Radiation 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.

The radiation stress tensor describes the additional forcing due to the presence of the waves, which changes the mean depth-integrated horizontal momentum in the fluid layer. As a result, varying radiation stresses induce changes in the mean surface elevation (wave setup) and the mean flow (wave-induced currents).

For the mean energy density in the oscillatory part of the fluid motion, the radiation stress tensor is important for its dynamics, in case of an inhomogeneous mean-flow field.

The radiation stress tensor, as well as several of its implications on the physics of surface gravity waves and mean flows, were formulated in a series of papers by Longuet-Higgins and Stewart in 1960–1964.

Radiation stress derives its name from the analogous effect of radiation pressure for electromagnetic radiation.

Tautuku Beach Almost Tropical
Breaking waves on beaches induce variations in radiation stress, driving longshore currents. The resulting longshore sediment transport shapes the beaches, and may result in beach erosion or accretion.

Physical significance

The radiation stress – mean excess momentum-flux due to the presence of the waves – plays an important role in the explanation and modeling of various coastal processes:[1][2][3]

  • Wave setup and setdown – the radiation stress consists in part of a radiation pressure, exerted at the free surface elevation of the mean flow. If the radiation stress varies spatially, as it does in the surf zone where the wave height reduces by wave breaking, this results in changes of the mean surface elevation called wave setup (in case of an increased level) and setdown (for a decreased water level);
  • Wave-driven current, especially a longshore current in the surf zone – for oblique incidence of waves on a beach, the reduction in wave height inside the surf zone (by breaking) introduces a variation of the shear-stress component Sxy of the radiation stress over the width of the surf zone. This provides the forcing of a wave-driven longshore current, which is of importance for sediment transport (longshore drift) and the resulting coastal morphology;
  • Bound long waves or forced long waves, part of the infragravity waves – for wave groups the radiation stress varies along the group. As a result, a non-linear long wave propagates together with the group, at the group velocity of the modulated short waves within the group. While, according to the dispersion relation, a long wave of this length should propagate at its own – higher – phase velocity. The amplitude of this bound long wave varies with the square of the wave height, and is only significant in shallow water;
  • Wave–current interaction – in varying mean-flow fields, the energy exchanges between the waves and the mean flow, as well as the mean-flow forcing, can be modeled by means of the radiation stress.

Definitions and values derived from linear wave theory

One-dimensional wave propagation

For uni-directional wave propagation – say in the x-coordinate direction – the component of the radiation stress tensor of dynamical importance is Sxx. It is defined as:[4]

where p(x,z,t) is the fluid pressure, is the horizontal x-component of the oscillatory part of the flow velocity vector, z is the vertical coordinate, t is time, z = −h(x) is the bed elevation of the fluid layer, and z = η(x,t) is the surface elevation. Further ρ is the fluid density and g is the acceleration by gravity, while an overbar denotes phase averaging. The last term on the right-hand side, ½ρg(h+η)2, is the integral of the hydrostatic pressure over the still-water depth.

To lowest (second) order, the radiation stress Sxx for traveling periodic waves can be determined from the properties of surface gravity waves according to Airy wave theory:[5][6]

where cp is the phase speed and cg is the group speed of the waves. Further E is the mean depth-integrated wave energy density (the sum of the kinetic and potential energy) per unit of horizontal area. From the results of Airy wave theory, to second order, the mean energy density E equals:[7]

with a the wave amplitude and H = 2a the wave height. Note this equation is for periodic waves: in random waves the root-mean-square wave height Hrms should be used with Hrms = Hm0 / 2, where Hm0 is the significant wave height. Then E = ​116ρgHm02.

Two-dimensional wave propagation

For wave propagation in two horizontal dimensions the radiation stress is a second-order tensor[8][9] with components:

With, in a Cartesian coordinate system (x,y,z):[4]

where and are the horizontal x- and y-components of the oscillatory part of the flow velocity vector.

To second order – in wave amplitude a – the components of the radiation stress tensor for progressive periodic waves are:[5]

where kx and ky are the x- and y-components of the wavenumber vector k, with length k = |k| = kx2+ky2 and the vector k perpendicular to the wave crests. The phase and group speeds, cp and cg respectively, are the lengths of the phase and group velocity vectors: cp = |cp| and cg = |cg|.

Dynamical significance

The radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Here, the depth-integrated dynamical conservation equations are given, but – in order to model three-dimensional mean flows forced, or interacting with, surface waves – a three-dimensional description of the radiation stress over the fluid layer is needed.[10]

Mass transport velocity

Propagating waves induce a – relatively small – mean mass transport in the wave propagation direction, also called the wave (pseudo) momentum.[11] To lowest order, the wave momentum Mw is, per unit of horizontal area:[12]

which is exact for progressive waves of permanent form in irrotational flow. Above, cp is the phase speed relative to the mean flow:

with σ the intrinsic angular frequency, as seen by an observer moving with the mean horizontal flow-velocity v while ω is the apparent angular frequency of an observer at rest (with respect to 'Earth'). The difference kv is the Doppler shift.[13]

The mean horizontal momentum M, also per unit of horizontal area, is the mean value of the integral of momentum over depth:

with v(x,y,z,t) the total flow velocity at any point below the free surface z = η(x,y,t). The mean horizontal momentum M is also the mean of the depth-integrated horizontal mass flux, and consists of two contributions: one by the mean current and the other (Mw) is due to the waves.

Now the mass transport velocity u is defined as:[14][15]

Observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth (h+η) is made.

Mass and momentum conservation

Vector notation

The equation of mean mass conservation is, in vector notation:[14]

with u including the contribution of the wave momentum Mw.

The equation for the conservation of horizontal mean momentum is:[14]

where u ⊗ u denotes the tensor product of u with itself, and τw is the mean wind shear stress at the free surface, while τb is the bed shear stress. Further I is the identity tensor, with components given by the Kronecker delta δij. Note that the right hand side of the momentum equation provides the non-conservative contributions of the bed slope ∇h,[16] as well the forcing by the wind and the bed friction.

In terms of the horizontal momentum M the above equations become:[14]

Component form in Cartesian coordinates

In a Cartesian coordinate system, the mass conservation equation becomes:

with ux and uy respectively the x and y components of the mass transport velocity u.

The horizontal momentum equations are:

Energy conservation

For an inviscid flow the mean mechanical energy of the total flow – that is the sum of the energy of the mean flow and the fluctuating motion – is conserved.[17] However, the mean energy of the fluctuating motion itself is not conserved, nor is the energy of the mean flow. The mean energy E of the fluctuating motion (the sum of the kinetic and potential energies satisfies:[18]

where ":" denotes the double-dot product, and ε denotes the dissipation of mean mechanical energy (for instance by wave breaking). The term is the exchange of energy with the mean motion, due to wave–current interaction. The mean horizontal wave-energy transport (u + cgE consists of two contributions:

  • u E : the transport of wave energy by the mean flow, and
  • cg E : the mean energy transport by the waves themselves, with the group velocity cg as the wave-energy transport velocity.

In a Cartesian coordinate system, the above equation for the mean energy E of the flow fluctuations becomes:

So the radiation stress changes the wave energy E only in case of a spatial-inhomogeneous current field (ux,uy).


  1. ^ Longuet-Higgins & Stewart (1964,1962).
  2. ^ Phillips (1977), pp. 70–81.
  3. ^ Battjes, J. A. (1974). Computation of set-up, longshore currents, run-up and overtopping due to wind-generated waves (Thesis). Delft University of Technology. Retrieved 2010-11-25.
  4. ^ a b Mei (2003), p. 457.
  5. ^ a b Mei (2003), p. 97.
  6. ^ Phillips (1977), p. 68.
  7. ^ Phillips (1977), p. 39.
  8. ^ Longuet-Higgins & Stewart (1961).
  9. ^ Dean, R.G.; Walton, T.L. (2009), "Wave setup", in Young C. Kim (ed.), Handbook of Coastal and Ocean Engineering, World Scientific, pp. 1–23, ISBN 981-281-929-0.
  10. ^ Walstra, D. J. R.; Roelvink, J. A.; Groeneweg, J. (2000), "Calculation of wave-driven currents in a 3D mean flow model", Proceedings of the 27th International Conference on Coastal Engineering, Sydney: ASCE, pp. 1050–1063, doi:10.1061/40549(276)81
  11. ^ Mcintyre, M. E. (1981), "On the 'wave momentum' myth", Journal of Fluid Mechanics, 106: 331–347, Bibcode:1981JFM...106..331M, doi:10.1017/S0022112081001626
  12. ^ Phillips (1977), p. 40.
  13. ^ Phillips (1977), pp. 23–24.
  14. ^ a b c d Phillips (1977), pp. 61–63.
  15. ^ Mei (2003), p. 453.
  16. ^ By Noether's theorem, an inhomogeneous medium – in this case a non-horizontal bed, h(x,y) not a constant – results in non-conservation of the depth-integrated horizontal momentum.
  17. ^ Phillips (1977), pp. 63–65.
  18. ^ Phillips (1977), pp. 65–66.


Primary sources
Further reading
  • Mei, Chiang C. (2003), The applied dynamics of ocean surface waves, Advanced series on ocean engineering, 1, World Scientific, ISBN 9971-5-0789-7
  • Phillips, O. M. (1977), The dynamics of the upper ocean (2nd ed.), Cambridge University Press, ISBN 0-521-29801-6

The ADCIRC model is a high-performance, cross-platform numerical ocean circulation model popular in simulating storm surge, tides, and coastal circulation problems.

Originally developed by Drs. Rick Luettich and Joannes Westerink,

the model is developed and maintained by a combination of academic, governmental, and corporate partners, including the University of North Carolina at Chapel Hill, the University of Notre Dame, and the US Army Corps of Engineers.

The ADCIRC system includes an independent multi-algorithmic wind forecast model and also has advanced coupling capabilities, allowing it to integrate effects from sediment transport, ice, waves, surface runoff, and baroclinicity.

Airy wave theory

In fluid dynamics, Airy wave theory (often referred to as linear 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.Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states – giving a description of the wave kinematics and dynamics of high-enough accuracy for many purposes. Further, several second-order nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. Airy wave theory is also a good approximation for tsunami waves in the ocean, before they steepen near the coast.

This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the wave height to water depth (for waves in shallow water), and wave height to wavelength (for waves in deep water).

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.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

Chlamydomonas nivalis

Chlamydomonas nivalis is a unicellular red-coloured photosynthetic green alga that is found in the snowfields of the alps and polar regions all over the world. They are one of the main algae responsible for causing the phenomenon of watermelon snow, where patches of snow appear red or pink and emit a ripe watermelon odour upon disturbance. The first account of microbial communities that form red snow was made by Aristotle. Researchers have been active in studying this organism for over 100 years.

Although C. nivalis is closely related to Chlamydomonas reinhardtii, the environmental conditions each species inhabits are very different. C. nivalis can be found in mountains, snowfields, and polar regions around the world. The habitat of C. nivalis subjects the cells to environmental extremes including limited nutrients, low temperatures, and intense sunlight. In comparison with the mesophilic C. reinhardtii, C. nivalis has special mechanisms that allow it to be cryotolerant and survive on rock surfaces as well as in soil, meltwater, and snow. Secondary carotenoids, a thick cell wall, and particles on the cell wall are some characteristics that protect the cyst from light, drought, and radiation stress. Although the seasonal mobile to dormant life cycle of C. nivalis is complex, it also helps the algae exploit its niche and survive unfavourable conditions. As a result, C. nivalis is one of the best known and studied snow algae. When taking account of the photoprotective effect of its secondary carotenoid, astaxanthin, among the other adaptive mechanisms to its extreme habitat, it can be understood how C. nivalis became so dominant in microbial snow algae communities. Green motile offspring are produced in the spring and throughout the summer. They develop into red dormant cysts, the stage where this organism spends most of its life cycle, as the winter season begins and remain a cyst until the spring.

This alga is an interesting organism for researchers in various fields to study due to its possible role in lowering global albedo, ability to survive in extreme environments, and production of commercially relevant compounds. Additionally, its life cycle is still being studied today in an effort to better understand this organism and amend previous classification errors.

Index of physics articles (R)

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.

Infragravity wave

Infragravity waves are surface gravity waves with frequencies lower than the wind waves – consisting of both wind sea and swell – thus corresponding with the part of the wave spectrum lower than the frequencies directly generated by forcing through the wind.

Infragravity waves are ocean surface gravity waves generated by ocean waves of shorter periods. The amplitude of infragravity waves is most relevant in shallow water, in particular along coastlines hit by high amplitude and long period wind waves and ocean swells. Wind waves and ocean swells are shorter, with typical dominant periods of 1 to 25 s. In contrast, the dominant period of infragravity waves is typically 80 to 300 s, which is close to the typical periods of tsunamis, with which they share similar propagation properties including very fast celerities in deep water. This distinguishes infragravity waves from normal oceanic gravity waves, which are created by wind acting on the surface of the sea, and are slower than the generating wind.

Whatever the details of their generation mechanism, discussed below, infragravity waves are these subharmonics of the impinging gravity waves.

Technically infragravity waves are simply a subcategory of gravity waves and refer to all gravity waves with periods greater than 30 s. This could include phenomena such as tides and oceanic Rossby waves, but the common scientific usage is limited to gravity waves that are generated by groups of wind waves.

The term "infragravity wave" appears to have been coined by Walter Munk in 1950.

List of finite element software packages

This is a list of software packages that implement the finite element method for solving partial differential equations.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

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.

Physical oceanography

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

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

Radiation-induced cancer

The evidence indicates that of all cancer-related deaths, almost 25–30% are due to tobacco, as many as 30–35% are linked to diet, about 15–20% are due to infections, and the remaining percentage are due to other factors like ionizing radiation, stress, physical activity, environmental pollutants etc. Additionally, the vast majority of non-invasive cancers are non-melanoma skin cancers caused by ultraviolet radiation. Ultraviolet's position on the electromagnetic spectrum is on the boundary between ionizing and non-ionizing radiation. Non-ionizing radio frequency radiation from mobile phones, electric power transmission, and other similar sources have been described as a possible carcinogen by the World Health Organization's International Agency for Research on Cancer, but the link remains unproven.Exposure to ionizing radiation is known to increase the future incidence of cancer, particularly leukemia. The mechanism by which this occurs is well understood, but quantitative models predicting the level of risk remain controversial. The most widely accepted model posits that the incidence of cancers due to ionizing radiation increases linearly with effective radiation dose at a rate of 5.5% per sievert. If the linear model is correct, then natural background radiation is the most hazardous source of radiation to general public health, followed by medical imaging as a close second.

Range Software

Range Software is finite element analysis software package.

Rip current

A rip current, often simply called a rip (or misleadingly rip tide), is a specific kind of water current which can occur near beaches with breaking waves. A rip is a strong, localized, and narrow current of water which moves directly away from the shore, cutting through the lines of breaking waves like a river running out to sea, and is strongest near the surface of the water.Rip currents can be hazardous to people in the water. Swimmers who are caught in a rip current and who do not understand what is going on, and who may not have the necessary water skills, may panic, or exhaust themselves by trying to swim directly against the flow of water. Because of these factors, rips are the leading cause of rescues by lifeguards at beaches, and rips are the cause of an average of 46 deaths by drowning per year in the United States.

A rip current is not the same thing as undertow, although some people use the term incorrectly when they often mean a rip current. Contrary to popular belief, neither rip nor undertow can pull a person down and hold them under the water. A rip simply carries floating objects, including people, out beyond the zone of the breaking waves.

Stokes wave

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.

Stokes' wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations.

While, in the strict sense, Stokes wave refers to progressive periodic waves of permanent form, the term is also used in connection with standing waves and even for random waves.

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".

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.

Wave setup

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 (during the shoaling process). 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.

Wave setup can be mathematically modeled by considering the variation in radiation stress (Longuet-Higgins & Stewart 1962). Radiation stress is the tensor of excess horizontal-momentum fluxes due to the presence of the waves.

Wave–current interaction

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.

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

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.

Ocean zones
Sea level


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