## Mixing times

The intensity of turbulence required to entrain phytoplankton covers almost 2 orders of magnitude: non-motile algae with sinking rates of ~40 ¡m s-1 are effectively dispersed through turbulence fields where u* < 600 ¡m s-1 (Eq. 2.19), whereas u* > 50mms-1 is sufficient to disperse the least entrainable buoyant plankters (ws > 1mms-1) (see Section 2.6.1). As has already been suggested, the intensity of mixing is often less important to the alga than is the vertical depth through which it is mixed. The depth of water through which phytoplankton is randomised can be approximated from the Wedderburn equation (2.34). Putting W = 1 and u* > 0.6 ms-1, the numerator is equivalent to >0.36 for each 1000 m of horizontal distance, L. Dividing out gravity, the product, Apw (hm)2 solves at ~0.037kgm-1. This is equivalent to an average density gradient of >0.04 kg m-3 m-1 per km across a lake for a 1-m mixed layer, ~0.01 for a 2-m layer, ~0.0045 for a 3-m layer, and so on. The weaker is the average density gradient, then the greater is the depth of entrainment likely to be. The limiting condition is the maximum penetration of turbulent dissipation, unimpeded by density constraints. Where a density difference blocks the free passage of entraining turbulence, the effective floor of the layer of entrainment is defined by a significant local steepening of the vertical density gradient. Reynolds' (1984a) consideration of the entrainment of diatoms, mostly having sinking rates, ws, substantially less than 40 ¡m s-1, indicated that the formation of local density gradients of >0.02 kg m-3 m-1 probably coincided with the extent of full entrainment, that is, in substantial agreement with the above averages. For the highly buoyant cyanobacterial colonies, however, disentrainment will occur in much stronger levels of turbulence and from mixed layers bounded by much weaker gradients.

Assumption of homogeneous dispersion of particles fully entrained in the actively mixed layer allows us to approximate the average velocity of their transport and, hence, the average time of their passage through the mixed layer. In their elegant development of this topic, Denman and Gargett (1983) showed that the average time of travel (tm) through a mixed layer unconstrained by physical boundaries or density gradients corresponds to:

Because, in this instance, both hm and u* are directly scaled to the wind speed, U (Eqs. 2.5, 2.11), tm is theoretically constant. Interpolation into Eq. (2.35) of entries in Table 2.2 in respect of Bodensee permit its solution at 1416 s. The probability of a complete mixing cycle is thus 2 x 1416 s (~47 minutes).

In the case of the wind-mixed layer of a small or shallow basin, or one bounded by a density gradient, the timescale through the layer is inversely proportional to the flux of turbulent kinetic energy:

Following this logic, a wind of 8 ms-1 may be expected to mix a 20-m epilimnion in 2000 s (33 minutes) but a 2-m layer in just 200 s (3.3 minutes). A wind speed of 4 m s-1 would take twice as long in either case.

These approximations are among the most important recent derivations pertaining to the environment of phytoplankton. They have a profound relevance to the harvest of light energy and the adaptations of species to maximise the opportunities provided by turbulent transport (see Chapter 3).

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