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.

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 20$\degree$C. The respiration rate is adjusted to the local water temperature using the relationship:

$\rho_a=\rho_{a,20}*1.047^{(T_{water}-20)}$ 7:3.1.17

where $\rho_a$ is the local respiration rate of algae (day$^{-1}$ or hr$^{-1}$), $\rho_{a,20}$ is the local algal respiration 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).

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 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:

$\mu_{a,20}=\mu_{max}*FL*FN*FP$ 7:3.1.3

where $\mu_{a,20}$ is the local specific algal growth rate at 20°C (day$^{-1}$ or hr$^{-1}$), $\mu_{max}$ is the maximum specific algal growth rate (day$^{-1}$ or hr$^{-1}$), $FL$ 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:

7:3.1.7

where is the photosynthetically-active light intensity at a depth below the water surface (MJ/m-hr), is the photosynthetically-active solar radiation reaching the ground/water surface during a specific hour on a given day (MJ/m-hr), is the light extinction coefficient (m), and is the depth from the water surface (m). Substituting equation 7:3.1.7 into equation 7:3.1.6 and integrating over the depth of flow gives:

7:3.1.8

where is the algal growth attenuation factor for light for the water column, is the half-saturation coefficient for light (MJ/m-hr), is the photosynthetically-active solar radiation reaching the ground/water surface during a specific hour on a given day (MJ/m-hr), is the light extinction coefficient (m), and is the depth of water in the channel (m). Equation 7:3.1.8 is used to calculated for hourly routing. The photosynthetically-active solar radiation is calculated:

7:3.1.9

where is the solar radiation reaching the ground during a specific hour on current day of simulation (MJ m h), and is the fraction of solar radiation that is photosynthetically active. The calculation of is reviewed in Chapter 1:1. The fraction of solar radiation that is photosynthetically active is user defined.

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:

7:3.1.10

where is the fraction of daylight hours, is the daylight average photosynthetically-active light intensity (MJ/m-hr) and all other variables are defined previously. The fraction of daylight hours is calculated:

7:3.1.11

where is the daylength (hr). is calculated:

7:3.1.12

where is the fraction of solar radiation that is photosynthetically active, is the solar radiation reaching the water surface in a given day (MJ/m), and is the daylength (hr). Calculation of and are reviewed in Chapter 1:1.

The light extinction coefficient, , is calculated as a function of the algal density using the nonlinear equation:

7:3.1.13

where is the non-algal portion of the light extinction coefficient (), is the linear algal self shading coefficient (, is the nonlinear algal self shading coefficient , is the ratio of chlorophyll to algal biomass ( chla/mg alg), and is the algal biomass concentration (mg alg/L).

Equation 7:3.1.13 allows a variety of algal, self-shading, light extinction relationships to be modeled. When , no algal self-shading is simulated. When and , linear algal self-shading is modeled. When and are set to a value other than 0, non-linear algal self-shading is modeled. The Riley equation (Bowie et al., 1985) defines and .

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.

7:3.1.14

where is the algal growth limitation factor for nitrogen, is the concentration of nitrate in the reach (mg N/L), is the concentration of ammonium in the reach (mg N/L), and is the Michaelis-Menton half-saturation constant for nitrogen (mg N/L).

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

7:3.1.15

where is the algal growth limitation factor for phosphorus, is the concentration of phosphorus in solution in the reach (mg P/L), and is the Michaelis-Menton half-saturation constant for phosphorus (mg P/L).

The Michaelis-Menton half-saturation constant for nitrogen and phosphorus define the concentration of N or P at which algal growth is limited to 50% of the maximum growth rate. Users are allowed to set these values. Typical values for range from 0.01 to 0.30 mg N/L while will range from 0.001 to 0.05 mg P/L.

Once the algal growth rate at 20C is calculated, the rate coefficient is adjusted for temperature effects using a Streeter-Phelps type formulation:

7:3.1.16

where is the local specific growth rate of algae (day or hr), is the local specific algal growth rate at 20C (day or 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:

$\Delta orgP_{str}=(\alpha_2*\rho_a*algae-\beta_{P,4}*orgP_{str}-\sigma_5*orgP_{str})*TT$ 7:3.3.1

where $\Delta orgP_{str}$ is the change in organic phosphorus concentration (mg P/L), $\alpha_2$ is the fraction of algal biomass that is phosphorus (mg P/mg alg biomass), $\rho_a$ is the local respiration or death rate of algae (day$^{-1}$ or hr$^{-1}$), $algae$ is the algal biomass concentration at the beginning of the day (mg alg/L), $\beta_{P,4}$ is the rate constant for mineralization of organic phosphorus (day$^{-1}$ or hr$^{-1}$), $orgP_{str}$ is the organic phosphorus concentration at the beginning of the day (mg P/L), $\alpha_5$ is the rate coefficient for organic phosphorus settling (day$^{-1}$ or hr$^{-1}$), 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 amount of ammonium (NH) in the stream may be increased by the mineralization of organic nitrogen and diffusion of ammonium from the streambed sediments. The ammonium concentration in the stream may be decreased by the conversion of NH to NO or the uptake of NH by algae. The change in ammonium for a given day is:

7:3.2.4

where is the change in ammonium concentration (mg N/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 constant for biological oxidation of ammonia nitrogen (day

The amount of oxygen that can be dissolved in water is a function of temperature, concentration of dissolved solids, and atmospheric pressure. An equation developed by APHA (1985) is used to calculate the saturation concentration of dissolved oxygen:

$Ox_{sat}=exp[-139.34410+\frac{1.575701*10^5}{T_{wat,K}}-\frac{6.642308*10^7}{T_{wat,K}^2}+\frac{1.243800*10^{10}}{T_{wat,K}^3}-\frac{8.621949*10^{11}}{T_{wat,K}^4}]$ 7:3.5.3

where $Ox_{sat}$ is the equilibrium saturation oxygen concentration at 1.00 atm (mg O$_2$/L), and $T_{wat,K}$ is the water temperature in Kelvin (273.15+$\degree$C).

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 20C. The CBOD deoxygenation rate is adjusted to the local water temperature using the relationship:

7:3.4.2

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

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

7:3.4.3

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

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

The amount of soluble, inorganic phosphorus in the stream may be increased by the mineralization of organic phosphorus and diffusion of inorganic phosphorus from the streambed sediments. The soluble phosphorus concentration in the stream may be decreased by the uptake of inorganic P by algae. The change in soluble phosphorus for a given day is:

7:3.3.4

where is the change in solution phosphorus concentration (mg P/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 benthos (sediment) source rate for soluble P (mg P/m-day or mg P/m-hr), is the depth of water in the channel (m), is the fraction of algal biomass that is phosphorus (mg P/mg alg biomass), is the local growth rate of algae (day or hr), is the algal biomass concentration at the beginning of the day (mg alg/L), and

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.

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.

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

$\kappa_2=\kappa_{2,20}*1.024^{(T_{water}-20)}$ 7:3.5.4

where $\kappa_2$ is the reaeration rate (day$^{-1}$ or hr$^{-1}$), $\kappa_{2,20}$ is the reaeration 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).

Numerous methods have been developed to calculate the reaeration rate at 20$\degree$C, $\kappa_{2,20}$. 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).

Reareation will occur when water falls over a dam, weir, or other structure in the stream. To simulate this form of reaeration, a “structure” command line is added in the watershed configuration file (.fig) at every point along the stream where flow over a structure occurs.

The amount of reaeration that occurs is a function of the oxygen deficit above the structure and a reaeration coefficient:

$\Delta Ox_{str}=D_a-D_b=D_a(1-\frac{1}{rea})$ 7:3.5.10

where $\Delta Ox_{str}$ is the change in dissolved oxygen concentration (mg O$_2$/L), $D_a$ is the oxygen deficit above the structure (mg O$_2$/L), $D_b$ is the oxygen deficit below the structure (mg O$_2$/L), and $rea$ is the reaeration coefficient.

The oxygen deficit above the structure, , is calculated:

7:3.5.11

where is the equilibrium saturation oxygen concentration (mg O/L), and is the dissolved oxygen concentration in the stream (mg O/L).

Butts and Evans (1983) documents the following relationship that can be used to estimate the reaeration coefficient:

7:3.5.12

where is the reaeration coefficient, is an empirical water quality factor, is an empirical dam aeration coefficient, is the height through which water falls (m), and is the average water temperature (C).

The empirical water quality factor is assigned a value based on the condition of the stream:

= 1.80 in clean water

= 1.60 in slightly polluted water

= 1.00 in moderately polluted water

= 0.65 in grossly polluted water

The empirical dam aeration coefficient is assigned a value based on the type of structure:

= 0.70 to 0.90 for flat broad crested weir

= 1.05 for sharp crested weir with straight slope face

= 0.80 for sharp crested weir with vertical face

= 0.05 for sluice gates with submerged discharge

Table 7:3-5: SWAT+ input variables used in in-stream oxygen calculations.

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

7:3.2.10

where is the change in nitrate concentration (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), is the fraction of algal nitrogen uptake from ammonium pool, is the fraction of algal biomass that is nitrogen (mg N/mg alg biomass), is the local growth rate of algae (day or hr), is the algal biomass concentration at the beginning of the day (mg alg/L), and 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.