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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.

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

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.

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.^{ [1] } 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*,^{ [2] } 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).^{ [3] }

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.^{ [4] } 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.

- ↑ 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. - ↑ Kartashova (2007)
- ↑ Kartashova (2009)
- ↑ Kartashova, E. (2010).
*Nonlinear Resonance Analysis*. Cambridge University Press. ISBN 978-0-521-76360-8.

**Spectroscopy** is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency, predominantly in the electromagnetic spectrum, though matter waves and acoustic waves can also be considered forms of radiative energy; recently, with tremendous difficulty, even gravitational waves have been associated with a spectral signature in the context of LIGO and laser interferometry. Spectroscopic data are often represented by an emission spectrum, a plot of the response of interest as a function of wavelength or frequency.

In fluid dynamics, **turbulence** or **turbulent flow** is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

In fluid dynamics, **gravity waves** are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which gives rise to wind waves.

**Computational fluid dynamics** (**CFD**) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.

A physical quantity is said to have a **discrete spectrum** if it takes only distinct values, with gaps between one value and the next.

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

**Four-wave mixing** (FWM) is an intermodulation phenomenon in non-linear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave mixing can be compared to the intermodulation distortion in standard electrical systems. It is a parametric nonlinear process, in that the energy of the incoming photons is conserved. FWM is a phase-sensitive process, in that the efficiency of the process is strongly affected by phase matching conditions.

The **Stationary wavelet transform** (SWT) is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform (DWT). Translation-invariance is achieved by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of in the th level of the algorithm. The SWT is an inherently redundant scheme as the output of each level of SWT contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of N in the wavelet coefficients. This algorithm is more famously known as "*algorithme à trous*" in French which refers to inserting zeros in the filters. It was introduced by Holschneider et al.

A **direct numerical simulation (DNS)** is a simulation in computational fluid dynamics in which the Navier–Stokes equations are numerically solved without any turbulence model. This means that the whole range of spatial and temporal scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales, up to the integral scale , associated with the motions containing most of the kinetic energy. The Kolmogorov scale, , is given by

In fluid dynamics, **turbulence kinetic energy** (**TKE**) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations.

For a pure wave motion in fluid dynamics, the **Stokes drift velocity** is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation.

In fluid dynamics, a **Tollmien–Schlichting wave** is a streamwise unstable wave which arises in a bounded shear flow. It is one of the more common methods by which a laminar bounded shear flow transitions to turbulence. The waves are initiated when some disturbance interacts with leading edge roughness in a process known as receptivity. These waves are slowly amplified as they move downstream until they may eventually grow large enough that nonlinearities take over and the flow transitions to turbulence.

The **Reynolds number** is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow, and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.

**Internal tides** are generated as the surface tides move stratified water up and down sloping topography, which produces a wave in the ocean interior. So internal tides are internal waves at a tidal frequency. The other major source of internal waves is the wind which produces internal waves near the inertial frequency. When a small water parcel is displaced from its equilibrium position, it will return either downwards due to gravity or upwards due to buoyancy. The water parcel will overshoot its original equilibrium position and this disturbance will set off an internal gravity wave. Munk (1981) notes, "Gravity waves in the ocean's interior are as common as waves at the sea surface-perhaps even more so, for no one has ever reported an interior calm."

**Magnetohydrodynamic turbulence** concerns the chaotic regimes of magnetofluid flow at high Reynolds number. Magnetohydrodynamics (MHD) deals with what is a quasi-neutral fluid with very high conductivity. The fluid approximation implies that the focus is on macro length-and-time scales which are much larger than the collision length and collision time respectively.

The **Kolmogorov–Zurbenko (KZ) filter** was first proposed by A. N. Kolmogorov and formally defined by Zurbenko. It is a series of iterations of a moving average filter of length *m*, where *m* is a positive, odd integer. The KZ filter belongs to the class of low-pass filters. The KZ filter has two parameters, the length *m* of the moving average window and the number of iterations *k* of the moving average itself. It also can be considered as a special window function designed to eliminate spectral leakage.

**Multiscale turbulence** is a class of turbulent flows in which the chaotic motion of the fluid is forced at different length and/or time scales. This is usually achieved by immersing in a moving fluid a body with a multiscale, often fractal-like, arrangement of length scales. This arrangement of scales can be either passive or active

In continuum mechanics, an **energy cascade** refers to the transfer of energy from large scales of motion to the small scales or a transfer of energy from the small scales to the large scales. This transfer of energy between different scales requires that the dynamics of the system is nonlinear. Strictly speaking, a cascade requires the energy transfer to be local in scale, evoking a cascading waterfall from pool to pool without long-range transfers across the scale domain.

- 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. - Nazarenko, Sergey (2011).
*Wave Turbulence*. Springer-Verlag. ISBN 978-3642159411. - Kartashova, E. (2006). "A model of laminated turbulence".
*JETP Letters*.**83**(7): 283–287. arXiv: physics/0512014 . Bibcode:2006JETPL..83..283K. doi:10.1134/S0021364006070058. - Kartashova, E. (2007). "Exact and quasi-resonances in discrete water wave turbulence".
*Physical Review Letters*.**98**(21): 214502 (4 pp.). arXiv: math-ph/0701077 . Bibcode:2007PhRvL..98u4502K. doi:10.1103/PhysRevLett.98.214502. - Zakharov, V.E.; Korotkevich, A.O.; Pushkarev, A.N.; Dyachenko, A.I. (2005). "Mesoscopic wave turbulence".
*JETP Letters*.**82**(8): 487–491. arXiv: physics/0508155 . doi:10.1134/1.2150867. - Kartashova, E. (2009). "Discrete wave turbulence".
*EPL*.**87**(4): 44001 (5 pp.). arXiv: 0907.4406 . Bibcode:2009EL.....8744001K. doi:10.1209/0295-5075/87/44001.

- Newell, A.C.; Rumpf, B. "Wave turbulence".
*Annual Review of Fluid Mechanics*.**43**: 59–78. Bibcode:2011AnRFM..43...59N. doi:10.1146/annurev-fluid-122109-160807.

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