Appendix F Processes: inorganic nitrogen

F.1 Sediment ammonium and nitrate flux

Ammonium and nitrate are exchanged between the water column and sediments via specification of separate sediment fluxes. In both cases, 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 these nitrogenous species, the WQ Module can be parameterised to allow for this if required.

The user specified rates of ammonium and nitrate fluxes (which can be spatially varying) are modified by overlying ambient dissolved oxygen concentration (together with respective user specified half saturation oxygen concentrations) and water temperature. These modifications are simulated via Michaelis-Menten and Arrhenius models, respectively, as per Equation (F.1). \[\begin{equation} \left.\begin{aligned} \href{AppDiags.html#WQDiagAmmSedFlx}{F_{sed\langle computed\rangle}^{NH_4}} =& F_{sed}^{NH_4} \times \frac{K_{sed-O_2}^{NH_4}}{K_{sed-O_2}^{NH_4} + \left[DO\right]} \times \hphantom{\text{ab}} \left[\theta_{sed}^{NH_4}\right]^{\left(T-20\right)} \\ \\ \href{AppDiags.html#WQDiagNitSedFlx}{F_{sed\langle computed\rangle}^{NO_3}} =& F_{sed}^{NO_3} \times \underbrace{\frac{\left[DO\right]}{K_{sed-O_2}^{NO_3} + \left[DO\right]}}_{\text{Influence of oxygen}} \times \underbrace{\vphantom{\frac{\left[DO\right]}{K_{sed-O_2}^{NO_3} + \left[DO\right]}} \left[\theta_{sed}^{NO_3}\right]^{\left(T-20\right)}}_{\text{Influence of temperature}} \end{aligned}\right\} \tag{F.1} \end{equation}\] \(F_{sed}^{NH_4}\) and \(F_{sed}^{NO_3}\) are the user specified ammonium and nitrate sediment fluxes at 20\(^o\)C without the influence of dissolved oxygen, \(\left[DO\right]\) is the overlying dissolved oxygen concentration, \(K_{sed-O_2}^{NH_4}\) and \(K_{sed-O_2}^{NO_3}\) are the user specified half saturation concentrations of dissolved oxygen for ammonium and nitrate sediment flux, \(\theta_{sed}^{NH_4}\) and \(\theta_{sed}^{NO_3}\) are the corresponding temperature coefficients, and \(T\) is ambient water temperature.

As per silicate sediment flux (see Section E.1), the above equations lead to linearly varying ambient ammonium and nitrate concentrations in the demonstration model when dissolved oxygen concentration, half saturation oxygen concentrations and ambient temperature are set to be constant.

A more realistic environmental setting has sediment ammonium and nitrate fluxes 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 (in the dissolved oxygen constituent model, with \(K_{sed-O_2}^{O_2}\) = 4 mg/L) and each simulation using a the same value for both \(K_{sed-O_2}^{NH_4}\) and \(K_{sed-O_2}^{NO_3}\). The ammonium and nitrate sediment flux rates were both specified as 140 mg/m\(^2\)/d, and temperature effects were turned off. The predicted temporal evolution of water column ammonium and nitrate concentrations is provided in Figure F.1. Use the “play” button or drag the slider to see how different half saturation concentrations change ambient concentrations (ordinate) in time (abscissa).

Figure F.1: Move the slider to see the effect of changing the \(K_{sed-O_2}^{NH_4}\) (and \(K_{sed-O_2}^{NO_3}\) - they are set to be equal to each other in this example) values on ambient ammonium and nitrate concentrations. Ambient dissolved oxygen is drawn down in time

Figure F.1 illustrates the different oxygen responses of ammonium and nitrate sediment fluxes, as reflected in the different formulations within Equation (F.1). Importantly, the figure demonstrates that as ambient oxygen concentrations reach zero near 60 hours (see Figure D.2 with the slider bar set to \(K_{sed-O_2}^{O_2}\) = 4 mg/L), nitrate sediment flux ceases and the resultant ambient concentrations plateau. The converse is true to ammonium. One implication of this behaviour is that in environmental systems being simulated using the WQ Module, the rate of delivery of ammonium mass to the water column will generally increase under low dissolved oxygen conditions (e.g. waters isolated underneath a strong lacustrine a thermocline), whilst the corresponding nitrate mass fluxes will reduce under the same conditions. This is the expected and observed behaviour of environmental systems.

The rate of ammonium and nitrate flux is also related to ambient water temperature, via the Arrhenius model in Equation (F.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 for both ammonium and nitrate set to 4 mg/L. All temperature coefficients were set to 1.05. The results are provided in Figure F.2. Use the “play” button or drag the slider to see how different ambient temperatures change the ambient ammonium and nitrate concentrations (ordinate) in time (abscissa).

Figure F.2: Move the slider to see the effect of changing the ambient water temperature on ammonium and nitrate concentrations, with ambient dissolved oxygen drawdown

The figure shows that as ambient water temperature increases, the rate of increases of both ammonium and nitrate concentrations follow suit. Again, nitrate concentrations plateau once background dissolved oxygen concentrations reach zero, however ammonium concentrations continue to rise throughout the simulation. This is expected behaviour.

F.2 Nitrification

Nitrification is the pelagic biological oxidation of ammonium to nitrate. The equations representing this process (including intermediates) are as follows.

\[\begin{equation} \left.\begin{aligned} 2NH_4^{+} + 3O_2 \rightarrow & 2NO_2^{-} + 4H^+ + 2H_2O \\ \\ 2NO_2^{-} + O_2 \rightarrow & 2NO_3^- \end{aligned}\right\} \tag{F.2} \end{equation}\]

Dissolved oxygen is consumed by nitrification of ammonium, in a molar ratio of 1:2 N:O\(_2\) (i.e. 2 moles of ammonium consume 4 moles of diatomic oxygen). Nitrite is an intermediate product of nitrification, and simulation of this quantity is possible within the WQ Module although not currently activated. Contact support@tuflow.com if this capability is required.

The pelagic nitrification rate is computed within the WQ Module via Equation (F.3).

\[\begin{equation} R_{nitrif\langle computed\rangle}^{NH_4} = R_{nitrif}^{NH_4} \times \underbrace{\frac{\left[ DO \right]}{K_{nitrif-O_2}^{NH_4} + \left[DO\right]}}_{\text{Influence of oxygen}} \times \underbrace{ \vphantom{\frac{\left[ DO \right]}{K_{nitrif-O_2}^{NH_4} + \left[DO\right]}} \left[\theta_{nitrif}^{NH_4}\right]^{\left(T-20\right)}}_{\text{Influence of temperature}} \tag{F.3} \end{equation}\]

\(R_{nitrif}^{NH_4}\) is the user specified ammonium nitrifcation rate at 20\(^o\)C without the influence of dissolved oxygen, \(\left[DO\right]\) is the ambient dissolved oxygen concentration, \(K_{nitrif-O_2}^{NH_4}\) and \(\theta_{nitrif}^{NH_4}\) are the half saturation concentration of dissolved oxygen and temperature coefficient for nitrification of ammonium respectively (both are user specified), and \(T\) is ambient water temperature.

The nitrification rate from Equation (F.3) is multiplied by ambient ammonium concentration to compute the flux of ammonium to nitrate at each model timestep in each model cell via Equation (F.4).

\[\begin{equation} \href{AppDiags.html#WQDiagNitrif}{F_{nitrif\langle computed\rangle}^{NH_4}} = R_{nitrif\langle computed\rangle}^{NH_4} \times \left[ NH_4 \right] \tag{F.4} \end{equation}\]

Equation (F.3) reveals a linear relationship between \(R_{nitrif\langle computed\rangle}^{NH_4}\) and both (constant) dissolved oxygen concentration and half saturation oxygen concentration. In dissolved oxygen conditions away from oxygen sinks other than nitrification itself (such as sediments), this leaves ambient temperature as the key external determinant of the computed nitrification rates.

A suite of demonstration model simulations were executed to illustrate this, with each simulation having its own temperature, and a constant nitrification rate \(R_{nitrif}^{NH_4}\) of 0.1/day. Ammonium and nitrate initial conditions were always set to 4 and 0 mg/L respectively, \(K_{nitrif-O_2}^{NH_4}\) to 4 mg/L and \(\theta_{nitrif}^{NH_4}\) was set to 1.05. In order to illustrate only the workings of nitrification, all sediment fluxes were set to zero. The predicted temporal evolution of water column ammonium and nitrate concentrations in these simulations are provided in Figure F.3. Corresponding dissolved oxygen concentrations are also included to demonstrate the impact of nitrification on water column oxygen. Use the “play” button or drag the slider to see how different ambient temperatures change the ambient ammonium, nitrate and dissolved oxygen concentrations (ordinate) in time (abscissa).

Figure F.3: Move the slider to see the effect of changing ambient water temperature on the concentrations of ammonia, nitrate and oxygen

The figure demonstrates the expected behaviour, where ammonium is consumed and nitrate produced. The drawdown in oxygen presented in the figure is due only to nitrification.

The figure also reveals the mass conserving behaviour of the WQ Module: the decrease in ammonium (N) concentration is the same as the corresponding increase in nitrate (N) concentration for each of the different temperature simulations. This confirms the 1:1 ratio of nitrogen conversion from ammonium to nitrate in moles (Equation (F.2)), and hence concentration in an otherwise closed system.

The drawdown in oxygen also demonstrates the WQ Module mass balance. In the 12\(^o\)C case for example:

• The change in dissolved oxygen concentration is 4.37 mg/L
• Converting this to moles of O\(_2\) gives 0.1366 moles O\(_2\) (= 4.37/32)
• The change in ammonium (or nitrate) concentration is 0.955 mg/L
• Converting this to moles of N gives 0.0682 moles of N (= 0.955/14)

The number of moles of dissolved oxygen consumed (0.137) is twice that the number of moles of N transformed from ammonium to nitrate (0.0682), to two decimal places. This is consistent with Equation (F.2).

A full WQ Module mass balance is presented in Appendix S.

F.3 Denitrification

Denitrification is the pelagic biological reduction of nitrate to free nitrogen gas, \(N_2\). This typically occurs as a result of microbial oxygen stripping from nitrate in low dissolved oxygen environments. It occurs through a series of intermediate stages, but is typically represented as a single-step process with a governing denitrification rate. This rate is computed within the WQ Module via either Equation (F.5) or (F.6). These are identical other than their treatment of the influence of dissolved oxygen on denitrification, which is either Michaelis-Menten or exponential, respectively. Unlike nitrification, denitrification includes the influence of nitrate concentration in both rate calculations, again in a Michaelis-Menten form.

\[\begin{equation} R_{denit\langle computed\rangle}^{NO_3} = R_{denit}^{NO_3} \hphantom{\text{abc}} \times \hphantom{\text{abc}} \frac{K_{denit-O_2-MM}^{NO_3}}{K_{denit-O_2-MM}^{NO_3} + \left[DO\right]} \times \left[\theta_{denit}^{NO_3}\right]^{\left(T-20\right)} \times \frac{\left[NO_3\right]}{K_{denit-NO_3}^{NO_3} + \left[NO_3\right]} \tag{F.5} \end{equation}\]

\[\begin{equation} R_{denit\langle computed\rangle}^{NO_3} = R_{denit}^{NO_3} \hphantom{\text{abc}} \times \hphantom{\text{abc}} \underbrace{\vphantom{\frac{\left[NO_3\right]}{K_{denit-NO_3}^{NO_3} + \left[NO_3\right]}}{\rm e}^{\left[\frac{-\left[DO\right]}{K_{denit-O_2-exp}^{NO_3}}\right]}}_{\text{Influence of oxygen}} \times \hphantom{\text{abcd}} \underbrace{\vphantom{\frac{\left[NO_3\right]}{K_{denit-NO_3}^{NO_3} + \left[NO_3\right]}}\left[\theta_{denit}^{NO_3}\right]^{\left(T-20\right)}}_{\text{Influence of temperature}} \times \underbrace{\frac{\left[NO_3\right]}{K_{denit-NO_3}^{NO_3} + \left[NO_3\right]}}_{\text{Influence of nitrate}} \tag{F.6} \end{equation}\]

\(R_{denit}^{NO_3}\) is the user specified nitrate denitrification rate at 20\(^o\)C and without the influence of dissolved oxygen, and \(\left[DO\right]\) and \(\left[NO_3\right]\) are the ambient dissolved oxygen and nitrate concentrations respectively. If the Michaelis-Menten oxygen model is used then \(K_{denit-O2-MM}^{NO_3}\) is the half saturation concentration of oxygen for denitrification. If the exponential model is used then \(K_{denit-O2-exp}^{NO_3}\) is the dissolved oxygen concentration that normalises (i.e. non-dimensionalises) the ambient oxygen concentration used in Equation (F.6). Equation (F.5) is the default. Finally, \(\theta_{denit}^{NO_3}\) is the temperature coefficient for denitrification of nitrate, \(T\) is ambient water temperature, and \(K_{denit-NO_3}^{NO_3}\) is the Michaelis-Menten half saturation nitrate concentration that influences denitrification in both Equations (F.5) and (F.6). \(K_{denit-NO_3}^{NO_3}\) is not user definable, but hardwired to a value of 0.07 mg/L of nitrate (5.0 mmol/m\(^3\)).

The computed denitrification rate from Equation (F.5) or (F.6) is then multiplied by ambient nitrate concentration to compute the flux of nitrate to free nitrogen gas at each model timestep in each model cell via Equation (F.7).

\[\begin{equation} \href{AppDiags.html#WQDiagDenitrif}{F_{denit\langle computed\rangle}^{NO_3}} = R_{denit\langle computed\rangle}^{NO_3} \times \left[ NO_3 \right] \tag{F.7} \end{equation}\]

Unlike nitrification, denitrification does not produce or consume dissolved oxygen. It does, however, consume nitrate and therefore presents a feedback loop in Equations (F.5) through (F.7).

The demonstration model was executed over two suites of identical simulations that varied temperature from one simulation to the next, with the first suite using Equation (F.5) and the second suite using Equation (F.6). Only denitrification was turned on, and constant values for \(R_{denit}^{NO_3}\) (1.5/day - a deliberately very large value was chosen to demonstrate trends), \(K_{denit-O_2}^{NO_3}\) (4 mg/L) and \(K_{denit-NO_3}^{NO_3}\) (15.5 mg/L) were set. Temperature coefficients were set to 1.05. The predicted temporal evolution of water column nitrate concentrations in these simulation pairs are provided in Figure F.4.

Figure F.4: Move the slider to see the effect of changing temperature on the concentrations of nitrate, where two different denitrification models are used

The figure demonstrates the expected behaviour, where nitrate is consumed in a non-linear manner. The different models produce different drawdown rates, also as expected. In this example, these differences are substantial.

The same two suites of demonstration models were also used to illustrate the influence of varying \(K_{denit-O_2}^{NO_3}\) (instead of temperature) on denitrification. All set ups from the models used above were retained, other than setting ambient water temperature to constant at 20 \(^o\)C, and allowing \(K_{denit-O_2}^{NO_3}\) to vary. The predicted temporal evolution of water column nitrate concentrations in these demonstration model simulation pairs are provided in Figure F.5.

Figure F.5: Move the slider to see the effect of changing \(K_{denit-O_2}^{NO_3}\) on the concentrations of nitrate, where two different denitrification models are used

Substantial differences in the prediction of nitrate concentrations is again observed, especially at lower half saturation concentrations.

F.4 Anaerobic oxidation of ammonium

Anaerobic oxidation of ammonium (also referred to as anammox) is the microbial conversion of ammonium and nitrite to free nitrogen gas in low dissolved oxygen environments. Oxygen for this process is stripped from nitrite. The equation representing this process is as follows.

\[\begin{equation} NH_4^{+} + NO_2^{-} \rightarrow N_2 + 2H_2O \tag{F.8} \end{equation}\] \(NH_4^{+}\) and \(NO_2^{-}\) are ammonium and nitrite, respectively, and N\(_2\) is free nitrogen gas. In the absence of simulating nitrite, the WQ Module assumes that nitrite and nitrate are related (only at dissolved oxygen concentrations below 0.1 mg/L) as follows.

\[\begin{equation} \left[NO_2\right] = \left[NO_3\right] \times \left( 1.0 - \frac{\left[ DO \right]}{0.1 + \left[ DO \right]}\right) \tag{F.9} \end{equation}\] \(\left[NO_2\right]\), \(\left[NO_3\right]\) and \(\left[DO\right]\) are the ambient nitrite, ammonium and dissolved oxygen concentrations, respectively. The anammox flux to free nitrogen gas N at each model timestep in each model cell that has an ambient dissolved oxygen concentration less than 0.1 mg/L is computed as per Equation (F.10).

\[\begin{equation} \href{AppDiags.html#WQDiagAnammox}{F_{anmx\langle computed\rangle}^{N_2}} = k_{anmx}^{N_2} \times \frac{\left[NH_4\right]}{K_{anmx-NH_4}^{N_2} + \left[NH_4\right]} \times \frac{\left[NO_2\right]}{K_{anmx-NO_2}^{N_2} + \left[NO_2\right]} \tag{F.10} \end{equation}\] \(k_{anmx}^{N_2}\) is the user specified anaerobic volumetric mass oxidation rate without the influence of ammonium or nitrite, \(K_{anmx-NH_4}^{N_2}\) and \(K_{anmx-NO_2}^{N_2}\) are the Michaelis-Menten half saturation ammonium and nitrite concentrations for anammox respectively, and \(\left[NH_4\right]\) and \(\left[NO_2\right]\) are the ambient ammonium and nitrite concentrations, respectively. For ambient conditions that have dissolved oxygen concentrations greater than 0.1 mg/L, the anammox rate is set to zero.

F.5 Dissimilatory reduction of nitrate to ammonium

Dissimilatory reduction of nitrate to ammonium (also referred to as DRNA) is the microbial conversion of nitrate to ammonium. Dissolved oxygen is not consumed in this process. The WQ Module assumes the number of moles of nitrate-N consumed is the same as the number of ammonium-N moles produced. The DRNA rate is computed within the WQ Module via Equation (F.11).

\[\begin{equation} R_{DRNA\langle computed\rangle}^{NO_3} = R_{DRNA}^{NO_3} \times \frac{K_{DRNA-O_2}^{NO_3}}{K_{DRNA-O_2}^{NO_3} + \left[DO\right]} \tag{F.11} \end{equation}\]

\(R_{DRNA}^{NO_3}\) is the user specified rate of dissimilatory reduction of nitrate to ammonium without the influence of oxygen, \(K_{DRNA-O_2}^{NO_3}\) is the Michaelis-Menten half saturation dissolved oxygen concentration for DRNA, and \(\left[DO\right]\) is the ambient dissolved oxygen concentration.

The computed DRNA rate from Equation (F.11) is multiplied by ambient nitrate concentration to compute the flux to ammonium at each model timestep in each model cell as per Equation (F.12).

\[\begin{equation} \href{AppDiags.html#WQDiagDRNA}{F_{DRNA\langle computed\rangle}^{NO_3}} = R_{DRNA\langle computed\rangle}^{NO_3} \times \left[NO_3\right] \tag{F.12} \end{equation}\]

Although not recommended, the influence of oxygen in all the processes described in sections F.1 to F.5 can be removed by setting the command oxygen == off in the nitrogen model block. This switch applies to sediment flux, nitrification and denitrification. Anammox and DRNA will be switched off entirely if this command is issued.

F.6 Wet deposition of inorganic nitrogen

Wet deposition of inorganic nitrogen includes the pluvial delivery of both ammonium and nitrate to the water column. A global total inorganic nitrogen concentration in rainfall is specifiable, together with the proportion of that total that is nitrate. From this, the WQ Module computes ammonium and nitrate concentrations (Equations (F.13) and (F.14), respectively).

\[\begin{equation} \left[{NH_4}\right]_{rain} = \left[{TN}\right]_{rain} \times (1-f_{TN}^{NO_3}) \tag{F.13} \end{equation}\]

\[\begin{equation} \left[{NO_3}\right]_{rain} = \left[{TN}\right]_{rain} \times f_{TN}^{NO_3} \tag{F.14} \end{equation}\]

\(\left[{TN}\right]_{rain}\) and \(f_{TN}^{NO_3}\) are the concentration of total nitrogen in rain and the fraction of that total that is nitrate, respectively. The WQ Module separately multiplies \(\left[{NH_4}\right]_{rain}\) and \(\left[{NO_3}\right]_{rain}\) by each timestep’s rainfall depth and corresponding model surface area(s) to compute the wet mass flux of ammonium and nitrate (Equations (F.15) and (F.16), respectively). These masses are added to the uppermost model layer cells and corresponding changes to ambient concentrations computed.

\[\begin{equation} F_{atm-wet}^{NH_4} = \left[{NH_4}\right]_{rain} \times \text{Timestep rainfall depth} \times \text{Model area(s)} \tag{F.15} \end{equation}\]

\[\begin{equation} F_{atm-wet}^{NO_3} = \left[{NO_3}\right]_{rain} \times \text{Timestep rainfall depth} \times \text{Model area(s)} \tag{F.16} \end{equation}\]

If rainfall depth is specified as being spatially variable (i.e. by application of multiple atmospheric boundary conditions within TUFLOW FV), then \(F_{atm-wet}^{NH_4}\) and \(F_{atm-wet}^{NO_3}\) will correspondingly vary spatially.

F.7 Dry deposition of inorganic nitrogen

Dry deposition of nitrogen includes the delivery of both ammonium and nitrate to the water column from atmospheric fallout. This fallout is typically thought of as being associated with dust, with the dry fallout dissolving on entry to the water column to become dissolved inorganic nitrogen. A single constant daily deposition rate of total nitrogen is specifiable, and the WQ Module splits this into separate ammonia and nitrate rates using the same fraction applied to the wet deposition total inorganic nitrogen concentration (Section F.6).

\[\begin{equation} R^{NH_4}_{atm-dry} = R^{TN}_{atm-dry} \times (1-f_{TN}^{NO_3}) \tag{F.17} \end{equation}\]

\[\begin{equation} R^{NO_3}_{atm-dry} = R^{TN}_{atm-dry} \times f_{TN}^{NO_3} \tag{F.18} \end{equation}\]

\(R^{TN}_{atm-dry}\) and \(f_{TN}^{NO_3}\) are the user specified rate of dry total inorganic nitrogen deposition, and the fraction that is nitrate, respectively. These individual rates \(R^{NH_4}_{atm-dry}\) and \(R^{NO_3}_{atm-dry}\) are multiplied by model timestep and the global model surface area to compute the dry mass flux of ammonium and nitrate (Equations (F.19) and (F.20), respectively). These masses are added to the uppermost model layer cells and corresponding changes to ambient concentrations computed.

\[\begin{equation} F_{atm-dry}^{NH_4} = R^{NH_4}_{atm-dry} \times \text{Timestep} \times \text{Global model area} \tag{F.19} \end{equation}\]

\[\begin{equation} F_{atm-dry}^{NO_3} = R^{NO_3}_{atm-dry} \times \text{Timestep} \times \text{Global model area} \tag{F.20} \end{equation}\]

This constant dry mass flux for each of ammonium and nitrate is applied globally.

The total atmospheric depositional flux of dissolved inorganic nitrogen (noting that dry atmospheric flux is assumed to dissolve on entry to the water column) is as per Equation (F.21).

\[\begin{equation} \href{AppDiags.html#WQDiagDINAtmFlx}{F_{atm-din\langle computed \rangle}^{N}} = F_{atm-wet}^{NH_4} + F_{atm-wet}^{NO_3} + F_{atm-dry}^{NH_4} + F_{atm-dry}^{NO_3} \tag{F.21} \end{equation}\]