Abstract

Water temperature is an important indicator for biodiversity and ecosystem sustainability. In this study, a simplified equilibrium temperature model was incorporated into the CE-QUAL-W2 (W2) model. This model is easy to implement, needing fewer meteorological variables and no parameter calibration. The model performance was evaluated using observed data from four stations on the Lower Minnesota River. Results show that the simplified equilibrium temperature model performed as well as the original equilibrium temperature model and the term-by-term process model for water temperature predictions with the values of the coefficient of determination (R2), Nash–Sutcliffe Efficiency (NSE), and Percent Error (PE) in the accepted range (R2 = 0.974, NSE = 0.972, PE = 1.377%). The impact of the water temperature on carbonaceous biochemical oxygen demand (CBOD) concentrations under three different water temperature models was evaluated, and results show that the monthly averaged CBOD concentrations of the simplified equilibrium temperature model were almost the same as that of the term-by-term approach. For all the four calibration stations, the simplified equilibrium temperature approach performs better than the other two models for dissolved oxygen simulation (R2 = 0.791, NSE = 0.65, PE = 7.596%), which indicates that the simplified equilibrium temperature model can be a potential tool to simulate water temperature for water quality modelling.

INTRODUCTION

Water temperature is an important indicator to determine the overall health of aquatic ecosystems. It plays a significant role in many chemical and biological processes present in rivers. All aquatic species have a specific water temperature range that they can tolerate and dramatic changes in water temperature may have adverse impacts on the habitat of aquatic species (Caissie et al. 2007). For example, variability in water temperature affects the trait-mediated survival of a newly settled coral reef fish as warmer water leads to faster larval and juvenile growth and shorter pelagic larval durations (Grorud-Colvert & Sponaugle 2011). A fish species could perish due to osmoregulatory dysfunction if weekly stream temperature drops below a threshold temperature (Mohseni et al. 1998).

Many water temperature models have been developed over the past years, which can be categorized into deterministic models and statistical models (Caissie 2006). Deterministic models are generally based on heat budget calculations while statistical models use regression techniques to correlate water temperature with meteorological or other physical variables, such as air temperature, solar radiation and flow discharge. Statistical models have been used tremendously in water temperature predictions because of their relative simplicity and minimal data requirement, such as linear regression models (Krider et al. 2013), non-linear regression models (Mohseni et al. 1998), and stochastic models (Cho & Lee 2012). These statistical models are useful tools for water temperature predictions, however, several drawbacks exist, as was summarized in Benyahya et al. (2007). For example, linear regression models are less appropriate when the assumption of a linear relationship cannot be verified, and stochastic models are appropriate when residuals are stationary. Moreover, the impact of watershed hydrological conditions are not included in these statistical models (Ficklin et al. 2012). Therefore, the statistical models may not be reliable when interpreting and predicting the impact of environmental and anthropological drivers, such as climate and land use change.

Deterministic water temperature models simulate the spatial and temporal change of water temperature based on energy balances of heat fluxes and water mass balance in a water body (Hebert et al. 2011). Heat fluxes are calculated at the water–air and water–sediment interfaces, and heat exchange at the water–air interface is generally more influential than that at the water–sediment interface (Caissie et al. 2007). At the water–air interface, heat fluxes can be calculated based on a full energy balance approach, which includes incident and reflected short/longwave radiations, evaporative heat loss, and heat conduction. Various deterministic water temperature models, such as the QUAL2 K model (Chapra et al. 2012), LARSIM-WT model (Haag & Luce 2008), and CE-QUAL-W2 (W2) model (Cole & Wells 2011), have been widely applied in different water bodies.

The W2 model is a two-dimensional, laterally averaged, longitudinal/vertical, hydrodynamic and water quality model. This model has been continuously developed and maintained by the US Army Corps of Engineers (USACE) and Portland State University since the late 1980s (Martin 1988; Cole & Wells 2011). Since its first release in 1986, the W2 model has been successfully applied to hundreds of rivers, lakes, reservoirs and estuaries throughout the world for thermal and water quality investigations (Debele et al. 2008; Afshar et al. 2011; Noori et al. 2015; Brown et al. 2016). The latest version of the W2 model can simulate the complete eutrophication process in water bodies, including carbon, nitrogen and phosphorus cycles, dissolved oxygen (DO), and benthic sediment diagenesis modelling (Zhang et al. 2015). Additionally, a mercury simulation module was integrated into the model to simulate mercury cycling in water bodies (Zhu et al. 2017). Water temperature simulation is the basis for eutrophication and mercury modelling. Currently, the W2 model employs two approaches to compute water–air heat exchange: the term-by-term process method and equilibrium temperature approach (Cole & Wells 2011), which are quite complex. The primary objective of this paper is to incorporate a simplified equilibrium model into the W2 model for water temperature prediction. The model performance for the three different water temperature models and the impact of the water temperature on carbonaceous biochemical oxygen demand (CBOD) concentrations were evaluated against observed water temperature and water quality data from the Lower Minnesota River (LMR).

MATERIALS AND METHODS

Study area

The study area (longitude: −93.64° W ∼ −93.15° W; latitude: 44.69° N ∼ 44.90° N) covers the lower 40 miles of the Minnesota River, locating within the seven-county Twin Cities Metropolitan Area. The river reach extends from Jordan to St Paul, Minnesota, which is the confluence with the Mississippi River. The LMR has been listed as impaired due to low levels of DO and high levels of turbidity (MPCA 2008) because of water pollution. The Minnesota's third and fourth largest wastewater treatment plants (Blue Lake and Seneca) discharge to this river reach. Meanwhile, the LMR receives permitted discharge from several other facilities, notably storm water discharge from the Minneapolis/St Paul International Airport and cooling-water discharge from the Black Dog Generating Plant, a power generating plant owned and operated by Xcel Energy. Figure 1 is a detailed map of the study area, including major tributaries, wastewater treatment plants, power plants, and airport outfalls.

Figure 1

Study area of the LMR (Smith et al. 2010).

Figure 1

Study area of the LMR (Smith et al. 2010).

CE-QUAL-W2 water temperature model

Surface heat exchange can be formulated as a term-by-term process using explicit adjacent cell transport computation as long as the integration time step is shorter than or equal to the frequency of the meteorological data. Surface heat exchange processes depending on water surface temperatures are computed using previous time-step data and therefore lag from transport processes by the integration time step. Term-by-term surface heat exchange is computed as: 
formula
(1)
where Hn is the net rate of heat exchange across the water surface (W m−2), Hs is incident shortwave solar radiation (W m−2), Ha is incident longwave radiation (W m−2), Hsr is reflected shortwave solar radiation (W m−2), Har is reflected longwave radiation (W m−2), Hbr is back radiation from the water surface (W m−2), He is evaporative heat loss (W m−2), Hc is heat conduction (W m−2). The detailed description of each term can be found in Cole & Wells (2011).
Since some of the terms in the term-by-term heat balance equation are surface temperature dependent and others are measurable or computable input variables, the most direct route is to define an equilibrium temperature (Te) as the temperature at which the net rate of surface heat exchange is zero. The equilibrium temperature is defined as a hypothetical water temperature at which the net heat flux is zero. The net heat input is assumed to be proportional to the difference between the water temperature and the equilibrium temperature: 
formula
(2)
where ρw is the density of water (kg m−3), Cpw is the specific heat capacity of water (J kg−1 °C −1), H is water depth (m), t is time, Tw is water temperature (°C), Te is the equilibrium temperature (°C), KT is the overall heat exchange coefficient (W m−2 °C −1).

In the original W2 model, all approximations of the individual surface heat exchange terms enter into the evaluation of the coefficient of surface heat exchange and the equilibrium temperature, and the whole process is very complex, needing many iterations to get KT and Te (Cole & Wells 2011). Meanwhile, meteorological variables, such as solar radiation, wind speed, dew point temperature and air temperature, are needed as input. There are some parameters for calibration as well (Cole & Wells 2011).

In this study, a simplified equilibrium temperature approach was integrated into the W2 model. Compared with the original equilibrium temperature model, this approach is easy to implement, and needs only solar radiation, wind speed and dew point temperature as input meteorological variables. Also, there are no parameters for calibration. This approach has also been incorporated into the Soil and Water Assessment Tool, and it performed well for stream temperature prediction (Du et al. 2017).

The overall heat exchange coefficient KT can be calculated from the empirical relationships that include wind velocity, dew point temperature and initial water temperature (Brady et al. 1969; Edinger et al. 1974): 
formula
(3a)
 
formula
(3b)
 
formula
(3c)
where Tdew is the dew point temperature (°C), wind is the wind speed (m s−1). Both Tdew and wind are input meteorological data of the W2 model. The equilibrium temperature can be calculated by the empirical relationship of the overall heat exchange coefficient, the dew point temperature and the solar radiation (Brady et al. 1969): 
formula
(4)
where Hs (W m−2) is the gross rate of shortwave solar radiation, which is also input meteorological data of the W2 model.

Model development

A W2 model has previously been developed for use in establishing goals for load reduction of point and non-point sources and evaluation of management scenarios to improve current water quality conditions in the LMR, and the equilibrium temperature approach has been used to investigate river temperature (Smith et al. 2010). In this study, the simplified equilibrium temperature approach was employed, the term-by-term method, and simulation results of the original W2 model were used as comparison. The model was set up and validated using data collected by the metropolitan council during 2001–2004 periods. The LMR was modeled with 90 longitudinal segments, with length varying from 134.0 to 2,321.4 m (Smith et al. 2010). The monitoring stations chosen for comparison of modeled and observed data are locations at River Miles (RM) 25.1, 14.3, 8.5, and 3.5 from the downstream mouth (Figure 1). Their corresponding segments in the W2 model are labeled as segments 23, 46, 68 and 83. These four stations were located from upstream to downstream reflecting river temperature conditions in different parts of the LMR (Figure 1). Additional information for the four river temperature monitoring stations can be found in Table 1. The coefficient of determination (R2), Nash–Sutcliffe Efficiency (NSE), and Percent Error (PE) were used to evaluate the model performance. The average NSE value of the four monitoring stations was chosen as the objective function for the river temperature calibration, and parameters were adjusted to obtain the maximum average NSE value. The average values of R2 and PE were also calculated to evaluate the performance: 
formula
(5a)
 
formula
(5b)
 
formula
(5c)
Table 1

Detailed information for the four river temperature stations in the LMR

StationsRiver milesNumber of water temperature samples
Minnesota River at Shakopee 25.1 118 
Minnesota River at Savage 14.3 117 
Minnesota River at Black Dog 8.5 112 
Minnesota River at Ft Snelling 3.5 115 
StationsRiver milesNumber of water temperature samples
Minnesota River at Shakopee 25.1 118 
Minnesota River at Savage 14.3 117 
Minnesota River at Black Dog 8.5 112 
Minnesota River at Ft Snelling 3.5 115 

where OVi is the observed value, OVmean is the averaged observed value for the simulation time period, MVi is the simulated value, MVmean is the averaged simulated value for the simulation time period.

RESULTS AND DISCUSSION

Performance of the simplified equilibrium temperature model

The detailed hydrodynamic calibration process can be found in Smith et al. (2010). For the simplified equilibrium temperature model, the input information needed is meteorological variables, including solar radiation, wind speed, and dew point temperature. To evaluate the performance of the simplified equilibrium temperature model, we performed a comparison of the model simulation results with observed water temperature (Figure 2). The results of the simplified equilibrium temperature model are in good agreement with the observed data for all four stations. Furthermore, the simulated results of the simplified equilibrium temperature model were compared with those of the original equilibrium temperature model, and the term-by-term process model in W2. Table 2 shows the results of statistical performance for the three different water temperature models. For all the four stations, the simplified equilibrium temperature approach performs as well as the other two models. The original equilibrium water temperature model had average NSE, R2 and PE as 0.976, 0.978 and −1.560% respectively. The term-by-term water temperature model had average NSE, R2 and PE as 0.977, 0.978 and −0.712% respectively. The simplified equilibrium water temperature model had average NSE, R2 and PE as 0.972, 0.974 and 1.377% respectively. Generally, the original equilibrium water temperature model and the term-by-term water temperature model tend to overestimate water temperature, and the simplified equilibrium water temperature model tends to underestimate water temperature. Figure 3 presents the simulation results of these three approaches (Minnesota River at Shakopee).

Figure 2

Comparison of modelled and observed water temperature in the LMR.

Figure 2

Comparison of modelled and observed water temperature in the LMR.

Table 2

Water temperature statistics for the three water temperature models for the four stations in the LMR

StationNSER2PE (%)
Original equilibrium temperature model 
 Minnesota River at Shakopee 0.962 0.963 −0.056 
 Minnesota River at Savage 0.975 0.976 −0.890 
 Minnesota River at Black Dog 0.984 0.986 −2.418 
 Minnesota River at Ft Snelling 0.983 0.986 −2.874 
 Average 0.976 0.978 −1.560 
Term-by-term temperature model 
 Minnesota River at Shakopee 0.962 0.963 0.632 
 Minnesota River at Savage 0.976 0.977 −0.087 
 Minnesota River at Black Dog 0.985 0.985 −1.546 
 Minnesota River at Ft Snelling 0.985 0.986 −1.846 
 Average 0.977 0.978 −0.712 
Simplified equilibrium temperature model 
 Minnesota River at Shakopee 0.959 0.961 2.071 
 Minnesota River at Savage 0.969 0.973 2.161 
 Minnesota River at Black Dog 0.979 0.981 0.738 
 Minnesota River at Ft Snelling 0.979 0.981 0.539 
 Average 0.972 0.974 1.377 
StationNSER2PE (%)
Original equilibrium temperature model 
 Minnesota River at Shakopee 0.962 0.963 −0.056 
 Minnesota River at Savage 0.975 0.976 −0.890 
 Minnesota River at Black Dog 0.984 0.986 −2.418 
 Minnesota River at Ft Snelling 0.983 0.986 −2.874 
 Average 0.976 0.978 −1.560 
Term-by-term temperature model 
 Minnesota River at Shakopee 0.962 0.963 0.632 
 Minnesota River at Savage 0.976 0.977 −0.087 
 Minnesota River at Black Dog 0.985 0.985 −1.546 
 Minnesota River at Ft Snelling 0.985 0.986 −1.846 
 Average 0.977 0.978 −0.712 
Simplified equilibrium temperature model 
 Minnesota River at Shakopee 0.959 0.961 2.071 
 Minnesota River at Savage 0.969 0.973 2.161 
 Minnesota River at Black Dog 0.979 0.981 0.738 
 Minnesota River at Ft Snelling 0.979 0.981 0.539 
 Average 0.972 0.974 1.377 
Figure 3

Comparison of simplified ET, ET, and TERM water temperature approaches (ET: Equilibrium Temperature, TERM: Term-by-term in the W2 model, Simplified ET: Simplified Equilibrium Temperature).

Figure 3

Comparison of simplified ET, ET, and TERM water temperature approaches (ET: Equilibrium Temperature, TERM: Term-by-term in the W2 model, Simplified ET: Simplified Equilibrium Temperature).

Effect of water temperature simulation on water quality modelling

Water temperature simulation has an impact on water quality modelling in the W2 model since water temperature affects chemical reaction rates and oxygen saturation concentration. The W2 model uses an exponential equation to correct chemical reaction rates based on the simulated water temperature (Cole & Wells 2011): 
formula
(6)
where k(T) is the reaction rate at a local temperature (d−1), k20 is the reaction rate at 20 °C (d−1), θ is the temperature correction coefficient, and Tw is water temperature simulated by the W2 model (°C).

To investigate the impact of different water temperature simulations on water quality concentration simulations, the simulated CBOD concentrations of Minnesota River at Shakopee and Ft Snelling by three different water temperature models were output and analyzed. CBOD was divided into six groups, with decay rates at 20 °C being 0.0345, 0.0322, 0.0294, 0.0495, 0.0257 and 0.0315 d−1 respectively (Smith et al. 2010). Figure 4 shows the monthly average of simulated CBOD concentrations. At Shakopee and Ft Snelling, CBOD concentrations of the original equilibrium temperature approach were generally lower than the other two models (Figure 4), which may be induced by its overestimation of water temperature. The monthly averaged CBOD concentrations of the simplified equilibrium temperature model were almost the same as those of the term-by-term approach in the W2 model at the two stations.

Figure 4

Monthly variations of CBOD concentrations simulated by the three different water temperature models in the year 2002 (ET: Equilibrium Temperature, TERM: Term-by-term in the W2 model, Simplified ET: Simplified Equilibrium Temperature).

Figure 4

Monthly variations of CBOD concentrations simulated by the three different water temperature models in the year 2002 (ET: Equilibrium Temperature, TERM: Term-by-term in the W2 model, Simplified ET: Simplified Equilibrium Temperature).

Table 3 shows the results of statistical performance for the three different water temperature models. Generally, all these model approaches tend to underestimate DO concentrations. The original equilibrium water temperature model had average NSE, R2 and PE as 0.634, 0.777 and 7.699% respectively. The term-by-term water temperature model had average NSE, R2 and PE as 0.633, 0.783 and 7.794% respectively. The simplified equilibrium water temperature model had average NSE, R2 and PE as 0.650, 0.791 and 7.596% respectively. For all the four stations, the simplified equilibrium temperature approach performs better than the other two models for DO simulation, which indicates that the simplified equilibrium temperature model can be a potential tool to simulate water temperature for water quality modelling.

Table 3

DO statistics for the three water temperature models for the four stations in the LMR

StationNSER2PE (%)
Original equilibrium temperature model 
 Minnesota River at Shakopee 0.744 0.800 4.264 
 Minnesota River at Savage 0.508 0.704 9.328 
 Minnesota River at Black Dog 0.690 0.843 8.171 
 Minnesota River at Ft Snelling 0.594 0.762 9.033 
 Average 0.634 0.777 7.699 
Term-by-term temperature model 
 Minnesota River at Shakopee 0.746 0.804 4.296 
 Minnesota River at Savage 0.506 0.708 9.366 
 Minnesota River at Black Dog 0.686 0.847 8.290 
 Minnesota River at Ft Snelling 0.593 0.771 9.223 
 Average 0.633 0.783 7.794 
Simplified equilibrium temperature model 
 Minnesota River at Shakopee 0.752 0.809 4.291 
 Minnesota River at Savage 0.539 0.727 9.123 
 Minnesota River at Black Dog 0.710 0.859 7.996 
 Minnesota River at Ft Snelling 0.597 0.767 8.975 
 Average 0.650 0.791 7.596 
StationNSER2PE (%)
Original equilibrium temperature model 
 Minnesota River at Shakopee 0.744 0.800 4.264 
 Minnesota River at Savage 0.508 0.704 9.328 
 Minnesota River at Black Dog 0.690 0.843 8.171 
 Minnesota River at Ft Snelling 0.594 0.762 9.033 
 Average 0.634 0.777 7.699 
Term-by-term temperature model 
 Minnesota River at Shakopee 0.746 0.804 4.296 
 Minnesota River at Savage 0.506 0.708 9.366 
 Minnesota River at Black Dog 0.686 0.847 8.290 
 Minnesota River at Ft Snelling 0.593 0.771 9.223 
 Average 0.633 0.783 7.794 
Simplified equilibrium temperature model 
 Minnesota River at Shakopee 0.752 0.809 4.291 
 Minnesota River at Savage 0.539 0.727 9.123 
 Minnesota River at Black Dog 0.710 0.859 7.996 
 Minnesota River at Ft Snelling 0.597 0.767 8.975 
 Average 0.650 0.791 7.596 

CONCLUSIONS

Water temperature is an important indicator for biodiversity and ecosystem sustainability. In this paper, a simplified equilibrium temperature model was incorporated into the W2 model. The simplified equilibrium temperature model was applied to the LMR. Results show that the simplified equilibrium temperature model performed as well as the original equilibrium temperature model and the term-by-term process model for water temperature predictions. Additionally, the impact of the water temperature on water quality was analyzed through the variations of CBOD and DO concentrations under three different water temperature models. Results show that CBOD concentrations of the original equilibrium temperature approach were generally lower than the other two models, which may be induced by its overestimation of water temperature. The monthly averaged CBOD concentrations of the simplified equilibrium temperature model were almost the same as that of the term-by-term approach in the W2 model. For all the four stations, the simplified equilibrium temperature approach performs better than the other two models for DO simulation, which indicates that the simplified equilibrium temperature model can be a potential tool to simulate water temperature for water quality modelling.

ACKNOWLEDGEMENTS

This work was jointly funded by the National Key R&D Program of China (2016YFC0401506) and the Projects of the National Natural Science Foundation of China (51679146, 51479120). The original Minnesota River's W2 Model was developed and obtained from the technical report (Smith et al. 2010).

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