# Appendix J Processes: phyto limitation

## J.1 General

Several functions are available within the WQ Module that allow for modification of computed phytoplankton rates in response to environmental conditions. These functions were mentioned in passing in previous Appendices and are called limitation functions because they describe how temperature, nitrogen, phosphorus, silicate, light and salinity potentially limit phytoplankton rates. Their computed values are always dimensionless and between zero and one, and are applied multiplicatively to modify initially computed rates. Limitation functions of zero completely shut off initially computed rates, and conversely, limitation functions of one do not affect these rates whatsoever. Limitation functions that are valued between zero and one linearly affect computed rates as per Equation (I.2).

The key principle of limitation functions is that they provide a quantitative comparison of phytoplanktonic rate requirements with ambient conditions, and then modify phytoplankton behaviour accordingly. For example, an initially computed \(R_{prod\langle computed\rangle}^{phy}\) may make a productivity demand on ambient nitrogen that exceeds instantaneous nitrogen availability. In this case, primary production will need to be reduced to reflect this nitrogen scarcity. This is affected via the multiplicative application to the initial primary productivity rate of a (computed) nitrogen limitation \(L_{nit}^{phy}\) that is less than one. A similar principle applies to temperature, phosphorus, silicate, light and salinity limitation functions: they modify computed phytoplanktonic rates based on a relative assessment of these rates to instantaneous ambient environmental conditions.

There are multiple limitation functions available within the WQ Module for each of temperature, nitrogen, phosphorus, silicate, light and salinity. These are described below.

## J.2 Temperature

Temperature limitation is often considered a key influencer of phytoplankton dynamics. The WQ Module offers the following temperature limitation functions.

### J.2.1 None

This model includes no temperature limitation to the computed primary productivity rate. The temperature limitation is set to 1.0 for all temperatures.

### J.2.2 Standard

This is a generic yet customisable temperature limitation function. It comprises three piecewise continuous sub functions: \[\begin{equation} L_{T}^{phy} = \begin{cases} \left[\theta_{prod}^{phy}\right]^{\left(T-20\right)} & \text{$T \le T_{std}^{phy}$} \\ \\ f(T_{opt}^{phy},T_{std}^{phy},T_{max}^{phy}) & \text{$T_{std}^{phy} \lt T \le T_{max}^{phy}$} \\ \\ 0 & \text{$T_{max}^{phy} \lt T$} \end{cases} \tag{J.1} \end{equation}\]

\(\theta_{prod}^{phy}\) is the temperature coefficient for primary productivity, \(f\) is a sub function of \(T_{opt}^{phy}\), \(T_{std}^{phy}\) and \(T_{max}^{phy}\), which are the optimal, standard and maximum water temperatures respectively, and \(T\) is ambient water temperature.

The function \(f\) comprises a two stage algorithm:

**Stage 1.**Preprocess \(T_{opt}^{phy}\), \(T_{std}^{phy}\) and \(T_{max}^{phy}\) in a once-off iterative calculation to generate a secondary suite of parameters that are then used in Stage 2 computations**Stage 2.**Use ambient environmental conditions, \(\theta_{prod}^{phy}\), \(T_{std}^{phy}\) and the parameters generated in Stage 1 to compute the temperature limitation function at every water quality timestep and computational cell throughout a simulation

**Stage 1**
The intent of Stage 1 is to use \(T_{opt}^{phy}\), \(T_{std}^{phy}\) and \(T_{max}^{phy}\) to iteratively compute three parameters \(aTn\), \(bTn\) and \(kTn\) that describe a curve fitting of \(L_{T}^{phy}\) between \(T_{std}^{phy}\) and \(T_{max}^{phy}\) (i.e. function \(f\) in Equation (J.1)). The conditions for this curve fitting ensure that \(L_{T}^{phy}\) is piecewise continuous:

\[\begin{equation} L_{T}^{phy} = \begin{cases} \left[\theta_{prod}^{phy}\right]^{\left(T-20\right)} & \text{$T = T_{std}^{phy}$} \\ \\ 0 & \text{$T = T_{max}^{phy}$} \end{cases} \tag{J.2} \end{equation}\]

To do so, the WQ Module uses Equation (J.3) to compute the intermediate quantity \(G\) by iterating on \(kTn\), with the \(kTn\) seed set to 6.0.

\[\begin{equation} \left.\begin{aligned} G = &\left(kTn \times \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{opt}^{phy}\right)} \times \left[\theta_{prod}^{phy}\right]^{\left(T_{max}^{phy} - 20.0 \right)}\right) - \ldots \\ \\ &\left(\left[\theta_{prod}^{phy}\right]^{\left(T_{opt}^{phy} - 20.0 \right)} \times \left( \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{max}^{phy}\right)} - \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{std}^{phy}\right)} \right)\right) \end{aligned}\right\} \tag{J.3} \end{equation}\]

The iteration target is \(|G| \le 0.5\). If this target is not met on a given iteration, a new value of \(kTn\) is computed as per Equation (J.4).

\[\begin{equation} kTn = kTn - \frac{G}{\nabla G} \tag{J.4} \end{equation}\]

\(\nabla G\) is computed as per Equation (J.5).

\[\begin{equation} \left.\begin{aligned} \nabla G =& \left(\left(\left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{opt}^{phy}\right)} \times \left[\theta_{prod}^{phy}\right]^{\left(T_{max}^{phy} - 20.0 \right)}\right) \times \left(1.0 + kTn \times T_{opt}^{phy} \times ln\left(\theta_{prod}^{phy}\right)\right)\right) - \ldots \\ \\ &\left(\left[\theta_{prod}^{phy}\right]^{\left(T_{opt}^{phy} - 20.0 \right)} \times ln\left(\theta_{T}^{phy}\right) \times \left(T_{max}^{phy} \times \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{max}^{phy}\right)} - T_{std}^{phy} \times \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{std}^{phy}\right)}\right)\right) \end{aligned}\right\} \tag{J.5} \end{equation}\]

Once the iteration target is met (i.e. \(|G| \le 0.5\)), the final value of \(kTn\) is used to compute \(aTn\) and \(bTn\) as per Equation (J.6).

\[\begin{equation}
\left.\begin{aligned}
aTn &= \frac{-ln\left(\frac{\left[\theta_{prod}^{phy}\right]^{\left(T_{opt}^{phy}-20\right)}}{kTn \times \left[\theta_{prod}^{phy}\right]^{\left(kTn \times T_{opt}^{phy}\right)}}\right)}{kTn \times ln\left(\theta_{prod}^{phy}\right)}
\\ \\
bTn &= \left[\theta_{prod}^{phy}\right]^{kTn \times \left(T_{std}^{phy}-aTn\right)}
\end{aligned}\right\}
\tag{J.6}
\end{equation}\]

All of \(aTn\), \(bTn\) and \(kTn\) are fixed at the end of Stage 1 for repeated use in Stage 2. and used in Stage 2 at every timestep and model cell to compute \(L_{T}^{phy}\) in the water temperature range between \(T_{std}^{phy}\) and \(T_{max}^{phy}\).

**Stage 2**
The intent of Stage 2 is to use \(aTn\), \(bTn\) and \(kTn\) to describe a curve for \(L_{T}^{phy}\) between \(T_{std}^{phy}\) and \(T_{max}^{phy}\). This curve is computed as per Equation (J.7).

\[\begin{equation} L_{T}^{phy} = \left[\theta_{prod}^{phy}\right]^{\left(T-20\right)} - \left[\theta_{prod}^{phy}\right]^{kTn \times \left(T-aTn\right)} + bTn \tag{J.7} \end{equation}\]

The complete piecewise continuous function for the generic temperature limitation function \(L_T^{phy}\) is provided in Equation (J.8).

\[\begin{equation} L_{T}^{phy} = \begin{cases} \left[\theta_{prod}^{phy}\right]^{\left(T-20\right)} & \text{$T \le T_{std}^{phy}$} \\ \\ \left[\theta_{prod}^{phy}\right]^{\left(T-20\right)} - \left[\theta_{prod}^{phy}\right]^{kTn \times \left(T-aTn\right)} + bTn & \text{$T_{std}^{phy} \lt T \le T_{max}^{phy}$} \\ \\ 0 & \text{$T_{max}^{phy} \lt T$} \end{cases} \tag{J.8} \end{equation}\]

Depending on the user specification of \(T_{std}\) and \(T_{opt}\), this limitation function can have a value greater than 1.0. This is acceptable, and can be interpreted as temperature effects actively increasing phytoplankton growth. \(L_{T}^{phy}\) is the only limitation function (other than limitation functions that are applied to enhance respiration due to salinity - see Sections J.7.2.2, J.7.3.2 and J.7.4.2) that can vary outside the range 0.0 to 1.0.

The form of \(L_{T}^{phy}\) with varying \(T\) depends on three user specified (or default) temperatures and their derived quantities \(aTn\), \(bTn\) and \(kTn\). As such, the following two figures present \(L_{T}^{phy}\) varying with \(T\) and at constant \(T_{max}^{phy}\) = 25\(^o\)C, with the following conditions:

- Figure J.1:
- Line 1: \(T_{std}^{phy}\) held constant at 10\(^o\)C, with the slider varying \(T_{opt}^{phy}\)
- Line 2: \(T_{std}^{phy}\) held constant at 18\(^o\)C, with the slider varying \(T_{opt}^{phy}\)

- Figure J.2:
- Line 1: \(T_{opt}^{phy}\) held constant at 10\(^o\)C, with the slider varying \(T_{std}^{phy}\)
- Line 2: \(T_{opt}^{phy}\) held constant at 18\(^o\)C, with the slider varying \(T_{std}^{phy}\)

The WQ log file reports the computed temperature limitation function for each phytoplankton group as columnar data. Users can copy and paste these data into plotting software (such as Microsoft Excel) to view each temperature imitation function and therefore ensure that its form is as intended.

## J.3 Nitrogen

Nitrogen offers two limitation functions, one for each of the basic and advanced phytoplankton constituent models. These are described separately following. If nitrogen fixing is implemented for a particular phytoplankton group, then the calculations following are not implemented and \(L_{nit}^{phy}\) is set to one.

### J.3.1 Basic phytoplankton constituent model

If internal phytoplankton nutrients are not simulated dynamically, Equation (J.9) is used to compute \(L_{nit}^{phy}\).

\[\begin{equation} L_{nit}^{phy} = \frac{\left(\left[N\right]_{avail} - \left[N\right]_{min}^{phy}\right)}{\left(\left[N\right]_{avail} - \left[N\right]_{min}^{phy}\right) + K_{lim-N}^{phy}} \tag{J.9} \end{equation}\]

\(\left[N\right]_{avail}\) is the available ambient nitrogen pool on which phytoplankton can draw for primary productivity, and is the sum of inorganic (ammonium and nitrate) nitrogen. \(\left[N\right]_{min}^{phy}\) is the nitrogen concentration below which phytoplankton is no longer permitted to uptake nitrogen. \(K_{lim-N}^{phy}\) is the half saturation nitrogen concentration for phytoplankton uptake.

All these parameters are considered by the WQ Module to be combined quantities, i.e. they are not specific to any particular nitrogen species, but rather, to nitrogen-N concentrations across all species uptaken. In the case of the basic phytoplankton constituent model, this is simply the sum of ammonium-N and nitrate-N concentrations.

Equation (J.9) has that when \(K_{lim-N}^{phy}\) is equal to the difference between ambient nitrogen concentration and \(\left[N\right]_{min}^{phy}\), i.e. when

\[\begin{equation} K_{lim-N}^{phy} = \left[N\right]_{avail} - \left[N\right]_{min}^{phy} \tag{J.10} \end{equation}\]

then \(L_{nit}^{phy}\) = 0.5. The form of \(L_{nit}^{phy}\) with varying \(\left[N\right]_{min}^{phy}\) and \(K_{lim-N}^{phy}\) is provided in Figure J.3, for constant \(\left[N\right]_{avail}\) = 1.0 mg/L.

The parameter \(K_{lim-N}^{phy}\) is the same as that specified in the computation of the the fraction of ammonium that is taken up with primary production (Appendix K.2.1) in the basic phytoplankton constituent model (Equation (K.3)). As such its setting should be considered in terms of its impacts on both nitrogen limitation and uptake processes.

### J.3.2 Advanced phytoplankton constituent model

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentrations are greater than the specified (or default) minimum, then Equation (J.11) is used to compute \(L_{nit}^{phy}\).

\[\begin{equation} L_{nit}^{phy} = \frac{X_{N-C-max}^{phy} \times \left(1.0 - \frac{X_{N-C-min}^{phy} \times \left[PHY\right] }{\left[IN\right]}\right)}{X_{N-C-max}^{phy} - X_{N-C-min}^{phy}} \tag{J.11} \end{equation}\]

\(X_{N-C-min}^{phy}\) and \(X_{N-C-max}^{phy}\) are the specified (or default) minimum and maximum internal nitrogen to Chl *a* (or carbon if using mmm units) ratios, respectively and \(\left[PHY\right]\) and \(\left[IN\right]\) are the current internal Chl *a* (or carbon) and nitrogen concentrations in the computational cell being considered, respectively. Substituting the definition for the minimum nitrogen to Chl *a* (or carbon) ratio \(X_{N-C-min}^{phy}=\left[IN\right]_{min}/\left[PHY\right]\) (where \(\left[PHY\right]\) is in either mgl or mmm units) into Equation (J.11) and rearranging, reveals the form of \(L_{nit}^{phy}\) as per Equation (J.12).

\[\begin{equation} L_{nit}^{phy} = \frac{X_{N-C-max}^{phy} \times \left(1.0 - \frac{\left[IN\right]_{min}} {\left[IN\right]}\right)}{X_{N-C-max}^{phy} - X_{N-C-min}^{phy}} \tag{J.12} \end{equation}\]

\(\left[IN\right]_{min}\) is the minimum internal nitrogen concentation corresponding to the current phytoplankton group \(\left[PHY\right]\). Equation (J.12) demonstrates that:

- If an internal nitrogen concentration is equal to \(\left[IN\right]_{min}\), then \(L_{nit}^{phy}\) is zero. This means that no internal nitrogen is available for primary production in the current timestep and that nitrogen is therefore completely preventing growth
- If an internal nitrogen concentration is equal to \(\left[IN\right]_{max}\), then \(L_{nit}^{phy}\) is one. This means that maximum internal nitrogen is available for primary production in the current timestep and that nitrogen is therefore placing no limit on growth

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentration is less than the specified (or default) minimum, then phytoplankton will look to external (i.e. ambient water column) nitrogen pools to support primary productivity. The associated limitation function is therefore computed in the same manner as Equation (J.9) in Appendix J.3.1. If this is the case, then the parameters \(K_{lim-N}^{phy}\) and \(\left[N\right]_{min}^{phy}\) are used.

This same parameter is also used in the computation of the nitrogen that is taken up with primary production (Appendix K.2.2) in the advanced phytoplankton constituent model (Equation (K.4)). As such its setting should be considered in terms of its impacts on both nitrogen limitation and uptake processes in the advanced phytoplankton constituent model.

## J.4 Phosphorus

Phosphorus offers two limitation functions, one for each of the basic and advanced phytoplankton constituent models. These are described separately following.

### J.4.1 Basic phytoplankton constituent model

If internal phytoplankton nutrients are not dynamically simulated, Equation (J.13) is used to compute \(L_{phs}^{phy}\).

\[\begin{equation} L_{phs}^{phy} = \frac{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right)}{\left(\left[P\right]_{avail} - \left[P\right]_{min}^{phy}\right) + K_{lim-P}^{phy}} \tag{J.13} \end{equation}\]

\(\left[P\right]_{avail}\) is the available phosphorus pool on which phytoplankton can draw for primary productivity, and is simply FRP. \(\left[P\right]_{min}^{phy}\) is the phosphorus concentration below which phytoplankton no longer is permitted to uptake phosphorus \(K_{lim-P}^{phy}\) is the half saturation phosphorus concentration for phytoplankton uptake.

Equation (J.13) has that when \(K_{lim-P}^{phy}\) is equal to the difference between ambient phosphorus concentration and \(\left[P\right]_{min}^{phy}\), i.e. when

\[\begin{equation} K_{lim-P}^{phy} = \left[P\right]_{avail} - \left[P\right]_{min}^{phy} \tag{J.14} \end{equation}\]

then \(L_{phs}^{phy}\) = 0.5. The form of \(L_{phs}^{phy}\) with varying \(\left[P\right]_{min}^{phy}\) and \(K_{lim-P}^{phy}\) is provided in Figure J.4, for constant \(\left[P\right]_{avail}\) = 1.0 mg/L.

### J.4.2 Advanced phytoplankton constituent model

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentrations are greater than the specified (or default) minimum, then Equation (J.15) is used to compute \(L_{phs}^{phy}\).

\[\begin{equation} L_{phs}^{phy} = \frac{X_{P-C-max}^{phy} \times \left(1.0 - \frac{X_{P-C-min}^{phy} \times \left[PHY\right] }{\left[IP\right]}\right)}{X_{P-C-max}^{phy} - X_{P-C-min}^{phy}} \tag{J.15} \end{equation}\]

\(X_{P-C-min}^{phy}\) and \(X_{P-C-max}^{phy}\) are the specified (or default) minimum and maximum internal phosphorus to Chl *a* (or carbon if using mmm units) ratios, respectively and \(\left[PHY\right]\) and \(\left[IP\right]\) are the current internal Chla *a* (or carbon) and phosphorus concentrations in the computational cell being considered, respectively. Substituting the definition for the minimum phosphorus to Chl *a* (or carbon) ratio \(X_{P-C-min}^{phy}=\left[IP\right]_{min}/\left[PHY\right]\) (where \(\left[PHY\right]\) is in either mgl or mmm units) into Equation (J.15) and rearranging, reveals the form of \(L_{phs}^{phy}\) as per Equation (J.16).

\[\begin{equation} L_{phs}^{phy} = \frac{X_{P-C-max}^{phy} \times \left(1.0 - \frac{\left[IP\right]_{min}} {\left[IP\right]}\right)}{X_{P-C-max}^{phy} - X_{P-C-min}^{phy}} \tag{J.16} \end{equation}\]

\(\left[IP\right]_{min}\) is the minimum internal phosphorus concentation corresponding to the current \(\left[PHY\right]\). Equation (J.16) demonstrates that:

- If an internal phosphorus concentration is equal to \(\left[IP\right]_{min}\), then \(L_{phs}^{phy}\) is zero. This means that no internal phosphorus is available for primary production in the current timestep and that phosphorus is therefore completely preventing growth
- If an internal phosphorus concentration is equal to \(\left[IP\right]_{max}\), then \(L_{phs}^{phy}\) is one. This means that maximum internal phosphorus is available for primary production in the current timestep and that phosphorus is therefore placing no limit on growth

If internal phytoplankton nutrients are simulated dynamically, and phytoplankton concentrations are less than the specified (or default) minimum, then phytoplankton will look to external (i.e. ambient water column) phosphorus pools to support primary productivity. The associated limitation function is therefore computed in the same manner as Equation (J.13) in Appendix J.4.1. If this is the case, then the parameters \(K_{lim-P}^{phy}\) and \(\left[P\right]_{min}^{phy}\) are used.

This same parameter is also used in the computation of the phosphorus that is taken up with primary production (Appendix K.3.2) in the advanced phytoplankton constituent model (Equation (K.8)). As such its setting should be considered in terms of its impacts on both phosphorus limitation and uptake processes in the advanced phytoplankton constituent model.

## J.5 Silicate

Equation (J.17) is used to compute \(L_{sil}^{phy}\), if phytoplankton is configured to take up silicate. Specifying silicate limitation parameters will trigger the WQ Module to take up silicate (Section 4.7.3.5.1.6).

\[\begin{equation} L_{sil}^{phy} = \frac{\left(\left[Si\right] - \left[Si\right]_{min}\right)}{\left(\left[Si\right] - \left[Si\right]_{min}\right) + K_{lim-Si}^{phy}} \tag{J.17} \end{equation}\] \(\left[Si\right]\) is the ambient silicate concentration. \(\left[Si\right]_{min}\) is the silicate concentration below which phytoplankton no longer is permitted to uptake silicate and \(K_{lim-Si}^{phy}\) is the half saturation silicate concentration for phytoplankton uptake.

\(K_{lim-Si}^{phy}\) is specified as a silicate concentration. When this concentration is equal to the difference between the relevant ambient concentration and \(\left[Si\right]_{min}\), i.e. when

\[\begin{equation} K_{lim-Si}^{phy} = \left[Si\right] - \left[Si\right]_{min} \tag{J.18} \end{equation}\]

then \(L_{sil}^{phy}\) = 0.5. The form of \(L_{sil}^{phy}\) with varying \(\left[Si\right]_{min}\) and \(K_{lim-Si}^{phy}\) is provided in Figure J.5, for constant \(\left[Si\right]_{avail}\) = 1.0 mg/L.

## J.6 Light

Phytoplankton require photosynthetically active radiation (PAR) in order to affect primary productivity. The required PAR may not always be available due to a range of environmental factors and as such may limit phytoplankton growth. In addition, some phytoplankton groups may suffer photoinhibition, which is a reduction in the ability to photosynthesise as a result of damage incurred by greater levels of radiation than can be tolerated. This is also referred to as light-induced damage, and in general terms means that increasing light intensity experienced by a phytoplankton group does not translate into a monotonic increase in the light limitation function: primary productivity can decrease with increasing available light where photoinhibition occurs. A saturating light intensity \(I_S\) is specified for those models that include photoinhibition (rather then a half saturation light for limitation \(I_K\) specified for models that exclude photoinhibition) and \(I_S\) can be interpreted as the light intensity at which the limitation function has its peak, usually at a value of 1.0. Either side of this \(I_S\) light intensity, the limitation function decreases toward zero.

The WQ Module offers a number of different models that deploy combinations of these processes to estimate phytoplankton light limitation functions. These are described following.

### J.6.1 Basic

This model includes phytoplankton light limitation but with no photoinhibition. Limitation is therefore purely due to insufficient light availability. The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.19).

\[\begin{equation} L_{lght}^{phy} = 1.0 - \left[ \int_{\frac{-PAR_b}{I_{K-bas}}}^{\infty} \! \left( \frac{e^{-t}}{t} \right)\, \mathrm{d}t - \int_{\frac{-PAR_t}{I_{K-bas}}}^{\infty} \! \left( \frac{e^{-t}}{t} \right)\, \mathrm{d}t\right] \times \left[ \vphantom{\int_{\frac{-PAR_b}{I_{K-bas}}}^{\infty} \! \left( \frac{e^{-t}}{t} \right)\, \mathrm{d}t} \frac{1}{\text{max}(k_{bio}\times dz,10^{-3})}\right] \tag{J.19} \end{equation}\]

\(PAR_b\) and \(PAR_t\) are photosynthetically available radiation at the bottom and top faces of a model cell, respectively, \(I_{K-bas}\) is the half saturation constant for light limitation of growth, \(\text{k}_\text{bio}\) is the local change in light extinction coefficient due to self shading and \(dz\) is cell vertical thickness.

### J.6.2 Monod

This model includes phytoplankton light limitation but with no photoinhibition, as per Monod (1950). Limitation is therefore purely due to insufficient light availability. The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.20).

\[\begin{equation} L_{lght}^{phy} = \frac{\frac{PAR_c}{I_{K-mon}}}{1 + \frac{PAR_c}{I_{K-mon}}} \tag{J.20} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{K-mon}\) is the half saturation constant for light limitation of growth. Equation (J.20) has that when \(PAR_c\) is equal to \(I_{K-mon}\), \(L_{lght}^{phy}\) = 0.5.

### J.6.3 Steele

This model includes phytoplankton light limitation with photoinhibition, as per Steele (1962). The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.21).

\[\begin{equation} L_{lght}^{phy} = \frac{PAR_c}{I_{S-ste}} \times e^{\left(1.0 - \frac{PAR_c}{I_{S-ste}}\right)} \tag{J.21} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{S-ste}\) is the saturating light intensity. Equation (J.21) has that when \(PAR_c\) is equal to \(I_{S-ste}\), \(L_{lght}^{phy}\) = 1.0, i.e., the light intensity at which no light limitation or photoinhibition occurs. The form of Equation (J.21) is presented in Figure J.6, with the abscissa spanning a typical range of instantaneous photosynthetically available radiation.

### J.6.4 Webb

This model includes phytoplankton light limitation with no photoinhibition, as per Webb (1974). The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.22).

\[\begin{equation} L_{lght}^{phy} = 1.0 - e^{-\left(\frac{PAR_c}{I_{K-web}}\right)} \tag{J.22} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{K-web}\) is the half saturation constant for light limitation of growth. The form of Equation (J.22) is presented in Figure J.7, with the abscissa spanning a typical range of instantaneous photosynthetically available radiation.

### J.6.5 Jassby

This model includes phytoplankton light limitation with no photoinhibition, as per Jassby and Platt (1976). The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.23).

\[\begin{equation} L_{lght}^{phy} = tanh\left(\frac{PAR_c}{I_{K-jas}}\right) \tag{J.23} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{K-jas}\) is the half saturation constant for light limitation of growth. The form of Equation (J.23) is presented in Figure J.8, with the abscissa spanning a typical range of instantaneous photosynthetically available radiation.

### J.6.6 Chalker

This model includes phytoplankton light limitation with no photoinhibition, as per Chalker (1980). The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.24).

\[\begin{equation} L_{lght}^{phy} = \frac{e^{1.5\left(\frac{PAR_c}{I_{K-cha}}\right)}-1.0}{e^{1.5\left(\frac{PAR_c}{I_{K-cha}}\right)}+0.5} \tag{J.24} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{K-cha}\) is the half saturation constant for light limitation of growth. The form of Equation (J.24) is presented in Figure J.9, with the abscissa spanning a typical range of instantaneous photosynthetically available radiation.

### J.6.7 Klepper

This model includes phytoplankton light limitation with photoinhibition, as per Klepper et al. (1988) and Ebenhoh et al. (1997). The limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.25).

\[\begin{equation} L_{lght}^{phy} = \frac{7.0\times\left(\frac{PAR_c}{I_{S-kle}}\right)}{1.0 + 5.0\times\left(\frac{PAR_c}{I_{S-kle}}\right) + \left(\frac{PAR_c}{I_{S-kle}}\right)^2} \tag{J.25} \end{equation}\]

\(PAR_c\) is the photosynthetically available radiation at the centre of a model cell and \(I_{S-kle}\) is the saturating light intensity. The form of Equation (J.25) is presented in Figure J.10, with the abscissa spanning a typical range of instantaneous photosynthetically available radiation.

### J.6.8 Integrated

This model includes phytoplankton light limitation with photoinhibition, and is the integrated form of the Steele (1962) model provided in Equation (J.21). In this form, the limitation function \(L_{lght}^{phy}\) is computed within a model cell as per Equation (J.26).

\[\begin{equation} L_{lght}^{phy} = \frac{e^{\left(1.0 - \frac{PAR_b}{I_{S-int}}\right)} - e^{\left(1.0 - \frac{PAR_t}{I_{S-int}}\right)}}{\text{k}_\text{bio} \times dz} \tag{J.26} \end{equation}\]

\(PAR_b\) and \(PAR_t\) are photosynthetically available radiation at the bottom and top faces of a model cell, respectively, \(I_{S-int}\) is the saturating light intensity, \(\text{k}_\text{bio}\) is the local change in light extinction coefficient due to self shading and \(dz\) is cell vertical thickness. In most cases, this model should produce similar results to the Steele model in section J.6.3.

## J.7 Salinity

Phytoplankton can exhibit sensitivity to salinity. Where activated, the WQ Module captures these sensitivities via multiplicative application of a salinity limitation function as either a reduction of primary productivity (Equation (I.4)) or an increase of respiration (Equation (I.7)). Application of these rates is mutually exclusive for a given phytoplankton group: limitation can be applied either to primary productivity or respiration, but not both. One way to conceptualise this is as follows:

- A salinity limitation \(L_{sal-pp}^{phy}\) applied to primary productivity is always between zero and one and will result in a reduction in primary productivity. Corresponding respiration rates will remain unaffected by salinity and therefore by Equation (I.1), phytoplankton biomass should be expected to decrease as a result
- A salinity limitation \(L_{sal-r}^{phy}\) applied to respiration is always greater than one and will result in an enhancement of respiration. Corresponding primary productivity rates will remain unaffected by salinity and therefore by Equation (I.1), phytoplankton biomass should be expected to decrease as a result, noting however that exudation (which is another loss term) will continue and is related to primary productivity

The salinity limitation models available within the WQ Module are described following. All can be applied to both primary productivity and respiration, other than the estuarine limitation function which is applicable only to productivity.

### J.7.1 None

This model includes no salinity limitation to either of the computed primary productivity or respiration rates. The salinity limitation set to 1.0 for all salinities.

### J.7.2 Freshwater

This model is typically used for phytoplankton groups that have a preference for fresh water. As such, the salinity limitation is set to 1.0 for all salinities greater than zero and up to a specified (or default) optimal salinity. The general expression for the limitation function above this optimal salinity is presented in Equation (J.27), and this expression is applied to both primary productivity and respiration. Its specific applications to these two processes are described in the following subsections.

\[\begin{equation} L_{sal-r}^{phy} = L_{sal-pp}^{phy} = \begin{cases} 1.0 & \text{$S \le S_{opt-fresh}^{phy}$} \\ \\ \left(\frac{\left(L^{phy}_{max-fresh}-1.0\right)\times S^2 - \left(\left(L^{phy}_{max-fresh}-1.0\right) \times 2.0 \times S_{opt-fresh}^{phy}\right)\times S}{\left(S_{max-fresh}^{phy} - S_{opt-fresh}^{phy}\right)^2}\right) + \ldots \\ \\ \text{$ $} \ldots \left(\frac{\left(\left(L^{phy}_{max-fresh}-1.0\right) \times \left(S_{opt-fresh}^{phy}\right)^2\right)}{\left(S_{max-fresh}^{phy} - S_{opt-fresh}^{phy}\right)^2}\right) + 1.0 & \text{$S_{opt-fresh}^{phy} \lt S$}\\ \\ \end{cases} \tag{J.27} \end{equation}\]

\(S_{opt-fresh}^{phy}\) and \(S_{max-fresh}^{phy}\) are the optimal and maximum salinities respectively, \(L^{phy}_{max-fresh}\) is the value of the salinity limitation function at \(S_{max-fresh}^{phy}\) and \(S\) is ambient salinity.

#### J.7.2.1 Primary productivity

In order to apply this limitation function to primary productivity, \(L^{phy}_{max-fresh}\) is set such that \[\begin{equation} 0.0 \le L^{phy}_{max-fresh} \le 1.0 \tag{J.28} \end{equation}\] In this case, and with increasing salinity, the limitation function decreases from 1.0 at \(S_{opt-fresh}^{phy}\) to a specified (or default) limitation function value at a specified (or default) maximum salinity. If \(L^{phy}_{max-fresh}\) is not zero, then the function continues downward to zero at salinities greater than \(S_{max-fresh}^{phy}\). When the function reaches zero it remains there.

The simplest application of this limitation function is to set \(L^{phy}_{max-fresh}\) as zero at \(S_{max-fresh}^{phy}\). This will ensure that no primary productivity occurs above a maximum salinity of \(S_{max-fresh}^{phy}\) (this approach is reflected in the naming of \(L^{phy}_{max-fresh}\) and \(S_{max-fresh}^{phy}\)). For example, if a particular phytoplankton group was known to feel the effects of salinity on primary productivity above a salinity of 5 g/L, and to entirely cease producing at salinities greater than 10 g/L, then the following would be set by the user:

- \(S_{opt-fresh}^{phy}\) = 5.0 g/L
- \(S_{max-fresh}^{phy}\) = 10.0 g/L, and
- \(L^{phy}_{max-fresh}\) = 0.0

Another example is the case where \(L^{phy}_{max-fresh}\) is not zero at \(S_{max-fresh}^{phy}\). For example, if a particular phytoplankton group was known to feel the effects of salinity on primary productivity above a salinity of 2 g/L, and to have a 75% limitation on productivity at a salinity of 10 g/L, then the following would be set by the user:

- \(S_{opt-fresh}^{phy}\) = 2.0 g/L
- \(S_{max-fresh}^{phy}\) = 10.0 g/L, and
- \(L^{phy}_{max-fresh}\) = 0.25

Using these values initially, but varying \(S_{max-fresh}^{phy}\), the form of Equation (J.27) is presented in Figure J.12. The salinity at which the limitation function falls to zero is greater than \(S_{max-fresh}^{phy}\), but is always 0.25 at \(S_{max-fresh}^{phy}\).

#### J.7.2.2 Respiration

In order to apply this limitation function to respiration, \(L^{phy}_{max-fresh}\) is set such that \[\begin{equation} 1.0 \lt L^{phy}_{max-fresh} \tag{J.29} \end{equation}\] In this case, and with increasing salinity, the limitation function increases from 1.0 at \(S_{opt-fresh}^{phy}\) to a specified (or default) limitation function value at a specified (or default) salinity. The function then increases indefinitely.

For example, if a particular phytoplankton group was known to feel the effects of salinity on respiration above a salinity of 5 g/L, such that at a salinity of 10 g/L its respiration rate doubled, then the following would be set by the user:

- \(S_{opt-fresh}^{phy}\) = 5.0 g/L
- \(S_{max-fresh}^{phy}\) = 10.0 g/L, and
- \(L^{phy}_{max-fresh}\) = 2.0

Using these values initially but varying \(S_{max-fresh}^{phy}\), the form of Equation (J.27) is presented in Figure J.13.

### J.7.3 Marine

This model is typically used for phytoplankton groups that have a preference for marine (saline) water. As such, the salinity limitation is set to 1.0 for all salinities greater a specified (or default) optimal salinity. The general expression for the limitation function below this optimal salinity is presented in Equation (J.30), and this expression is applied to both primary productivity and respiration. Its specific applications to these two processes are described in the following subsections.

\[\begin{equation} L_{sal-r}^{phy} = L_{sal-pp}^{phy} = \begin{cases} \left(\frac{\left(L^{phy}_{zero-marine}-1.0\right)\times S^2}{\left(S_{opt-marine}^{phy}\right)^2}\right) - \left(2.0 \times \frac{\left(L^{phy}_{zero-marine}-1.0\right)\times S}{\left(S_{opt-marine}^{phy}\right)}\right) + L^{phy}_{zero-marine} & \text{$S \le S_{opt-marine}^{phy}$}\\ \\ 1.0 & \text{$S_{opt-marine}^{phy} \lt S$} \\ \\ \end{cases} \tag{J.30} \end{equation}\]

\(S_{opt-marine}^{phy}\) is the optimal salinity , \(L^{phy}_{zero-marine}\) is the value of the salinity limitation function at zero salinity and \(S\) is ambient salinity.

#### J.7.3.1 Primary productivity

In order to apply this limitation function to primary productivity, \(L^{phy}_{zero-marine}\) is set such that \[\begin{equation} 0.0 \le L^{phy}_{zero-marine} \le 1.0 \tag{J.31} \end{equation}\] In this case, and with decreasing salinity, the limitation function decreases from 1.0 at \(S_{opt-marine}^{phy}\) to a specified (or default) limitation function value at zero salinity, \(L^{phy}_{zero-marine}\).

The simplest application of this limitation function is to set \(L^{phy}_{zero-marine}\) to zero. This will ensure that no primary productivity occurs at zero salinity. For example, if a particular phytoplankton group was known to feel the effects of salinity on primary productivity below a salinity of 20 g/L, and to entirely cease producing at salinities of 0 g/L, then the following would be set by the user:

- \(S_{opt-marine}^{phy}\) = 20.0 g/L, and
- \(L^{phy}_{zero-marine}\) = 0.0

Using these values initially but varying \(S_{opt-marine}^{phy}\), the form of Equation (J.30) is presented in Figure J.14.

Another example is the case where \(L^{phy}_{zero-marine}\) is not zero. For example, if a particular phytoplankton group was known to feel the effects of salinity on primary productivity below a salinity of 10 g/L, and to have a 75% limitation on productivity at a salinity of 0 g/L (i.e. still produce in completely frsh water), then the following would be set by the user:

- \(S_{opt-marine}^{phy}\) = 10.0 g/L, and
- \(L^{phy}_{zero-marine}\) = 0.25

Using these values initially, but varying \(S_{zero-marine}^{phy}\), the form of Equation (J.30) is presented in Figure J.15.

#### J.7.3.2 Respiration

In order to apply this limitation function to respiration, \(L^{phy}_{zero-marine}\) is set such that \[\begin{equation} 1.0 \lt L^{phy}_{zero-marine} \tag{J.32} \end{equation}\] In this case, and with decreasing salinity, the limitation function increases from 1.0 at \(S_{opt-marine}^{phy}\) to a specified (or default) limitation function value at zero salinity.

For example, if a particular phytoplankton group was known to feel the effects of salinity on respiration below a salinity of 5 g/L, such that at a salinity of 0 g/L its respiration rate doubled, then the following would be set by the user:

- \(S_{opt-marine}^{phy}\) = 5.0 g/L
- \(L^{phy}_{zero-marine}\) = 2.0

Using these values initially but varying \(S_{zero-marine}^{phy}\), the form of Equation (J.30) is presented in Figure J.16.

### J.7.4 Mixed

This model is typically used for phytoplankton groups that have a preference for mid range salinity waters. As such, the salinity limitation is set to 1.0 between specified (or default) optimal and maximum salinities. The general expression for the limitation function above and below this range is presented in Equation (J.33), and this expression is applied to both primary productivity and respiration. Its specific applications to these two processes are described in the following subsections.

\[\begin{equation} L_{sal-r}^{phy} = L_{sal-pp}^{phy} = \begin{cases} \left(\frac{\left(L^{phy}_{zero-mix}-1.0\right)\times S^2}{\left(S_{opt-mix}^{phy}\right)^2}\right) - \left(2.0 \times \frac{\left(L^{phy}_{zero-mix}-1.0\right)\times S}{\left(S_{opt-mix}^{phy}\right)}\right) + L^{phy}_{zero-mix} & \text{$S \le S_{opt-mix}^{phy}$} \\ \\ 1.0 & \text{$S_{opt-mix}^{phy} \lt S \le S_{max-mix}^{phy}$} \\ \\ \left(\frac{\left(L^{phy}_{zero-mix}-1.0\right)\times \left(S_{max-mix}^{phy} + S_{opt-mix}^{phy} - S\right)^2}{\left(S_{opt-mix}^{phy}\right)^2}\right) - \ldots \\ \ldots \left(2.0 \times \frac{\left(L^{phy}_{zero-mix}-1.0\right)\times \left(S_{max-mix}^{phy} + S_{opt-mix}^{phy} - S\right)}{\left(S_{opt-mix}^{phy}\right)}\right) + L^{phy}_{zero-mix} & \text{$S_{max-mix}^{phy} \lt S \le \left(S_{max-mix}^{phy} + S_{opt-mix}^{phy}\right)$} \\ \\ L^{phy}_{zero-mix} & \text{$\left(S_{max-mix}^{phy} + S_{opt-mix}^{phy}\right) \lt S$} \end{cases} \tag{J.33} \end{equation}\]

\(S_{opt-mix}^{phy}\) and \(S_{max-mix}^{phy}\) are the optimal and maximum salinities, respectively, \(L^{phy}_{zero-mix}\) is the value of the salinity limitation function at salinities of zero and greater than (\(S_{opt-mix}^{phy} + S_{max-mix}^{phy}\)) and \(S\) is ambient salinity.

#### J.7.4.1 Primary productivity

In order to apply this limitation function to primary productivity, \(L^{phy}_{zero-mix}\) is set such that \[\begin{equation} 0.0 \le L^{phy}_{zero-mix} \le 1.0 \tag{J.34} \end{equation}\] In this case, the limitation function:

- Decreases at salinities below \(S_{opt-mix}^{phy}\) to \(L^{phy}_{zero-mix}\) at a salinity of zero, and
- Decreases at salinities above \(S_{max-mix}^{phy}\) to \(L^{phy}_{zero-mix}\) at a salinity equal to the sum of the specified optimal and maximum salinities

The simplest application of this limitation function is to set \(L^{phy}_{zero-mix}\) to zero. This will ensure that no primary productivity occurs both at zero salinity and above a salinity of \(S_{opt-mix}^{phy}\) + \(S_{max-mix}^{phy}\). For example,if a particular phytoplankton group was known to prefer salinities between 5 and 10 g/L, then the following would be set by the user:

- \(S_{opt-mix}^{phy}\) = 5.0 g/L
- \(S_{max-mix}^{phy}\) = 10.0 g/L
- \(L^{phy}_{zero-mix}\) = 0.0

Using these values initially but varying \(S_{opt-mix}^{phy}\), the form of Equation (J.33) is presented in Figure J.17. The absolute range between \(S_{opt-mix}^{phy}\) and \(S_{max-mix}^{phy}\) is held constant at 5.0 g/L for ease of comparison, so \(S_{max-mix}^{phy}\) increases at the same rate as \(S_{opt-mix}^{phy}\) when the slider moves.

Another example is the case where \(L^{phy}_{zero-mix}\) is not zero. For example,if a particular phytoplankton group was known to prefer salinities between 5 and 10 g/L and have a 60% limitation on productivity at zero salinity, then the following would be set by the user:

- \(S_{opt-mix}^{phy}\) = 5.0 g/L
- \(S_{max-mix}^{phy}\) = 10.0 g/L
- \(L^{phy}_{zero-mix}\) = 0.4

Using these values initially but varying \(S_{opt-mix}^{phy}\), the form of Equation (J.33) is presented in Figure J.18. The absolute range between \(S_{opt-mix}^{phy}\) and \(S_{max-mix}^{phy}\) is again held constant at 5.0 g/L for ease of comparison, so \(S_{max-mix}^{phy}\) increases at the same rate as \(S_{opt-mix}^{phy}\) when the slider moves.

#### J.7.4.2 Respiration

In order to apply this limitation function to respiration, \(L^{phy}_{zero-mix}\) is set such that \[\begin{equation} 1.0 \lt L^{phy}_{zero-mix} \tag{J.35} \end{equation}\] In this case, the limitation function:

- Increases at salinities below \(S_{opt-mix}^{phy}\) to \(L^{phy}_{zero-mix}\) at a salinity of zero, and
- Increases at salinities above \(S_{max-mix}^{phy}\) to \(L^{phy}_{zero-mix}\) at a salinity equal to the sum of the specified optimal and maximum salinities

For example,if a particular phytoplankton group was known to prefer salinities between 5 and 10 g/L but feel the effects of salinity outside this range such that at a salinity of zero its respiration rate doubled, then the following would be set by the user:

- \(S_{opt-mix}^{phy}\) = 5.0 g/L
- \(S_{max-mix}^{phy}\) = 10.0 g/L
- \(L^{phy}_{zero-mix}\) = 2.0

Using these values initially but varying \(S_{opt-mix}^{phy}\), the form of Equation (J.33) is presented in Figure J.19. The absolute range between \(S_{opt-mix}^{phy}\) and \(S_{max-mix}^{phy}\) is again held constant at 5.0 g/L for ease of comparison, so \(S_{max-mix}^{phy}\) increases at the same rate as \(S_{opt-mix}^{phy}\) when the slider moves.

### J.7.5 Estuarine

This model is typically used for phytoplankton groups that have a preference for brackish (typically estuarine) waters. As such, the salinity limitation is set to one at a specified (or default) optimal mid range salinity. The general expression for the limitation function above and below this salinity is presented in Equation (J.36), and this expression is applied only to primary productivity.

\[\begin{equation} L_{sal}^{phy} = \begin{cases} e^{P^{phy}_{est}\left(S-S_{opt-est}^{phy}\right)} \times \left( \frac{\left(S_{max-est}^{phy}-S\right)}{\left(S_{max-est}^{phy}-S_{opt-est}^{phy}\right)}\right)^{P^{phy}_{est}\left(S_{max-est}^{phy}-S_{opt-est}^{phy}\right)} & \text{$S \le S_{max-est}^{phy}$} \\ \\ 0.0 & \text{$S_{max-est}^{phy} \lt S$} \\ \\ \end{cases} \tag{J.36} \end{equation}\]

\(S_{opt-est}^{phy}\) and \(S_{max-est}^{phy}\) are the optimal and maximum salinities, respectively, \(P^{phy}_{est}\) is the power coefficient and \(S\) is ambient salinity.

The intention of applying this limitation function is that it be applied to a relatively narrow band of mid range (estuarine or brackish) salinities. For example, if a particular phytoplankton group was known to produce optimally at a salinity of 15 g/L but had its growth completely suppressed less than 5 g/L and greater than 20 g/L, then the following would be set by the user:

- \(S_{opt-est}^{phy}\) = 15.0 g/L
- \(S_{max-est}^{phy}\) = 20.0 g/L
- \(P^{phy}_{est}\) = 1.6 (or greater)

Using these values but varying \(P^{phy}_{est}\), the form of Equation (J.36) is presented in Figure J.20. \(S_{opt-est}^{phy}\) and \(S_{max-est}^{phy}\) are held constant at 15.0 g/L and 20 g/L, respectively. The slider starts at a \(P^{phy}_{est}\) value of 0.2, and the key feature to note is that the salinity limitation function distribution tightens as \(P^{phy}_{est}\) increases.