Appendix: Formulation of Macroalgal Growth Model

The model state variables are:

[NO3] - nitrate concentrations in estuarine intertidal water (kg m−3), ‘corrected’ for macroalgal uptake;
[NH4] - ammonium concentrations in estuarine intertidal water (kg m−3), ‘corrected’ for macroalgal uptake;
Ns - macroalgal stored nitrogen (kg m−2); and
Nf - macroalgal assimilated nitrogen (kg m−2).

Macroalgal growth equations

Growth is modeled as a two-stage process:

1. uptake, in which nitrogen is taken into an internal reservoir;

2. assimilation, which occurs during the photosynthetic process in which nitrogen becomes fixed into the plant structure.

The uptake step is light independent; hence, the macroalgae are assumed to be able to absorb nutrients in the dark, but are unable to grow until light is available.

The key state variables are the amount of nitrogen per area stored by the macroalgae after uptake Ns (kg m−2), and the amount of assimilated nitrogen per unit area Nf. Let Qmin (kg N kg−1 dry wt) be the ratio (assumed fixed) between assimilated nitrogen and dry weight biomass (B):

(1)

then the internal nitrogen concentration of stored nitrogen per mass of dry weight, Q (kg N kg−1 dw) is:

(2)

The equation for the uptake into the internal storage Ns (kg m−2) is:

(3)

where and are rates of uptake of the indicated nitrogen species from the water, and the last term on the right-hand side represents fixing of nitrogen via photosynthetic growth. Uptake can only occur when the macroalgae are immersed in water and the limiter is defined such that:

(4)

where h (m) is the water depth with respect to Mean Sea Level (positive down) and η(t) is the time-varying tidal elevation. Thus, is 1 when the location is wet and 0 otherwise. Uptake from water is assumed to be governed by the Michaelis-Menten form:

(5)

for the two nitrogen species (nitrate or ammonium), where [N] is the water concentration of the given nitrogen species, VmaxN (kg N (kg dw)−1 d−1) is the maximum specific uptake rate for N, and kN (kg m−3) is the half-saturation constant. The internal concentration is constrained to lie between maximum and minimum values by the limiter function ():

(6)

Growth causes the assimilation of stored nitrogen into the plant structure via:

(7)

where represents growth-limiting factors, is the macroalgal renewal (or turnover /mortality) rate, ψ is an enhanced mortality dependent on tidal bed stress, and Nmin represents a specified minimum biomass that the nitrogen biomass is not allowed to fall below and acts as an ‘initial’ seeding value to begin the seasonal growth cycle. The bed stress mortality function is specified empirically as:

(8)

where τmax is the maximum tidal bed stress at a point over a spring neap cycle. The function ψ increases macroalgal mortality for τmax> τc at a rate determined by the parameter.

Following Solidoro et al. (1997), a multiplicative formulation for the effect of multiple limiting factors is used. Thus, the growth limitation function is taken to be a product of individual factors:

(9)

representing limitation due to internal nitrogen concentration (), temperature (), light (), salinity (), and biomass density (), respectively. As in Solidoro et al. (1997) and Martins & Marques (2002), growth is inhibited if internal concentrations fall below maximum nitrogen values via:

(10)

where, following Solidoro et al. (1997), kc = (Qmin – k) with k being a half-saturation constant for internal concentration-dependent growth. The growth rate dependence on temperature T (°C) is assumed to be given by an ‘S’ curve (Solidoro et al., 1997):

(11)

where T0 (°C) is a transition temperature below which significant inhibition of growth occurs, and Tr (°C) controls the range over which the transition between limitation and non-limitation occurs. For intertidal regions, it is not clear if the relevant temperature should be the sea or air temperature (or some combination of the two). In the simulations, T is taken to refer to the water temperature.

A standard formulation (e.g., Martins & Marques, 2002) for the dependence of growth rate on light intensity, including photo-inhibition effects, is:

(12)

where I (µmol photon m−2 s−1) is the light intensity and I0 (µmol photon m−2 s−1) is a critical intensity value. Light intensity at depth h (m) is:

I = Iinc Exp(−kt h)(13)

where Iinc is the surface light intensity and kt (m−1) is the attenuation coefficient. Sanders et al. (1997) fitted a curve for kt based on measurements at relatively high suspended load in the Ouse estuary. At lower turbidity, an extrapolation down to a clear water value of 0.4 m−1 is made, assuming a linear dependence on suspended concentration C (mg l−1), to give:

(14)

Above a critical biomass density, an empirical limiter curtails growth simulating the effect of self-shading and physical space constraints.

Concentrations of nitrogen in the water

Reference levels of dissolved nitrogen concentrations in water need to be specified independently and must be those that would pertain to the situation in which macroalgal growth is absent. For the Medway calculations reported on here, they were supplied from a TELEMAC model calculation.

For intertidal regions, the equations for ammonium concentration [NH4] (kg m−3) and nitrate concentration [NO3] (kg m−3) are:

(15)

(16)

where the [.]ref denotes reference concentrations that the solution moves toward in the absence of uptake by macroalgae. The rate at which this occurs is termed the refresh rate λR (d−1) and will determine the equilibrium biomass achieved. The macroalgal uptake rate functions are UNH4, UNO3. Note that the biomass per unit area B is divided by an average water depth hw to be dimensionally correct. Conceptually the macroalgae are envisaged to be spread out evenly over the depth of water. The depth hw is taken as the average depth over a spring-neap cycle that the macroalgae experiences when wet. Use of the instantaneous value of h+η caused numerical problems due to rapid fluctuations in water concentrations in very shallow conditions. The quantity Y represents possible input of ammonium from the bed due to the decay of organic material. In this study, it was set to zero.

Assignment of Model Parameters

In this section, the particular set of values adopted for the model parameters are described and related to the literature of previous experimental and field measurements. Parameters that could not be derived from the literature or those that are estuary specific (e.g., the ‘refresh’ rate) were fixed by calibrating with measurements for the Medway or from a previous study undertaken in Langstone harbour (Cefas, 2003).

Maximum uptake rates VmaxN (d-1)

The maximum uptake rate is usually denoted Vmax and for a given nitrogen species is generally given as a rate per dry weight, i.e., kg N (kg dw biomass)−1 d−1. A number of experiments giving VmaxN were summarized in Solidoro et al. (1997). These gave a range for VmaxNH4 from 0.017 to 0.125 kg N (kg dw)−1 d−1. From a smaller number of reported experiments, the range for VmaxNO3 was 0.0096 to 0.04 kg N (kg dw)−1 d−1. Solidoro et al. (1997) use the high end of the reported range taking VmaxNH4 = 0.125 kg N (kg dw)−1 d−1 and VmaxNO3 = 0.028 kg N (kg dw)−1 d−1. Values chosen in this study were VmaxNH4 = 0.1 kg (kg dw)−1 d−1 and VmaxNO3 = 0.03 kg N (kg dw)−1 d−1.

Half saturation constants KN (kg m−3)

Values for KN from a number of experiments were summarized in Solidoro et al. (1997). These give KNH4 in the range 0.7 × 10−4 to 7.0 × 10−4 kg m−3 and KNO3 in the range 2.5 × 10−4 to 5.0 × 10−4 kg m−3. In this study, the following values were used: KNH4 = 7.0 × 10−4 kg m−3 and KNO3 = 3.0 × 10−4 kg m−3.

Internal nitrogen limits Qmin, Qmax and kc

After reviewing literature Solidoro et al. (1997) used the values Qmin = 0.01 kg N (kg dw)−1, Qmax = 0.04 kg N (kg dw)−1 and kc = 0.008 kg N (kg dw)−1. These were adopted in this study.

Maximum growth rate (d−1)

This represents the fastest rate at which macroalgal biomass can increase. Following Solidoro et al. (1997), a value of 0.5 d−1 was used.

Turnover rate (d−1)

Previous models have assigned values to this parameter by calibration with observations. Solidoro et al. (1997) take a constant = 0.03 d−1. Eilola and Stigebrandt (2001) use = 0.04 d−1 based on the modeling of Oberg (1999). A re-analysis of the Owen and Stuart (1983) field data carried out by Trimmer et al. (2000) indicated higher turnover rates of the order 0.1 to 0.2 d−1. These studies also suggested a rate that increased through spring to a peak in summer. Based on this a maximum summer value of λd = 0.1 d−1 was adopted, with a linear increase starting from half this value during the initial period of spring growth.

Critical bed stress τc and δ

A value of τc = 0.5 N m−2 was set by comparing predicted distributions with observations. The sensitivity parameter δ = 0.05 N m−2 was set to cause a rapid increase in mortality for τ > τc.

Minimum bed biomass N0

Eilola and Stigebrandt (2001) used a value equivalent to an N biomass of 0.02 g m−2. A similar value of 0.01 g m−2 was adopted for this study.

Critical Temperatures T0 and Tr

Based on the three-year average of observed temperatures for a nearby location (Langstone harbour; Pye, 2000), values of T0 = 12 °C and Tr = 1 °C were used.

Critical intensity I0

A value of 500 µmol photons m−2s−1 was used following Martins & Marque (2002).

Refresh rate

This was used as a calibration parameter to obtain the observed magnitude of biomass in the Medway. For these calculations a value of 0.25 d−1 was used.

References

Cefas, 2003. Investigation of factors controlling the presence of macroalgae in some estuaries of the Southern Water Region. Centre for Environment, Fisheries & Aquaculture Science. Contract C1642.

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Martins, I. & J.C. Marques, 2002. A model for the growth of opportunistic macroalgae (Enteromorpha spp.) in tidal estuaries. Estuarine, Coastal and Shelf Science 55: 247–257.

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Owens N.J.P. & W.D.P. Stewart, 1983. Enteromorpha and the cycling of nitrogen in a small estuary. Estuarine Coastal and Shelf Science 17: 287–296.

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