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.

Figure J.6: Move the slider to see the effect of changing \(I_S\) on the computed light limitation function

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.

Figure J.7: Move the slider to see the effect of changing \(I_K\) on the computed light limitation function

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.

Figure J.8: Move the slider to see the effect of changing \(I_K\) on the computed light limitation function

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.

Figure J.9: Move the slider to see the effect of changing \(I_K\) on the computed light limitation function

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.

Figure J.10: Move the slider to see the effect of changing \(I_S\) on the computed light limitation function

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.