BALTICA Volume 23 Number 1 June 2010 : 13-24

Mixing by Langmuir circulation in shallow lagoons

Irina Chubarenko, Boris Chubarenko, Elena Esiukova, Henning Baudler

Chubarenko, I., Chubarenko, B., Esiukova, E., Baudler, H., 2010. Mixing by Langmuir circulation in shallow lagoons. Baltica, 23 (1), 13-24. Vilnius. ISSN 0067-3064.

Abstract Field measurements and observations in shallow basins (Vistula Lagoon and Darss–Zingst Bodden Chain, the Baltic Sea) are reported, revealing characteristics of Langmuir circulation (LC) patterns during moderate winds. A system of large–scale rolls with horizontal axes is shown to be different in the open ocean as compared to shallow areas. CTD horizontal tows across the windrows, GPS registration of the cell’ width, videotape recording were used. Regular patterns of windrows, marking the roll-shaped circulation cells, develop within 5–10 min after the wind onset. The most probable distance between streaks is about double the local water depth, so that the width–to–depth ratio for the rolls is equal to 1; 78% of the rolls have a ratio of width to depth from 0.65 to 1.6, with peak values at 0.75, 1, 1.2, 1.4. It is shown, that in a shallow basin the pattern of windrows is fully developed, and the growth of the roll’ diameters are limited by the depth of the basin. Thorough analysis of video–records of the rows’ breakdown and reconstruction has revealed four possible kinds of Y-junctions, whilst in deep areas only one of them is reported to prevail. One more major difference of shallow water LC is caused by the presence of shores: eventually, the wind, waves and water current in a lagoon propagate in different directions. This makes the streak lines curved, and they drift in the direction of the cross–current component of the Stokes wave transport. Mathematical analysis of the behaviour of suspended particles has revealed, that the flow within the LC transports particles of different size and buoyancy along different trajectories, making mixing more effective.

Keywords water mixing, Langmuir circulation, shallow water, Darss–Zingst Bodden Chain, Vistula lagoon.

Irina Chubarenko [[email protected]], Boris Chubarenko [[email protected]], Elena Esiukova [elena_esi- [email protected]], P. P. Shirshov Institute of Oceanology of Russian Academy of Sciences, Atlantic Branch, prospect Mira, 1, 236000 Kaliningrad, Russia; Henning Baudler [[email protected]], Biological Station Zingst, Rostock University, Mühlenstrasse, 27, 18374 Ostseebad Zingst, Germany. Manuscript submitted 21 January 2010; ac- cepted 12 April 2010.

INTRODUCTION subject of the present study––is the roll–structured wind–wave mixing in upper layer, well known as Water mixing, induced by wind and waves, is of the Langmuir circulation. In shallow closed basin it utmost importance for shallow lagoon water dynamics appeared to be much different from the open ocean and water quality. Since influence of both bottom case, being limited by water depth and deflected by and boundaries is very significant, the resulting local currents. We report our field measurements flow structure is much different from the principal and observations in two shallow Baltic lagoons – the picture, well known for deep and unbounded open Vistula Lagoon and Darss–Zingst Bodden Chain, ocean. Before the all, general wind driven flow is carried out during summer measurement campaigns often deflected by boundaries. Then, since basin is of 2000–2006 years. closed, compensating currents occur inevitably along On windy days, one can often observe on water the bottom depressions. And the third––and the main surface of large basins long parallel streaks of foam,

13 flotsam, algae, and called windrows. They have un- doubtedly been familiar to sailors since long ago and even used to categorize wind speed (e.g., Thorpe 2004). These windrows make visible a pattern of parallel pairs of large alternate left–handed and right–handed horizontal rolls, or circulatory cells, called Langmuir circulation (hereinafter, LC). The phenomenon is named after Irving Langmuir, who noticed during an- Atlantic crossing in 1927 that Sargasso weed aligned into nearly regular rows when wind exceeded 5 to 10 m/s, and that, when the wind suddenly shifted 90 degrees, the lines reformed within 20 minutes (Lang- muir 1938). This phenomenon is now considered as one of turbulent processes in upper layers of large water bodies, driven by wind and waves, influential in producing and maintaining the uniform surface mixed layer and in driving dispersion (see Leibovich Fig. 1. Sketch of the Langmuir circulation. Important to note 1983). Even though Langmuir himself continued his that the distance between convergence zones on the surface investigations in smaller basin (Lake George, NY), equals to two roll’ diameters. In lower part forces are shown, acting on the sediment particle suspended by the flow: the and established the essential kinematics of alternating dynamic pressure force Fflow, the Archimedean force FArch and horizontal roll–vortices aligned with the wind there, the gravity force mg; vector sum of the latter two gives the the most serious attention was paid historically to LC buoyancy force B of the particle. features in the open ocean. In order to emphasise the features of LC, which are most important for our study, Perhaps here is the reason why this phenomenon is not we just list here the most famous investigations. So, sufficiently investigated in such areas; a few references Stommel (1949) calculated particle trajectories based still can be mentioned here––laboratory work of Mat- on idealized roll–vortices, and showed that particles, sunaga and Uzaki (2004), and field measurements on which sink (such as phytoplankton), or rise (such as 15 m deep oceanic shelf, reported by Gargett and Wells micro–bubbles), are trapped within the cores of the (2007). Lack of information had motivated us to per- vortices. Laboratory experiments of Faller (1971), form a special field measurement campaign in shallow Faller and Caponi (1978) showed that both wind and and semi–enclosed areas––the Darss–Zingst Bodden surface waves are required to produce the rolls. As- Chain and the Vistula Lagoon of the Baltic Sea. Appar- saf et al. (1971) used aerial photography to observe ently, natural conditions in brackish lagoons are very streak patterns on the ocean surface, and reconfirmed much favourable for the instrumental investigations the existence of multiple scales, as noted originally of the specific features of LC: their waters have some in Langmuir (1938); three scales were seen in several vertical salinity and temperature gradients, with more photos, separated by just under an order of magnitude cold and saline sea waters at the bottom and freshened and ranging from a few to hundreds of meters between riverine waters at the surface. When LC arises in such streaks. Some attempts to explain how the circulations a system, it can easily be registered, for example, by are generated have been done by Garrett (1976), Craik CTD-towing in surface layer: water salinity and tem- (1977), and Leibovich (1977). perature will vary coherently, showing the temperature Theory suggests (Leibovich 1983), that LC is pro- minimum/salinity maximum at the divergence zones, duced by the interaction of wave orbital motions with and the temperature maximum/salinity minimum in depth–varying upper layer current. Consider for clarity zones of convergence. Further statistical data analysis the simplest case, when wind and waves go in the same shows the distribution of line spacing, amplitude of direction, what is typical for the open sea conditions. temperature and salinity variations etc. Here, for the problem under investigation, we use mostly the data of Taken separately, both wind and waves generate water GPS and echo–sounder, since they are in full agreement flows directed down–wind: the wind–induced shear with the results of statistical analysis of CTD data, but flow, decaying with depth, and the wave–induced are easier to handle with. Stokes’ drift. The resulting down–wind flow, however, becomes unstable and breaks down into long rolls, Field site and measurement aligned with the wind––what we see as appearance of techniques windrows on water surface (Fig. 1). In a shallow and closed basin, the features of LC Field measurements and observations of LC have been (like roll’ spacing, direction, lifetime) are influenced performed in Vistula Lagoon (VL) and Darss–Zingst by local bottom topography and depth, coast geometry, Bodden Chain (DZBC) during summer field campaigns particular water density profile, mutual orientation in years 2001-2006 using small scientific boats of wind and coastline, probable near–shore currents. Ecolog of Atlantic Branch of P. P. Shirshov Institute

14 Fig. 2. Measurement sites: shallow Vistula Lagoon and Darss-Zingst Bodden Chain in the Baltic sea. of Oceanology of Russian Academy of Sciences sortechnik GmbH and Idronaut). The CTD probe was and Gamarus of Field Biological Station Zingst of fixed aside, so that measurements have been performed Rostock University (Fig. 2). These coastal lagoons in undisturbed zone from the moving boat with a spa- have limited water exchange with the Baltic Sea (they tial resolution of CTD measurements of 0.5–1.0 m. are of the restricted and chocked hydrological types, The second technique used GPS and echo–sounding: correspondingly; Chubarenko et al. 2005). Maximum coordinates of crossings of the streaks by the boat length of 100 (60) km and average depth of 2.7 (2) were registered by GPS simultaneously with the lo- meters, respectively, allow winds easily mix water cal depth echo–sounding, while the boat was moving column down to the very bottom. The observations perpendicularly to the windrows. Thirdly, 45 min. long were performed in central parts of both lagoons at the movie was taken in order to analyze the dynamics of depths of about 4 and 2.5 m, under moderate winds row’ reconstruction process. of 3–6 up to 9 m s-1 and stable heating conditions Both, the CTD records were rather easy to interpret, (sunshine; air temperature was higher than water and the pattern displayed on the water surface was well temperature). Topography in these areas is very pronounced; that enabled clear identification of the LC signal. Even though the lagoon undergoes significant favourable for the formation of large fields of long wind mixing, water near the bottom is slightly colder regular windrows (see Fig. 1): even and flat bottom and saltier than at the surface (due to connection with and distant shores with low sides. the sea). Langmuir’ vortexes, arising in this weakly Several measurement techniques were applied to stratified environment, transport upward from the bot- register the features of LC. First, horizontal towing tom more saline/cold water (in upwelling zones), while in subsurface layer (5–15 cm) across the streaks was more fresh/warm water from the surface is carried carried out by CTD probes (CTD–90M, ASD Sen- down (in zones of down welling). Zone of upwelling

15 Fig. 3. Water temperature T, salinity S and specific density σ obtained by horizontal sub-surface towing of CTD-probe across Langmuir circulation cells in Vistula Lagoon, 21.07.2001. Wind speed 6-9 m s-1; local depth 2-3 m; distance between lines of the flow convergence (indicated by arrows) is 4-5 m. is associated with flow divergence at the surface (and Observations flow convergence at the bottom, see Fig. 1), while down welling zone is associated with surface flow Threshold wind speed and line spacing convergence (bottom divergence). Horizontal CTD towing through this structure (at any depth) registers Surface streaking is generally reported to occur when a characteristic regular ‘kissing’ behaviour of tem- wind speed exceeds some threshold value, most perature and salinity curves: maximum temperature frequently placed at 3 m s-1 (Thorpe 2004). Under corresponds to minimum salinity and vice versa. To thermally unstable conditions, streaking was observed illustrate this, Fig. 3 displays data of towing of CTD at lower wind speeds. Theoretical considerations probe at the depth of 5–10 cm across the LC cells (Leibovich 1977), numerical modelling (Levina et al. formed in Vistula Lagoon on 21 July 2001 under wind 2000) and laboratory experiments show that helical 6–9 m s-1. Local depth is about 2.5 m. Arrows point structures of various scales are inherent feature of at the surface convergence zones (windrows; down wind–wave–induced flow. They appear how only welling regions). Distance between convergence lines wind–induced current and waves emerge at the surface. As far as we are aware, the minimum scale, observed is 4–5 m, with the temperature / salinity / density dif- in laboratory, is 1–2 mm (Thorpe 2004). With time ference between the convergence and divergence zones -3 of wind/wave action and further development of LC, of about 0.005 °C / 0.001 psu / 0.001 g cm . Data of small scale instabilities (smaller rolls) are swallowed CTD towing had agreed very well with the data of GPS by bigger ones, so that distance between streaks and registration. Since the latter is easier to handle with, intensity of helical motion increase (Levina et al. we have preferred them in presentation. 2000). Observers in field use to detect the surface

16 streaking visually, what means the availability of some visualize the picture of streaks during the observations, tracers at the surface, which make the pattern visible. we are inclined to the latter explanation. Therefore, the value of minimum required wind speed The results of measurements of the distance be- is necessarily subjective. Thorpe (2004) suggested that tween the windrows (undimentionalised by the local the strength of surface–sweeping currents in LC scales water depth of 2 m), observed in August 2004 in flat with the speed, and in field there is some minimum and shallow lagoon of Darss–Bodden (see Fig. 1) under sweeping speed required to overcome the random steady SSW wind of 8–10 m s-1, are presented (Fig. surface speed fluctuations that resist organization of 4). Water and air temperature were 21.3 °C/25 °C, surface tracers into rows. correspondingly. The boat had crossed the windrows In the observations in DZBC in August 2004, perpendicularly, and the observer registered by GPS the we were lucky to monitor the very process of wind coordinates of 161 crossing (total length of the traverse strengthening (from 1 to 9 m s-1) and appearance of was more than 800 m). Later, they were recalculated LC. Under weak winds of 2–3 m s-1, ripples and small into the distances between the points, which varied waves appeared at water surface, however, we could from 2 to 12 m (with the accuracy of GPS registra- tion of 0.1 m). The resulting distribution has many not detect any streaks, even scattering over the surface peaks. Since the roll diameter is roughly one half of additional tracers––long wide lines of small pieces of -1 the measured distance, the results show the peak roll paper. Under winds of 4–5 m s , narrow and weak diameters at 1.5 m, 1.8–2 m, 2.4–2.5 m, 2.9 m, 3.4 m, windrows became at last detectable. Spacing between 3.9 m. Since the local depth D was 1.8–2 m, the ratio them was surprisingly large: 4–5 m, practically the of the roll’ width to the roll’ depth is (0.75–1.9):1, same distance, as persisted later on, when in three -1 with the main peak at the circular cell with proportion hours winds became much stronger (up to 8–9 m s ). 1:1. 78% of roll diameters fall in range from 0.65:1 to This means, that ether (i) flow structure with smaller 1.5:1. Only in 2.5 % of cases the roll width was smaller rolls existed for some period, but the observer could than 0.65 D, whilst in 20 % of cases it was larger than not recognize them because they are too weak, or (ii) 1.9 D. In the latter, high percentage include the cases, the process of energy transfer from smaller to larger when the observer had simply missed some streaks scales proceeds too fast, “swallowing” of smaller rolls in between, if they were weak. The mean distance and permanent change of the spacing between lines between windrows is 5.1 m (calculated for the whole causes variations in surface currents too often, so that set) and 4.2 m (for the most probable 78% of cases); floating material has not enough time to be trapped into this statistical parameter, however, has rather limited more or less defined rows. In a view of our efforts to sense for the given problem.

Fig. 4. Distribution curve for dimensionless distance between the windrows, as registered by GPS in Darss-Zingst Bodden Chain. Total number of counts 161; the length of the traverse is 818 m; local depth 1.8-2 m.

17 The distribution curves from three such experiments first formula is not applicable (it would give L1≈25 m are displayed together: the described above (in DZBC, for the wind speed of about 5 m s-1), while the second presented in Fig. 4 in more details), and in VL (the other one gives L2≈4.6 m, what is within the measured range. two; Fig. 5). The curves are the best fit (polynomial, Even though we believe, that the diameter of Langmuir 5 power) of the experimental data, with the reliability rolls is prescribed by water depth, available for mix- of approximation not less than R2=0.93. Each of data ing, these estimates at least show that open ocean LC sets from VL had 80 experimental points with the is much different from one in shallow water. total length of transverse of 480/490 m. Horizontal Observations of Langmuir circulation in the North axis is the same as in Fig. 4: the ratio of the distance Sea by Graham and Hall (1997) showed the ratio of

Fig. 5. Distribution curves from three experiments: (1) - in Darss-Zingst Bodden Chain (the same experiment as in Fig. 4, which for convenience is reproduced here with thin line); (2) and (3) - in Vistula Lagoon. The curves are the best fit of the experimental data (R2>0.93). between windrows at the surface to the local water windrow spacing to the water depth to be between depth. The figure shows, that in all three experiments 0.3 and 0.5. These values are believed to be typical of the distributions are much alike, with the main peak at fetch limited, shallow seas where cells may not be fully the roll diameters, close to the local depth. In VL, the developed (Thorpe 2004). The values for these ratios, peak spacing is 10% smaller, what may be explained reported by Leibovich (1983), are between 0.66 and by the existence of water stratification near the bottom 1.66, what is very close to our observations. Asaf et al. (the rolls are smaller, because do not reach the very (1971) found the spacing of the largest streak scales bottom). to be 280 m, with a mixed-layer depth of 200 m or L/ In the open ocean, Faller and Woodcock (1964) D=1.4 (in our observations – 1.5). found that there was a significant correlation between the mean crosswind separation of the windrows, L, and Orientation of windrows, their twist and drift the wind speed, W, taking it to be L=(4.8 s) × W (m), while Graham and Hall (1997) found the relation to be In order to estimate the deflection of the streaks’ L=(0.68 s) × W + 1.2 (m), using measurements taken in direction from the wind direction, we have used kind of a shallow sea. For our shallow water measurements, the a weather vane of a size of ca. 30 cm x 50 cm. During

18 the time of observations (≈6 h), the windrows were the cross-wind component of the Stokes drift. For the found to deviate significantly from the wind direction: Fig.6 this means, that near the coast, windrows should up to 30° in both directions (both to the right and to the experience the onshore drift. left from the direction of wind). It happens because in a close basin water currents are much influenced by Formation times and persistence coastlines, and, finally, are significantly deflected from Observers in the ocean usually estimate the time of the direction of wind (Chubarenko, Chubarenko 2002; formation of the LC in 15–20 min. We have found it Chubarenko et al. 2004). An impressive example from difficult to obtain it in our observations, because either coastal zone of the Curonian Lagoon is given (Fig. wind speed increased with time (or it was impossible 6). Living dunes of the lagoon side of the Curonian to say when was ‘the onset’ of wind), or wind was Spit supply by their sand a large shoal of Curonian strong enough, so that LC were just present from Lagoon with the depths of 20–50 cm (on the photo, the beginning to the end of measurements. Rough see Fig. 6). They stretch out of the shore up to several estimation is, that both in VL and in DZBC, regular picture of windrows develops very fast: within 10 min. after the wind onset. As it was mentioned above, we haven’t observed the very process of developing of LC: it had just gradually become apparent in its final spacing, and then persisted for a long time. Windrows may twist and curve, however, are rather stable, and it was impossible to give any characteristic length of them. Variations and change of the structure goes through the formation of so–called Y–junctions (Thorpe 1992). Y–junctions Although LC was usually regarded as regular and steady, the windrows are often twisted and subject to amalgamation one with another. Observations of Farmer and Li (1995) and Thorpe (1992, 2004), -1 Fig. 6. Langmuir circulation in coastal zone of the Curonian performed in sea in winds >12 m s indicated for Lagoon (the Baltic Sea). Windrows are curved by along- the first time that the convergences, as delineated the-shore current. by lines of bubble clouds, can merge into so–called ‘Y–junctions’ (because it looks like the letter Y), predominantly pointing downwind, with an angle hundreds of meters, providing perfect conditions for between the pair at the junction of about 30o. They the formation of LC. Fine net of convergence lines wrote that the physical reason for this could be the between Langmuir’ rolls and regular picture of surface growth of roll’ diameters with growing wind fetch. ripples develop over this submerged plateau even Among just a few observations published about this under the weakest winds. Important is (see Fig. 6) reorganization of lines, there is no information about sleek-lines amalgamation (i.e., the divergence zone that the orientation of windrows change considerably amalgamation). In our observations in lagoons, we near the shore, because wind induced current here is have paid special attention to the process of surface transformed into the coastal jet. The same mechanism pattern reorganization (see also Chubarenko, Baudler works over the entire area of the lagoon: water current, (2006). generated by direct wind shear, interacts with another kinds of current, most often––topographically induced ones (gradient currents, coastal jets etc.). As the result, water current is not of the same direction as wind, hence, the windrows, generated by the currents, are not parallel to the wind either, but mainly following the current direction. There is another complication in this problem. In open areas, typically, wind and waves propagate in the same direction. In shallow or coastal areas, where waves ‘feel’ the bottom, they experience refraction, and wave fronts tend to become parallel to isobaths (in particular, to the shore line, see Fig. 6). Thus, current and waves are not parallel. This situation was inves- Fig. 7. Four principally possible variants of amalgamation/ destruction of Langmuir circulation cells: both convergence tigated analytically by Cox (1997). He showed that and divergence lines can merge together, with narrow end of rolls, aligned with the current, drift in the direction of the resulting Y-junction pointing either up- or downwind.

19 8 8 8 The life–timeThe life–time of Y–junction of Y–junction in our observations in our observations was around was 2–5around min., 2–5 i.e. min., after i.e. 2.5 after min. 2.5 after min. its afterformation its formation The life–time of Y–junction in our theobservations regularthe pictureregular was around inpicture given 2–5 in area given min., was areai.e. re–established after was 2.5re–established min. again. after Fromits again. formation the From board, the the board, observer the observer was able was to monitorable to monitor the the the regular picture in given area was re–establishedoriginationorigination again. and Fromdevelopment and the development board, of athe certain observer of a Y–junctioncertain was Y–junction able (usually to monitor (usually the singlethe the one, single sometimes one, sometimes – two junctions – two junctions at once, at once, In order to understand the kinematics of rolls be- itself to embrace the boat, so that it was always stay- origination and development of a certain Y–junctionnever more)never (usually during more) the theseduring single ca. these one,5 min.; ca.sometimes then,5 min.; the then, – regular two the junctions regularpicture atwaspicture once, re–established was re–established for another for another5–10 min. 5–10 min. neath water surface when one observes the described ing in foam line (convergence zone). Neighboring above merging of windrowsnever more) (convergence during these zones) ca. 5 min.; on then,convergence the regular zonesRather picture often often,Rather was became Y’s re–establishedoften, was twisted Y’s caused was and forbycaused formed theanother very by the 5–10boat. very min.Almost boat. Almostall the time all the (when time anchored), (when anchored), the boat the was boat naturally was naturally it, consider the process in more detail (see Fig. 7). If Y–junction (either in front or behind the ship). In all turned by wind to the position, roughly parallel to windrows; and one of the windrows adjusted itself to embrace to consider four neighbouringRather rolls often, in regular Y’s was picture caused suchby the re–constructions,turned very boat.by wind Almost to thethe all numberposition, the time of roughly (whenrolls increased, anchored), parallel to thewindrows; boat was and naturally one of the windrows adjusted itself to embrace of LC as it is shownturned in the by left wind upper to the panel position, of Fig. roughly 7; what parallel theis logically boat, to windrows;the so boat, thatcorrect: itso wasand that the onealways itboat was of the hadstayingalways windrows disturbed stayingin foam adjusted the inline foam (convergence itself line to(convergence embrace zone). Neighboring zone). Neighboring convergence convergence zones often zones often there are two convergence zones (windrows) and one existed regular structure and large rolls broke out into divergence zone (sleekthe boat, line) so in that between. it was Windalways blows staying smaller in foambecame ones. line became (convergencetwisted twistedand formed zone). and formedY–junction Neighboring Y–junction (either convergence in(either front inorzones frontbehind often or thebehind ship). the In ship). all such In allre–constructions, such re–constructions, the number the number ‘into picture’. The becameamalgamation twisted of and windrows formed Y–junctionon the (eitherof rolls in front ofincreased, rolls or behindincreased, what the is ship).whatlogically isIn logically allcorrect: such re–constructions,correct:the boat the ha dboat disturbed ha thed disturbednumber the existed the existedregular structureregular structure and large and rolls large broke rolls broke water surface means that instead of four rolls in the Discussion beginning one has ofafter rolls reconstruction increased, what two isrolls logically only, correct:out the into boatout smaller ha intod disturbed smallerones. theones. existed regular structure and large rolls broke but their size is twice larger. Thus, in the most often The described observations provide an opportunity out into smaller ones. observed ‘classical oceanic case’ of (convergence) Y– to get better insight into both (i) development of LC junction pointing downwind, the outside rolls gradually with timeDISCUSSION andDISCUSSION (ii) mixing in presence of LC in shallow (i. e. when the observer moves downwind) swallows basins. Below we show that theory of Craik and up the internal onesDISCUSSION and grow in size. This way, the Leibovich and theory of the developed turbulence lead presence of (convergence) Y–junctions at the surface to a veryThe similar describedThe and described complementaryobservations observations provide results, providean the opportunity former an opportunity to get better to get insight better intoinsight both into (i) bothdevelopment (i) development of LC with of LC time with time is an indicator of increase of roll’ diameters. providing physical background of the roll’ appearance, In principle, notThe this described single one, observations but four kinds provide of anand opportunity theand latter (ii) andmixing showingto get(ii) mixingbetterin thepresence wayinsight in presenceof of theirinto LC bothin developmentof shallow LC (i) indevelopment shallow basins. Bebasins. lowof LC weBe with lowshow wetime that show theory that of theory Craik of and Craik Leibovich and Leibovich and theory and theory merging of convergenceand (ii) and mixing divergence in presence lines are of posLC- in shallowwith time.of basins.the The developedof finalBe thelow developedpicture, we turbulence show predicted turbulencethat leadtheory by tothe oflead atheory Craikvery to asimilarofand very Leibovich similarand complementary and theorycomplementary results, results,the former the formerproviding providing physical physical sible in such system (see Fig. 7). They are: the developed turbulence, is greatly like that observed (i) when two divergenceof the developed zones (sleek turbulence lines) merge lead toin a ourverybackground measurements similarbackground and of theincomplementary shallow roll’of the appearance, lagoons, roll’ appearance,results, what and allowsthe the formerand latter the showproviding lattering show the physical ingway the of waytheir of development their development with time. with The time. final The final together – one observes the “Y” of divergence revealing the specific features of mixing by LC in background of the roll’ appearance, and thepicture, latter picture,predicted show ingpredicted by the the way theoryby ofthe their oftheory the development developed of the developed turbulence, with time. turbulence, isThe greatly final is greatlylike that like observed that observed in our measurements in our measurements in in zones, what can naturally be called “d–Y”; shallow areas as compared to deep ones. (ii) when two convergencepicture, predicted zones flow by the together theory – of the developedshallow turbulence,shallow lagoons, lagoons, what is greatly allows what like revealingallows that revealing observed the specific the in ourspecific features measurements features of mixing of inmixingby LC in by shallow LC in shallow areas as areas compared as compared to deep to deep the well known “classical” Y–junction is for­ ones. med, whichshallow can be calledlagoons, more what exactly allows now revealing as Fully theones. developedspecific features picture of mixing by LC in shallow areas as compared to deep the “c–Y”;ones. (iii) splitting up of one convergence zone into two In shallow areas and near the coasts, Langmuir rolls ones gives the inverted “c–Y”; arise readily,Fully developed Fullyas a result developed picture of instability picture of wind–induced shear flow in the presence of depth variable surface (iv) and splittingFully up of developed the divergence picture zone gives the In shallowIn shallow areas and areas near and the near coasts, the Langmuircoasts, Langmuir rolls arise rolls readily, arise readily,as a result as aof result instability of instability of wind–induced of wind–induced shear shear inverted “d–Y”. Stokes’ drift. Theory of Craik and Leibovich (Craik The observationIn in shallowDZBC revealed areas and that near all the typescoasts, Langmuir1977;flow Leibovich rolls in flowthe arise presence 1977,in readily,the 1983)presence of asdepth ahas result of variabledemonstrated depth of instability variable surface the surfaceStokes’ of wind–induced Stokes’drift. Theory drif sheart. Theoryof Craik of and Craik Leib andovich Leib (Craikovich 1977; (Craik Leibovich 1977; Leibovich of Y–junctions exist in the same time in LC picture in reasons and mechanism of the roll formation, but flow in the presence of depth variable surface1977, Stokes’ 1983)1977, drif has 1983)t. demonstratedTheory has demonstratedof Craik the and reasons Leibthe reasonsovichand mechanism (Craik and mechanism 1977; of theLeibovich roll of theformation, roll formation, but it cannot but it predictcannot thepredict parameter, the parameter, a shallow basin. Kinematically this means, that rolls it cannot predict the parameter, most suitable for observation––the rolls’ spacing. In order to explain must both merge together1977, 1983) (in some has locations)demonstrated and thedis- reasons andmost mechanism suitablemost suitable for of observation––the the forroll observation––the formation, rolls’ but spacing.it rolls’cannot spacing. Inpredict order In theto order explainparameter, to explainthe formation the formation of Langmuir of Langmuir cells, Craik cells, and Craik and the formation of Langmuir cells, Craik and Leibovich integrate (in another locations) at the same time. Often, most suitable for observation––the rolls’have spacing. introduced InLeibovich order into to the explainhave equation introduced the formation of water into the ofmotion Langmuirequation of cells, water Craik motion and (the Navier–Stokes equation) an additional force – the if one convergence zone vanished in “c–junction”, Leibovich have introduced into the equation of water motion (the Navier–Stokes equation) an additional force – the (the Navier–Stokes equation)r an additionalr force – the next after it (theLeibovich neighbouring have introducedone) convergence into the equation of water motionr (the Navier–Stokesr equation) an additional force – the the “vortex “vortexforce” force”= F× = U ×, curl whichU , iswhich the is the vector product of the wave Stokes drift, Us(z), and the vorticity zone is very well pronounced, but the next after it (in “vortex force” F U s curlsU , which is the vector product of the wave Stokes drift, Us(z), and the vorticity r r vector product of the wave Stokes drift, U (z), and some time) will be the “d–junction”F. =WeU have× curl notU yet s “vortex force” s , whichthe isvorticity ωthe= vectorcurlU ω =product,curlU associated of, , associatedassociated the with wave the Stokes withwith wind–induced thethe drift, wind–inducedwind– Us(z) current., and the current. vorticity explanation for this; from the point of view of roll’ induced current. ω = Along with the theory of LC, the mechanism similar to the above described was investigated in theory of structure, the both junctionscurlU mean, associated the same: with disap the wind–induced- Along withcurrent.Along the theory with theof LC, theory the ofmechanism LC, the mechanism simi- similar to the above described was investigated in theory of pearance of one divergence zone and one convergence Along with the theory of LC, larthe tomechanism thethe developed abovethe similar describeddeveloped turbulence, to the was turbulence, above investigatedor so–called described or so–called indynamo–theory wastheory investigated dynamo–theory (see, in fortheory (see, example, offor example, Levina etLevina al. 2000). et al. There, 2000). the There, quantity the quantity of of zone (see Fig. 7). of the developed turbulence, or so–called dynamo– The life–time ofthe Y–junction developed in ourturbulence, observations or so–called was theory dynamo–theory (see, for example, (see, for Levinaexample, et Levinaal. 2000). et al.There, 2000). There, the quantity of helicity of the flow, s, was introduced= as s×= U × curlU dr to describe the presence in it of lots of small chaotic around 2–5 min., i.e. after 2.5 min. after its formation the quantityhelicity ofof helicitythe flow, of s, the was flow, introduced s, was introduced as s ∫ U curlU∫ dr to describe the presence in it of lots of small chaotic the regular picture in given area was re–established helicity of the flow, s, was introduced as s = ∫ U ×whirls.curlU Numericaldr to to describe describe modelling the the presence presenceof the flow in in it possessing of lots of smallthe helicity, chaotic had demonstrated (Levina et al. 2000) that the again. From the board, the observer was able to it of lotswhirls. of small Numerical chaotic modelling whirls. Numerical of the flow modelling possessing the helicity, had demonstrated (Levina et al. 2000) that the monitor the originationwhirls. and Numerical development modelling of a certain of the flowof thepossessingdevelopment flow possessingdevelopment the helicity,of turbulence the of hadhelicity, turbulence demonstrated leads had to leadsdemonstratedthe growth (Levina to the ofgrowth et the al. whir2000) of thel’ sizes,that whir the l’to sizes, their toamalgamation their amalgamation and strengthening, and strengthening, until until Y–junction (usually the single one, sometimes – two (Levina et al. 2000) that the development of turbu- junctions at once, neverdevelopment more) during of turbulence these ca. leads5 min.; to thelence growth leads of tothe the whir growthl’ sizes, of tothe their whirl’ amalgamation sizes, to their and strengthening, until then, the regular picture was re–established for another amalgamation and strengthening, until the only gyre is 5–10 min. formed, whose dimensions are limited by the scale of Rather often, Y’s was caused by the very boat. the numerical domain only (so–called hydrodynamic Almost all the time (when anchored), the boat was α–effect). In particular, this theory was successfully naturally turned by wind to the position, roughly par- applied to an explanation of typhoon and tornado allel to windrows; and one of the windrows adjusted generation in highly turbulent atmosphere.

20 The similarity is obvious: the “helicity” in the picture of windrows is already fully developed (the theory of developed turbulence and the “vortex force” rolls have reached the size, limited by water depth), of Craik and Leibovich introduce in fact the same term and now the rolls, growing in size, are balanced in the equation of motion (Chubarenko, Baudler 2006). with the rolls, splitting into smaller ones (in inverted There is a difference as well, and deeply physical one: Y–junctions). in the “helicity” the vector product is taken from the velocity vector and its own vorticity, whilst in the Transportation of material of different buoyancy “vortex force” the same combination is made from physically independent quantities – the velocity U of s In order to disclose specific features of mixing by the Stokes wave drift, and the vorticity w = curlU of the decreasing with depth wind-induced current. Thus, LC in shallow areas, consider the influence of LC the physical meaning now is: the nonlinear interaction onto transport of particles of different buoyancy. It between the surface wave drift and wind–induced has been known for many years that LC causes an current supplies the flow with an additional helicity. increased concentration of motile or buoyant algae; it Now, following Levina et al. (2000), the generated enhances mixing, transports organisms between high by turbulence small whirls are to grow in size, to and low light levels, promotes patchiness of swimming 10 interflow in bigger ones and give them their energy. or buoyant algae, and affects their exposure to light Finally, numerical simulations predict the structure of and pollution. Concentration of buoyant particles, the maximum possible size, limited by just external sediments, oil droplets, orTransportation bubbles within of LC material varies of different buoyancy geometry of the problem. both in time in space. Thus, if in a system there is a source of helicity, the It was shown howeverIn (see order Thorpe to disclose 2004) thatspecific sys- features of mixing by LC in shallow areas, consider the influence of LC onto transport small whirls with time will flow together and form one tem of currents, produced ofby particles such coherent of different structures, buoyancy. It has been known for many years that LC causes an increased concentration of big revolving cell, which occupies all allowed space. limits rather than favours the spreading of floating Turning back to a shallow basin, we find out that it is material, especially in transversemotile direction.or buoyant It canalgae; easily it enhances mixing, transports organisms between high and low light levels, promotes the depth of it, what limits the growth of the Langmuir be observed for the material, floating on the surface: on cells. We see, that the cells are to grow naturally, and patchiness of swimming or buoyant algae, and affects their exposure to light and pollution. Concentration of reaching the windrows, it remains trapped until time the final (“developed”) size of rolls should be (roughly, buoyant particles, sediments, oil droplets, or bubbles within LC varies both in time in space. since the flow is turbulent) equal to the depth of the ba- when the local circulation breaks up, when adjacent sin. The picture we observed in field (see Fig. 1)––with cells split or merge together. BeneathIt wasthe surface, shown howeverwhere (see Thorpe 2004) that system of currents, produced by such coherent structures, different particles have different buoyancy, the process mainly equidistant streaks––can be then classified as limits rather than favours the spreading of floating material, especially in transverse direction. It can easily be a “fully developed” LC, since the cells can no longer of mixing is more complicated. Stommel (1949) has increase their size being restricted by water depth. In calculated particle trajectoriesobserved based for the on material, idealized floating on the surface: on reaching the windrows, it remains trapped until time when the the ocean, the size of rolls can increase together with (steady) roll–vortices, and showed that the particles local circulation breaks up, when adjacent cells split or merge together. Beneath the surface, where different the thickness of the upper mixed layer, so, the picture which slowly sink (such as phytoplankton) or slowly of windrows is never “fully developed”. rise (such as micro–bubbles)particles should have be trapped different within buoyancy, the process of mixing is more complicated. Stommel (1949) has calculated the cores of the vortices (so–calledparticle trajectories Stommel retention based on idealized (steady) roll–vortices, and showed that the particles which slowly sink (such Y–junctions in fully developed LC field zone). Later, Bees et al. (1998) calculated numerically the steady cross–sectionalas streamlines phytoplankton) for the or particles slowly rise (such as micro–bubbles) should be trapped within the cores of the vortices (so– The kinematics of Y–junctions supports our suggestion of neutral, slightly positivecalled and slightly Stommel negative retention buoy zone).- Later, Bees et al. (1998) calculated numerically the steady cross–sectional that, in contrast to the deep areas, the LC in a shallow ancy, and demonstrated that the process is much more basin becomes fully developed soon after the beginning complicated. He found, forstreamlines example, thatfor thefor particlesparticles of neutral, slightly positive and slightly negative buoyancy, and demonstrated that the of the wind action. Indeed, in the ocean, observer of positive buoyancy, thereprocess are two is muchqualitatively more complicated. dif- He found, for example, that for particles of positive buoyancy, there are two report typically the Y–junctions, delineated by clouds ferent behaviours: some particles are trapped in closed of air bubbles and foam (i.e., convergence zones), with orbits at some distance belowqualitatively the surface different (Stommel behaviours: some particles are trapped in closed orbits at some distance below the surface the narrow end of the Y’s overwhelmingly pointing retention zone), whereas others(Stommel accumulate retention at thezone), lines whereas others accumulate at the lines of convergence at the fluid surface, and they downwind (Thorpe 2004). This means (see Figs 7, of convergence at the fluid surface, and they cannot 8), that with increasing wind fetch (or time of the change one type of motioncannot to another. change one type of motion to another. action), four cells (rolls) merge into two rolls. Thus, Not only the initial position ofNot a particle only the is imporinitial -position of a particle is important, but also its density and size. In order to reveal more the process indicates: the rolls grow in size. The fact, tant, but also its density and size. In order to reveal that only the “c–junctions” are reported in the ocean, more details of the motiondetails of particlesof the motion of different of particles of different buoyancy, consider the forces driving the motion of some small but not the “d–junctions” (which are the same merging of four cells into two cells), we can explain only by buoyancy, consider the forces driving the motion of particle within LC cell (bottom of Fig. 1). These forces are the gravity force Fg , the Archimedean buoyancy force difficulties in identification: the “d–Y” is manifested as some small particle within LC cell (bottom of Fig. 1). chain of sleeks (flattened spots) on the water surface, These forces are the gravity force Fg , the Archime- F m and are really difficult to observe. In shallow basins, dean buoyancy force FArchF Archand andthe dynamicthe dynamic pressure pressure force flow . The second Newton’ low applied to the particle of mass p , volume we have registered all four possible kinds of Y–junction force Fflow . The second Newton’ low applied to the (presented in Fig. 7): they demonstrated, that the particle of mass, volume Vp and density ρ p gives r r r r 1 m p ⋅a = FArch + m p ⋅ g + Fflow , (1) 21 r r where a is the particle acceleration, FArch = ρ w ⋅Vp ⋅ g is the Archimedean force, ρw is the water density, r and Fflow is the dynamic pressure force, exerted on the particle by the flow. The sum of the first two forces, we r r r r obtain the buoyancy force B = Fg − FArch = (ρ p − ρw ) ⋅Vp ⋅ g , which can be directed both upward (for floating material) and downward (for sediment particles).

The third force (force of the dynamic pressure) is proportional to the difference of the Lc flow velocity u Lc

and the particle velocity u p . It pushes the particle in the flow direction, if it moves slower than the water does, but

1 Formulae are typed in Microsoft Equation 3.0. 10 10 10 10 Transportation of material of different buoyancy Transportation of materialTransportation of different of material buoyancy of different buoyancy TransportationIn order to disclose of materialspecific features of different of mixing buoyancy by LC in shallow areas, consider the influence of LC onto transport In order to disclose specificIn order features to disclose of mixing specific by features LC in shallow of mixing areas, by LCconsider in shallow the in areas,fluence consider of LC onto the intransportfluence of LC onto transport Inof orderparticles to disclose of different specific buoyancy. features It ofhas mixing been known by LC forin shallowmany years areas, that consider LC causes the inanfluence increased of LC concentration onto transport of of particles of differentof particlesbuoyancy. of It different has been buoyancy. known for It many has been years known that LC for causes many yearsan increased that LC concentration causes an increased of concentration of ofmotile particles or buoyant of different algae; buoyancy. it enhances It hasmixing, been transpor known tsfor organisms many years between that LC high causes and anlow increased light levels, concentration promotes of motile or buoyant algae;motile it enhances or buoyant mixing, algae; transpor it enhancests organisms mixing, transporbetweents high organisms and low between light levels, high anpromotesd low light levels, promotes motilepatchiness or buoyant of swimming algae; itor enhances buoyant algae,mixing, and transpor affectsts their organisms exposure between to light high and an pollution.d low light Concentration levels, promotes of patchiness of swimmingpatchiness or buoyant of swimming algae, and or affects buoyant their algae, exposure and affects to light their and exposure pollution. to Concentration light and pollution. of Concentration of patchinessbuoyant particles, of swimming sediments, or buoyant oil droplets, algae, orand bubbles affects within their exposure LC varies to both light in and time pollution. in space. Concentration of buoyant particles, sediments,buoyant particles,oil droplets, sediments, or bubbles oil withindroplets, LC or varies bubbles both within in time LC in varies space. both in time in space. buoyant Itparticles, was shown sediments, however oil (see droplets, Thorpe or 2004) bubbles that within system LC of varies currents, both produced in time in by space. such coherent structures, It was shown howeverIt (seewas shownThorpe however 2004) that (see system Thorpe of 2004)currents, that produced system of by currents, such coherent produced structures, by such coherent structures, limits ratherIt was than shown favours however the spreading (see Thorpe of floating 2004) that materi systemal, especially of currents, in transverseproduced bydirection. such coherent It can easily structures, be limits rather than favourslimits the rather spreading than favours of floating the spreadingmaterial, especiallyof floating in materi transverseal, especially direction. in Ittransverse can easily direction. be It can easily be limitsobserved rather for thanthe material, favours the floating spreading on the of surface: floating on materi reachial,ng especially the windrows, in transverse it remains direction. trapped It untilcan easilytime when be the observed for the material,observed floating for theon material,the surface: floating on reachi on theng thesurface: windrows, on reachi it remainsng the windrows, trapped until it remains time when trapped the until time when the localobservedlocal circulation circulation for breaks the material,breaks up, when up, floating when adjacent adjacent on thecells surface: cells split splitor on merge orreachi merge together.ng thetogether. windrows, Beneath Beneath the it remains surface, the surface, trapped where where different until different time when the local circulation breaks up, when adjacent cells split or merge together. Beneath the surface, where different 11 localparticles circulation have different breaks buoyancy,up, when adjacent the process cells of split mixing or merge is more together. complicated. Beneath Stommel the surface, (1949) where has calculateddifferent particles have differentparticles buoyancy, have the different process buoyancy, of mixing the is moreprocess complicated. of mixing isStommel more complicated. (1949) has calculatedStommel (1949) has calculated particleparticlesparticle trajectories trajectories have different based based on buoyancy, idealized on idealized (steady)the process (steady) roll–vortices, of roll–vortices, mixing isand more showedand complicated. showed that thethat particles Stommelthe particles which (1949) which slowly has slowly calculated sink sink(such (such particle trajectories based on idealized (steady) roll–vortices, and showed thatdrags the it particles back if it which moves slowly faster sinkthan (such the flow. This way, floating material in the LC surface convergence (down as phytoplankton)particleas phytoplankton) trajectories or slowly or based slowly rise on (such riseidealized (such as micro–bubbles) (steady)as micro–bubbles) roll–vortices, should should be and trapped beshowed trapped within that within thethe particlescores the cores of whichthe of vortices the slowly vortices (so– sink (so– (such as phytoplankton) or slowly rise (such as micro–bubbles) should be trappedwelling within zones)the cores is carriedof the vortices down, while (so– sediment particles in the bottom convergence zones are lifted up (see Fig. 1). calledascalled Stommelphytoplankton) Stommel retention retention or slowlyzone). zone). Later,rise Later,(such Bees asBees et micro–bubbles) al. et (1998) al. (1998) calculated calculatedshould numerically be trappednumerically withinthe steady the the steady corescross–sectional cross–sectional of the vortices (so– called Stommel retention zone). Later, Bees et al. (1998) calculated numericallyCommonly, the steady this cross–sectionalrelation is assumed as streamlinescalledstreamlines Stommel for the for particles theretention particles of zone). neutral, of neutral, Later, slightly Beesslightly positive et al.positive (1998) and slightlyand calculated slightly negative numericallynegative buoyancy, buoyancy, the andsteady demonstratedand cross–sectional demonstrated that thethat the streamlines for the particles of neutral, slightly positive and slightly negative buoyancy, and demonstrated that the 2 streamlinesprocess is much for the more particles complicated. of neutral, He slightlyfound, for positive example, and thatslightly for particlesnegative ofbuoyancy, positive buoyancy,and demonstrated there are that two the Fflow =1 2 ρ w ⋅v ⋅ S ⋅Cx (Re) , process is much moreprocess complicated. is much He more found, complicated. for example, He thatfound, for forparticles example, of positive that for buoyancy,particles of there positive are two buoyancy, there are two processqualitatively is much different more complicated.behaviours: some He found, particles for areexample, trapped that in forclosed particles orbits of at positivesome distance buoyancy, below there the aresurface two= − qualitatively differentqualitatively behaviours: differentsome particles behaviours: are trapped some inparticles closed areorbits trapped at some in closed distance orbits below at somethewhere surface distance v ubelowLc u thep is surface the particle-flow relative velocity, S is the cross–flow particle area and C x (Re) qualitatively(Stommel retention different zone), behaviours: whereas some others particles accumulate are tr atapped the lines in closed of convergence orbits at some at the distance fluid surface, below the and surface they (Stommel retention zone),(Stommel whereas retention others zone), accumulate whereas at othersthe lines accumulate of convergence at the linesat the of fluid convergence surface,represents and at the theythe fluid particle surface, resistance and they coefficient. (Stommelcannot change retention one type zone), of whereasmotion to others another. accumulate at the lines of convergence at the fluid surface, and they cannot change one typecannot of motion change to one another. type of motion to another. Since in shallow water body the mixing by LC reaches the very bottom, it plays a certain role in sediment cannot changeNot only one the type initial of motion position to of another. a particle is important, but also its density and size. In order to reveal more Not only the initial positionNot only of thea particle initial positionis important, of a particlebut also is its important, density and but size. also In its order densityre-suspension. to revealand size. more Let In orderus apply to reveal the equation more (1), as for example, to the vertical motion of a sediment particle in the LC details ofNot the only motion the initialof particles position of differentof a particle buoyancy is important,, consider but the also forces its density driving and the size. motion In order of some to reveal small more details of the motion detailsof particles of the of motion different of particlesbuoyancy of, consider different the buoyancy forces driving, consider the the motion forces of bottomdriving some smallconvergence the motion of zone some (see small Fig. 1) leads to the expression for the acceleration a of a particle in the form: details of the motion of particles of different buoyancy, consider the forces driving the motion of some small particle within LC cell (bottom of Fig. 1). These forces are the gravity force Fg , the Archimedean buoyancy force particle within LC cell (bottom of Fig. 1). These forces are the gravity force Fg , the Archimedean buoyancy force particle within LC cell (bottom of Fig. 1). These forces are the gravity force Fg , the Archimedean buoyancy force ⎛ ⎞ 2 ρw ρ w v ⋅ S ⋅Cx particle within LC cell (bottom of Fig. 1). These forces are the gravity force Fg , the Archimedean buoyancy force a = ⎜ −1⎟⋅ g + ⋅ (2), FArch and the dynamic pressure force Fflow . The second Newton’ low applied to the particle of mass m p , volume ⎜ ρ ⎟ ρ FArch and the dynamic pressure force Fflow . The second Newton’ low applied to the particle of mass m p , volume ⎝ p ⎠ p 2Vp FArch and the dynamic pressure force Fflow . The second Newton’ low applied to the particle of mass m p , volume FArch and the dynamic pressure force Fflow . The second Newton’ low applied to the particle of mass m p , volume Vp and density ρ p gives Vp and density ρ p gives where the ratio of a particle volume to its cross sectional area V S = h represents in fact the characteristic Vp and density ρ p gives V ρ r r r r p and density p gives r r r r r 1 r m p ⋅a = FArch + m p ⋅ gr + Fflow , 1 (1) r the negativesize buoyancy of the particle. will stop The the analysis upward of motion the expression and (2) shows that: (i) particle acceleration depends on its density ( m p ⋅a = FArch + m p ⋅ g +⋅Fflow= , (1)+ ⋅ + 1 r r m p ar FrArch m p g Fdriveflow , the (1) particle back to the bottom. Since less–dense mr ⋅ar= F + m ⋅ gr + F , (1)1 p Arch r p r flow (smallerρ ρ )), and shape tiny and (smaller size ( hC ), particlesh ); (ii) gotthe higherbigger are the density and the size of a particle, the smaller is its where a is the particle acceleration, FArch = ρ w ⋅Vp ⋅ g is the Archimedeanr force,ρ wp is the water density,px p where a is the particle acceleration, FArch = ρ w ⋅Vp ⋅ g is the Archimedean force, w is the water density,ρ wherewhere a ais is the the particle particler acceleration, acceleration, Fr Arch = ρ w ⋅Vp ⋅ g is theaccelerations Archimedean and force, higher upwardw is the velocitieswater density,, they are r = ρ ⋅ ⋅ ρ r where a is the particleis the acceleration, Archimedean F force,Arch w isV ptheg water is the density, Archimedean and trapped force, byacceleration.w theis the upwelling water Thus, density, flow from for the longer very beginning periods of of vertical motion, lighter and smaller sediment particles get bigger and Fflow is the dynamicr pressure force, exerted on the particle by the flow. The sum of the first two forces, we and Fflow is the dynamic pressureFflow force, is the exerted dynamic on the pressure particle force, by the exerted flow. The on thesum of the first two forces, we r and Fflow is the dynamic pressure force, exerted on the particle bytime. the flow. Thesevertical The small sum lifting particles of the accelerations first pass two longer forces, and ways ob wetain in water, higher vertical velocities. F particler r byr the flow. The sum of the first two forces and flow is the dynamicr pressurer r force, exerted on the particler r by the flow. The sumfollowing of the first the circulartwo forces, flow we of LC for a longer portion obtain the buoyancy force B = Fg − FArch =r(ρ p r− ρwr) ⋅Vp ⋅ g , which we obtain can be the rdirected both upward (forWhile floating accelerating upward, the particles move together with flow of upwelling, and thus get lesser obtain the buoyancy force B = Fg − FArch = (ρ p −=ρw ) ⋅−Vp ⋅ g ,= whichρ − canρ be⋅ directed⋅ bothof the upward cell, while(for floating more heavy and big pieces of sedi- obtain the buoyancyrbuoyancyr rforce,force BwhichFg canF Archbe directedr( p bothw ) upwardVp g , which can be directed both upward (for floating obtain the buoyancy force B = F − F = (ρ − ρ ) ⋅V ⋅ g , which can be directedment both escape upwardvelocities the circle(for relative floating earlier. to Thisit. This conclusion makes smaller is if full the upward force of dynamic pressure. Since sediment particles have material)material) and downwardand downward (for sediment(for(for sediment floatingg particles).Arch particles). material)p andw downwardp (for sediment material) andparticles). downward (for sediment particles). agreement with the result of laboratory experiments by material)The and third downward force (force (for sedimentThe of the third dynamic particles).force (forcepressure) of the is proportionaldynamic pressure) to the differenceis Dethleff of the alwayset al.Lc (2009).flow negative velocity buoyancy u Lc and the upward force Fflow becomes smaller and smaller, the balance between them will The third force (force of the dynamic pressure) is proportional to the difference of the Lc flow velocity u Lc u Theproportional third force to (force the difference of the dynamic of the pressure) Lc flow isvelocity proportional Thus, to the the difference existence of of the the Lc LC flow not just velocity maintains Lc 11 the The third force (force of the dynamic pressure) is proportional to the difference of thebe Lcreached flow atvelocity some11 depth,u Lc acceleration will become zero, then downward directed, and finally the negative and the particle velocity u pu. L c It andpushes the particlethe particle velocity in theu flowp . It pushesdirection, the ifparticle it moves sedimentsslower than in thesuspended water does, state, but but really mixes different and the particle velocity u p . It pushes the particleu in the flow direction, if it moves slower than the water does, but and the particlein the flowvelocity direction,p . It pushes if it moves the particleslower thanin the the flow water direction, fractions if it ofmoves it by slower forcing11 than different the water particles does, to but follow 1 u ρ 1 and Formulae the particle are typed velocitydrags in Microsoft itp does,back. It pushesEquation ifbut it dragsmoves the 3.0. itparticle fasterback if thain it thenmoves the flow flow. faster direction, This than way, theif it floatingflow. moves materisloweral than buoyancyin the the LC water willsurface does, stop convergence butthe upward motion(down and drive the particle back to the bottom. Since less–dense (smaller p ) and Formulae are typed in1 Microsoft Equation 3.0. different trajectories. The same idea can be applied to 11 drags it back if it moves fasterFormulae tha Thisnare the typed way,flow. in floating ThisMicrosoft way, material Equation floating in 3.0. materithe LCal surfacein the LC conver surface- convergence (down 1 welling zones) is carried down, while sediment particles in the bottoma floating convergence material: zones particle are liftedwith significant up (see Fig. positive1). Formulae are typed in Microsoftgence Equation (down 3.0.welling zones) is carried down, while buoyancytiny will (smaller just stay h onp ) the particles water surfacegot higher (in surfaceaccelerations and higher upward velocities, they are trapped by the dragswelling it back zones) if it is moves carried faster down, tha whilensediment the sedimentflow. particlesThis particles way, in floating the in thebottom materibottom convergenceal convergence in the LC zonessurface zones convergence are lifted up (down(see Fig. 1). Commonly, this dragsrelation it backis assumed if it moves as faster than the flow. Thisconvergence way, floating zone) materi or describeal in the smallLC surface circles convergence near this (down are lifted up (see Fig. 1). Commonly, this relation is upwelling flow for longer periods of time. These small particles pass longer ways in water, following the circular wellingCommonly, zones) this is carriedrelation down, is assumed while assediment particles in the bottom convergence zones are2position, lifted up while (see less Fig. buoyant 1). ones will be carried down assumed wellingas zones) is carried down,F whileflow = 1sediment2 ρ w ⋅v particles⋅ S ⋅Cx (Re) in the, bottom convergence zones are lifted up (see Fig. 1). Commonly, this relation is assumed as F =1 2 ρ ⋅v 2 ⋅ S ⋅C (Re) , by the flowflow into of LCthe forwater a longer body. portion of the cell, while more heavy and big pieces of sediment escape the circle earlier. Commonly,flow this relationw is assumedx as One more aspect of water mixing and transport wherewhere v = u − u isis thethe particle-flowparticle-flow2 relative relative velocity, ve- S is the cross–flow particle area and C (Re) FLc =1p 2 ρ ⋅v ⋅ S ⋅C (Re) , This conclusion is if full agreementx with the result of laboratory experiments by Dethleff et al. (2009). = − locity, S is theflow cross–floww particle xarea and C (Re) in the presence 2of LC in shallow lagoon in compari- where v uLc u p is the particle-flow relative velocity, S is the cross–flowx particleF area=1 and2 ρ C⋅xv(Re)⋅ S ⋅C (Re) , representsrepresents the particle the particleresistance resistance coefficient. coefficient. sonflow to that inw an Thus,open theoceanx existence follows of from the LCthe notabove just maintains the sediments in suspended state, but really mixes different = − representswhere the vparticleuLc resistanceu p is the coefficient.particle-flowSince in shallowrelative velocity,water body S is thethe cross–flowmixing by particleLC described area and differencesC x (Re) in the picture of Y–junctions. Since in shallow water bodyv = u the− mixingu by LC reaches the very bottom,fractions it ofplays it by a forcingcertain roledifferent in sediment particles to follow different trajectories. The same idea can be applied to a reaches the verywhere bottom, it Lcplays pa iscertain the particle-flow role in Asrelative it was velocity, shown forS is the the first cross–flow time by particle Thorpe area (2004), and C x (Re) representsSince the particle in shallow resistance water bodycoefficient.sediment the mixing re-suspension. by LC reaches Let us the apply very the bottom, equation it plays (1), a certain rolefloating in sediment material: particle with significant positive buoyancy will just stay on the water surface (in surface re-suspension. Letrepresents us apply the the particle equation resistance (1), as for coefficient. example, to thethe verticaldispersion motion of the of materiala sediment across particle the LCin the depends LC re-suspension. Let us apply the equationas for example,(1), as for to example, the vertical to the motion vertical of motiona sediment of a sedimenton cell’s particle life time, in the thus LC on formation of Y–junctions. Since in shallow waterbottom body convergence the mixing byzone LC (see reaches Fig. 1)the leads very to bottom, the expression it plays a for certain the acceleration roleconvergence in sediment a of zone)a particle or describe in the form: small circles near this position, while less buoyant ones will be carried down by the particle in the LCSince bottom in shallowconvergence water zone body (see the Fig.mixing If by for LC example reaches thesome very material bottom, of it positiveplays a certainbuoyancy role in sediment re-suspension.bottom convergence Let us applyzone (seethe equation Fig. 1) leads (1), asto forthe example,expression to forthe the vertical acceleration motion aof of a asediment particle particlein the form: in the LC 1) leads to the expression for the acceleration a of a was initiallyflow trappedinto the waterinto two body. convergence lines at particle inre-suspension. the form: Let us apply⎛ theρ equation⎞ (1),ρ as for2 ⋅ example,⋅ to the vertical motion of a sediment particle in the LC bottom convergence zone (see Fig. 1) leads to the expression for the acceleration⎜ w ⎟ a of a wparticleva certainS inC thex distance form: one from the other, after the re– ⎛ ⎞ 2a = −1 ⋅ g + ⋅ (2), One more aspect of water mixing and transport in the presence of LC in shallow lagoon in comparison to bottomρw convergenceρ w vzone⋅ S ⋅(see⎜Cρx Fig.⎟ 1) leadsρ to theconstruction expression (thefor the roll’ acceleration amalgamation), a of a particlethe material in the form: a = ⎜ −1⎟⋅ g + ⋅ ⎝ p (2),⎠ p 2Vp ⎛⎜ ρ ⎞⎟ ρ 2 2V is carriedthat into in onean open line. ocean Thus, follows the dispersion from the of above the described differences in the picture of Y–junctions. As it was shown ⎝ρw p ⎠ ρ w p v ⋅ S ⋅Cp x a = ⎜ −1⎟⋅ g + ⋅ (2), ⎛ ρ material⎞ isρ reduced.2 ⋅ Such⋅ qualitative analysis of the ⎜ ⎟ (2) ⎜ w ⎟ w v S Cx where theρ ratiop of a particleρ p volume2Vp to its crossa sectional= −area1 ⋅ gV+S = h⋅ represents (2),in fact the characteristic ⎝ ⎠ ⎜ ρ behaviour⎟ ρof the materials with different buoyancy in where the ratio of a particle volume to its cross sectional area V S = h represents⎝ p in fact⎠ the characteristicp 2Vp size of the particle. The analysis of the expression (2) shows that:amalgamating (i) particle acceleration and disintegrating depends rolls on its shows, density that ( the size of wherethe particle. the ratio The of analysis a particle ofwhere volumethe expression the to ratio its cross of (2) a showssectionalparticle that: volume area (i) Vpartic toS its=leh crossacceleration represents sec- independsmaterial fact the onin characteristic itsthe density ocean undergoes( the described above ρ where the ratio of a particle volume to its cross sectional area V S = h represents in fact the characteristic p ), shapetional and area size V ( CS x=, hhp );represents (ii) the bigger in fact are the the character density- andtwo the preferredsize of a particle, ways of the cell smaller re–organization is its (c–Y and sizeρ of the particle. The analysish of the expression (2) shows that: (i) particle acceleration dependsd–Y). At on the its samedensity time, ( the distribution of variants of p ), shape and size ( C x , p ); (ii)istic the sizebigger sizeof theare of particle. thethe densityparticle. The and analysisThe the analysis size of theof of aexpression particle,the expression the smaller (2) shows is its that: (i) particle acceleration depends on its density ( acceleration. Thus, from the very beginning of vertical motion, lighterthe rolls and re–construction smaller sediment in shallows particles isget equiprobable. bigger ρ h (2) shows that: (i) particle acceleration depends on its acceleration.p ), shape and Thus, size (fromC x , thep ); very (ii) thebeginning bigger areof vertical the density motion, and lighterthe size and of asmaller particle, sediment the smallerFinally, particles is as its compared get bigger to the open ocean LC conditions, vertical densitylifting accelerations ( ρ ),), shapeshape andand sizesizeobtain ((CC higher,,hh );); verti (ii)(ii) thecalthe biggervelocities.bigger are the density and the size of a particle, the smaller is its p xx pp the dispersion of the material with positive buoyancy is acceleration.vertical lifting Thus, accelerations from the very and beginningobaretain the higher density of vertical verti andcal the motion, velocities. size of lighter a particle, and smallerthe smaller sediment is particles get bigger While accelerating upward, the particles move together withhigher flow in shallows,of upwelling, whilst and for thus the get negatively–buoyant lesser its acceleration.acceleration. Thus, fromThus, the from very the beginning very beginning of verti- of vertical motion, lighter and smaller sediment particles get bigger vertical liftingWhile accelerations accelerating andupward, obtain the higher particles verti movecal velocities. together with flow of upwelling, andmaterial thus get the lesser dispersion is smaller (since it is trapped velocitiescal relative motion, to lighter it. This and makes smaller smaller sediment the upward particles force get of dynamic pressure. Since sediment particles have vertical lifting accelerations and obtain higher vertiintocal bottom velocities. convergence zones). velocitiesWhile relative accelerating to it. This upward, makesbigger the smaller particles vertical the upwardmove lifting together forceaccelerations of with dynamic flow and of pressure.obtain upwelling, higher Since and sediment thus get particleslesser have always negative buoyancyWhile and acceleratingthe upward force upward, F the becomes particles smaller move together and smaller, with flowthe balance of upwelling, between and them thus will get lesser velocities relative to it. This makes verticalsmaller velocities.the upward force of dynamic pressure.flow Since sediment particles have always negative buoyancy and the upward force Fflow becomes smaller and smaller, the balanceConclu betweension thems will be reached Whileat somevelocities accelerating depth, accelerationrelative upward, to it. will This the become makesparticles zersmaller o,move then the downward upward force directed, of dynamic and finally pressure. the negative Since sediment particles have alwaysbe reached negative at some buoyancy depth, and acceleration the upwardtogether will force with become Fflowflow zer becomesofo, thenupwelling, downwardsmaller and and thus directed, smaller, get lesser and the finallybalance the between negative them will TheF Langmuir circulation is the phenomenon widely buoyancyvelocities will stopalways relative the upward negative to it. This motion buoyancy makes and smaller driveand the the the upward particle upward force back toflow the becomes bottom. Sincesmaller less–dense and smaller, (smaller the balance ρ ) and between them will observed both in the ocean and in shallows.p This be reached at some depth, accelerationforce will of becomedynamic zer pressure.o, then downward Since sediment directed, particles and finally the negative ρ buoyancy will stop the upward motion andbe drive reached the particle at some back depth, to theacceleration bottom. Since will become less–denseprocess zero, (smallerinduces then downward thep ) verticaland directed, currents, and finallywhich the grow negative have always negative buoyancy and the upward force buoyancy will stop the upwardtiny motion(smaller and hp drive) particles the particle got higher back accelerations to the bottom. and Since higher less–dense upwardin size, (smaller velocitiesamalgamate ρ, they) and into are trappedbigger byand the bigger rolls, becomes smaller and smaller, the balance be- p tiny (smaller hp ) particles got higherFflow accelerationsbuoyancy and will higher stop the upward upward velocities motion ,and they drive areoccupying trappedthe particle by allthe back the to space the bottom. (depth) Since allowed less–dense by input (smaller ρ p ) and upwellingtween flow them for longerwill be periodsreached ofat sometime. depth,These accelerationsmall particles pass longer ways in water, following the circular h energetic. In the ocean the ‘allowed’ space is limited by tinyupwelling (smaller flow p ) for particles longer got periods higher willof accelerationstime. become These zero, small and then higherparticles downward upward pass directed, longer velocities waysand, finally they in water, are trappedthe following depth by of the the the circular upper mixed layer, which grows with flow of LC for a tinylonger (smaller portion h pof) particlesthe cell, whilegot higher more accelerationsheavy and big and pieces higher of sediment upward velocitiesescape the, they circle are earlier. trapped by the upwellingflow of LC flow for for a longer longer portion periods of of the time. cell, These while small more particles heavy and pass big longer pieces ways of sediment in water, escape following the circle the circular earlier. This conclusion upwellingis if full agreement flow for longerwith the periods result ofof time.laboratory These experiments small particles by Dethleffpass longer et al. ways (2009). in water, following the circular flowThis of conclusion LC for a longer is if full portion agreement of the22 with cell, thewhile result more of heavylaboratory and bigexperiments pieces of bysediment Dethleff escape et al. the(2009). circle earlier. Thus, theflow existence of LC offor the a longer LC not portion just maintains of the cell, the while sediments more inheavy suspended and big state, pieces but of really sediment mixes escape different the circle earlier. This conclusionThus, the is ifexistence full agreement of the LCwith not the just result maintains of laboratory the sediments experiments in suspended by Dethleff state, et butal. (2009).really mixes different fractions of it byThis forcing conclusion different is particles if full agreement to follow withdifferent the result trajectories. of laboratory The same experiments idea can beby appliedDethleff to et a al. (2009). fractionsThus, of it the by existenceforcing different of the LC particles not just to maintains follow different the sediments trajectories. in suspended The same state, idea but can really be applied mixes to different a floating material: particleThus, with thesignificant existence positive of the LCbuoyancy not just will maintains just stay the on se thediments water insurface suspended (in surface state, but really mixes different fractionsfloating ofmaterial: it by forcing particle different with significant particles topositive follow buoyancy different trajectories.will just stay The on thesame water idea surface can be (inapplied surface to a convergence zone)fractions or describe of it bysmall forcing circles different near this particles position, to followwhile lessdifferent buoyant trajectories. ones will Thebe carried same idea down can by be the applied to a floatingconvergence material: zone) particle or describe with significant small circles positive near thisbuoyancy position, will while just stayless onbuoyant the water ones surface will be (in carried surface down by the flow into the waterfloating body. material: particle with significant positive buoyancy will just stay on the water surface (in surface convergenceflow into the zone) water or body. describe small circles near this position, while less buoyant ones will be carried down by the One moreconvergence aspect of water zone) mixing or describe and transport small circles in the near presence this position, of LC in while shallow less lagoon buoyant in onescomparison will be carriedto down by the flow into Onethe water more body.aspect of water mixing and transport in the presence of LC in shallow lagoon in comparison to that in an open oceanflow intofollows the waterfrom thebody. above described differences in the picture of Y–junctions. As it was shown that in Onean open more ocean aspect follows of water from mixing the above and transport described in differences the presence in ofthe LC picture in shallow of Y–junctions. lagoon in comparisonAs it was shown to One more aspect of water mixing and transport in the presence of LC in shallow lagoon in comparison to that in an open ocean follows from the above described differences in the picture of Y–junctions. As it was shown that in an open ocean follows from the above described differences in the picture of Y–junctions. As it was shown time, mostly due to the very LC–work. The diameters References of the rolls grow simultaneously with the thickness of the upper mixed layer, favouring roll amalgamation Assaf, G., Gerald, R., Gordon, A. L., 1971. Some mecha- rather than destruction. In shallow areas, the LC is nisms of oceanic mixing revealed in aerial photographs. limited by the bottom depth, so some equilibrium state Journal of Geophysical Research 76, 6550-6572. is reached soon, with the diameter of rolls prescribed by Bees, M. A., Mezic, I., McGlade, J., 1998. Planktonic the local water depth. In this state, roll’ amalgamation interactions and chaotic advection in Langmuir circu- and destruction are balanced. Observations show, lation. Mathematics and Computers in Simulation 44, that the picture of windrows in shallow area becomes 527-544. fully developed soon after the beginning of the wind Chubarenko, I.P., Chubarenko, B.V., 2002. General water action (5–10 min). However, even in fully developed dynamics of the Vistula Lagoon. Environmental and structure, the rolls continue amalgamation/destruction Chemical Physics 24(4), 213-217. process, which causes the formation of the Y–junctions Chubarenko, B. V., Koutitonski, V. G., Neves, R., Umgiesser, of four kinds at the water surface. G., 2004. Modelling concept (Chapter 6). In Gonenc, I. In the fully developed picture of the windrows in a E., Wolflin, J. (Eds), Costal Lagoons: Ecosystem Proc- shallow basin, the most probable are the rolls of regular esses and Modelling for Sustainable Use and Develop- circular shape (what is in full agreement with Thorpe ments, CRC Press, Boca Raton, London, New York, (2004), who termed them to be of ‘square shape’). The Washington D. C., 231-306. distribution of the roll’ diameters has other obvious Chubarenko, B. V., Chubarenko, I. P., Baudler, H., 2005. peaking values, where the ratio of the horizontal roll’ Comparison of Darss–Zingst Bodden Chain and Vistula diameter to the roll’ depth is 0.75, 1.2 and 1.4. In a Lagoon (Baltic Sea) in a view of hydrodynamic numeri- whole, 78% of the roll’ diameters in shallow basin are cal modelling. Baltica 18 (2), 56-67. in range 0.65–1.5 of water depth. Chubarenko, O. B., Baudler, H., 2006. Dispersion of mate- Analysis of the motion of particles of different rial in a shallow and deep basins limited by Langmuir buoyancy in the LC showed that they experience differ- circulation. Issledovano v Rossii, 566-577. http://zhurnal. ent forcing in the flow, get different accelerations and ape.relarn.ru/articles /2006/056e.pdf velocities, and finally describe different trajectories. Craik, A. D. D., 1977. The generation of Langmuir circu- Thus, the existence of LC not just maintains sediments lations by an instability mechanism. Journal of Fluid in suspended state and entrains the floating material Mechanics 81, 209-223. into the water body, but really mixes different frac- Cox, S. M., 1997. Onset of Langmuir circulation when shear tions of them by forcing different particles to follow flow and Stokes drift are not parallel. Fluid Dynamics different paths. Research 19, 149-167. To sum up, we conclude that LC is a very important Dethleff, D., Kempema, E. W., Koch, R., Chubarenko, I., environmental phenomenon, because it essentially 2009. On the helical flow of Langmuir circulation – influences the dispersion and transportation of dis- approaching the process of suspension freezing. Cold solved and suspended matter, especially in shallow Region Science and Technology 56 (1), 50-57. basins. In spite of its evident importance, LC is still Faller, A. J., Woodcock, A. H., 1964. The spacing of win- not yet represented (or even parameterized) in envi- drows of Sargassum in the ocean. Journal of Marine ronmental models: neither in global circulation nor in Research 22, 22-29. climate modelling, nor even in operational models, Faller, A. J., 1971. Oceanic turbulence and Langmuir cir- including small scale models for prediction of oil spill culations. Annual Review of Ecology and Systematics propagation. 2, 201-236. Faller, A. J., Caponi, E. A., 1978. Laboratory studies of wind–driven Langmuir circulations. Journal of Geo- Acknowledgements physical Research 83, 3617-3633. Farmer, D. M., Li, M., 1995. Patterns of bubble clouds Financial support of AAA of Rostock University and organized by Langmuir circulation. Journal of Physical Russian Foundation for Basic Research (grant № 10- Oceanography 25, 1425-1440. 05-00540) is kindly acknowledged. We are thankful to Garrett, C., 1976. Generation of Langmuir circulations by our colleagues Dr. A. Humbert (Darmstadt University surface waves – A feedback mechanism. Journal of of Technology), V. Reif (Biological Station Zingst), V. Marine Research 34, 117-130. Chugaevitch (Kaliningrad State University) for fruitful Gargett A, Wells, JR, 2007. Langmuir turbulence in shallow collaboration during field measurement champagnes. water. Part 1. Observations. Journal of Fluid Mechanics Discussion of tank experiments on suspension 576, 27-61. freezing with D. Dethleff (Institute for Polar Ecology, Graham, A., Hall, A.J., 1997. The horizontal distribution Kiel) and E. Kempema (University of Wyoming, of bubbles in a shallow sea. Continental Shelf Research USA) motivated to write the equations of motion of 17, 1051-1082. suspended particles. Authors are grateful to reviewers Langmuir, I., 1938. Surface motion of water induced by for helpful comments. wind. Science 87, 119-123.

23 Leibovich, S., 1977. On the evolution of the system of wind Matsunaga, N., Uzaki, K.-I., 2004. The role of wind waves drift currents and Langmuir circulations in the ocean. on the formation and development of Langmuir circula- Part 1. Theory and averaged current. Journal of Fluid tions in a shallow water region. Coastal Engineering, Proceedings 29th Int. Conf., Lisbon, Portugal, 19-24 Mechanics 79, 715-743. Sept. 2004, 1, 542-554. Leibovich, S., 1983. The form and dynamics of Langmuir Stommel, H., 1949. Trajectories of small bodies sinking circulation. Annual Review of Fluid Mechanics 15, slowly through convection cells. Journal of Marine 391-427. Research 8, 24-29. Levina, G. V., Moiseev, V. V., Rutkevich, P. B., 2000. Hydro- Thorpe, S. A., 1992. The breakup of Langmuir circulation and the instability of an array of vortices. Journal of dynamic alpha–effect in a convective system. Advances Physical Oceanography 22, 350-360. in Fluid Mechanics, Nonlinear Instability, Chaos and Thorpe, S. A., 2004. Langmuir Circulation. Annual Review Turbulence, V. 2, 111-161. of Fluid Mechanics 36, 55-79.

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