Programa de Protección Social al Adulto Mayor

One of the problems associated with simplifying the model is that some important complexity has been taken away. One of the main concerns is that the nutrient satiation effect on P has been removed. The parameters used in the LES-NPZ model suggest that a phytoplankton cell reaches a satiated state if it resides in a nitrate concentration ofN ≈3. In such a concentration, growth will reach its maximum efficiency. With the nitrate flux condition set toQ= 130, the ambient nutrient concentration hits this con- centration after 6-7 days. This means that any comparisons between the correlations must be made in the first week of simulation time if one wants to use the aggregation dynamic equation. When the < P > field is completely satiated with nutrients, the phytoplankton will essentially see the entire nutrient field as homogeneous. However, this does not mean that the correlations will be useless at these times. As the correla- tions have the same period as the population cycles themselves, it will give historical information about which quantities are stimulated by the aggregations, and so can be used as a guide for what the mechanisms are behind the said aggregations.

The analysis of these correlation terms must be treated with a certain degree of caution as data is only recorded at certain specific sampling times. Monitoring these correla- tions continuously during a simulation would exceed computational storage capacity

limits. The correlation statistics exhibit rapid fluctuations, on time scales much faster than the sampling time. Consequently the sampled data gives a smoother impression of the evolution of the correlation statistics than is actually the case. Therefore, we can apply certain smoothing techniques to the correlation data to get a general view of the dynamics of each term without any great loss of information. This smoothing will be a 5-point moving average of the form,

f_is = 1 5 i+2 X j=i−2 fj

where fj is the data andfis is the smoothed data, aroundt=ti.

When the mean phytoplankton concentration falls to a very low level i.e. < P >≈0, the aggregation intensity measure behaves erratically. We are not interested in popula- tion fluctuations about a very low background level, rather we are looking for evidence of concentration aggregations against a backdrop of medium to high < P > levels. Hence in this investigation we will focus on behaviour when < P >0. We can apply a simple filter, which effectively disregards all the terms in equations (5.4 - 5.7) at times when < P >≈ 0. For example, take the aggregation intensity, I, we can apply the following I =I×min 1, < P > 0.1 2! , (5.8)

which means that if< P >goes below a value of 0.1, then the aggregation intensity will be penalised. The same process will be applied to the biological doubles and triples, as well as the flow terms.

Figure 5.3 shows the evolution of the aggregation intensity taken from three differ- ent simulations associated with µmax_P = (4.32,8.64, 17.28) days−1. Each simulation was driven by the same wind stress value, U∗ = 3.5×10−3ms−1. One can see in fig- ure 5.3 the same characteristic bimodal structure (for each period) to the aggregation

intensity evolution, as highlighted in figure 5.2. The key question is, what is driving these sudden surges in aggregation intensity during the growth and decay phases of the mean concentration field< P >? From the analysis of equation 5.7, one would expect aggregation intensity growth to be promoted when the biological doubles and triples are large compared with the flow terms. This ratio between the biological terms and the physical terms can be expressed by

Biology Flow = 2 (1−I) < P > (a < P 0_N0 _{>−b < P}0_Z0_{>) +} 2 < P >2 a < P 02_N0_{>−b < P}02_Z0_> 1 < P >2 ∂ < w0P02_> ∂z + 2 < w0P0> < P >2 ∂ < P > ∂z − 2I < P > ∂ < w0P0> ∂z . (5.9)

Figure 5.4 shows a plot of the evolution of the ratio of biological double and triple terms to the flow terms (equation 5.9), taken at the initial few days of the simulation to avoid analysing satiated< P > fields. This equation assumes there is no significant contribution from biological diffusion, this is fair as diffusion is a very weak process rel- atively and particularly within the time frame of the biological population cycles (≈a couple of days), will most likely yield negligible contribution towards spatial formation. The filtering mechanism was also in operation so the P concentration was also taken when < P >>0.1. There is a distinct increase in the scale of ratio 5.9 with increasing

µmax_P , with the highest value of µmax_P giving a ratio an order of magnitude larger at some times in figure 5.4(a) compared to figure 5.4(c). This suggests that with increas- ingµmax_P , biological processes become more dominant, which is what might be expected.

The lower growth rate case (figure 5.4(a)), corresponds to a regime in which the biolog- ical correlations are comparable in magnitude to the mixing terms associated with the flow. So for growth rates at or aboveµmax_P ≥4.32 days−1, one would expect significant fluctuations in the P field to become apparent. This is a very important aspect in aggregation formation as it is quite obvious that once the physical mixing dominates, the chance of a strong heterogeneous large scale patch being formed are small. It is therefore an assertion that any maximum growth rates larger thanµmax_P = 4.32 days−1

are very likely to induce aggregations and anything below will not be strong enough for aggregations to manifest themselves.

The ratios in this section are based on one particular wind stress value of 3.5 ×

10−3 ms−1, and the depth from which the statistics in figure 5.4 were derived is below the penetration depth calculated in chapter 4. This was calculated from

Z 0 ˆ z < w2 > dz = Z zˆ zML < w2 > dz, (5.10)

where the penetration depth zpen was defined to be twice the mid-energy depth ˆz,

i.e. zpen = 2ˆz (zML is the mixed layer depth). It seems highly likely that biological

aggregations will start to manifest themselves below the penetration depth provided

µmax

P > 4 days

−1 _{and the mixed layer is not fully penetrated, which was the reason} for the choice of wind stress used in this section. The correlations between zpen and

(a)µmaxP = 4.32 days −1 (b) µmaxP = 8.64 days −1 (c) µmax P = 17.28 days −1

Figure 5.3: Aggregation intensity vs. mean phytoplankton concentration for varying

µmax_P . The dashed line shows the evolution of the mean phytoplankton concentration (RHS scale) and the solid line shows the evolution of the aggregation intensity,I (LHS scale). This was taken atz ≈26m which was the optimum aggregation depth for this simulation.

(a)µmaxP = 4.32 days −1 (b) µmaxP = 8.64 days −1 (c) µmax P = 17.28 days −1

Figure 5.4: Ratio of biological dependant correlations and flow dependant correlations taken at a the first week in simulation time (before phytoplankton are satiated with nutrient) with varying µmax_P . The dashed line shows the evolution of the mean phy- toplankton concentration (RHS scale) and the solid line shows the evolution of the aggregation intensity, I (LHS scale). This was taken at z≈26m which was the opti- mum aggregation depth for this simulation.