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

Using these values initially but varying $S_{max-fresh}^{phy}$, the form of Equation (J.27) is presented in Figure J.11.

Figure J.11: Move the slider to see the effect of changing $S_{max-fresh}^{phy}$ on the computed salinity limitation function. $S_{opt-fresh}^{phy}$ = 5.0 g/L and $L^{phy}_{max-fresh}$ = 0.0 and are fixed

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 $S_{max-fresh}^{phy}$.

Figure J.12: Move the slider to see the effect of changing $S_{max-fresh}^{phy}$ on the computed salinity limitation function. $S_{opt-fresh}^{phy}$ = 2.0 g/L and $L^{phy}_{max-fresh}$ = 0.25 and are fixed

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.

Figure J.13: Move the slider to see the effect of changing $S_{max-fresh}^{phy}$ on the computed salinity limitation function. $S_{opt-fresh}^{phy}$ = 5.0 g/L and $L^{phy}_{max-fresh}$ = 2.0 and are fixed

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.

Figure J.14: Move the slider to see the effect of changing $S_{opt-marine}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-marine}$ = 0.0 and is fixed

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.

Figure J.15: Move the slider to see the effect of changing $S_{opt-marine}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-marine}$ = 0.25 and is fixed

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.

Figure J.16: Move the slider to see the effect of changing $S_{opt_marine}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-marine}$ = 2.0 and is fixed

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.

Figure J.17: Move the slider to see the effect of changing $S_{opt-mix}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-mix}$ = 0.0 and is fixed. $S_{max-mix}^{phy}$ is set to be $S_{opt-mix}^{phy}$ + 5.0 g/L

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.

Figure J.18: Move the slider to see the effect of changing $S_{opt-mix}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-mix}$ = 0.4 and is fixed. $S_{max-mix}^{phy}$ is set to be $S_{opt-mix}^{phy}$ + 5.0 g/L

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.

Figure J.19: Move the slider to see the effect of changing $S_{opt-mix}^{phy}$ on the computed salinity limitation function. $L^{phy}_{zero-mix}$ = 2.0 and is fixed. $S_{max-mix}^{phy}$ is set to be $S_{opt-mix}^{phy}$ + 5.0 g/L

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.

Figure J.20: Move the slider to see the effect of changing $P^{phy}_{est}$ on the computed salinity limitation function. $S_{opt-est}^{phy}$ and $S_{max-est}^{phy}$ are set to be 15 and 20 g/L, respectively, and are fixed