# Appendix G Processes: inorganic phosphorus

## G.1 Sediment FRP flux

Filterable reactive phosphorus (FRP) is exchanged between the water column and sediments via specification of sediment fluxes. This flux is most commonly out of the sediments, i.e. a positive specification of sediment flux. Although it is rare that sediments act as sinks of FRP, WQ Module can be parameterised to allow for this if required.

The user specified rate of FRP flux (which can be spatially varying) is modified by overlying ambient dissolved oxygen concentration (together with a user specified half saturation oxygen concentration) and water temperature. These modifications are simulated via Michaelis-Menten and Arrhenius models, respectively, as per Equation (G.1). \[\begin{equation} \href{AppDiags.html#WQDiagFRPSedFlx}{F_{sed\langle computed\rangle}^{FRP}} = F_{sed}^{FRP} \times \underbrace{\frac{K_{sed-O_2}^{FRP}}{K_{sed-O_2}^{FRP} + \left[DO\right]}}_{\text{Influence of oxygen}} \times \underbrace{\vphantom{\frac{K_{sed-O_2}^{FRP}}{K_{sed-O_2}^{FRP} + \left[DO\right]}} \left[\theta_{sed}^{FRP}\right]^{\left(T-20\right)}}_{\text{Influence of temperature}} \tag{G.1} \end{equation}\] \(F_{sed}^{FRP}\) is the user specified FRP sediment flux at 20\(^o\)C without the influence of dissolved oxygen, \(\left[DO\right]\) is the overlying dissolved oxygen concentration, \(K_{sed-O_2}^{FRP}\) is the user specified half saturation concentration of dissolved oxygen for FRP sediment flux, \(\theta_{sed}^{FRP}\) is the corresponding temperature coefficient, and \(T\) is ambient water temperature.

As per silicate sediment fluxes (see Section E.1), the above equation leads to linearly varying ambient FRP concentrations in the demonstration model when dissolved oxygen concentration, half saturation oxygen concentration and temperature are set to be constant.

A more realistic environmental setting has sediment FRP flux occurring against a background of oxygen concentration drawdown. The demonstration model has been used to illustrate this via execution of a suite of simulations that experience dissolved oxygen drawdown, with \(K_{sed-O_2}^{O_2}\) = 4 mg/L and each simulation using a different \(K_{sed-O_2}^{FRP}\). The FRP sediment flux rate was specified as 400 mg/m\(^2\)/d and temperature effects were turned off. The predicted temporal evolution of water column FRP concentrations is provided in Figure G.1. Use the “play” button or drag the slider to see how different half saturation concentrations change ambient concentrations.

The rate of FRP flux is also related to ambient water temperature, via the Arrhenius model in Equations (G.1). To demonstrate this, the same model above (with time varying dissolved oxygen concentration) was executed at a range of ambient temperatures, but constant half saturation concentrations for oxygen set to 4 mg/L. All temperature coefficients were set to 1.05. The results are provided in Figure G.2. Use the “play” button or drag the slider to see how different ambient temperatures change ambient concentrations.

The figure shows that as ambient water temperature increases, the rate of increase of FRP concentrations follow suit. This is expected behaviour.

## G.2 Wet deposition of inorganic phosphorus

Wet deposition of inorganic phosphorus includes the pluvial delivery of filterable reactive phosphorus to the water column. A global FRP concentration in rainfall is specifiable. The WQ Module multiplies this concentration by each timestep’s rainfall depth and corresponding model surface area(s) to compute the wet mass flux of FRP over each timestep (Equation (G.2)).

\[\begin{equation} F_{atm-wet}^{FRP} = \left[{FRP}\right]_{rain} \times \text{Timestep rainfall depth} \times \text{Model area} \tag{G.2} \end{equation}\]

\(\left[{FRP}\right]_{rain}\) is the specified rainfall concentration of FRP. This mass is added to the uppermost model layer cells and corresponding changes to ambient FRP concentrations computed.

If rainfall depth is specified as being spatially variable (i.e. by application of multiple atmospheric boundary conditions within TUFLOW FV), then \(F_{atm-wet}^{FRP}\) will correspondingly vary spatially.

## G.3 Dry deposition of inorganic phosphorus

Dry deposition of inorganic phosphorus includes the delivery of adsorbed FRP to the water column from atmospheric dust fallout. A constant daily mass loading rate of adsorbed FRP is specifiable, and the WQ Module multiples this rate by model timestep and the global model surface area to compute the mass flux of adsorbed FRP (Equation (G.3)). This mass is added to the uppermost model layer cells and corresponding changes to ambient concentrations computed. Dry deposition is only included in calculations if adsorbed FRP is simulated.

\[\begin{equation} F_{atm-dry}^{FRPads} = R^{FRPads}_{atm-dry} \times \text{Timestep} \times \text{Global model area} \tag{G.3} \end{equation}\]

\(R^{FRPads}_{atm-dry}\) is the user specified rate of adsorbed FRP atmospheric fallout. One (constant) mass flux is applied globally. This dry deposition is only applied if FRP adsorption is simulated, and if this is the case, then this introduction of dry deposited FRP is assigned to the adsorbed FRP computed variable. If adsorbed FRP is not included in a simulation, then dry deposition of FRP will be ignored.

The total atmospheric depositional flux of inorganic phosphorus is as per Equation (G.4).

\[\begin{equation} \href{AppDiags.html#WQDiagDIPAtmFlx}{F_{atm-dip\langle computed \rangle}^{P}} = F_{atm-wet}^{FRP} + F_{atm-dry}^{FRPads} \tag{G.4} \end{equation}\]

## G.4 Adsorption and desorption

At each timestep and within each active computational cell, the WQ Module distributes the total FRP mass between dissolved FRP and adsorbed FRP if the phosphorus adsorption constituent model is used. Total FRP is conserved via Equation (G.5).

\[\begin{equation} \left[FRP\right]_{tot} = \left[FRPads\right] + \left[FRP\right] \tag{G.5} \end{equation}\]

\(\left[FRP\right]_{tot}\) is the total FRP available for distribution between dissolved and adsorbed states, and \(\left[FRPads\right]\) and \(\left[FRP\right]\) are the ambient adsorbed and dissolved FRP concentrations, respectively.

The distribution between adsorbed and dissolved FRP concentrations is computed within the WQ Module via either a linear or quadratic model. These offer alternative approaches to simulating the distribution of total FRP between adsorbed and free states.

### G.4.1 Linear model

Equation (G.6) is the linear adsorption model, and the default.

\[\begin{equation} \left.\begin{aligned} \left[FRPads\right] &= \left(\frac{K_{ads-L}^{FRP} \times \left[SS\right]}{1 + K_{ads-L}^{FRP} \times \left[SS\right]}\right) \times \left[FRP\right]_{tot}\\ \\ \left[FRP\right] &= \left(\frac{1}{1 + K_{ads-L}^{FRP} \times \left[SS\right]}\right) \times \left[FRP\right]_{tot} \end{aligned}\right\} \tag{G.6} \end{equation}\]

\(\left[SS\right]\) is the sum of suspended sediment concentrations of sediment fractions that have been designated as able to adsorb FRP and \(K_{ads-L}^{FRP}\) is the linear sorption partitioning coefficient.

Figure G.3 provides an indicative representation of the general form of the linear relationship, for assumed total FRP (0.1 mg/L) and SS (5mg/L) concentrations.

As a guide, equation (G.6) has that if \(K_{ads-L}^{FRP} \times \left[SS\right]\) has a value of 1, then an equal (50:50) proportioning of free and adsorbed FRP will result. This guide can be useful if ambient suspended sediment concentrations are known - even approximately - in advance.

### G.4.2 Quadratic model

Equation (G.7) is the quadratic adsorption model.

\[\begin{equation} \left.\begin{aligned} \left[FRPads\right] &= \frac{1}{2} \times \left[\left(\left[FRP\right]_{Tot} + \frac{1}{K_{ads-Q}^{FRP}} + \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right) - \text{C} \right] \\ \\ \left[FRP\right] &= \frac{1}{2} \times \left[\left(\left[FRP\right]_{Tot} - \frac{1}{K_{ads-Q}^{FRP}} - \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right) + \text{C} \right] \\ \\ \text{C} &= \sqrt{\left(\left[FRP\right]_{Tot} + \frac{1}{K_{ads-Q}^{FRP}} - \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right)^2 + \left(\frac{4.0\times \left[SS\right] \times Q^{FRP}_{max}}{K_{ads-Q}^{FRP}}\right)} \end{aligned}\right\} \tag{G.7} \end{equation}\]

\(\left[SS\right]\) is again the sum of suspended sediment concentrations of sediment fractions that have been designated as able to adsorb FRP, \(K_{ads-Q}^{FRP}\) is the ratio of adsorption and desorption rate coefficients and \(Q^{FRP}_{max}\) is the maximum adsorption capacity of suspended sediment, for FRP.

Figure G.4 provides an indicative representation of the general form of the quadratic relationship, again for assumed total FRP (0.1 mg/L) and SS (5mg/L) concentrations.

The parameters used to prepare the above figures were deliberately chosen to illustrate the different behaviour that can result from these two models. This underscores the need to make careful decisions regarding the selection of an adsorption model.

## G.5 Settling

At each timestep and within each active computational cell, the WQ Module computes the mass of adsorbed FRP that has settled either through the water column or onto the bed, using the settling velocity provided by the user. The limitations of this approach are recognised and future releases of the WQ Module will link this process directly to the settling and resuspension calculations of the TUFLOW FV ST Module.