Langmuir circulation

Last updated
Langmuir circulation Langmuir Circulation.jpg
Langmuir circulation
White streaks in this lagoon are due to the Langmuir circulation. Rodeo Lagoon From Trail.JPG
White streaks in this lagoon are due to the Langmuir circulation.
These lines of sargassum can stretch for miles along the surface. The clumps of floating algae are often concentrated by the strong winds and wave action associated with the Gulf Stream. Lines of sargassum Sargasso Sea.jpg
These lines of sargassum can stretch for miles along the surface. The clumps of floating algae are often concentrated by the strong winds and wave action associated with the Gulf Stream.

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. [1] 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. [2]

Contents

Theory

The driving force of these circulations is an interaction of the mean flow with wave averaged flows of the surface waves. Stokes drift velocity of the waves stretches and tilts the vorticity of the flow near the surface. The production of vorticity in the upper ocean is balanced by downward (often turbulent) diffusion . For a flow driven by a wind characterized by friction velocity the ratio of vorticity diffusion and production defines the Langmuir number [2]

where the first definition is for a monochromatic wave field of amplitude , frequency , and wavenumber and the second uses a generic inverse length scale , and Stokes velocity scale . This is exemplified by the Craik–Leibovich equations [3] which are an approximation of the Lagrangian mean. [4] [5] In the Boussinesq approximation the governing equations can be written

where

In the open ocean conditions where there may not be a dominant length scale controlling the scale of the Langmuir cells the concept of Langmuir Turbulence is advanced. [6]

Observations

The circulation has been observed to be between 0°–20° to the right of the wind in the northern hemisphere [7] and the helix forming bands of divergence and convergence at the surface. At the convergence zones, there are commonly concentrations of floating seaweed, foam and debris along these bands. Along these divergent zones, the ocean surface is typically clear of debris since diverging currents force material out of this zone and into adjacent converging zones. At the surface the circulation will set a current from the divergence zone to the convergence zone and the spacing between these zones are of the order of 1–300 m (3–1,000 ft). Below convergence zones narrow jets of downward flow form and the magnitude of the current will be comparable to the horizontal flow. The downward propagation will typically be in the order of meters or tenths of meters and will not penetrate the pycnocline. The upwelling is less intense and takes place over a wider band under the divergence zone. In wind speeds ranging from 2–12 m/s (6.6–39.4 ft/s) the maximum vertical velocity ranged from 2–10 cm/s (0.79–3.94 in/s) with a ratio of down-welling to wind velocities ranging from −0.0025 to −0.0085. [8]

Biological effects

Higher windward velocity in convergent zones of Langmuir cells Windward velocity profile in Langmuir cells.jpg
Higher windward velocity in convergent zones of Langmuir cells

Langmuir circulations (LCs), which are counter-rotating cylindrical roll vortices in the upper ocean, have significant role in vertical mixing. Though they are transient and their strength as well as direction depend on wind and wave properties, they facilitate mixing of nutrients and affect the distribution of marine organisms like plankton in the upper mixed layer of ocean. [9] [10] The wind-generated roll vortices create regions where organisms of different buoyancy, orientation and swimming behavior can aggregate, resulting in patchiness. [11] Indeed, LC can produce significant aggregation of algae during events like red tide. [10] Theoretically, LC size increases with the wind speed unless limited by density discontinuities by pycnocline. But the visibility of surface effects of LC could be limited by the breaking waves during strong winds that disperse the materials present at the surface. [12] So, the surface effects of LC are more likely to be visible during winds stronger than critical wind speed of 3 m/s but not too strong.

PIV vectors of counter rotating vortices PIV vectors of counter rotating vortices.jpg
PIV vectors of counter rotating vortices

Moreover, previous studies have shown that organisms and materials can aggregate at different regions within LC like downwelling current in convergent zone, upwelling current in divergent zone, retention zone in LC vortex and region between upwelling and downwelling zones. [13] Similarly, LC are found to have higher windward surface current in convergent zones due to jet like flow. This faster moving convergent region in water surface can enhance the transport of organisms and materials in the direction of wind.

Effect on plants

Distribution of particles within Langmuir cells Aggregation of particles within Langmuir Circulation.tif
Distribution of particles within Langmuir cells

In 1927, Langmuir saw the organized rows of Sargassum natans while crossing the Sargasso Sea in the Atlantic Ocean. Unlike active swimmers like animals and zooplankton, plants and phytoplankton are usually passive bodies in water and their aggregation are determined by the flow behavior. In windrows, concentrated planktonic organisms color the water and indicate the presence of LC. There has been observation of greater variability in plankton tows collected along the wind direction than samples collected perpendicular to the wind. [14] And one of the reason for such variation could be due to LC that results convergence (high sample) or in between (low sample) zones in alongwind tow. [15] Similarly, such converging effect of LC has also been observed as high chlorophyll zone at about 100 m in Lake Tahoe which could be due to oblique towing through LC. [16] In addition, Sargassum get carried from surface to benthos in downwelling zone of LC and can lose buoyancy after sinking at depth for enough time. [17] Some of the plants that are usually observed floating in water could get submerged during high wind conditions due to downwelling current of LC. Besides, LC could also lead to patchiness of positively buoyant dinoflagellates (including toxic red tide organisms) during blooms. [18] Moreover, the negatively buoyant phytoplankters which would sink slowly in still water has been observed to get retained in euphotic zone which may be due to suspension created by vertical convection cells. [19] [20]

Furthermore, a broader study on the Langmuir supercells in which the circulation can reach the seafloor observed the aggregation of macroalgae Colpomenia sp. in the sea floor of shallow waters (~5 m) in Great Bahama Bank due to local wind speed of around 8 to 13 m/s. [21] Such LC could be responsible for transport of carbon biomass from shallow water to deep sea. This effect was evident as the concentration of the algae were found to reduce dramatically after the occurrence of LC as observed from ocean color satellite imagery (NASA) during the period of the study. Such aggregation of negatively buoyant macroalgae on sea floor is similar to windrows of positively buoyant particles on water surface due to LC.

Effect on animals

While plants have passive reaction to LC, animals can react to both the LC, presence of plant/food aggregation and light. One of such observation was the adaptation of Physalia to windrows containing entangling Sargassum . Physalia tend to drift across the windrows which also increased food or zooplankter availability in divergent zones. [22]

Moreover, studies in Lake Mendota have shown good correlation between Daphnia pulex concentration and the appearance of foam lines. Similarly, significant differences were observed in catches of Daphnia hyaline when sampling in and out of foamlines in South Wales lake, with greater number appearing in divergent zone. [23] Such distribution of particles and animals can be described using mathematical model developed by Stommel that suggested area of retention on upwelling zone for sinking particles and on downwelling zone for positively buoyant particles. [15] Actually, the zooplankton could become trapped in upwelling zones to a point where animals are stimulated to swim downwards. [24] A more detailed model was later developed by Stavn describing the zooplankton aggregation where the animal orientation, dorsal light reaction and current velocity determined their region of concentration in either downwelling (due to slow current), upwelling (due to high current) and in between latter two zones (due to intermediate currents). [11] There has been further improvement in such models like the modification of Stommel's model by Titman & Kilham in order to consider the difference in maximum downwelling and upwelling velocities [25] and by Evans & Taylor that discussed the instability of Stommel's regions due to varying swimming speed with depth which produced spiral trajectories affecting accumulation region. [10]

Nevertheless, high concentration of planktonic organisms within LC can attract birds and fish. Schools of White Bass Roccus chrysops were observed feeding upon Daphnia along the foam track. [26] In contrast, lesser Flamingoes Phoeniconaias minor were observed feeding on bubble lines containing concentrated blue-green algae. [27] Similarly, medusae were found to aggregate in linear pattern (average spacing of 129 m) parallel with wind in the Bering Sea which could be due to large LCs. [12] Such aggregation can affect the feeding and predation of medusae.

Effect on surface tension

High concentration of surfactants (surface-active substances) produced by phytoplanktons can result higher Marangoni stress in converging regions in LC. Numerical simulation suggest that such Marangoni stress due to surfactant can increase the size of vortical structures, vertical velocity and remixing of water and biological/chemical components in the local region compared to that without surfactant. [28]

Finally, more theoretical and experimental investigations are needed to confirm the significance of LC.

Related Research Articles

The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid. The governing equation is:

<span class="mw-page-title-main">Baroclinity</span> Measure of misalignment between the gradients of pressure and density in a fluid

In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic flow is one in which the density depends on both temperature and pressure. A simpler case, barotropic flow, allows for density dependence only on pressure, so that the curl of the pressure-gradient force vanishes.

In fluid mechanics or more generally continuum mechanics, incompressible flow refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero.

<span class="mw-page-title-main">Internal wave</span> Type of gravity waves that oscillate within a fluid medium

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance, the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the wind stress exerted on the surface of the open ocean and the vertically integrated meridional (north-south) transport of ocean water.

<span class="mw-page-title-main">Eddy (fluid dynamics)</span> Swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime

In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid behind the obstacle flows into the void creating a swirl of fluid on each edge of the obstacle, followed by a short reverse flow of fluid behind the obstacle flowing upstream, toward the back of the obstacle. This phenomenon is naturally observed behind large emergent rocks in swift-flowing rivers.

<span class="mw-page-title-main">Ekman transport</span> Net transport of surface water perpendicular to wind direction

Ekman transport is part of Ekman motion theory, first investigated in 1902 by Vagn Walfrid Ekman. Winds are the main source of energy for ocean circulation, and Ekman transport is a component of wind-driven ocean current. Ekman transport occurs when ocean surface waters are influenced by the friction force acting on them via the wind. As the wind blows it casts a friction force on the ocean surface that drags the upper 10-100m of the water column with it. However, due to the influence of the Coriolis effect, the ocean water moves at a 90° angle from the direction of the surface wind. The direction of transport is dependent on the hemisphere: in the northern hemisphere, transport occurs at 90° clockwise from wind direction, while in the southern hemisphere it occurs at 90° anticlockwise. This phenomenon was first noted by Fridtjof Nansen, who recorded that ice transport appeared to occur at an angle to the wind direction during his Arctic expedition during the 1890s. Ekman transport has significant impacts on the biogeochemical properties of the world's oceans. This is because it leads to upwelling and downwelling in order to obey mass conservation laws. Mass conservation, in reference to Ekman transfer, requires that any water displaced within an area must be replenished. This can be done by either Ekman suction or Ekman pumping depending on wind patterns.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation.

In physical oceanography and fluid dynamics, the wind stress is the shear stress exerted by the wind on the surface of large bodies of water – such as oceans, seas, estuaries and lakes. Stress is the quantity that describes the magnitude of a force that is causing a deformation of an object. Therefore, stress is defined as the force per unit area and its SI unit is the Pascal. When the deforming force acts parallel to the object's surface, this force is called a shear force and the stress it causes is called a shear stress. When wind is blowing over a water surface, the wind applies a wind force on the water surface. The wind stress is the component of this wind force that is parallel to the surface per unit area. Also, the wind stress can be described as the flux of horizontal momentum applied by the wind on the water surface. The wind stress causes a deformation of the water body whereby wind waves are generated. Also, the wind stress drives ocean currents and is therefore an important driver of the large-scale ocean circulation. The wind stress is affected by the wind speed, the shape of the wind waves and the atmospheric stratification. It is one of the components of the air–sea interaction, with others being the atmospheric pressure on the water surface, as well as the exchange of energy and mass between the water and the atmosphere.

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.

<span class="mw-page-title-main">Radiation stress</span> Term in physical oceanography

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 oceanography, Ekman velocity – also referred as a kind of the residual ageostrophic velocity as it deviates from geostrophy – is part of the total horizontal velocity (u) in the upper layer of water of the open ocean. This velocity, caused by winds blowing over the surface of the ocean, is such that the Coriolis force on this layer is balanced by the force of the wind.

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.

Conservative temperature is a thermodynamic property of seawater. It is derived from the potential enthalpy and is recommended under the TEOS-10 standard as a replacement for potential temperature as it more accurately represents the heat content in the ocean.

<span class="mw-page-title-main">River plume</span> Mix of fresh river water and seawater

A river plume is a freshened water mass that is formed in the sea as a result of mixing of river discharge and saline seawater. River plumes are formed in coastal sea areas at many regions in the World. River plumes generally occupy wide, but shallow sea surface layer bounded by sharp density gradient. The area of a river plume is 3-5 orders of magnitude greater than its depth, therefore, even small rivers with discharge rates ~1–10 m/s form river plumes with horizontal spatial extents ~10–100 m. Areas of river plumes formed by the largest World rivers are ~100–1000 km2. Despite relatively small volume of total freshwater runoff to the World Ocean, river plumes occupy up to 21% of shelf areas of the World Ocean, i.e., several million square kilometers.

<span class="mw-page-title-main">Southern Caribbean upwelling system</span> Low latitude tropical upwelling system

The Southern Caribbean Upwelling system (SCUS) is a low latitude tropical upwelling system. Where; due to multiple environmental and bathymetric conditions water from the deep sea is forced to the surface layers of the ocean. The SCUS is located at about 10°N on the southern coast of the Caribbean sea basin off Colombia, Venezuela, and Trinidad.

<span class="mw-page-title-main">Relative wind stress</span> A summary of wind stress. A very important concept for large scale ocean circulation models.

Relative wind stress is a shear stress that is produced by wind blowing over the surface of the ocean, or another large body of water. Relative wind stress is related to wind stress but takes the difference between the surface ocean current velocity and wind velocity into account. The units are Newton per meter squared or Pascal . Wind stress over the ocean is important as it is a major source of kinetic energy input to the ocean which in turn drives large scale ocean circulation. The use of relative wind stress instead of wind stress, where the ocean current is assumed to be stationary, reduces the stress felt over the ocean in models. This leads to a decrease in the calculation of power input into the ocean of 20–35% and thus, results in a different simulation of the large scale ocean circulation.

Eddy pumping is a component of mesoscale eddy-induced vertical motion in the ocean. It is a physical mechanism through which vertical motion is created from variations in an eddy's rotational strength. Cyclonic (Anticyclonic) eddies lead primarily to upwelling (downwelling) in the Northern Hemisphere and vice versa in the Southern hemisphere. It is a key mechanism driving biological and biogeochemical processes in the ocean such as algal blooms and the carbon cycle.

Atmospheric circulation of a planet is largely specific to the planet in question and the study of atmospheric circulation of exoplanets is a nascent field as direct observations of exoplanet atmospheres are still quite sparse. However, by considering the fundamental principles of fluid dynamics and imposing various limiting assumptions, a theoretical understanding of atmospheric motions can be developed. This theoretical framework can also be applied to planets within the Solar System and compared against direct observations of these planets, which have been studied more extensively than exoplanets, to validate the theory and understand its limitations as well.

References

  1. ↑ Open University (2001), Ocean Circulation (2nd ed.), Butterworth-Heinemann, ISBN   9780750652780
  2. 1 2 Thorpe, S.A. (2004), "Langmuir circulation", Annual Review of Fluid Mechanics, 36: 55–79, Bibcode:2004AnRFM..36...55T, doi:10.1146/annurev.fluid.36.052203.071431
  3. ↑ Craik, A.D.D.; Leibovich, S. (1976), "A Rational model for Langmuir circulations", Journal of Fluid Mechanics, 73 (3): 401–426, Bibcode:1976JFM....73..401C, doi:10.1017/S0022112076001420, S2CID   18089261
  4. ↑ Andrews, D.G.; McIntyre, M.E. (1978), "An exact theory of nonlinear waves on a Lagrangian-mean flow", Journal of Fluid Mechanics, 89 (4): 609–646, Bibcode:1978JFM....89..609A, doi:10.1017/S0022112078002773, S2CID   4988274
  5. ↑ Leibovich, S. (1980), "On wave-current interactions theories of Langmuir circulations", Journal of Fluid Mechanics, 99 (4): 715–724, Bibcode:1980JFM....99..715L, doi:10.1017/S0022112080000857, S2CID   14996095
  6. ↑ McWilliams, J.; Sullivan, P.; Moeng, C. (1997), "Langmuir turbulence in the ocean", Journal of Fluid Mechanics, 334 (1): 1–30, Bibcode:1997JFM...334....1M, doi:10.1017/S0022112096004375, S2CID   76650016
  7. ↑ Stewart, Robert H. (2002), Introduction To Physical Oceanography (Fall 2002 ed.)
  8. ↑ Leibovich, S. (1983), "The form and dynamics of Langmuir circulations", Annual Review of Fluid Mechanics, 15: 391–427, Bibcode:1983AnRFM..15..391L, doi:10.1146/annurev.fl.15.010183.002135
  9. ↑ Kukulka, T.; Plueddemann, A. J.; Trowbridge, J. H.; Sullivan, P. P. (2009-05-28). "Significance of Langmuir circulation in upper ocean mixing: Comparison of observations and simulations". Geophysical Research Letters. 36 (10): L10603. Bibcode:2009GeoRL..3610603K. doi:10.1029/2009gl037620. hdl: 1912/3380 . ISSN   0094-8276. S2CID   55791721.
  10. 1 2 3 Evans, Geoffrey T.; Taylor, F. J. R. (1980). "Phytoplankton accumulation in Langmuir cells1: Phytoplankton in Langmuir cells". Limnology and Oceanography. 25 (5): 840–845. doi:10.4319/lo.1980.25.5.0840.
  11. 1 2 Stavn, Robert Hans (1971). "The Horizontal-Vertical Distribution Hypothesis: Langmuir Circulations and Daphnia Distributions1". Limnology and Oceanography. 16 (2): 453–466. Bibcode:1971LimOc..16..453S. doi: 10.4319/lo.1971.16.2.0453 .
  12. 1 2 Hamner, William M.; Schneider, David (1986). "Regularly spaced rows of medusae in the Bering Sea: Role of Langmuir circulation1". Limnology and Oceanography. 31 (1): 171–176. Bibcode:1986LimOc..31..171H. doi:10.4319/lo.1986.31.1.0171. S2CID   6201912.
  13. ↑ Stavn, Robert Hans (1971). "The Horizontal-Vertical Distribution Hypothesis: Langmuir Circulations and Daphnia Distributions1". Limnology and Oceanography. 16 (2): 453–466. Bibcode:1971LimOc..16..453S. doi: 10.4319/lo.1971.16.2.0453 .
  14. ↑ Nees, J. C. (1949). PhD Thesis, University of Wisconsin, Madison.
  15. 1 2 Stommel, H. (1949). Trajectories of small bodies sinking slowly through convection cells. J. Mar. Res., 8, 24-9.
  16. ↑ Powell, T. M.; Richerson, P. J.; Dillon, T. M.; Agee, B. A.; Dozier, B. J.; Godden, D. A.; Myrup, L. O. (1975-09-26). "Spatial Scales of Current Speed and Phytoplankton Biomass Fluctuations in Lake Tahoe". Science. 189 (4208): 1088–1090. Bibcode:1975Sci...189.1088P. doi:10.1126/science.189.4208.1088. ISSN   0036-8075. PMID   17800160. S2CID   38185886.
  17. ↑ Johnson, David L.; Richardson, Philip L. (1977). "On the wind-induced sinking of Sargassum". Journal of Experimental Marine Biology and Ecology. 28 (3): 255–267. doi:10.1016/0022-0981(77)90095-8. ISSN   0022-0981.
  18. ↑ Ryther, J. H. (1955). Ecology of autotrophic marine dinoflagellates with reference to red water conditions. In: The luminescence of biological systems (Johnson, F. H. (Ed.)), Am. Assoc. Adv. Sci., pp. 387-413
  19. ↑ Koppen, W. (1921). Annln Hydrogr. Berl., 49, 170-3.
  20. ↑ Smayda, T. J. (1970). Suspension and sinking of phytoplankton in the sea. Oceanogr. Mar. Biol. Ann. Rer., 8, 353-414.
  21. ↑ Dierssen, H. M.; Zimmerman, R. C.; Drake, L. A.; Burdige, D. J. (2009-02-18). "Potential export of unattached benthic macroalgae to the deep sea through wind-driven Langmuir circulation". Geophysical Research Letters. 36 (4): L04602. Bibcode:2009GeoRL..36.4602D. doi: 10.1029/2008GL036188 . ISSN   0094-8276.
  22. ↑ Woodcock, A. H. (1944). A theory of surface water motion deduced from the wind-induced motion of the Physalia. J. Mar. Res., 5, 196--205.
  23. ↑ George, D. G. & Edwards, R. W. (1973). Daphnia distribution within Langmuir circulation. Lirnnol. Oceanogr, 18, 798-800.
  24. ↑ Hutchinson, G. E. (1967). A treatise on Limnology, Vol. 2, New York. John Wiley and Sons, lll5pp.
  25. ↑ Titman, David; Kilham, Peter (1976). "Sinking in freshwater phytoplankton: Some ecological implications of cell nutrient status and physical mixing processes1". Limnology and Oceanography. 21 (3): 409–417. Bibcode:1976LimOc..21..409T. doi:10.4319/lo.1976.21.3.0409. hdl: 2027.42/109909 . ISSN   0024-3590.
  26. ↑ McNaught, Donald C.; Hasler, Arthur D. (1961). "Surface Schooling and Feeding Behavior in the White Bass, Roccus Chrysops (Rafinesque), in Lake Mendota". Limnology and Oceanography. 6 (1): 53–60. Bibcode:1961LimOc...6...53M. doi: 10.4319/lo.1961.6.1.0053 . ISSN   0024-3590.
  27. ↑ Tuite, C. H. (1981). "Standing crop densities and distribution of Spirulina and benthic diatoms in East African alkaline saline lakes". Freshwater Biology. 11 (4): 345–360. doi:10.1111/j.1365-2427.1981.tb01266.x. ISSN   0046-5070.
  28. ↑ Botte, Vincenzo; Mansutti, Daniela (August 2012). "A numerical estimate of the plankton-induced sea surface tension effects in a Langmuir circulation". Mathematics and Computers in Simulation. 82 (12): 2916–2928. doi:10.1016/j.matcom.2012.07.014.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.