and to save markedly computer resources. Peculiarities of the numeric methods are also discussed.

A particle model for three-dimensional Langmuir collapse simulation

Abstract The numerical simulation of the final stage of Langmuir collapse requires three-dimensional kinetic consideration and thus is at the limit of todays computer capability. We have managed to solve this problem by making systematic use of the theoretically known fundamental physical properties of a collapsing cavity, and by taking every possible measure to match the physical parallelism of the problem with the architecture of the multiprocessor complex used. The pecularities of the numerical model, the software and hardware implementation principles of this model, and also a number of the results of the simulation are presented.

A noncollisional mechanism for plasma species separation

Abstract A noncollisional mechanism for the separation of ion species moving through a plasma is proposed. The results of a numerical simulation prove the possibility of anomalous quick spatial separation of ion species on the scale of about 200 Debye radii.