The purpose of this section is not to demonstrate that sinking particles are lost from turbulent, surface-mixed layers but to provide the basis of estimating the rate of loss. The converse, how slowly they are lost, is the essence of adaptation to planktic survival. The development here is rather briefer than that in Reynolds (1984a), as its principles are now broadly accepted by plankton scientists. Its physical basis is rather older, owing to Dobbins (1944) and Cordoba-Molina et al. (1978). Smith (1982) considered its application to plankton. Interestingly, empirical validation of the theory comes from using plankton algae in laboratory-scale measurements.

Let us first take the example of a completely static water column, of height hw (in m), open at the surface with a smooth bottom, to which small inert, uniform particles are added at the top. Supposing their density exceeds that of the water, that they satisfy the laminar-flow condition of the Stokes equation and sink through the water column at a predictable velocity, ws (in m s-1), then the time they take to settle out from the column is t! = hw/ws (in s). If a large number (N0, m-3) of such particles are initially distributed uniformly through the water column, after which its static condition is immediately restored, they would settle out at the same rates but, depending on the distance to be travelled, in times ranging from zero to t. The last particle will not settle in a time significantly less than t', which continues to represent the minimum period in which the column is cleared of particles. At any intermediate time, t, the proportion of particles settled is given by N0 ws t/hw. The number remaining in the column (Nt) is approximately

Let us suppose that the column is now instantaneously and homogeneously mixed, such that the particles still in the column are redistributed throughout the column but those that have already settled into the basal boundary layer are not resuspended. This action reintroduces particles (albeit now more dilute) to the top of the column and they recommence their downward trajectory. Obviously, the time to complete settling is now longer than t! (though not longer than 2t').

The process could be repeated, each time leaving the settled particles undisturbed but redistributing the unsettled particles on each occasion. If, within the original period, t!, m such mixings are accommodated at regular intervals, separating quiescent periods each t'/m in duration. The general formula for the population remaining in suspension after the first short period is derived from Eq. (2.20):

Nt/m = No(1 - wst'/mhw) (2.21) After the second, it will be

Nf = No(1 - wst'/mhw)m (2.22) Because t = hw/ws, Eq. (2.22) simplifies to

As m becomes large, the series tends to an exponential decay curve

where e is the base of natural logarithms (~2.72). Solving empirically,

This derivation is instructive in several respects. The literal interpretation of Eq. (2.25) is that repeated (i.e. continuous) mixing of a layer should be expected still to retain 36.8% of an initial population of sinking particles at the end of a period during which particles would have

Figure 2.17

The number of particles retained in a continuously mixed supension compared to the retention of the same particles in a static water column of identical height. Redrawn from Reynolds (1984a).

The number of particles retained in a continuously mixed supension compared to the retention of the same particles in a static water column of identical height. Redrawn from Reynolds (1984a).

left the same layer had it been unmixed. Moreover, the time to achieve total elimination (te) is an asymptote to infinity but we may deduce that the time to achieve 95% or 99% elimination is (respectively) calculable from te/t' = log e0.05/log 0.368 = 3.0

Using this approach, the longevity of suspension can be plotted (Fig. 2.17). It takes three times longer for 95% of particles to escape a mixed layer than were the same depth of water left unmixed. It may also be noted that the number of mixings does not have to be vast to achieve this effect. Substituting in Eq. (2.23), if m = 2, Nf = 0.25 N0; if m = 5, Nt = 0.33 N0; if m = 20, Nt = 0.36 N0.

The formulation does not predict the value of the base time, t. However, it is abundantly evident from the rest of this chapter that, the smaller is the specific sinking rate, ws, the longer will be t for any given column of length, hm. Further, the greater is the mixed depth, then the longer is the period of maintenance. Any tendency towards truncation of the mixed depth, hm, will accelerate the loss rate of any species that does not have, or can effect, an absolutely slow rate of sinking.

This confirms the adaptive significance of a slow intrinsic sinking rate. It has much less to do with delaying settlement directly but, rather, through the extension it confers to average residence time within an actively-mixed, entraining water layer. The relationship also provides a major variable in the population ecology of phytoplankton, because mixed layers may range from hundreds of metres in depth down to few millimetres and, in any given water body, the variability in mixed-layer depth may occur on timescales of as little as minutes to hours.

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