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In fluid dynamics, **wave–current interaction** is the interaction between surface gravity waves and a mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected.

In physics and engineering, **fluid dynamics** is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and **hydrodynamics**. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,

In fluid dynamics, the fluid flow is often decomposed into a **mean flow** – and deviations from the mean. The averaging can be done either in space or in time, or by ensemble averaging.

For depth-integrated and phase-averaged flows, the quantity of primary importance for the dynamics of the interaction is the wave radiation stress tensor.

In mathematics, an **integral** assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function f of a real variable x and an interval [*a*, *b*] of the real line, the **definite integral**

In physics and mathematics, the **phase** of a periodic function of some real variable is the relative value of that variable within the span of each full period.

In colloquial language, an **average** is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an **average value**.

Wave–current interaction is also one of the possible mechanisms for the occurrence of rogue waves, such as in the Agulhas Current. When a wave group encounters an opposing current, the waves in the group may pile up on top of each other which will propagate into a rogue wave.^{ [1] }^{ [2] }

**Rogue waves** are unusually large, unexpected and suddenly appearing surface waves that can be extremely dangerous, even to large ships such as ocean liners.

The **Agulhas Current** is the western boundary current of the southwest Indian Ocean. It flows down the east coast of Africa from 27°S to 40°S. It is narrow, swift and strong. It is suggested that it is the largest western boundary current in the world ocean, with an estimated net transport of 70 Sverdrups, as western boundary currents at comparable latitudes transport less — Brazil Current, Gulf Stream, Kuroshio.

Peregrine (1976) identifies five major sub-classes within wave–current interaction:

- interaction of waves with a large-scale current field, with slow – as compared to the wavelength – two-dimensional horizontal variations of the current fields;
- interaction of waves with small-scale current changes (in contrast with the case above), where the horizontal current varies suddenly, over a length scale comparable with the wavelength;
- the combined wave–current motion for currents varying (strongly) with depth below the free surface;
- interaction of waves with turbulence; and
- interaction of ship waves and currents, such as in the ship's wake.

In physics, a **field** is a physical quantity, represented by a number or tensor, that has a value for each point in space-time. For example, on a weather map, the surface temperature is described by assigning a real number to each point on a map; the temperature can be considered at a fixed point in time or over some time interval, if one wants to study the dynamics of temperature change. A surface wind map, assigning a vector to each point on a map that describes the wind velocity at that point, would be an example of a 1 dimensional tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank 1 tensor field, and the full description of electrodynamics can be formulated in terms of two interacting vector fields at each point in space-time, or as a single rank 2 tensor field theory.

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.

In physics, **length scale** is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem. Scientific reductionism says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales. The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the renormalization group.

- ↑ Dysthe, Kristian; Krogstad, Harald E.; Müller, Peter (2008), "Oceanic rogue waves",
*Annual Review of Fluid Mechanics*,**40**: 287–310, Bibcode:2008AnRFM..40..287D, doi:10.1146/annurev.fluid.40.111406.102203 - ↑ Peregrine (1976)

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.

**Physical oceanography** is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

A **capillary wave** is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.

In fluid dynamics, **wind waves**, or **wind-generated waves**, are surface waves that occur on the free surface of bodies of water. They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.

In fluid dynamics, **vortex-induced vibrations** (VIV) are motions induced on bodies interacting with an external fluid flow, produced by – or the motion producing – periodical irregularities on this flow.

**Horizontal convective rolls**, also known as **horizontal roll vortices** or **cloud streets**, are long rolls of counter-rotating air that are oriented approximately parallel to the ground in the planetary boundary layer. Although horizontal convective rolls, also known as cloud streets, have been clearly seen in satellite photographs for the last 30 years, their development is poorly understood, due to a lack of observational data. From the ground, they appear as rows of cumulus or cumulus-type clouds aligned parallel to the low-level wind. Research has shown these eddies to be significant to the vertical transport of momentum, heat, moisture, and air pollutants within the boundary layer. Cloud streets are usually more or less straight; rarely, cloud streets assume paisley patterns when the wind driving the clouds encounters an obstacle. Those cloud formations are known as von Kármán vortex streets.

In physical oceanography, **Langmuir circulation** consists of a series of shallow, slow, counter-rotating vortices at the ocean's surface aligned with the wind. These circulations are developed when wind blows steadily over the sea surface. Irving Langmuir discovered this phenomenon after observing windrows of seaweed in the Sargasso Sea in 1927. Langmuir circulations circulate within the mixed layer; however, it is not yet so clear how strongly they can cause mixing at the base of the mixed layer.

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 **Stokes wave** is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the **Stokes expansion** – obtained approximate solutions for non-linear wave motion.

**Howell Peregrine** was a British applied mathematician noted for his contributions to fluid mechanics, especially of free surface flows such as water waves, and coastal engineering.

In fluid dynamics, **Airy wave theory** gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

In fluid dynamics, the **Coriolis–Stokes force** is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the wind stress.

In fluid dynamics, the **radiation stress** is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

In fluid dynamics, **wave setup** is the increase in mean water level due to the presence of breaking waves. Similarly, **wave setdown** is a wave-induced decrease of the mean water level before the waves break. For short, the whole phenomenon is often denoted as wave setup, including both increase and decrease of mean elevation. This setup is primarily present in and near the coastal surf zone. Besides a spatial variation in the (mean) wave setup, also a variation in time may be present – known as surf beat – causing infragravity wave radiation.

In continuum mechanics, the **generalized Lagrangian mean** (**GLM**) is a formalism – developed by D.G. Andrews and M.E. McIntyre – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flow field, but appointed to fixed Eulerian coordinates.

In continuum mechanics, **wave action** refers to a conservable measure of the wave part of a motion. For small-amplitude and slowly varying waves, the **wave action density** is:

In continuum mechanics, Whitham's **averaged Lagrangian** method – or in short **Whitham's method** – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system.

**Owen Martin Phillips** was a U.S. physical oceanographer and geophysicist who spent most of his career at the Johns Hopkins University.

In fluid dynamics, **Green's law** describes the evolution of non-breaking surface gravity waves propagating in shallow water of gradually varying depth and width. The law is named after George Green. In its simplest form, for wavefronts and depth contours parallel to each other, it states:

In fluid dynamics, the **Craik–Leibovich (CL) vortex force** describes a forcing of the mean flow through wave–current interaction, specifically between the Stokes drift velocity and the mean-flow vorticity. The CL vortex force is used to explain the generation of Langmuir circulations by an instability mechanism. The CL vortex-force mechanism was derived and studied by Sidney Leibovich and Alex D.D. Craik in the 1970s and 80s, in their studies of Langmuir circulations.

- Bretherton, F. P.; Garrett, C. J. R. (1968), "Wavetrains in inhomogeneous moving media",
*Proceedings of the Royal Society of London*, Series A, Mathematical and Physical Sciences,**302**(1471): 529–554, Bibcode:1968RSPSA.302..529B, doi:10.1098/rspa.1968.0034 - Bühler, O. (2014),
*Waves and mean flows*(2nd ed.), Cambridge University Press, ISBN 978-1-107-66966-6 - Peregrine, D. H. (1976), "Interaction of water waves and currents",
*Advances in Applied Mechanics*,**16**, Academic Press, pp. 9–117, ISBN 978-0-12-002016-4 - Phillips, O. M. (1977),
*The dynamics of the upper ocean*(2nd ed.), Cambridge University Press, ISBN 0-521-29801-6 - Craik, A. D. D. (1988),
*Wave interactions and fluid flows*, Cambridge University Press, ISBN 0-521-36829-4 - Prandle, D. (1992),
*Dynamics and exchanges in estuaries and the coastal zone*, Coastal and estuarine studies,**40**, American Geophysical Union, ISBN 0-87590-254-5 - Jonsson, I. G. (1990), "Wave–current interactions", in B. Le Méhauté and D. M. Hanes,
*Ocean Engineering Science*, The Sea,**9A**, Wiley Interscience, pp. 65–120, ISBN 0-471-11543-6

The **bibcode** is a compact identifier used by several astronomical data systems to uniquely specify literature references.

In computing, a **Digital Object Identifier** or **DOI** is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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