Examination of the full model 1 Forcing functions

A major complicating factor in examining the behaviour of the full model is its use of forcing functions to replicate the annual cycles of mixed{layer depth and solar irradiance. The former driven by an array of daily depth values, the latter by a series of standard astronomical formulae.

In one sense, forcing functions may be regarded as implicit representations of unmodelled processes. Where the exact character of these processes is poorly known, or where they can be accurately charac- terised by simple equations, this approach can be useful and can reduce model complexity. The seasonal cycle can then be regarded as the limit cycle produced by the full series of equations (those explicitly in the model, and those implicitly incorporated via the forcing functions). This cycle may be examined using Poincare sections to establish if the model exhibits any behaviour with a periodicity dierent to that of the forcing (Wiggins, 1990).

In another sense, the forcing functions can be regarded as parameters with continuously changing values. In the actual computer programs used to run the full model, this is essentially how the forcing functions are treated. These parameters create a series of attractors (one for each combination of forcing function values) onto which the model trajectory continually attempts to converge. In this view, unless the model is given sucient time to converge onto a particular attractor, the resultant trace in the model's phase space will be an unending transient (albeit one which may be repeated regularly with the period of the forcing). Note that \sucient time" here will be highly dependent on both the forcing functions themselves and the model equations inuenced by them.

In the context of elucidating the behaviour of the full model during the oscillations it produces during the summer months of OWS \India" runs, the forcing functions are best treated this latter way. Namely, treated as model parameters and examined to determine whether their range of annual variation includes regions in which the model behaves qualitatively dierently.

4.2.2 Fixed forcing studies

To examine its behaviour, runs of the full model were performed under OWS \India" conditions and allowed to equilibrate to the standard annual cycle found in previous work. Then, at specied days of the year the forcing functions were \frozen" to constant values and the model followed to establish the nature of the attractor at those values. Mixed{layer depth (M) was set to a constant value, the rate of change in depth of the mixed layer (h(t)) was set to zero, and the daily cycle of irradiance was exactly repeated on subsequent days.

0 200 400 600 800 1000 -600 -400 -200 0 200 Time (days)

Mixed-layer depth or Irradiance

Normal 0 200 400 600 800 1000 0 0.5 1 1.5 Time (days) Concentration 0 200 400 600 800 1000 -600 -400 -200 0 200 Time (days)

Mixed-layer depth or Irradiance

Day 75 0 200 400 600 800 1000 0 0.5 1 1.5 Time (days) Concentration 0 200 400 600 800 1000 -600 -400 -200 0 200 Time (days)

Mixed-layer depth or Irradiance

Day 136 0 200 400 600 800 1000 0 0.5 1 1.5 Time (days) Concentration 0 200 400 600 800 1000 -600 -400 -200 0 200 Time (days)

Mixed-layer depth or Irradiance

Day 197 0 200 400 600 800 1000 0 0.5 1 1.5 Time (days) Concentration

Figure 4.2: Seasonal cycles of forcing functions (left : mixed{layer depth, solid average daily irradiance, dashed) and plankton (right : phytoplankton, solid zooplankton, dashed) for simulations in which the forcing functions were locked at the labelled points in the annual cycle (see text for details). Mixed{layer depth in metres, irradiance in W m;2, concentrations in mmol N m;3.

Figure 4.2 shows some of the results obtained. A normal model trace is shown (row 1) plus the traces produced when the forcing functions are frozen at the winter mixed{layer depth maximum(day 75), just prior to the spring phytoplankton bloom (day 136), and at the summer mixed{layer depth minimum (day 197). In each example the values of the forcing functions and the populations of phytoplankton and zooplankton are shown.

In the case where the forcing functions are frozen at the winter mixed{layer depth maximum, the model nally converges on a steady equilibrium solution at low concentrations of all state variables (except nitrate, which converges on a value close to its subthermocline value). It is noticeable that this equilib- rium has a phytoplankton concentration approximately 5 times greater than the concentration observed on day 75 during a normal, forced simulation. This supports what was suggested earlier, namely that during forced runs the model solution is not on the attractor, at least not on day 75.

In the remaining cases presented, where the forcing functions are frozen in shallower mixed layer/higher irradiance regimes, stable equilibria are also found. However, unlike the winter case, the equilibria are marked by considerably lower nitrate and somewhat higher concentrations of the other state variables. When the forcing functions are frozen at pre{bloom levels (day 136) the bloom oscillation is followed by much smaller oscillations than usually observed in the summer. It is likely that the slightly lower phytoplankton growth rate is responsible. Curiously, the phytoplankton and zooplankton oscillations appear to be converging to an equilibrium around 0.5 mmol N m;3 by what would be the early winter. However, with the continuing depletion of nitrate (which, because of the high winter levels it rises to, was previously at almost unlimiting concentrations), the phytoplankton and zooplankton concentrations (as well as those of the other state variables) fall to a lower stable equilibrium.

A similar phenomenon occurs when the forcing functions are locked in the height of summer (day 197). However, because of the higher phytoplankton growth rates, nitrate is depleted faster and the \crash" towards lower concentrations occurs earlier. Concomitantly with the higher phytoplankton growth rates though, the summer oscillations observed are more similar to those observed in the forced case. In fact, because the xed summer minima forcing is even more favourable than the normal variable forcing, the oscillations are slightly more rapid and extreme.

In the winter then, it appears that high nitrate/low phytoplankton{zooplankton stable equilibria ex- ist, whilst in the summer, low nitrate/higher phytoplankton{zooplankton stable equilibria are found. The oscillations observed in the normal simulations would then appear to be transient, and reliant on the high quantities of nitrate entrained during the deep winter mixing. Summer entrainment through cross{thermocline mixing is unable to supply sucient nitrate to sustain bloom{level phytoplankton populations should summer conditions persist.