The hydraulic displacement and dispersion of phytoplankton is best approached by considering the case of algae in small lakes or tidal pools in which the volume is vulnerable to episodes of rapid flushing. Inflow is exchanged with the instantaneous lake volume and embedded plank-tic cells are removed in the outflowing volume that is displaced. In this instance, the algae thus removed from the water body are considered 'lost'. It may well be that the individuals thus 'lost' will survive to establish populations elsewhere. Indeed, this is an essential process of species dispersal. The balance of the original pop-ulaton that remains is, of course, now smaller and, occupying the similar volume of lake, on average, less concentrated. The predicted net rate of change in the depleting population may be offset or possibly compensated by the simultaneous rate of cell replication but, for the moment, we shall consider just the effects of biomass loss on the residual population. The essential question is whether the inflow simply replaces the original volume by direct displacement or by a flushing action, in which the inflow volume mixes extensively with the standing volume, displacing an equivalent volume of well-mixed water. In the latter case, the original volume and the algal population that it entrained will have been depleted less completely and will now be, on average, less dilute.

The mathematics of dilution are well established. Dilution of the standing volume and its suspended phytoplankton is described by an exponential-decay function. Until Uhlmann's (1971) consideration of the topic, there had been few attempts to express the dilution of phyto-plankton by displacement. He was quick to see and to exploit the opportunity to put losses in the same terms as recruitment (cf. Eq. 5.3) and simply sum the instantaneous exponents. As depletion rates to settlement and some forms of grazing succumb to analogous exponential functions, it is perhaps helpful to rehearse the logic that is invoked.

In the present case of dilution by wash-out, we suppose that a population of uniform, nonliving, isopycnic particles (N0) is fully entrained and evenly dispersed through the body of a brim-full impoundment of volume, V. The introduction of a volume of particle-free water, qs, in unit time t, displaces an equal volume into the outflow from the impoundment. Thus, the theoretical retention time of the impoundment, tr, is given by V/qs. The outflow volume will carry some of the suspended particles. From the initial population (No), particles will be removed in the proportion â€” qs /V. After a given short time step, t, the population remaining, Nt, is calculable from:

During a second time period, of identical length to the first, an equal proportion of the original might be removed but only if the remainder original population has not been intermixed with and diluted by the inflowing water. If there is mixing sufficient to render the residue uniform at t, then, at t2, we should have:

= N0(1 - qst/V)(1 - qst/V). Thus, after i such periods of length t,

and after one lake retention time (tw)

This is, of course, an exponential series, which quickly tends to

where e is the natural logarithmic base. Equation (6.5) has a direct mathematical solution (Ntw = 0.37N0) which predicts that, at the end of one theoretical retention time, the volume diluted by flushing retains 0.37 of the original population. Had the same volume simply been displaced by the inflow, the retained population would have fallen to zero. These possibilities represent the boundaries of probable dilution, lying between complete mixing with, and complete displacement by, the inflow volume.

Supposing the tendency is strongly towards the flushing of algae by inflow, Eq. (6.5) may be used to estimate the residual population at any given point in time, t, so long as the same rates of fluid exchange apply:

To now derive a term for the rate of change in the standing population that is attributable to outwash (rw in Eq. 6.1) is quite straightforward, provided that the time dimension of the inflow rate (s-1, d-1) is compatible with the other terms.

6.2.2 Dilution in the population ecology of phytoplankton

At first sight, the magnitude of hydraulic displacement rates (qs) relative to the scale of standing volume in large lakes and seas seems sufficiently trivial for outwash to be discounted as a critical factor in the population ecology of phyto-plankton. At the mesoscales of patch formation, where recruitment by growth is pitched against dilution through dispersion, it is possible to consider V as the patch size and qs as the rate of its horizontal diffusion as being critical to the maintenance of patchiness (the models of Kierstead and Slobodkin, 1953; Joseph and Sendner, 1958; see also Section 2.7.2). In lakes much smaller than 10 km2 in area, wherein the maintenance of large-scale developmental patches is largely untenable (that is, critical patch size usually exceeds the horizontal extent of the basin and any small-scale patchiness is very rapidly averaged) the proportion of qs to V assumes increasing importance. While tw > 100 days, small differences in hydraulic throughput may remain empirically inconsequential in relation to potential growth rate: rw is <0.01 d-1. Its doubling to 0.02 d-1 is still small in relation to the rates of growth that are possible. However, when the latter are themselves severely limited by environmental conditions, dilution effects can become highly significant. In the instance of Planktothrix agardhii considered in Section 5.4.5, performing at its best to grow under the mixed conditions of a temperate lake in winter, its replication rate of 0.16 d-1 will be insufficient to counter out-wash losses when tw < 7 days (rw > -0.16 d-1). If it is growing less well, the sensitivity to flushing clearly increases. Temperate lakes regularly experiencing retention times less than about 30 days seem not to support Planktothrix populations. In the persistently spring-flushed Mon-tezuma's Well, Arizona, studied by Boucher et al. (1984) (tw < 9 d), the distinctive phytoplankton comprises only fast-growing nanoplanktic species. In the English Lake District, some of the lakes have volumes that are small in relation to their largely impermeable, thin-soiled mountainous catchments, and which episodically shed heavy rainfall run-off (1.5--2.5 m annually). In Grasmere (tw ~ 24 d annual average but, instantaneously, ranging from 5 d to to), Reynolds and Lund (1988) showed that the phytoplankton had almost to recolonise the water column after wet weather, while it required a long dry summer for Anabaena to become established. Wet winters also keep phytoplankton numbers low but, in the relatively dry winter of 1973, the autumn maximum of Asterionella persisted to merge with the spring maximum.

The most highly flushed environments are rivers. Larger ones often do support an indigenous phytoplankton, usually in at least third- or fourth-order affluents, and sometimes in very great abundance (perhaps an order of magnitude greater concentration than in many lakes; I have an unpublished record of over 600 |g chla L-1 measured in the River Guadiana at Mourao, Portugal, under conditions of late summer flow). The ability of open-ended systems, subject to persistent unidirectional flow, to support plankton is paradoxical. It is generally supposed to be a function of the 'age' of the habitat (length of the river and the time of travel of water from source to mouth), for there is no way back for organisms embedded in the unidirectional flow. On the other hand, the wax and wane of specific populations in given rivers seem fully reproducible; they are scarcely stochastic events. Moreover, some detailed comparisons of the mean time of travel through plankton-bearing reaches of the River Severn, UK, with the downstream population increment would imply rates of growth exceeding those of the best laboratory cultures (Reynolds and Glaister, 1993). Downstream increases in the phytoplankton of the Rhine, as reported by de Ruyter van Steveninck et al. (1992), would require specific growth performances paralleling anything that could be imitated in the laboratory. It was also puzzling how the upstream inocula might be maintained and not be themselves washed out of a plankton-free river (Reynolds, 1988b).

These problems have been raised on many previous occasions and they have been subject to some important investigations and critical analyses (Eddy, 1931; Chandler, 1937; Welch, 1952; Whitton, 1975). However, it was not until relatively more recently that the accepted tenets advanced by the classical studies of (such as) Zacharias (1898), Kofoid (1903) and Butcher (1924) could be verified or challenged. New, quantitative, dynamic approaches to the study of the physics of river flow and on the adaptations and population dynamics of river plankton (pota-moplankton) developed quickly during the mid-1980s. These were reviewed and synthesised by Reynolds and Descy (1996) in an attempt to assemble a plausible theory that would explain the paradoxes about potamoplankton. The principal deduction is that rivers are actually not very good at discharging water. Not only is their velocity structure highly varied, laterally, vertically and longitudinally, but significant volumes (between 6% and 40%) may not be moving at all. A part of this non-flowing water is, depending on the size of the stream, explained in terms of boundary friction with banks and bed but a large part is immobilised in the so-called 'fluvial dead-zones' (Wallis et al., 1989; Carling, 1992). These structures are sufficiently tangible to be sensed remotely, either by their differentiated temperature or chlorophyll content (Reynolds et al., 1991; Reynolds, 1995b). They are delimited by shear boundaries across which fluid is exchanged with the main flow. The species composition of such plankton they may contain is hardly different from that of the main flow but the concentration may be significntly greater. It has also been shown that the enhancement factor is a function of the fluid exchange rate and the algal growth rate: the longer cells are retained, the greater is the concentration that can be achieved by growing species before they are exchanged with the flow (Reynolds et al., 1991). Each dead-zone has its own V and qs characteristics and its own dynamics. The analogy to a little pond, 'buried in the river' (Reynolds, 1994b), is not entirely a trite one. Reynolds and Glaister (1993) proposed a model to show how the serial effects of consecutive fluvial dead-zones contribute to the downstream recruitment of phytoplankton. The recruitment is, nevertheless, sensitive to changes in discharge, fluid exchange and turbidity. It proposes a persistent advantage to fast-growing r-selected opportunist (C-strategist) phytoplankton species or to process-constrained (CR-strategist) ruderals, as later confirmed by the categorisation of potamoplankters of Gosselain and Descy (2002).

The model does not cover the many other fates that may befall river plankton or influence its net dynamics, especially consumption by filter-feeding zooplankton and (significantly) zoobenthos, including large bivalves (Thorp and Casper, 2002; Descy et al., 2003). Neither does it cover explicitly the issue of the perennation of algal inocula. However, in reviewing the available literary evidence, Reynolds and Descy (1996) argued for the importance of the effective mero-plankty to centric diatoms whose life cycles conspicuously include benthic resting stages (Stephanodiscus and Aulacoseira, both common in potamoplankton generally, have proven survival ability in this respect). They also cited the remarkable studies of Stoyneva (1994) who demonstrated the role of macrophytes as shelters and substrata for many potamoplanktic chlorophyte species. The presence of such plants, in headwaters and in lateral dead-zones, provides a constant source of algae that can alternate between periphytic and planktic habitats. This is little different from the proposition advanced by Butcher (1932) over 60 years before. His prognoses about the sources of potamoplankton remain the best explanation to the inoculum paradox and the one aspect still awaiting quantitative verification.

Most phytoplankters are normally heavier than the water in which they are dispersed and, therefore, tend to sink through the adjacent medium. The settling velocity (ws) of a small planktic alga that satisfies the condition of laminar flow, without frictional drag (Re < 0.1), may be predicted by the modified Stokes equation (2.16; see Sections 2.4.1, 2.4.2):

where ds is the diameter of a sphere of identical volume to the alga, (pc â€” pw) is the difference between its average density and that of the water, and 0r is the coefficient of its form resistance owing to its non-sphericity; n is the viscosity of the water, and g is the gravitational force attraction. In completely still water, particles may be expected to settle completely through a column of water of height hw within a period of time (t )

not exceeding the quotient hw/ws (Section 2.6.2):

The point that has been made at length and on many previous occasions is that planktic algae do not inhabit a static medium but one that is subject to significant physical movement. Forcing of its motion by buoyancy, tide, wind, Coriolis effect is resisted by the viscosity of the water. This resistance is largely responsible for the characteristically turbulent motion that predominates in surface waters of the sea, in lakes and rivers (see Section 2.3). Moreover, the turbulent velocities so overwhelm the intrinsic velocities of phy-toplankton sinking that the organisms are effectively entrained and randomised through turbulent layers. However, it may be emphasised again that turbulent entrainment does not overcome the tendency of heavy particles to sink relative to the adjacent medium and, in boundary layers and at depths not pentrated by turbulence, particles are readily disentrained and more nearly conform to the behaviour expressed in Eq. (2.16).

Following Humphries and Imberger (1982), the criterion for effective entrainment (^) is set by Eq. (2.19) (i.e when ws < 15[(w')2]1/2) and is illustrated in Fig. 2.16. The depth of the mixed layer over which it applies (hm) may be calculated from the Monin-Obukhov and Wedderburn formulations. It may often be recognisable from the vertical gradient of density (Spw > 0.02 kg mâ€”3 mâ€”1) (Section 2.6.5) or, casually, from inspection of the vertical distribution of isotherms.

The estimation of sinking loss rates from a fully mixed water column (H) or a mixed layer (hm) applies logic analogous to the dilution by wash-out of a fully dispersed population of particles subject to leakage across its lower boundary. Moreover, as sinking loss is the reciprocal of prolonged suspension, it is relatively simple to adapt Eqs. (2.20-2.25) to the sequence of steps traced in Eqs. (6.3-6.6) and to be able to assert that population remaining in the column, hm, at the end of a period, t, of sustained and even sinking losses is predicted by:

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