In
continuum mechanics
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 mo ...
, wave turbulence is a set of
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s deviated far from
thermal equilibrium
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 ...
. Such a state is usually accompanied by
dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
. It is either
decaying turbulence or requires an external source of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
to sustain it. Examples are waves on a
fluid surface excited by
wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...
s or
ship
A ship is a large watercraft that travels the world's oceans and other sufficiently deep waterways, carrying cargo or passengers, or in support of specialized missions, such as defense, research, and fishing. Ships are generally distinguished ...
s, and waves in
plasma excited by
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
etc.
Appearance
External sources by some resonant mechanism usually excite waves with
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is e ...
and
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s in some narrow interval. For example, shaking a container with frequency ω excites surface waves
with frequency ω/2 (
parametric resonance, discovered by
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
).
When wave
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
s are small – which usually means that the wave is far from
breaking – only those waves exist that are directly excited by an external source.
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
Advocates for Informed Choice, doing business as, dba interACT or interACT Advocates for Intersex Youth, is a 501(c)(3) nonprofit organization using innovative strategies to advocate for the legal and human rights of children with intersex trai ...
. That leads to an excitation of waves with frequencies and wavelengths in wide intervals, not necessarily in resonance with an external source. In experiments with high shaking amplitudes one initially observes waves that are in
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscil ...
with one another. 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.
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
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
–Zakharov (KZ) energy spectra and have the form ''k''
−α, with ''k'' the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
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.
The subject of DWT, first introduced in , 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 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 ...
diagram).
In some wave turbulent systems both discrete and statistical layers of turbulence are observed ''simultaneously'', this wave turbulent regime have been described in 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
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
(i.e.
Wyld's diagrams), while NR-diagrams are suitable for representing finite resonance clusters in discrete regime and energy cascades in mesoscopic regimes.
Notes
References
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Further reading
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{{physical oceanography
Nonlinear systems
Water waves
Oceanography