Wave turbulence

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.


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.

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.

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.[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.

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 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.


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


Further reading

Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Breaking wave

In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour.

The most generally familiar sort of breaking wave is the breaking of water surface waves on a coastline. Wave breaking generally occurs where the amplitude reaches the point that the crest of the wave actually overturns—the types of breaking water surface waves are discussed in more detail below. Certain other effects in fluid dynamics have also been termed "breaking waves," partly by analogy with water surface waves. In meteorology, atmospheric gravity waves are said to break when the wave produces regions where the potential temperature decreases with height, leading to energy dissipation through convective instability; likewise Rossby waves are said to break when the potential vorticity gradient is overturned. Wave breaking also occurs in plasmas, when the particle velocities exceed the wave's phase speed.

Energy cascade

In continuum mechanics, an energy cascade involves the transfer of energy from large scales of motion to the small scales (called a direct energy cascade) or a transfer of energy from the small scales to the large scales (called an inverse energy cascade). 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 (only between fluctuations of nearly the same size), evoking a cascading waterfall from pool to pool without long-range transfers across the scale domain.

This concept plays an important role in the study of well-developed turbulence. It was memorably expressed in this poem by Lewis F. Richardson in the 1920s. Energy cascades are also important for wind waves in the theory of wave turbulence.

Consider for instance turbulence generated by the air flow around a tall building: the energy-containing eddies generated by flow separation have sizes of the order of tens of meters. Somewhere downstream, dissipation by viscosity takes place, for the most part, in eddies at the Kolmogorov microscales: of the order of a millimetre for the present case. At these intermediate scales, there is neither a direct forcing of the flow nor a significant amount of viscous dissipation, but there is a net nonlinear transfer of energy from the large scales to the small scales.

This intermediate range of scales, if present, is called the inertial subrange. The dynamics at these scales is described by use of self-similarity, or by assumptions – for turbulence closure – on the statistical properties of the flow in the inertial subrange. A pioneering work was the deduction by Andrey Kolmogorov in the 1940s of the expected wavenumber spectrum in the turbulence inertial subrange.

Gradient pattern analysis

Gradient pattern analysis (GPA) is a geometric computing method for characterizing geometrical bilateral symmetry breaking of an ensemble of symmetric vectors regularly distributed in a square lattice. Usually, the lattice of vectors represent the first-order gradient of a scalar field, here an M x M square amplitude matrix. An important property of the gradient representation is the following: A given M x M matrix where all amplitudes are different results in an M x M gradient lattice containing asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the amplitudes can modify the respective gradient pattern.

The original concept of GPA was introduced by Rosa, Sharma and Valdivia in 1999. Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and digital images.

Index of physics articles (W)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Index of wave articles

This is a list of Wave topics.

Kenneth M. Watson

Kenneth Marshall Watson (born September 7, 1921) is a theoretical physicist and physical oceanographer.Watson graduated in 1943 with BS in electrical engineering from Iowa State College. From 1943 to 1946 he was a researcher at the United States Naval Research Laboratory in Washington, D.C. During his work for the U.S. Navy he went to night school at George Washington University. He graduated from the University of Iowa with Ph.D. in 1948 with thesis The polarizability of the meson-charge cloud of a neutron in an external electrostatic field. He was from 1948 to 1949 an Atomic Energy Commission (AEC) Fellow at the Institute for Advanced Study and from 1949 to 1951 an AEC Fellow at the Berkeley Radiation Laboratory. He was from 1951 to 1954 an assistant professor of physics at Indiana University and from 1954 to 1957 an associate professor of physics at the University of Wisconsin, Madison. In 1953 he was elected a fellow of the American Physical Society. From 1957 to 1981 he was a staff member of Lawrence Berkeley National Laboratory, as well as a professor of physics at the University of California, Berkeley. In 1974 he was elected a member of the National Academy of Sciences. From 1981 to 1991 he was the director of the Marine Physical Laboratory, Scripps Institute of Oceanography, as well as a professor of physical oceanography at the University of California, San Diego. In 1991 he retired as professor emeritus. His doctoral students include Shang-keng Ma.Watson was an advisor to various United States organizations associated with the United States Department of Defense. In 1959 he worked with Marvin L. Goldberger, Keith Brueckner, and Murray Gell-Mann to join John A. Wheeler, Charles H. Townes, and others in forming the JASON group of government advisors. Watson remained in JASON until 1998. In 1971 he, with four others, formed the company Physical Dynamics, Inc. and then remained on the board of directors until 1981.He did research in the early 1950s on nuclear and pi meson physics, as well as quantum mechanical collision processes, and in the late 1950s on plasma physics

and controlled nuclear fusion.To quote Watson:

In the mid-1960’s I began a series of investigations, in collaboration with M. L. Goldberger of the observation of “entangled” quantum mechanical systems. We were concerned with sequential measurements and interference effects for correlated systems.

Watson did research in the early 1970s on atomic and molecular scattering and in the late 1970s on fluid mechanics related to oceanography. He worked in the early 1980s on applying methods of statistical mechanics to internal wave turbulence and in the early 1990’s on analyzing the coupling of surface and internal gravity waves.To quote Watson:

In the mid 1990’s my interest in nonlinear classical mechanics and ocean surface waves led to a study of capillary waves (few centimeter wavelengths) interacting with longer waves (10 cm to a meter wavelengths).

Ocean surface wave dynamics can be formulated as nonlinear interactions among a set of harmonic oscillators. The Hamiltonian formulation of this is mathematically very similar to the equations of classical and quantum mechanical field theory that I had encountered at the beginning of my career. I developed a canonical transformation technique which greatly simplified numerical integration of the equations. Calculations of the “long wave effect” agreed with observations of the radar scattering.

He married in 1946 and is the father of two sons. His father was Louis Erwin Watson (1884–1957) and his mother was Irene Marshall Watson (born 1886 in Roanoke, Illinois).

List of aircraft upset factors

The U.S. FAA lists factors of aircraft upset in the Airplane Upset Recovery Training Aid as follows:

Turbulence causes:

Clear air turbulence

Mountain wave turbulence




Wake turbulence

Aircraft icing

Systems anomalies:

Flight instruments

Autoflight systems

Flight control and other anomalies


Instrument cross-check

Adjusting attitude and power


Distraction from primary cockpit duties

Vertigo or spatial disorientation

Pilot incapacitation

Improper use of airplane automation

Pilot techniques

Pilot induced oscillation avoidance and recovery

Combination causes:

Swept-wing airplane fundamentals for pilots

Flight dynamics

Energy states

Load factor (flight mechanics)

Aerodynamic flight envelope

Aerodynamic causes:

Angle of attack and stall


Control surface fundamentals

Spoiler-type devices


Lateral and directional aerodynamic considerations

Angle of sideslip

Wing dihedral effects

Pilot-commanded sideslip

Crossover speed

Static stability

Maneuvering in pitch

Mechanics of turning flight

Lateral and directional maneuvering

Flight at extremely low airspeeds

High-altitude factors



Automation during high-altitude flight

Primary flight display airspeed indications

Human factors and high altitude upsets

Additional considerations:

Multi-engine flame out

Core lock

Engine rollback

Flight at extremely high speeds

Defensive, aggressive maneuvers

Situation awareness

Startle factor

Negative G-force

Use of full control inputs

Counter-intuitive factors

Previous training in non-similar airplanes

Engine performance in upset situation

Post-upset conditions

Luke's variational principle

In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the so-called mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. This is often used when modelling the spectral density evolution of the free-surface in a sea state, sometimes called wave turbulence.

Both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity.

Nonlinear Schrödinger equation

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 (except hypothetically, as in some early attempts in the 1970s, to explain the quantum measurement process). The 1D NLSE is an example of an integrable model.

In quantum mechanics, the 1D NLSE is a special case of the classical nonlinear Schrödinger field, which in turn is a classical limit of a quantum Schrödinger field. Conversely, when the classical Schrödinger field is canonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantum nonlinear Schrödinger equation″) that describes bosonic point particles with delta-function interactions — the particles either repel or attract when they are at the same point. In fact, when the number of particles is finite, this quantum field theory is equivalent to the Lieb–Liniger model. Both the quantum and the classical 1D nonlinear Schrödinger equations are integrable. Of special interest is the limit of infinite strength repulsion, in which case the Lieb–Liniger model becomes the Tonks–Girardeau gas (also called the hard-core Bose gas, or impenetrable Bose gas). In this limit, the bosons may, by a change of variables that is a continuum generalization of the Jordan–Wigner transformation, be transformed to a system one-dimensional noninteracting spinless fermions.The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) in their study of optical beams.

Multi-dimensional version replaces the second spatial derivative by the Laplacian. In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.

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.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Roman Glazman

Roman Evsey Glazman (June 26, 1948 – April 24, 2006) was a Russian American physicist and oceanographer.

Stephan Fauve

Stéphan Fauve (born December 20, 1955 in Paris) is a French physicist. He is a Professor at the École normale supérieure (ENS) in Paris, a member of the ENS Physics Laboratory.

After defending his thesis in 1984 under the direction of Albert Libchaber as a preparatory associate at the École normale supérieure, Stephan Fauve was successively Professor at the École normale supérieure de Lyon (1987-1997), then at the ENS in Paris since 1997. He is a member of the French Academy of Sciences, Physics section.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Vladimir E. Zakharov

Vladimir Evgen'evich Zakharov (Russian: Влади́мир Евге́ньевич Заха́ров; born August 1, 1939) 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.Zakharov was awarded the Dirac Medal in 2003 for his contributions to the theory of turbulence, with regard to the exact results and the prediction of inverse cascades, and for "putting the theory of wave turbulence on a firm mathematical ground by finding turbulence spectra as exact solutions and solving the stability problem, and in introducing the notion of inverse and dual cascades in wave turbulence."Vladimir Zakharov is also a poet. He has published several books of poetry in Russia and his works regularly appear in periodicals. A collection of his poetry in an English translation The Paradise for Clouds was published in the UK in 2009.

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wind-wave dissipation

Wind-wave dissipation or "swell dissipation" is process in which a wave generated via a weather system loses its mechanical energy transferred from the atmosphere via wind. Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, capillary gravity waves play an essential role in this effect, "wind waves" or "swell" are also known as surface gravity waves.

Ocean zones
Sea level

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