Laminated Wave Turbulence: Generic Algorithms III

Model of laminated wave turbulence allows to study statistical and discrete layers of turbulence in the frame of the same model. Statistical layer is described by Zakharov-Kolmogorov energy spectra in the case of irrational enough dispersion function. Discrete layer is covered by some system(s) of Diophantine equations while their form is determined by wave dispersion function. This presents a very special computational challenge - to solve Diophantine equations in many variables, usually 6 to 8, in high degrees, say 16, in integers of order $10^{16}$ and more. Generic algorithms for solving this problem in the case of {\it irrational} dispersion function have been presented in our previous papers. In this paper we present a new generic algorithm for the case of {\it rational} dispersion functions. Special importance of this case is due to the fact that in wave systems with rational dispersion the statistical layer does not exist and the general energy transport is governed by the discrete layer alone.Comment: submitted to IJMP

Cluster formation in mesoscopic systems

Graph-theoretical approach is used to study cluster formation in mesocsopic systems. Appearance of these clusters are due to discrete resonances which are presented in the form of a multigraph with labeled edges. This presentation allows to construct all non-isomorphic clusters in a finite spectral domain and generate corresponding dynamical systems automatically. Results of MATHEMATICA implementation are given and two possible mechanisms of cluster destroying are discussed

Cluster Dynamics of Planetary Waves

The dynamics of nonlinear atmospheric planetary waves is determined by a small number of independent wave clusters consisting of a few connected resonant triads. We classified the different types of connections between neighboring triads that determine the general dynamics of a cluster. Each connection type corresponds to substantially different scenarios of energy flux among the modes. The general approach can be applied directly to various mesoscopic systems with 3-mode interactions, encountered in hydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.Comment: 6 pages, 3 figs, EPL, publishe

A Model of Intra-seasonal Oscillations in the Earth atmosphere

We suggest a way of rationalizing an intra-seasonal oscillations (IOs) of the Earth atmospheric flow as four meteorological relevant triads of interacting planetary waves, isolated from the system of all the rest planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of IOs both in the North and South Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.Comment: 4 pages, 2 figs, Submitted to PR

Nonlinear resonances of water waves

In the last fifteen years, a great progress has been made in the understanding of the nonlinear resonance dynamics of water waves. Notions of scale- and angle-resonances have been introduced, new type of energy cascade due to nonlinear resonances in the gravity water waves have been discovered, conception of a resonance cluster has been much and successful employed, a novel model of laminated wave turbulence has been developed, etc. etc. Two milestones in this area of research have to be mentioned: a) development of the $q$-class method which is effective for computing integer points on the resonance manifolds, and b) construction of the marked planar graphs, instead of classical resonance curves, representing simultaneously all resonance clusters in a finite spectral domain, together with their dynamical systems. Among them, new integrable dynamical systems have been found that can be used for explaining numerical and laboratory results. The aim of this paper is to give a brief overview of our current knowledge about nonlinear resonances among water waves, and formulate three most important open problems at the end.Comment: 14 pages, 3 figures, to appear in DCDS, final version (small changes in the text, type errors corrected, some additional bibliographic items added

Time scales and structures of wave interaction

In this paper we give a general account of Wave Interaction Theory which by now consists of two parts: kinetic wave turbulence theory (WTT), using a statistical description of wave interactions, and the D-model recently introduced in \emph{Kartashova, PRE \textbf{86}: 041129 (2012)} describing interactions of distinct modes. Applying time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Sch\"{o}dinger equation, we give an overview of the structures appearing in Wave Interaction Theory, their time scales and characteristic times. We demonstrate that kinetic cascade and D-cascade are not competing processes but rather two processes taking place at different time scales, at different characteristic levels of nonlinearity and due to different physical mechanisms. Taking surface water waves as an example we show that energy cascades in this system occur at much faster characteristic times than those required by the kinetic WTT but can be described as D-cascades. As D-model has no special pre-requisites, it may be rewarding to re-evaluate existing experiments in other wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc. To appear in EP