During the day, algae increase the stream’s dissolved oxygen concentration via photosynthesis. At night, algae reduce the concentration via respiration. As algae grow and die, they form part of the in-stream nutrient cycle. This section summarizes the equations used to simulate algal growth in the stream.

Parameters which affect water quality and can be considered pollution indicators include nutrients, total solids, biological oxygen demand, nitrates, and microorganisms (Loehr, 1970; Paine, 1973). Parameters of secondary importance include odor, taste, and turbidity (Azevedo and Stout, 1974).

The SWAT+ in-stream water quality algorithms incorporate constituent interactions and relationships used in the QUAL2E model (Brown and Barnwell, 1987). The documentation provided in this chapter has been taken from Brown and Barnwell (1987). The modeling of in-stream nutrient transformations has been made an optional feature of SWAT+. To route nutrient loadings downstream without simulating transformations, the variable IWQ in the basin input (.bsn) file should be set to 0. To activate the simulation of in-stream nutrient transformations, this variable should be set to 1.

In aerobic water, there is a stepwise transformation from organic nitrogen to ammonia, to nitrite, and finally to nitrate. Organic nitrogen may also be removed from the stream by settling. This section summarizes the equations used to simulate the nitrogen cycle in the stream.

The phosphorus cycle is similar to the nitrogen cycle. The death of algae transforms algal phosphorus into organic phosphorus. Organic phosphorus is mineralized to soluble phosphorus which is available for uptake by algae. Organic phosphorus may also be removed from the stream by settling. This section summarizes the equations used to simulate the phosphorus cycle in the stream.

Chlorophyll $a$ is assumed to be directly proportional to the concentration of phytoplanktonic algal biomass.

$chla=\alpha_0*algae$ 7:3.1.1

where $chla$ is the chlorophyll a concentration (μg chla/L), $\alpha_0$ is the ratio of chlorophyll $a$ to algal biomass (μg chla/mg alg), and $algae$ is the algal biomass concentration (mg alg/L).

Growth and decay of algae/chlorophyll $a$ is calculated as a function of the growth rate, the respiration rate, the settling rate and the amount of algae present in the stream. The change in algal biomass for a given day is:

$\Delta algae=((\mu _a*algae)-(\rho_a*algae)-(\frac{\sigma_1}{depth}*algae))*TT$ 7:3.1.2

where $\Delta algae$ is the change in algal biomass concentration (mg alg/L), $\mu_a$ is the local specific growth rate of algae (day$^{-1}$ or hr$^{-1}$), $\rho_a$ is the local respiration or death rate of algae (day$^{-1}$ or hr$^{-1}$), $\sigma_1$ is the local settling rate for algae (m/day or m/hr), $depth$ is the depth of water in the channel (m), $algae$ is the algal biomass concentration at the beginning of the day (mg alg/L), and $TT$ is the flow travel time in the reach segment (day or hr). The calculation of depth and travel time are reviewed in Chapter 7:1.

The amount of organic nitrogen in the stream may be increased by the conversion of algal biomass nitrogen to organic nitrogen. Organic nitrogen concentration in the stream may be decreased by the conversion of organic nitrogen to NH or the settling of organic nitrogen with sediment. The change in organic nitrogen for a given day is:

7:3.2.1

where is the change in organic nitrogen concentration (mg N/L), is the fraction of algal biomass that is nitrogen (mg N/mg alg biomass), is the local respiration or death rate of algae (day or hr), is the algal biomass concentration at the beginning of the day (mg alg/L), is the rate constant for hydrolysis of organic nitrogen to ammonia nitrogen (day or hr), is the organic nitrogen concentration at the beginning of the day (mg N/L), is the rate coefficient for organic nitrogen settling (day or hr), and is the flow travel time in the reach segment (day or hr). The fraction of algal biomass that is nitrogen is user-defined. Equation 7:3.1.17 describes the calculation of the local respiration rate of algae. The calculation of travel time is reviewed in Chapter 7:1.

The user defines the local rate constant for hydrolysis of organic nitrogen to NH at 20C. The organic nitrogen hydrolysis rate is adjusted to the local water temperature using the relationship:

7:3.2.2

where is the local rate constant for hydrolysis of organic nitrogen to NH (day or hr), is the local rate constant for hydrolysis of organic nitrogen to NH at 20C (day or hr), and is the average water temperature for the day or hour (C).

The user defines the rate coefficient for organic nitrogen settling at 20C. The organic nitrogen settling rate is adjusted to the local water temperature using the relationship:

7:3.2.3

where is the local settling rate for organic nitrogen (day or hr), is the local settling rate for organic nitrogen at 20C (day or hr), and is the average water temperature for the day or hour (C).

Reaeration occurs by diffusion of oxygen from the atmosphere into the stream and by the mixing of water and air that occurs during turbulent flow.

The local settling rate of algae represents the net removal of algae due to settling. The user defines the local settling rate of algae at 20$\degree$C. The settling rate is adjusted to the local water temperature using the relationship:

$\sigma_1=\sigma_{1,20}*1.024^{(T_{water}-20)}$ 7:3.1.18

where $\sigma_1$ is the local settling rate of algae (m/day or m/hr), $\sigma_{1,20}$ is the local algal settling rate at 20$\degree$C (m/day or m/hr), and $T_{water}$ is the average water temperature for the day or hour ($\degree$C).

Table 7:3-1: SWAT+ input variables used in algae calculations.

The carbonaceous oxygen demand (CBOD) of the water is the amount of oxygen required to decompose the organic material in the water. CBOD is added to the stream with loadings from surface runoff or point sources. Within the stream, two processes are modeled that impact CBOD levels, both of which serve to reduce the carbonaceous biological oxygen demand as the water moves downstream. The change in CBOD within the stream on a given day is calculated:

$\Delta cbod=-(\kappa_1*cbod+ \kappa_3*cbod)*TT$ 7:3.4.1

where $\Delta cbod$ is the change in carbonaceous biological oxygen demand concentration (mg CBOD/L), $\kappa_1$ is the CBOD deoxygenation rate (day$^{-1}$ or hr$^{-1}$), $cbod$ is the carbonaceous biological oxygen demand concentration (mg CBOD/L), $\kappa_3$ is the settling loss rate of CBOD (day$^{-1}$ or hr$^{-1}$), and $TT$ is the flow travel time in the reach segment (day or hr). The calculation of travel time is reviewed in Chapter 7:1.

The user defines the carbonaceous deoxygenation rate at 20$\degree$C. The CBOD deoxygenation rate is adjusted to the local water temperature using the relationship:

$\kappa_1=\kappa_{1,20}*1.047^{(T_{water}-20)}$ 7:3.4.2

where $\kappa_1$ is the CBOD deoxygenation rate (day$^{-1}$ or hr$^{-1}$), $\kappa_{1,20}$ is the CBOD deoxygenation rate at 20$\degree$C (day$^{-1}$ or hr$^{-1}$), and $T_{water}$ is the average water temperature for the day or hour ($\degree$C).

The user defines the settling loss rate of CBOD at 20$\degree$C. The settling loss rate is adjusted to the local water temperature using the relationship:

$\kappa_3=\kappa_{3,20}*1.024^{(T_{water}-20)}$ 7:3.4.3

where $\kappa_3$ is the settling loss rate of CBOD (day$^{-1}$ or hr$^{-1}$), $\kappa_{3,20}$ is the settling loss rate of CBOD at 20$\degree$C (day$^{-1}$ or hr$^{-1}$), and $T_{water}$ is the average water temperature for the day or hour ($\degree$C).

Table 7:3-4: SWAT+ input variables used in in-stream CBOD calculations.

The amount of nitrate ($NO_3^-$) in the stream may be increased by the oxidation of $NO_2^-$. The nitrate concentration in the stream may be decreased by the uptake of $NO_3^-$ by algae. The change in nitrate for a given day is:

$\Delta NO3_{str}=(\beta_{N,2}*NO2_{str}-(1-fr_{NH4})*\alpha_1*\mu_a*algae)*TT$ 7:3.2.10

where $\Delta NO3_{str}$ is the change in nitrate concentration (mg N/L), $\beta_{N,2}$ is the rate constant for biological oxidation of nitrite to nitrate (day$^{-1}$ or hr$^{-1}$), $NO2_{str}$ is the nitrite concentration at the beginning of the day (mg N/L), $fr_{NH4}$ is the fraction of algal nitrogen uptake from ammonium pool, $\alpha_1$ is the fraction of algal biomass that is nitrogen (mg N/mg alg biomass), $\mu _a$ is the local growth rate of algae (day$^{-1}$ or hr$^{-1}$), $algae$ is the algal biomass concentration at the beginning of the day (mg alg/L), and $TT$ is the flow travel time in the reach segment (day or hr). The local rate constant for biological oxidation of nitrite to nitrate is calculated with equation 7:3.2.9 while the fraction of algal nitrogen uptake from ammonium pool is calculated with equation 7:3.2.7. Section 7:3.1.2.1 describes the calculation of the local growth rate of algae. The calculation of travel time is reviewed in Chapter 7:1.

Table 7:3-2: SWAT+ input variables used in in-stream nitrogen calculations.

The local specific growth rate of algae is a function of the availability of required nutrients, light and temperature. SWAT+ first calculates the growth rate at 20°C and then adjusts the growth rate for water temperature. The user has three options for calculating the impact of nutrients and light on growth: multiplicative, limiting nutrient, and harmonic mean.

The multiplicative option multiplies the growth factors for light, nitrogen and phosphorus together to determine their net effect on the local algal growth rate. This option has its biological basis in the mutiplicative effects of enzymatic processes involved in photosynthesis:

7:3.1.3

where is the local specific algal growth rate at 20°C (day or hr), is the maximum specific algal growth rate (day or hr), is the algal growth attenuation factor for light, is the algal growth limitation factor for nitrogen, and is the algal growth limitation factor for phosphorus. The maximum specific algal growth rate is specified by the user.

The limiting nutrient option calculates the local algal growth rate as limited by light and either nitrogen or phosphorus. The nutrient/light effects are multiplicative, but the nutrient/nutrient effects are alternate.

The algal growth rate is controlled by the nutrient with the smaller growth limitation factor. This approach mimics Liebig’s law of the minimum:

7:3.1.4

where is the local specific algal growth rate at 20°C (day or hr), is the maximum specific algal growth rate (day or hr), is the algal growth attenuation factor for light, is the algal growth limitation factor for nitrogen, and is the algal growth limitation factor for phosphorus. The maximum specific algal growth rate is specified by the user.

The harmonic mean is mathematically analogous to the total resistance of two resistors in parallel and can be considered a compromise between equations 7:3.1.3 and 7:3.1.4. The algal growth rate is controlled by a multiplicative relation between light and nutrients, while the nutrient/nutrient interactions are represented by a harmonic mean.

7:3.1.5

where is the local specific algal growth rate at 20°C (day or hr), is the maximum specific algal growth rate (day or hr), is the algal growth attenuation factor for light, is the algal growth limitation factor for nitrogen, and is the algal growth limitation factor for phosphorus. The maximum specific algal growth rate is specified by the user.

Calculation of the growth limiting factors for light, nitrogen and phosphorus are reviewed in the following sections.

Algal Growth Limiting Factor for Light.

A number of mathematical relationships between photosynthesis and light have been developed. All relationships show an increase in photosynthetic rate with increasing light intensity up to a maximum or saturation value. The algal growth limiting factor for light is calculated using a Monod half-saturation method. In this option, the algal growth limitation factor for light is defined by a Monod expression:

7:3.1.6

where is the algal growth attenuation factor for light at depth , is the photosynthetically-active light intensity at a depth below the water surface (MJ/m-hr), and is the half-saturation coefficient for light (MJ/m-hr). Photosynthetically-active light is radiation with a wavelength between 400 and 700 nm. The half-saturation coefficient for light is defined as the light intensity at which the algal growth rate is 50% of the maximum growth rate. The half-saturation coefficient for light is defined by the user.

Photosynthesis is assumed to occur throughout the depth of the water column. The variation in light intensity with depth is defined by Beer’s law:

For daily simulations, an average value of the algal growth attenuation factor for light calculated over the diurnal cycle must be used. This is calculated using a modified form of equation 7:3.1.8:

Algal Growth Limiting Factor for Nutrients

The algal growth limiting factor for nitrogen is defined by a Monod expression. Algae are assumed to use both ammonia and nitrate as a source of inorganic nitrogen.

The algal growth limiting factor for phosphorus is also defined by a Monod expression.

The amount of nitrite () in the stream will be increased by the conversion of to and decreased by the conversion of to . The conversion of to occurs more rapidly than the conversion of to , so the amount of nitrite present in the stream is usually very small. The change in nitrite for a given day is:

7:3.2.8

where is the change in nitrite concentration (mg N/L), is the rate constant for biological oxidation of ammonia nitrogen (day or hr), is the ammonium concentration at the beginning of the day (mg N/L), is the rate constant for biological oxidation of nitrite to nitrate (day or hr), is the nitrite concentration at the beginning of the day (mg N/L), and is the flow travel time in the reach segment (day or hr). The local rate constant for biological oxidation of ammonia nitrogen is calculated with equation 7:3.2.5. The calculation of travel time is reviewed in Chapter 7:1.

The rate constant for biological oxidation of nitrite to nitrate will vary as a function of in-stream oxygen concentration and temperature. The rate constant is calculated:

7:3.2.9

where is the rate constant for biological oxidation of nitrite to nitrate (day or hr), is the rate constant for biological oxidation of nitrite to nitrate at 20C (day or hr), is the dissolved oxygen concentration in the stream (mg O/L), and is the average water temperature for the day or hour (C). The second term on the right side of equation 7:3.2.9, , is a nitrification inhibition correction factor. This factor inhibits nitrification at low dissolved oxygen concentrations.

An adequate dissolved oxygen concentration is a basic requirement for a healthy aquatic ecosystem. Dissolved oxygen concentrations in streams are a function of atmospheric reareation, photosynthesis, plant and animal respiration, benthic (sediment) demand, biochemical oxygen demand, nitrification, salinity, and temperature. The change in dissolved oxygen concentration on a given day is calculated:

7:3.5.1

where is the change in dissolved oxygen concentration (mg O/L), is the reaeration rate for Fickian diffusion (day or hr), is the saturation oxygen concentration (mg O/L), is the dissolved oxygen concentration in the stream (mg O/L), is the rate of oxygen production per unit of algal photosynthesis (mg O/mg alg), is the local specific growth rate of algae (day or hr), is the rate of oxygen uptake per unit of algae respired (mg O/mg alg), is the local respiration or death rate of algae (day or hr), is the algal biomass concentration at the beginning of the day (mg alg/L), is the CBOD deoxygenation rate (day or hr), is the carbonaceous biological oxygen demand concentration (mg CBOD/L), is the sediment oxygen demand rate (mg O/(m.day) or mg O/(m.hr)), is the depth of water in the channel (m), is the rate of oxygen uptake per unit NH oxidation (mg O/mg N), is the rate constant for biological oxidation of ammonia nitrogen (day or hr), is the ammonium concentration at the beginning of the day (mg N/L), is the rate of oxygen uptake per unit oxidation (mg O/mg N), is the rate constant for biological oxidation of nitrite to nitrate (day or hr), is the nitrite concentration at the beginning of the day (mg N/L) and is the flow travel time in the reach segment (day or hr). The user defines the rate of oxygen production per unit algal photosynthesis, the rate of oxygen uptake per unit algal respiration, the rate of oxygen uptake per unit NH oxidation and rate of oxygen uptake per unit oxidation. Section 7:3.1.2.1 describes the calculation of the local growth rate of algae while equation 7:3.1.17 describes the calculation of the local respiration rate of algae. The rate constant for biological oxidation of NH is calculated with equation 7:3.2.5 while the rate constant for oxidation is determined with equation 7:3.2.9. The CBOD deoxygenation rate is calculated using equation 7:3.4.2. The calculation of depth and travel time are reviewed in Chapter 7:1.

The user defines the sediment oxygen demand rate at 20C. The sediment oxygen demand rate is adjusted to the local water temperature using the relationship:

7:3.5.2

where is the sediment oxygen demand rate (mg O/(m.day) or mg O/(m.hr)), is the sediment oxygen demand rate at 20C (mg O/(m.day) or mg O/(m.hr)), and is the average water temperature for the day or hour (C).

The amount of organic phosphorus in the stream may be increased by the conversion of algal biomass phosphorus to organic phosphorus. Organic phosphorus concentration in the stream may be decreased by the conversion of organic phosphorus to soluble inorganic phosphorus or the settling of organic phosphorus with sediment. The change in organic phosphorus for a given day is:

7:3.3.1

where is the change in organic phosphorus concentration (mg P/L), is the fraction of algal biomass that is phosphorus (mg P/mg alg biomass), is the local respiration or death rate of algae (day or hr), is the algal biomass concentration at the beginning of the day (mg alg/L), is the rate constant for mineralization of organic phosphorus (day or hr), is the organic phosphorus concentration at the beginning of the day (mg P/L), is the rate coefficient for organic phosphorus settling (day or hr), and is the flow travel time in the reach segment (day or hr). The fraction of algal biomass that is phosphorus is user-defined. Equation 7:3.1.17 describes the calculation of the local respiration rate of algae. The calculation of travel time is reviewed in Chapter 7:1.

The user defines the local rate constant for mineralization of organic phosphorus at 20C. The organic phosphorus mineralization rate is adjusted to the local water temperature using the relationship:

7:3.3.2

where is the local rate constant for organic phosphorus mineralization (day or hr), is the local rate constant for organic phosphorus mineralization at 20C (day or hr), and is the average water temperature for the day or hour (C).

The user defines the rate coefficient for organic phosphorus settling at 20C. The organic phosphorus settling rate is adjusted to the local water temperature using the relationship:

7:3.3.3

where is the local settling rate for organic phosphorus (day or hr), is the local settling rate for organic phosphorus at 20C (day or hr), and is the average water temperature for the day or hour (C).

The user defines the reaeration rate at 20C. The reaeration rate is adjusted to the local water temperature using the relationship:

7:3.5.4

where is the reaeration rate (day or hr), is the reaeration rate at 20C (day or hr), and is the average water temperature for the day or hour (C).

Numerous methods have been developed to calculate the reaeration rate at 20C, . A few of the methods are listed below. Brown and Barnwell (1987) provide additional methods.

Using field measurements, Churchill, Elmore and Buckingham (1962) derived the relationship:

7:3.5.5

where is the reaeration rate at 20C (day), is the average stream velocity (m/s), and is the average stream depth (m).

O’Connor and Dobbins (1958) incorporated stream turbulence characteristics into the equations they developed. For streams with low velocities and isotropic conditions,

7:3.5.6

where is the reaeration rate at 20C (day), is the molecular diffusion coefficient (m/day), is the average stream velocity (m/s), and is the average stream depth (m). For streams with high velocities and nonisotropic conditions,

7:3.5.7

where is the reaeration rate at 20C (day), is the molecular diffusion coefficient (m/day), is the slope of the streambed (m/m), and is the average stream depth (m). The molecular diffusion coefficient is calculated

7:3.5.8

where is the molecular diffusion coefficient (m/day), and is the average water temperature (C).

Owens et al. (1964) developed an equation to determine the reaeration rate for shallow, fast moving streams where the stream depth is 0.1 to 3.4 m and the velocity is 0.03 to 1.5 m/s.

7:3.5.9

where is the reaeration rate at 20C (day), is the average stream velocity (m/s), and is the average stream depth (m).

The local respiration or death rate of algae represents the net effect of three processes: the endogenous respiration of algae, the conversion of algal phosphorus to organic phosphorus, and the conversion of algal nitrogen to organic nitrogen. The user defines the local respiration rate of algae at 20C. The respiration rate is adjusted to the local water temperature using the relationship:

7:3.1.17

where is the local respiration rate of algae (day or hr), is the local algal respiration rate at 20C (day or hr), and is the average water temperature for the day or hour (C).