Aggregation Depth Dependence

To quantify how strong the level of aggregation is in a simulation, we first need to focus in on the depths at which aggregations are most prominent. Therefore, we need to define a depth for which aggregation is likely to reach its highest level. This will be known as the optimum aggregation, zopt. It is tempting to definezopt as the depth at

which I reaches its maximum. However, as discussed previously, maximum intensity is not a robust statistic and any definition of zopt based on such a maximum would

exhibit sharp fluctuations. It is much better to define zopt in terms of an averaged

aggregation intensity at each depth, as ephemeral aggregations will be smoothed out with this statistic. A (potential) phytoplankton aggregation will then be defined at the depth at which the average aggregation intensity, denoted by Iav(z), reaches its

maximum, i.e. atIav(zopt). This leads to the following mathematical definition forzopt,

Maxj "_PN i=0I(zj, ti) tN # = Maxj[Iav(zj)] =Iav(zopt). (5.11)

where tN is the time-scale of the simulation. Graphically, this is shown in figure 5.7.

Figure 5.8 shows profiles of Iav of low, medium and high wind stresses, where high

wind stress characterises a regime in which the boundary layer is penetrated and low wind stress characterises a regime in which Iav(z) profiles are not unimodal. It is ap-

Figure 5.7: Profile of the average aggregation intensity, normalised by its own maximum

Iav(z)

max(Iav(z)). zopt is displayed to show how the optimum aggregation depth is defined

graphically. This example was taken for a low wind stress value ofU∗ = 1.5×10−3 ms−1.

unique depth at which Iav reaches a definite maximum (figure 5.11(a)) (Note however,

there is no physical reason why the distribution should be bi-modal, in fact the levels of aggregation are very low (although not shown in these graphs) and so any inferences can be disregarded), whereas for medium wind stresses it is evident that the profiles are unimodal and have a distinct optimum aggregation depth (figure 5.11(b)). High wind stresses also have a distinct optimum aggregation depth, however the profile is beginning to show signs of spreading, where the thin peak, observed in the medium wind stress case, is becoming less distinct asU∗increases (figure 5.8(c)). This spreading of average aggregation intensity as U∗ increases is indicative that the boundary layer is starting to become extremely well mixed, across the whole of the boundary layer of depth 33 m prescribed here. Further increases in U∗ beyond those shown here would see a return to a completely uniformly mixed boundary layer in which no distinct zopt

value would be apparent. It can then be inferred that at over intermediate ranges of wind stress, phytoplankton aggregations will tend to thrive close to one distinct depth interval in the boundary layer and more importantly, this depth interval will be

relatively small. That is, one would expect to see aggregations in thin layers, a phe- nomena observed in many experimental trials (McManus et al., 2003; Menden-Deuer and Gr¨unbaum, 2006; Widder et al., 1999). This depth interval (or layer thickness) will be analysed later in this chapter.

Analysing the behaviour of optimum aggregation depth, zopt, with increasing wind

stress, two things are apparent. Firstly, and most obviously, zopt deepens as the wind

stress increases. Secondly, for large wind stress,zopt stops deepening and remains con-

stant. This behaviour is almost identical to the penetration depth behaviour, as can be seen in figure 5.9. One can see that the optimum aggregation depth is consistently beneath the penetration depth, zpen. As U∗ increases, the transitional band of low Re

number, quasi-laminar flow, between the penetration depth and the mixed layer base, becomes thinner, note that this laminar band is artificial, because it is established by the prescribed mixed layer base. The laminar band is artificial in the sense that the mixed layer depth is fixed in this model, with a no-slip boundary condition attached. If the mixed layer depth was doubled, the higher levels of turbulent mixing would likely stay at the same range of depths (down to 10-15m), but the band of laminar flow would increase dramatically. This means that an appropriate choice of mixed layer depth must be chosen and the pycnocline is an apt reference for this choice. The lam- inar layer thinning promotes a smaller interval of depths at which the phytoplankton are likely to flourish and hence form thin layers. This region of the boundary layer will be termed laminar stratified. As was remarked on in section 4.3.2, at a particular wind stress, the flow becomes more or less fully developed (i.e. active at all depths). This is clear from the levels of heterogeneity in figure 4.11, when the mixing levels increase to a level where phytoplankton concentrations are homogenised over the entire depth of the boundary layer. Now since the amount of wind stress needed to fully penetrate the boundary layer is reached at around U∗ = 4×10−3, adding more wind stress to the surface would not significantly increase the penetration depth as penetration has already been achieved (note, the penetration depth will never reach the bottom of

(a) low wind stress

(b) medium wind stress

(c) high wind stress

Figure 5.8: Average aggregation intensity profiles, Iav(z), normalised by their own

the boundary layer as this would imply a balance of < w2 > between the top half of the boundary layer and the bottom half, a phenomenon which cannot happen due to friction and energy dissipation). Hence, the start of a plateau in penetration depth happens whenU∗ hits a critical value (in this caseU∗= 4×10−3), when the boundary layer is fully penetrated.

Figure 5.9: Variation of optimum aggregation depth, zopt, a measure of the depth at

which lateral biological concentrations are most heterogeneous, with wind stress. The variation of the penetration depth zpen is also shown.

The region of laminar stratification represents a transitional flow between laminar flow and turbulent flow, where vertical currents are still prominent but lateral turbulent mixing is not. If the penetration depth could be thought of as a pseudo mixed layer depth, then planktonic aggregations brought on by means of laminar stratification, would appear in a similar region to that of the pycnocline (i.e. below the mixed layer). In this way, the stratified laminar layer has similar properties to that of the pycnocline, in that turbulent mixing is not present, but vertical currents are (Fernando, 1991). As the physics prescribing density variation is not present, only analogies to the pycnocline can be made. In fact if there was a pycnocline present in the model, it would be likely

that the thin layer aggregation would be enhanced due to the flow being even more quiescent in the laminar layer than what is observed in this model setup.