arXiv subjects

Elena Kartashova

Publications and source records attributed to Elena Kartashova.

At least 19 recordsLinked to original sources

arXiv2013

Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time $t = t_b$ has an asymptotic decay of $k^{-4/3}$, with corresponding energy spectrum decaying as $k^{-8/3}$. This spectrum is formed by the singularity of the form $(x-x_b)^{1/3}$ in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wave numbers if small dissipation or dispersion is accounted in the viscous Burgers or Korteweg-de Vries equations.

math-ph↗

arXiv2013

Universal breaking point asymptotic for energy spectrum of Riemann waves in weakly nonlinear non-dispersive media

In this Letter we study the form of the energy spectrum of Riemann waves in weakly nonlinear non-dispersive media. For quadratic and cubic nonlinearity we demonstrate that the deformation of an Riemann wave over time yields an exponential energy spectrum which turns into power law asymptotic with the slope being approximately -8/3 at the last stage of evolution before breaking. We argue, that this is the universal asymptotic behaviour of Riemann waves in any nonlinear non-dispersive medium at the point of breaking. The results reported in this Letter can be used in various non-dispersive media, e.g. magneto-hydro dynamics, physical oceanography, nonlinear acoustics.

physics.flu-dyn↗

arXiv2013

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 Schödinger 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 EPL

physics.flu-dyn↗

arXiv2013

Turbulence of capillary waves revisited

Kinetic regime of capillary wave turbulence is classically regarded in terms of three-wave interactions with the exponent of power energy spectrum being $ν=-7/4$ (two-dimensional case). We show that a number of assumptions necessary for this regime to occur can not be fulfilled. Four-wave interactions of capillary waves should be taken into account instead, which leads to exponents $ν=-13/6$ and $ν=-3/2$ for one- and two-dimensional wavevectors correspondingly. It follows that for general dispersion functions of decay type, three-wave kinetic regime need not prevail and higher order resonances may play a major role. Accepted for publication.

physics.flu-dyn↗

arXiv2012

A constructive method for computing generalized Manley--Rowe constants of motion

The Manley--Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley--Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a \emph{Mathematica} implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

physics.flu-dyn↗

arXiv2012

Energy transport in weakly nonlinear wave systems with narrow frequency band excitation

A novel discrete model (D-model) is presented describing nonlinear wave interactions in systems with small and moderate nonlinearity under narrow frequency band excitation. It integrates in a single theoretical frame two mechanisms of energy transport between modes, namely intermittency and energy cascade and gives conditions when which regime will take place. Conditions for the formation of a cascade, cascade direction, conditions for cascade termination, etc. are given and depend strongly on the choice of excitation parameters. The energy spectra of a cascade may be computed yielding discrete and continuous energy spectra. The model does not need statistical assumptions as all effects are derived from the interaction of distinct modes. In the example given -- surface water waves with dispersion function $ø^2=g\,k$ and small nonlinearity -- D-model predicts asymmetrical growth of side-bands for Benjamin-Feir instability while transition from discrete to continuous energy spectrum excitation parameters properly chosen yields the saturated Phillips' power spectrum $\sim g^2ø^{-5}$. D-model can be applied to the experimental and theoretical study of numerous wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc.

physics.flu-dyn↗

arXiv2011

Energy spectra of \textbf{2D} gravity and capillary waves with narrow frequency band excitation

In this Letter we present a new method, called chain equation method (CEM), for computing a cascade of distinct modes in a two-dimensional weakly nonlinear wave system generated by narrow frequency band excitation. The CEM is a means for computing the quantized energy spectrum as an explicit function of frequency $ø_0$ and stationary amplitude $A_0$ of excitation. The physical mechanism behind the generation of the quantized cascade is modulation instability. The CEM can be used in numerous \textbf{2D} weakly nonlinear wave systems with narrow frequency band excitation appearing in hydrodynamics, nonlinear optics, electrodynamics, convection theory etc. In this Letter the CEM is demonstrated with examples of gravity and capillary waves with dispersion functions $ø(k) \sim k^{1/2}$ and $ø(k) \sim k^{3/2}$ respectively, and for two different levels of nonlinearity $\eps=A_0k_0$: small ($\eps\sim 0.1$ to 0.25) and moderate ($\eps\sim 0.25$ to 0.4).

physics.flu-dyn↗

arXiv2011

Dynamical cascade generation as basic mechanism of Benjamin-Feir instability

A novel model of discretized energy cascade generated by Benjamin-Feir instability is presented. Conditions for appearance of direct and inverse cascades are given explicitly, as well as conditions for stabilization of the wave system due to cascade termination. These results can be used directly for explanation of available results of laboratory experiments and as basic forecast scenarios for planned experiments, depending on the frequency of an initially excited mode and steepness of its amplitude.

physics.flu-dyn↗

arXiv2011

A novel model of wave turbulence

A novel D-model of wave turbulence is presented which allows to reproduce in a single frame various nonlinear wave phenomena such as intermittency, formation and direction of energy cascades, possible growth of nonlinearity due to direct energy cascades, etc. depending on the initial state. No statistical assumptions are used, all effects are due to the behavior of distinct modes. Classical energy spectra $E_ø\sim ø^{-ν}, \ ν= \const >0,$ for dispersion function of the form $ ø\sim k^{\a}, \ \a>0$ are obtained as a particular case of a more general form of energy spectra: $E_ø \sim ø^{-ν}, \ 2+\a^{-1}\le ν\le 2(2+\a^{-1})$ where magnitude of $ν$ is defined by the parameters of the initial excitation. D-model is a generic model which can be expanded into a hierarchy of more refined models including dissipation, forcing, etc. D-model can be applied to the experimental and theoretical study of numerous wave turbulent systems appearing in hydrodynamics, nonlinear optics, electrodynamics, convection theory, etc.

physics.flu-dyn↗

arXiv2011

Computable Integrability

This is a preliminary version of the textbook on integrable systems. The work has been partly supported by Grant Nr.10/2006-RU, Austrian Academic Exchange Service ÖAD and Grant P20164-N18, Austrian Science Fund FWF

nlin.SI↗

arXiv2010

Discrete wave turbulence of rotational capillary water waves

We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are obtained. We also present the construction and investigation of resonance clustering. Some physical implications of the obtained results are discussed.

math-ph↗

arXiv2010

Resonance clustering in wave turbulent regimes: Integrable dynamics

Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in $k$-space, and 2) existence of "gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic}, where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.

nlin.SI↗

arXiv2009

Discrete Wave Turbulence

In this Letter we present discrete wave turbulence (DWT) as a counterpart of classical statistical wave turbulence (SWT). DWT is characterized by resonance clustering, not by the size of clusters, i.e. it includes, but is not reduced to, the study of low-dimensional systems. Clusters with integrable and chaotic dynamics co-exist in different sub-spaces of the $\mathbf{k}$-space. NR-diagrams are introduced, a handy graphical presentation of an arbitrary resonance cluster allowing to reconstruct uniquely dynamical system describing the cluster. DWT is shown to be a novel research field in nonlinear science, with its own methods, achievements and application areas.

physics.flu-dyn↗

arXiv2009

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.

nlin.SI↗

arXiv2009

Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

The influence of an underlying current on 3-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant 3-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows; 2) only positive constant vorticities can trigger the appearance of three-wave resonances; 3) the number of positive constant vorticities which do trigger a resonance is countable; 4) the magnitude of a positive constant vorticity triggering a resonance can not be too small.

physics.flu-dyn↗

arXiv2009

Effect of the dynamical phases on the nonlinear amplitudes' evolution

In this Letter we show how the nonlinear evolution of a resonant triad depends on the special combination of the modes' phases chosen according to the resonance conditions. This phase combination is called dynamical phase. Its evolution is studied for two integrable cases: a triad and a cluster formed by two connected triads, using a numerical method which is fully validated by monitoring the conserved quantities known analytically. We show that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of the clusters. Indeed, some effects are (i) to diminish the period of energy exchange $τ$ within a cluster by 20$%$ and more; (ii) to diminish, at time scale $τ$, the variability of wave energies by 25$%$ and more; (iii) to generate a new time scale, $T >> τ$, in which we observe considerable energy exchange within a cluster, as well as a periodic behaviour (with period $T$) in the variability of modes' energies. These findings can be applied, for example, to the control of energy input, exchange and output in Tokamaks; for explanation of some experimental results; to guide and improve the performance of experiments; to interpret the results of numerical simulations, etc.

nlin.SI↗

arXiv2008

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.

nlin.CD↗

arXiv2008

Dynamics of nonlinear resonances in Hamiltonian systems

It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We show that 1) most frequently met clusters are described by integrable dynamical systems, and 2) construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for systems with cubic Hamiltonian.

nlin.SI↗