where z is the absolute viscosity of the water (units: kg m-1 s-1). These various equations have been used to calculate that the turbulence generated in the open waters of the unstratified Bodensee (Lake of Constance) by winds of 5-20ms-1 drive a spectrum of eddies penetrating to depths of between 45-180 m, dissipating at rates (e) of between 1.4 x 10-8to2.2 x 10-7m2s-3 and culminating in eddy sizes (lm) of between 2.9 and 1.5 mm. In stratified lakes, where the density gradient acts as a barrier to penetration by weak eddies, and in shallow lakes, where the water column is unable to accommodate the unrestricted propagation of turbulence, the same driving energy must be dissipated within a smaller spatial extent and, hence, at a faster rate and to a smaller spatial limit. The energy of a 20 m s-1 wind applied to Lough Neagh (mean depth < 9 m), is calculated to be dissipated at ^4.3 x 10 6m2 s 3, i.e., at nearly twenty times the rate in Bodensee, under the same wind forcing, and culminating in an eddy size of ~0.7mm. In the most aggressively mixed estuaries and fluvial rapids, e may approach 5.5 x 10-4m2 s-3, with eddies as small as 0.2 mm across. The highest values observed in lakes are near the air-water interface and are of order 10-5 m2 s-3 but typical high values are of order 10-6m2 s-3.
The capacity of turbulent motion to entrain particles, including living organisms, depends broadly upon the magnitude of the relation of the turbulent velocity scale to the intrinsic rates of gravitational settling of the particles in water (ws, in ms-1). Whereas a stone always drops almost unimpeded through water, particles of clay (<5 mm in diameter but of similar density) persist in suspension for long periods and remain dispersed by the aggressive mixing evident in tidal estuaries. The sinking velocities of planktic diatoms such as Asterionella (ws < 10 mms-1), the cells of which form stellate colonies measuring 150-200 mm across, are so trivial compared to the shear velocity of turbulence generated at the lake surface by a wind of (say) 8 ms-1 (~1 x 10-2ms-1) that their entrainment is near complete (u/ws ~103); the algae go more or less where the flow takes them. In metalimnia, or in the boundary layer adjacent to the lake bottom, wherein turbulent velocities are abruptly depressed to below 10~3ms_1, the same diatoms are readily disentrained, sinking almost unimpeded. For smaller and less dense algae and bacteria, the threshold of disentrainment may be an order of magnitude or two smaller. Equally, the rather larger cells of some dino-flagellates, such as Ceratium, and the colonies of the Cyanobacterium, Microcystis, that aspire to rising velocities of <300 mms-1, have a correspondingly enhanced capacity to escape from weakly turbulent flows. The generally recognised criterion for the entrainment of particles is that C = ws/(15 u) < 1; based on the proportionality of the velocities of intrinsic settling and shear, however, entrainment can never be wholly complete. There is a persistent tendency for particles to be disentrained gradually into adjacent structures.
Physical Influences on the Size and Structure of Organisms
The scaling of turbulence, observations as to orders of magnitude of turbulence, and understandings of boundary layer fluid dynamics provide insight into a number of biological problems. For instance, in the context of evolutionary adaptation to physical characteristics of the aquatic medium, it is plain that the foliage of submerged macrophytic plants and algae require tensile strength rather than rigidity; live biomass is supported by the density of the water but turbulent stresses are accommodated by their typically, flexible, flat, much-dissected or filamentous leaves. Self-evidently, macroinvertebrates and fish must be adequately robust to maintain their integrity in the face of turbulence, some of which is, of course, generated by their own movements. In contrast, the often delicate and unstrengthened substance of plank-tic protistans, algae and cyanobacteria, most of which are smaller or much smaller than 1 mm across, is normally protected within the viscous range of the eddy spectrum. While they have no power to prevent their entrainment by turbulence, they rarely experience its physical stresses. When larger algae are exposed to intense turbulence, however, their sizes relative to the smallest eddies expose them to shear; mostly they are insufficiently robust to resist fragmentation and disruption.
Generally, the adaptive problem for small organisms is rather to cope with viscosity - especially in the context of motility and the foraging requirement of phagotrophic protists to encounter food. The analogy has been made of a person collecting bananas suspended in a swimming pool filled with a fluid of the consistency of molasses or setting concrete.
Reciprocating swimming strokes are of less use than are boring or grapple lines, favouring slender flagella over flat paddles. In contrast, mesozooplanktic crustaceans are of a size (in the range 0.2 to 2 mm) that coincides with the transitional scale, in which swimming and the generation of shear currents are feasible; these currents may entrain small particles towards the feeding apparatus where it may be collected as food. In the filter-feeding cladocerans (including Daphnia), the flattened, paddle-like abdominal limbs (phyllo-pods) beat rhythmically within a feeding chamber formed by the abdomen and the carapace: an inhalant current is thus drawn into a chamber, where the particles are strained out by bristle-like setae on the phyl-lopods, and carried in viscous flow to the mouth. To pursue the analogy, another person immersed in the same fluid might gain more satisfaction through collecting kidney beans from a vessel closer to the size of a bath tub. It is only by being big enough and strong enough to escape the drag of viscosity and to be able to generate turbulent currents that the rate of particle encounter can be significantly increased.
Open turbulence is also constrained in the vicinity of solid surfaces, for instance, at the bottom and sides bounding the flow field, as well as the bodies of aquatic animals. Boundary layers are regions of reduced flow adjacent to the solid surfaces, characterized by a gradient of velocities perpendicular to the surface. Depending upon current velocity in the main direction of flow, boundary gradients may be compressed to fractions of a millimeter or extend over several meters. Their presence influences the structural and behavioral adaptations of organisms in a number of ways. A well-known example is the dorsoventral flattening of Ecdyonurid mayfly larvae that can apply themselves to the surfaces of stones in fast flowing streams. By presenting a protrusion of barely 2 mm above the stone surface, animals are able to exploit the boundary layer as a physical refuge from the flow, whilst they graze on the diatoms and other algae attached to the surface. These algae are of a small size (generally <0.3 mm) so they also experience minimal shear stress; if they are sufficiently dense, their presence augments the boundary effect by extending it further into the water. It should be noted that boundary layers are not necessarily free of turbulence: even weak eddies potentially increase the delivery of food particles to the vicinity of benthic filter feeders. In the case of stream-dwelling larvae of the black fly, Simulium, the firm attachment and attitude of the abdomen ensure the optimum presentation of the filtering mouth parts to the flow.
Subject to other conditions, broader boundary layers, variously adjacent to stony or silty river beds or within beds of submerged macrophytic plants, offer refuges of weakened turbulence exploitable by algae, fungi and macroinvertebrates that together comprise the Aufwuchs communities. There is no exact equivalent word in English - but the substratum-specific subdivisions of epilithic, epipelic, and epiphytic associations together convey the analogous concept; the term periphyton is also close but it should refer strictly to the microhabitats and species associated with aquatic macrophytes. Periphyton and the plants in shallow water with which it is associated together constitute some of the most productive habitats in the biosphere, in some instances turning over more than 1.5 kg C m~2 annually.
Just as open-water turbulence dominates the environment of the planktic organisms in the near-surface layer, so the vertical extent of the entraining mixed layer has far-reaching consequences for the photosyn-thetic activity of phytoplankton. The vertical attenuation of light penetrating the surface, even in clear water, is such that significant net photosynthetic carbon fixation is severely restricted at depths greater than 60 m. At low angles of incidence and under cloud cover, this photic depth is further reduced. However, coloration of the water (due, for instance, to humic substances in solution) and the suspension of clay or other fine particles, as well as the presence of significant concentrations of phytoplankton, may restrict the photic depth to the order of a few meters or even centimeters only. In these instances, turbulent entrainment may well carry photosynthetic organisms to depths where net photosynthesis cannot be sustained. Under steady wind forcing, the probability is that the same chlorophyll-containing organisms will soon be carried back to the depth range wherein production may be resumed. Calculation of the proportionality of time during which net production is actually possible is complex, not least because, within limits, the photosynthetic apparatus itself is able to compensate to the average light exposure ('light adaptation'). However, the product of the day length and the ratio of the volumes of the photic and the wind-mixed layers (in large lakes, the ratio of their depths, hp/hm is convenient) is adequately illustrative. Ignoring, for the present, the variables determining the evaluation of hp and focusing on the variability in wind forcing on the water columns of deep, non-stratified lakes, there is an approximate proportionality among wind velocity, u* and hm, winds of 5-20 ms-1 being theoretically capable of driving turbulence to depths of about 44-177 m. For a mixed layer depth of 44 m and a dissipation rate of 10~7m2s~3, as would be found on a windy, cloudy day, it would require, probabilistically, an average of 42 minutes for an alga entrained in open turbulence to be transported through the entire mixed layer. In mixed columns truncated by pycno-clines or the bottom of the lake, the mixing time is proportional to the (constrained) depth. In mixed layers of the order of one meter in depth, mixing times are in the order of one minute.
The interaction between hm and hp can be used to approximate the instantaneous phytoplankton-carrying capacity of the mixed layer to support a productive phytoplankton. Whereas an otherwise clear mixed layer of 10 m in depth might support a chlorophyll concentration <150 mgm~3, deepening it to 40 m reduces the maximum supportable concentration to <20 mg m~3; mixing to 80 m takes the capacity to under 1 mg m~3. It follows that, in deep, temperate lakes like Bodensee, phytoplankton growth and net population increase in the winter months is weak, pending longer days, more surface heating and the onset of thermal stratification. Equally, as heat is lost in the autumnal months, so the stability of stratification and the resistance to wind mixing weaken, the mixed layer increases in depth and the downwelling light energy is increasingly diluted.
Mixed Layer Depth and the Maintenance of Non-motile Plankton
Weak winds and a shrinking mixed layer depth are not, however, an unmitigated benefit to phytoplankton. For non-motile plankton, as well as fine particles, the thickness of the entraining mixed layer has an important relationship to the intrinsic rates of their settlement (ws) and, thus, to the probability of their entry into the lower, less turbulent basal layer. Depletion from suspension is a first-order process, analogous to dilution; the rate of the decline in the suspended population is described by an exponent (—rS), equivalent to —(ws/hm). Plainly, the greater is the particle-specific ws, the greater is the dependence upon a deep mixed layer for its maintenance in suspension and the greater is its sensitivity to density stratification and the consequent contraction of hm. It may seem paradoxical that planktic diatoms such as Asterionella (ws typically in the range 0.3-1.0 m day—1) require a shrinking mixed layer to promote their growth but then have to grow fast enough to balance sinking loss rates from a shrinking mixed layer! A pycnocline reaching to within 3 m of the surface will impose a sinking loss rate of rS >—0.1 day—1, which must be countered by a comparable rate of cell recruitment (r') if a net collapse of the standing population is to be avoided. This is well within the capacity of the maximum performance of Asterionella but low temperature, shortage of nutrients and other losses (flow displacement, grazing by zooplankton) often make this hard to attain. Nevertheless, the onset of density stratification, especially in smaller lakes, is generally the principal correlative of the culmination of diatom abundance (as, for instance, in the so-called 'spring bloom') and settlement from suspension contributes prominently to their subsequent demise.
Not all plankton is heavier than water; diminution of the surface mixed layer promotes different strategies somewhat dependent upon the trophic status of the lake. In oligotrophic lakes, small, near-neutrally buoyant algae may proferate. In mesotrophic and eutrophic lakes, buoyant plankton may predominate. The most extreme case is also well known to freshwater ecolo-gists: under calm conditions, physical entrainment may be insufficient to prevent suitably buoyant Cyanobac-teria from accumulating at the water surface, where, if present in the water in significant amounts, they may constitute a striking surface scum. The exaggerated view of their abundance and the rapidity with which such 'water blooms' form have given them a notoriety among water users; that many of these are now known also to be toxic to livestock and to human consumers has increased the impetus to devise measures to better manage the frequency of their occurrence.
A major problem for planktic photoautotroph is to accumulate from the suspending water a sufficient stock of the essential nutrients to be able not just to maintain itself but to accumulate the material requirements of the next generation (effectively harvesting its own mass again of carbon, nitrogen, phosphorus, etc. within each generation). There are at least twenty elements required in the production of living cells, several of which (C, N, P) are typically regarded as being in short supply relative to the needs and, hence, likely to limit the capacity to support biomass. For autotrophic phytoplankton, these elements are obtained essentially from relatively simple compounds and ions, dissolved in the water. In most lakes, the main proximal sources of these elements (even the gaseous ones, like carbon dioxide) are dissolved in rainfall and, especially, inflowing streams draining the catchment. There is thus a tendency for supplies to fluctuate seasonally, and often quite uncoupled from the demands of production. Moreover, while the growth of phytoplank-ton removes nutrients from the water in the surface circulation, they are unlikely to be released again only after cells have died, including through consumption by animals and through progressive settlement of intact cells beyond the photic zone and the mixed layer. Other things being equal, the net flux of fecal pellets, detritus and disentrained phytoplankton is to greater depth. At the basin scale, there is an inevitable tendency for the resources of the lake to gravitate towards the bottom and for the surface waters to be depleted. Density stratification of the water masses only amplifies this segregation.
As a physical counter to vertical segregation, the mechanisms for transporting nutrient-rich water from depth may assume considerable biological importance. Increased wind action may expand the circulation of the surface mixed layer, depressing the pycnocline downwind, raising the shear stress of the return current and increasing the intensity of internal wave generation. Erosion and entrainment of deep, relatively nutrient-rich water into a deepened surface mixed layer can have a significant effect in refreshing nutrient availability and stimulating phyto-plankton growth. Such macroscale events may pass, with the system returning to something approaching the earlier physical state. Alternatively, they may persist and, aided by surface cooling and seasonally more frequent wind episodes, lead eventually to the breakdown of stratification and substantial vertical mixing throughout most of the water column.
Biological-physical interactions influence the nutrient relationships at the microscale of individual phyto-plankton cells. The resources present within healthy living cells are typically about one million times more concentrated than they are in the medium. This formidable gradient cannot be overcome by passive movement of molecules into the cell - they would move rapidly in the opposite direction - an elaborate apparatus and system of reactivity captures, retains and transports specific molecules from the exterior of the cell to the intracellular sites of deployment and assimilation. The operation involves kinases and the consumption of ATP; the cell expends significant energy in order to assimilate its essential resources.
Suspended in the relative vastness of the water mass, however, the phytoplankton cell can have no such influence on the external supply of nutrients or the frequency of the molecular encounters that it requires to satisfy its uptake demands. Within its immediate viscous environment, the cell is dependent upon diffusion of target molecules through the medium, in place of those taken up across the cell membrane. Delivery is governed by Fick's laws: the number of moles (n) of a solute that will diffuse across an area (a) per unit time, t, is a function of the gradient of solute concentration, C (i.e., dC/dx), and the coefficient of molecular diffusion of the substance (m). Then, introducing appropriate units, uptake per unit time is:
Taking a small, spherical alga like Chlorella (diameter, d — 4 x 10"6m; volume — 33 x 10"18m3; surface area — 50 x 10"12m2) and given that (i), for a small-sized molecule such as that of carbon dioxide, m — 10"9 m2 s"1, (ii), the thickness of the water layer adjacent from which CO2 molecules can be sequestered is equal to the cell radius, and (iii) that the carbon dioxide concentration in the water beyond is at air equilibrium (11 pmol l" , or 11 x 10"3 mol m ), eqn.  may be solved to show diffusion should sustain a rate of acquisition by the cell equivalent to 275 x 10"18 mol s . Given a cell carbon content of 0.63 x 10"12mol carbon and that the doubling requirement is the uptake of a further 0.63 x 10"12mol carbon, it can be calculated that eqn.  reveals a theoretical capability of fulfilling the demand within —2300 s, or just over 38min. This is rather less than the generation time (>9 h at 20 °C), so there is a considerable scope for CO2 depletion before it impinges on the growth rate of the cell.
For comparison, a turbulent velocity of 10"2ms"1 would deliver CO2 molecules at a the same concentration of 11 x 10"3mol m through an area equal to the area projected by the same Chlorella cell (-12.6 x 10"12 m2) at a rate of 1386 x 10"18 mol s"1. The capacity of open turbulence to fulfill a minimum frequency of encounter with nutrient molecules becomes especially important at very low resource concentrations. Taking its requirement for phosphorus instead of carbon (say, one hundredth the number of carbon atoms), the Chlorella mother cell has to accumulate some 0.006 x 10"12mol if the cell is to sustain adequately the next doubling of biomass. Like most freshwater phytoplankters, Chlorella has a high capacity for the uptake of phosphate (maximum — 13.5 x 10"18mol cell"1 s"1), with a sufficient affinity for phosphate molecules for the uptake to be half-saturated by an external concentration of soluble, reactive phosphorus of about 0.7 x 10"3 mol Pm"3 (about 20 mg P l"1). Given these conditions, the requirement could be met in less than 15 min). On the other hand, the external concentration required to sustain the maximum rate of growth (at 20 °C, 1.84 day"1) may be sequestered from an external concentration of just 6.3 x 10"6molP m, about 0.2 mg Pl"1, before the rate of growth may be supposed to become phosphorus-limited.
Ultimately, the rate of renewal of the local water and the delivery of fresh nutrients to the vicinity of the algal cell affects the capacity of molecular diffu-sivity to meet growth demands. Being generally smaller than the turbulent eddies, most phytoplankton cells are eventually reliant upon movement of themselves and their viscous packets of water relative to the flow field to realize the environmental capacity to support their growth. The effect of advective transport of a given flow field, relative to diffusion, on a phytoplankton cell can be calculated from the ratio of the respective dimensionless Peclet numbers (Pe) of particles phytoplankton cells, of diameter d, either sinking passively in turbulence-free water or transported advectively through the water.
where D is the appropriate diffusion coefficient in water. In the absence of turbulence, D is substituted for m. The calculation with respect to advective transport is difficult but solutions have been presented by Riebesell and Wolf-Gladrow (see Further Reading). Their calculations showed Pe values for smaller phytoplankton generally fall within the range 0.1-10, the larger values applying to larger algae moving more rapidly through the water. The Sherwood Number (Sh) expresses the ratio of the fluxes of nutrients arriving at the cell surface in the presence of motion with those of diffusion. For the small cells embedded deep within the eddy spectrum (Pe < 1), Sh is always close to 1, signifying that any effect of motion is marginal. For larger organisms (1 < Pe < 100), the non-linear dependence of the nutrient supply on advection is approximated by
The effect of turbulence on the nutrient supply to cells of varying diameters (d) is also scaled to the Peclet numbers, calculated from the turbulent shear rate dissipation rate, as
Then, in the range 0.01 < Pe < 100, the Sherwood scale conforms to:
These derivations are interesting in the context of selection of evolutionary adaptations. It is at once apparent that the absolutely greater nutrient requirements of larger cells are more difficult to glean from low concentrations of specific nutrients in the medium, without the intervention of significant levels of turbulence. The further deduction is made that this dependence leaves larger phytoplankters less likely than smaller ones to be able to fulfill all their resource requirements in unit time, except under strongly turbulent conditions. With a less prevalent constraint on the development of picoplanktic and nanoplanktic than on the growth of large algae, their frequently-observed dominance over microplankton in chronically nutrient-deficient oligotrophic systems finds a compelling physical explanation. There are other alternative, probably additive, reasons for the dominance of picophytoplankton, invoking the inability of most filter-feeding mesozooplankters to harvest them adequately from low concentrations. Interestingly, it generally requires the intervention of a macroscale physical event, such as storm deepening, to raise nutrient levels for a period long enough for larger algae to flourish. This principle, that deep turbulent embedding and passive delivery of nutrients provides a superior strategy for the survival of photoauto-trophs in an extensive, chronically nutrient-deficient medium, contrasts with what happens in the face of nutrient depletion in the epilimnion of a seasonally-stratified eutrophic lake. Here, the selective advantage passes to large-celled or large coenobial, motile, phy-toplankton that are the most readily disentrained from weak turbulence and are capable of undertaking vertical migrations to scavenge the residual nutrient resources of the water column.
This regulation by nutrient fluxes of the size selection of planktic organisms is an especially satisfying instance of demonstrable physical-biological interaction in open waters. It is clear that the interactions between fundamental organismic processes and the physical motion of the fluid in which they may be suspended do not stop with transport and entrainment. Physical processes play a large part in sustaining nutrient fluxes in chemically rarefied pelagic environments and in underpinning their relative exploitability as habitats of autotrophic microorganisms.
See also: Currents in Stratified Water Bodies 1: Density-Driven Flows; Currents in Stratified Water Bodies 2: Internal Waves; Currents in Stratified Water Bodies 3: Effects of Rotation; Currents in the Upper Mixed Layer and in Unstratified Water Bodies; Small-Scale Turbulence and Mixing: Energy Fluxes in Stratified Lakes.
Huisman J, van Oostveen P, and Weissing FJ (1999) Critical depth and critical turbulence: two different mechanisms for the development of phytoplankton blooms. Limnology and Oceanography 44: 1781-1787.
Hudson J, Schindler DW, and Taylor W (2000) Phosphate concentrations in lakes. Nature 406: 504-506.
Imberger J (1985) Thermal characteristics of standing waters: An illustration of dynamic processes. Hydrobiologia 125: 7-29.
Karp-Boss L, Boss E, and Jumars PA (1996) Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion. Oceanography and Marine Biology 34: 71-107.
Maclntyre S (1993) Vertical mixing in a shallow eutrophic lake -Possible consequences for the light climate of phytoplankton. Limnology and Oceanography 38: 798-817.
Mann KH and Lazier JRN (1991) Dynamics of Marine Ecosystems. Oxford, UK: Blackwell.
O'Brien KR, Ivey GN, Hamilton DP, et al. (2003) Simple mixing criteria for the growth of negatively buoyant phytoplankton. Limnology and Oceanography 48: 1326-1337.
Reynolds CS (1998) Plants in motion: Physical-biological interaction in the plankton. Coastal and Estuarine Studies 54: 535-560.
Reynolds CS (2006) Ecology of Phytoplankton. Cambridge, UK: Cambridge University Press.
Riebesell U and Wolf-Gladrow DA (2002) Supply and uptake of inorganic nutrients. In: Wiliams J.leB, Thomas DN, and Reynolds CS (eds.) Phytoplankton Productivity, pp. 109-140. Oxford, UK: Blackwell Science.
Rothschild BJ and Osborn TR (1988) Small-scale turbulence and plankton contact rates. Journal of Plankton Research 10: 465-474.
Vogel S (1994) Life in Mixing Fluids, 2nd edn. Princeton, NJ: Princeton University Press.
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