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 \(T_{std}^{phy}\)

Figure J.1: Move the slider to see the effect of changing \(T_{opt}^{phy}\) on the computed temperature limitation function

Figure J.2: Move the slider to see the effect of changing \(T_{std}^{phy}\) on the computed temperature limitation function

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.