simulating the 1-D MEDUSA in MarMOT is described in appendix A.
2.3
Generating the ensemble: Optimising the functional forms
Different marine biogeochemical models share common processes: the interaction between nutrients, phytoplankton, and zooplankton shown on figure 2.2. These processes could be modelled using different equations. For example multiple predator uptake has two functional forms that could be used to represent the process; using Michaelis-Menten expression or Holling type II (Fasham et al., 1990) and Holling type III (Ryabchenko et al., 1997). They have similar saturation curves despite being described by different mathematical equations. Structurally different analytical forms could be used interchangeably, given the uncertainty with which resource uptake by real organisms is measured (Fussmann and Blasius, 2005). Functions that show similar shapes to MEDUSA’s default functional forms are chosen to ensure that they describe the biogeochemical processes in a similar way.
The non-linear least square method is used to enhance the phenomenological similarity between the default functional forms. The parameters associated with the functions, apart from the maximum rate, are obtained by using the command ‘scipy.optimize.curve fit’, which uses non-linear least squares, in python. Here we made the maximum rate values for other functional forms similar to MEDUSA’s default parameter, so that these functions would become saturated at the same concen- tration. Different functional forms and their parameters are then embedded into the model code. The ensemble model has 128 members, which resulted from combining four phytoplankton nutrient uptake, two zooplankton grazing, four phytoplankton, and four zooplankton mortalities functions.
All possible combinations of functional forms that describe the main biogeochemical processes (such as nutrient uptake, grazing, and mortality, explained in section 2.3.1 to 2.3.3) can be generated as an ensem- ble, whereby each member contains a combination of functional forms similar to the default MEDUSA functions. These combinations are then embedded into the 1-D MEDUSA code. The same process func- tion is always used for both diatoms and non-diatoms, or mesozooplankton and microzooplankton. Each ensemble member has at least one functional form changed from the default functions. This provides a total number of 128 combinations, arising from 4 types of nutrient uptake, 4 phytoplankton mortality formulations, 2 types of zooplankton grazing, and 4 zooplankton mortalities.
2.3.1 Nutrient uptake ensemble
Alongside light, nutrient concentration limits the growth of phytoplankton. In MEDUSA the standard hyperbolic monod, hereafter Uh, function is used as the default function. The growth of cells monotoni-
Figure 2.2: Schematic diagrams of MEDUSA-1.0 adopted from Yool et al. (2011), red circles shows core biogeochemical process. These are represented by nutrient uptake by phytoplankton, zooplankton grazing, and plankton mortality, which could be parameterised by different equations
cally increases with ambient nutrient concentration, and halts when nutrients become scarce. If nutrient concentrations are high, the rate of uptake saturates. Other mathematical functions show similar proper- ties including (i) Sigmoidal (Fennel and Neumann, 2014) Us, (ii) the exponential (Ivlev, 1961), Ue, and
(iii) trigonometric functions (Jassby and Platt, 1976), Ut. All these functions include a shape-defining
parameter, k, which for monod and sigmoidal can be interpreted as a half saturation constant, and a max- imum uptake rate, described in equation 2.1. As explained in section 2.1, MEDUSA has silicon and iron nutrients as well as diatoms and non diatoms. The uptake of different phytoplankton types and nutrients use similar functions but different parameter values for k, summarised in Table 2.1, obtained by min- imising the sum squared difference with Uh. The nutrient uptake functions after optimization are shown
in Figure 2.3(a). The fit is done over the nutrient concentration ranging from 0.001 to 20 mmol m−3 for DIN and silicon. These are discretised into 1000 intervals. The difference in shape of the optimised functional forms are more obvious between 0.1 to 1 mmol N m−3.
2.3.2 Zooplankton grazing ensemble
In MEDUSA, both phytoplankton and zooplankton are grouped into ”small” and ”large” categories. The small zooplankton, represented by the microzooplankton, graze on non-diatoms and detritus, with the more nutrient rich, higher quality, non-diatoms preferred over detritus. Larger zooplankton, represented by mesozooplankton have a broader range of prey, including both microzooplankton and diatoms, which
2.3. Generating the ensemble: Optimising the functional forms 51
are higher quality food sources compared to non-diatoms and detritus.
Describing multiple grazing using mathematical forms can be done by defining the zooplankton grazing rate using the hyperbolic Michaelis-Menten or Holling type II, hereafter G2 expression and weighted
preference on the different food sources pn(Fasham et al., 1990). Suppose the specific grazing rate, G,
is described by:
G= g F
kg+ F
(2.4) where F is the total food with their preferences by grazers (paPa+ paPb), g is the maximum grazing rate,
and kg is the half-saturation constant for grazing. When Pa is grazed constantly, equation (2.4) is then
scaled into the zooplankton’s food preference by substituting the preference scaled prey concentration:
GPa= g
paPa
kg+ (paPa+ pbPb)
(2.5)
Since zooplankton preferences will change throughout the year, the assigned preference should change as a function of the food ratio. This could be achieved by defining the weighted preference, p∗aand p∗b:
p∗a= paPa paPa+ pbPb and p∗b= pbPb paPa+ pbPb (2.6)
Substituting p∗aand p∗bfor paand pb, in equation (2.5), grazing on Pa is described by:
GPa= g
paPa2
kg(paPa+ pbPb) + paPa2+ pbPb2
(2.7)
Another method of multiple grazing parameterisation is based on sigmoid Holling type III function, which is defined as:
G= gF
2
k2 g+ F2
(2.8) In the case of two resources, equation (2.8) for Pa uptake becomes:
GPa= g
paPa2
k2
g+ paPa2+ pbPb2
(2.9)
Equation (2.9) is the default functional form used in MEDUSA for zooplankton grazing. As shown in fig- ure 2.3(b) equation(2.7) and (2.9) have similar trends where grazing rate becomes constant as it reaches a certain phytoplankton concentration and a half saturation constant kg. During the fitting process, the
range of phytoplankton and microzooplankton concentrations used was 0.001 to 10 mmol m−3, discre- tised in 1000 intervals equally. At low zooplankton concentrations (between ∼0.01 to ∼0.5 mmol m−3) the Holling type III response has lower grazing rates than the hyperbolic, however as the phytoplankton concentration increases, the Holling type III curve has a more rapid increase in predation rate before
becoming saturated (Edwards and Yool, 2000) compared to the Holling type II, shown on Figure 2.3(c). Preferences for food types are kept the same as MEDUSA’s default parameters, with terms summarized in Table 2.1.
2.3.3 Plankton mortality ensemble
Alternative functions can describe the density-dependent mortality, and in this study the combinations of hyperbolic (ρh, ζh), linear (ρl, ζl), quadratic (ρq, ζq), and sigmoidal (ρs, ζs) functions to describe the
phytoplankton (ρ) and zooplankton (ζ ) mortalities are used (equations and abbreviations are shown on Table 2.1). Similar to grazing and nutrient uptake, the functional forms have different maximum rates for each plankton type. These maximum rates are made the same for all the different functions.
Of the four different mortality functions, linear and quadratic functions are most different in shape, as shown on Figure 2.3(c). Using the linear term is similar to constant removal of plankton at the same rate as the maximum mortality (µ). To make the linear function similar to the sigmoidal and hyperbolic functions, the maximum mortality rate is set so that the total loss integrated over the range of phytoplank- ton concentrations (calculated as the area below the function representing the total loss in linear terms, between 0.001 to 10 mmol m−3) is similar to that for the hyperbolic curve. The quadratic term, instead of asymptoting, continues to grow with plankton abundance. In order to keep this similar to other forms, after reaching a certain concentration the function is switched to linear, so that the rate plateaus at high abundance. For sigmoidal mortality, the default µ are not changed but the half-saturation constant, kM
is optimised. The optimised mortality functions are shown in Figure 2.3(c). The range of phytoplankton and zooplankton concentrations used during the fitting process was between 0.001 - 10 mmol m−3, and discretised within 1000 intervals equally. A distinctive feature of these functional forms after optimisa- tion is that the quadratic mortality rate remains low until phytoplankton concentration reaches 1.0 mmol m−3, and the linear function shows consistently high plankton mortality (Figure 2.3(c)).