Wave turbulence
Encyclopedia
Wave turbulence is a set of wave
s deviated far from thermal equilibrium
. Such state is accompanied by dissipation
. It is either decaying turbulence or requires external source of energy
to sustain it. Examples are waves on a fluid surface
excited by wind
s or ship
s, and waves in plasma
excited by electromagnetic waves etc.
External sources by some resonant mechanism usually excite waves with frequencies
and wavelength
s 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 amplitude
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. 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
, then 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.
. 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 wavefield 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
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 Feynmann-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.
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
s deviated far from thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
. Such state is accompanied by dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
. It is either decaying turbulence or requires external source of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
to sustain it. Examples are waves on a fluid surface
Free surface
In physics, a free surface is the surface of a fluid that is subject to constant perpendicular normal stress and zero parallel shear stress,such as the boundary between two homogenous fluids,for example liquid water and the air in the Earth's atmosphere...
excited by wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...
s or ship
Ship
Since the end of the age of sail a ship has been any large buoyant marine vessel. Ships are generally distinguished from boats based on size and cargo or passenger capacity. Ships are used on lakes, seas, and rivers for a variety of activities, such as the transport of people or goods, fishing,...
s, and waves in plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
excited by electromagnetic waves etc.
External sources by some resonant mechanism usually excite waves with frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
and wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It 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...
s 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
Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....
).
When wave amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
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. 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
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
, then 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. ZakharovVladimir E. Zakharov
Vladimir Evgen'evich Zakharov is a Soviet and Russian mathematician and physicist. He is currently Regents' Professor of mathematics at The University of Arizona and director of the Mathematical Physics Sector at the Lebedev Physical Institute...
. These solutions are called Kolmogorov–Zakharov (KZ) energy spectra and have the form k−α, with k the wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
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 wavefield 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
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 independent of amplitude.-Description:Generically two...
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 Feynmann-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.