# Wave turbulence

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In continuum mechanics, wave turbulence is a set of nonlinear waves deviated far from thermal equilibrium. Such a state is usually accompanied by dissipation. It is either decaying turbulence or requires an external source of energy to sustain it. Examples are waves on a fluid surface excited by winds or ships, and waves in plasma excited by electromagnetic waves etc.

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially uniform and temporally constant.

## Appearance

External sources by some resonant mechanism usually excite waves with frequencies and wavelengths in some narrow interval. For example, shaking container with the frequency ω excites surface waves with the frequency ω/2 (parametric resonance, discovered by Michael Faraday). When wave amplitudes are small – which usually means that the wave is far from breaking – only those waves exist that are directly excited by an external source.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light. In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Michael Faraday FRS was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

When, however, wave amplitudes are not very small (for surface waves: when the fluid surface is inclined by more than few degrees) waves with different frequencies start to interact. That leads to an excitation of waves with frequencies and wavelengths in wide intervals, not necessarily in resonance with an external source. It can be observed in the experiments with a high amplitude of shaking that initially the waves appear which are in resonance. Thereafter both longer and shorter waves appear as a result of wave interaction. The appearance of shorter waves is referred to as a direct cascade while longer waves are part of an inverse cascade of wave turbulence. In mechanical systems, resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.

## Statistical wave turbulence and discrete wave turbulence

Two generic types of wave turbulence should be distinguished: statistical wave turbulence (SWT) and discrete wave turbulence (DWT).

In SWT theory exact and quasi-resonances are omitted, which allows using some statistical assumptions and describing the wave system by kinetic equations and their stationary solutions – the approach developed by Vladimir E. Zakharov. These solutions are called Kolmogorov–Zakharov (KZ) energy spectra and have the form k−α, with k the wavenumber and α a positive constant depending on the specific wave system.  The form of KZ-spectra does not depend on the details of initial energy distribution over the wave field or on the initial magnitude of the complete energy in a wave turbulent system. Only the fact the energy is conserved at some inertial interval is important. Vladimir Evgen'evich Zakharov is a Soviet and Russian mathematician and physicist. He is currently Regents' Professor of mathematics at The University of Arizona, director of the Mathematical Physics Sector at the Lebedev Physical Institute, and is on the committee of the Stefanos Pnevmatikos International Award. Zakharov's research interests cover physical aspects of nonlinear wave theory in plasmas, hydrodynamics, oceanology, geophysics, solid state physics, optics, and general relativity. In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

The subject of DWT, first introduced in Kartashova (2006), are exact and quasi-resonances. Previous to the two-layer model of wave turbulence, the standard counterpart of SWT were low-dimensioned systems characterized by a small number of modes included. However, DWT is characterized by resonance clustering,  and not by the number of modes in particular resonance clusters – which can be fairly big. As a result, while SWT is completely described by statistical methods, in DWT both integrable and chaotic dynamics are accounted for. A graphical representation of a resonant cluster of wave components is given by the corresponding NR-diagram (nonlinear resonance diagram). 

In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude.

In some wave turbulent systems both discrete and statistical layers of turbulence are observed simultaneously, this wave turbulent regime have been described in Zakharov et al. (2005) and is called mesoscopic . Accordingly, three wave turbulent regimes can be singled out—kinetic, discrete and mesoscopic described by KZ-spectra, resonance clustering and their coexistence correspondingly.  Energetic behavior of kinetic wave turbulent regime is usually described by Feynman-type diagrams (i.e. Wyld's diagrams), while NR-diagrams are suitable for representing finite resonance clusters in discrete regime and energy cascades in mesoscopic regimes. Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate length. The scale of these materials can be described as being between the nanoscale size of a quantity of atoms and of materials measuring micrometres. The lower limit can also be defined as being the size of individual atoms. At the micrometre level are bulk materials. Both mesoscopic and macroscopic objects contain a large number of atoms. Whereas average properties derived from its constituent materials describe macroscopic objects, as they usually obey the laws of classical mechanics, a mesoscopic object, by contrast, is affected by fluctuations around the average, and is subject to quantum mechanics. In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. The scheme is named after its inventor, American physicist Richard Feynman, and was first introduced in 1948. The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. As David Kaiser writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations", and so "Feynman diagrams have revolutionized nearly every aspect of theoretical physics". While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory.

1. Zakharov, V.E.; Lvov, V.S.; Falkovich, G.E. (1992). Kolmogorov Spectra of Turbulence I – Wave Turbulence. Berlin: Springer-Verlag. ISBN   3-540-54533-6.
2. Kartashova, E. (2010). Nonlinear Resonance Analysis. Cambridge University Press. ISBN   978-0-521-76360-8.