arXiv subjects

Elena Kartashova

Publications and source records attributed to Elena Kartashova.

27 records · Page 2Linked to original sources

arXiv2008

Resonant interactions of nonlinear water waves in a finite basin

We study exact four-wave resonances among gravity water waves in a square box with periodic boundary conditions. We show that these resonant quartets are linked with each other by shared Fourier modes in such a way that they form independent clusters. These clusters can be formed by two types of quartets: (1) {\it angle-resonances} which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) {\it scale-resonances} which are much more rare but which are the only ones that can transfer energy between different scales. We find such resonant quartets and their clusters numerically on the set of 1000 x 1000 modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wavefield evolution. Finite box effects and associated resonant interaction among discrete wave modes appear to be important in most numerical and laboratory experiments on the deep water gravity waves, and our work is aimed at aiding the interpretation of the experimental and numerical data.

nlin.SI↗

arXiv2007

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.

nlin.CD↗

arXiv2007

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.

nlin.PS↗

arXiv2007

Exact and quasi-resonances in discrete water-wave turbulence

The structure of discrete resonances in water-wave turbulence is studied. It is shown that the number of exact 4-wave resonances is huge (hundreds million) even in comparatively small spectral domain when both scale and angle energy transport is taken into account. It is also shown that angle transport can contribute inexplicitly to scale transport. Restrictions for quasi-resonances to start are written out. The general approach can be applied directly to mesoscopic systems encountered in condensed matter (quantum dots), medical physics, etc.

math-ph↗

arXiv2007

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.

math-ph↗

arXiv2007

Quantifier elimination for approximate Beals-Kartashova factorization

The only known constructive factorization algorithm for linear partial differential operators (LPDOs) is Beals-Kartashova (BK) factorization \cite{bk2005}. One of the most interesting features of BK-factorization: at the beginning all the first-order factors are constructed and afterwards the factorization condition(s) should be checked. This leads to the important application area - namely, numerical simulations which could be simplified substantially if instead of computation with one LPDE of order $n$ we will be able to proceed computations with $n$ LPDEs all of order 1. In numerical simulations it is not necessary to fulfill factorization conditions exactly but with some given accuracy, which we call approximate factorization. The idea of the present paper is to look into the feasibility of solving problems of this kind using quantifier elinination by cylindrical algebraic decomposition.

math-ph↗

arXiv2006

Laminated Wave Turbulence: Generic Algorithms I

The model of laminated wave turbulence presented recently unites both types of turbulent wave systems - statistical wave turbulence (introduced by Kolmogorov and brought to the present form by numerous works of Zakharov and his scientific school since nineteen sixties) and discrete wave turbulence (developed in the works of Kartashova in nineteen nineties). The main new feature described by this model is the following: discrete effects do appear not only in the long-wave part of the spectral domain (corresponding to small wave numbers) but all through the spectra thus putting forth a novel problem - construction of fast algorithms for computations in integers of order $10^{12}$ and more. In this paper we present a generic algorithm for polynomial dispersion functions and illustrate it by application to gravity and planetary waves.

math-ph↗

arXiv2006

Fast Computation Algorithm for Discrete Resonances among Gravity Waves

Traditionally resonant interactions among short waves, with large real wave-numbers, were described statistically and only a small domain in spectral space with integer wave-numbers, discrete resonances, had to be studied separately in resonators. Numerical simulations of the last few years showed unambiguously the existence of some discrete effects in the short-waves part of the wave spectrum. Newly presented model of laminated turbulence explains theoretically appearance of these effects thus putting a novel problem - construction of fast algorithms for computation of solutions of resonance conditions with integer wave-numbers of order $10^3$ and more. Example of such an algorithm for 4-waves interactions of gravity waves is given. Its generalization on the different types of waves is briefly discussed.

math-ph↗

arXiv2006

A model of laminated wave turbulence

A model of laminated wave turbulence is presented. This model consists of two co-existing layers - one with continuous waves' spectra, covered by KAM theory and Kolmogorov-like power spectra, and one with discrete waves' spectra, covered by discrete classes of waves and Clipping method. Some known laboratory experiments and numerical simulations are explained in the frame of this model.

physics.flu-dyn↗