# Simulating Diurnal Variations of Water Temperature and Dissolved Oxygen in Shallow Minnesota Lakes

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Daily Year-Round Water Temperature Model

_{w}(z, t) is the water temperature in (°C), which is a function of depth (z) and time (t); A(z) (m

^{2}) is the horizontal area for each layer of water as a function of the depth, K

_{z}(m

^{2}/day) is the vertical turbulent heat diffusion coefficient, which is a function of depth and time; $\rho $C

_{p}(J/m

^{3}-°C) represents the heat capacity of water per unit volume, and H

_{w}(J/m

^{3}-day) is the sum of heat source and sink terms per unit volume of water. The determination of turbulent diffusion coefficients for lake temperature modeling has been discussed in detail by Fang [8]. In the regional daily water temperature MINLAKE model, the vertical heat diffusion coefficient K

_{z}(m

^{2}/day) for epilimnion and hypolimnion is calculated based on Equation (2):

_{s}is the surface area of the lake (km

^{2}), and N

^{2}is the Brunt-Vaisala stability frequency of the stratification (s

^{−2}). In the epilimnion, N

^{2}was set at a minimum value of 0.000075 [20].

^{2}-day) through the water surface during the open water seasons can be represented as:

_{sn}is net shortwave solar radiation, H

_{a}is net atmospheric longwave radiation, H

_{br}is longwave back radiation from the water to the atmosphere, H

_{c}is heat conduction/convection, and H

_{e}is the evaporation through the surface water.

_{s}

_{(i)}and H

_{s}

_{(i+1)}are shortwave solar radiation reaching the top and bottom of a water layer i (kcal/m

^{2}-day), k is an attenuation coefficient of water (1/m), and Δz is the thickness of the water layer (m). Equations (5) and (6) represent atmospheric longwave radiation and back radiation, respectively, where H

_{a}is atmospheric radiation (kcal/m

^{2}-day), ${\u03f5}_{a}$ is the atmospheric emissivity directly related to air temperature and cloud cover, T

_{aa}is the atmospheric absolute temperature (°K), σ is the Stefan–Boltzmann constant [11.7 × 10

^{−8}cal/(cm

^{2}°K

^{4}day)], and H

_{br}is back radiation from the water surface. For back radiation estimation, the emissivity of water surface ${\u03f5}_{ws}$ is set constant (0.97), and the water temperature of the top mixed layer is used as T

_{as}. Evaporation heat loss is one of the most complicated parts of water temperature calculations. Equations (7) and (8) represent heat transfer through evaporation (H

_{e}) and conduction (H

_{c}) in W/m

^{2}, where T

_{swv}and T

_{av}are the virtual temperatures [21] of the water surface and the air, respectively, in °K; e

_{s}

_{,}and e

_{air}are saturated and actual vapor pressure in millibars, and W

_{z}is the wind speed at two meters above the surface (m/s). In Equation (8), the constant coefficient 0.61 is the Bowen ratio, and T

_{sw}and T

_{air}are surface water temperature and air temperature (°C), respectively. The total heat absorbed in a water layer, HQ(i) (kcal/day) is calculated as:

_{max}. H

_{sn}

_{(i)}is the shortwave solar radiation reached the top surface of a water layer, A

_{(i)}and A

_{(i+1)}are the areas of the top and bottom surfaces of the water layer, respectively, and Δz

_{i}is the depth of the water layer.

#### 2.2. Daily Dissolved Oxygen Model

_{z}(m

^{2}/day) is the vertical turbulent diffusion coefficient for DO as a function of depth and time, and S(z, t) represents the sum of all the source and sink terms of DO (mg/L-day). The DO source and sink equation is given by:

_{s}could be a source or sink term, while the main sinks of the DO are the respiration processes in the water body R(z, t), sediment oxygen demand S

_{SOD}(z, t), and carbonaceous oxygen demand and nitrogenous oxygen demand represented together as S

_{BOD}

_{(z,t)}:

_{g}is the photosynthetic oxygen production rate (mg O

_{2}(mg Chl-a)

^{−1}h

^{−1}) in the MINLAKE model when biomass in a lake is represented using Chlorophyll-a concentrations (abbreviated as Chl-a). When the conditions of nutrients, sunlight, and water temperature are favorable, algal blooms may occur. With any of these conditions are limiting, algal growth will be restricted. This multiplicative relationship [23] is presented in Equation (13): k

_{gT}includes temperature correction on the photosynthetic oxygen production rate, k

_{gL}is the light limitation determined using the two-parameter Haldane kinetics equation to describe light-limited growth and inhabitation [24], and the limitation due to nutrients is directly represented or linked to Chl-a concentrations. Pmax in Equation (14) is the maximum photosynthesis rate for oxygen production in mg O

_{2}(mg Chl-a)

^{−1}h

^{−1}. Reaeration is simulated based on the gas-transfer theory presented by Holley [25] using k

_{e}, the bulk surface oxygen transfer velocity (m/day) in Equation (15), where A

_{s}is the lake surface area (m

^{2}), C

_{1}(mg/L) and V(1) are the oxygen concentration and water volume in the surface layer, and C

_{s}is the DO saturation concentration (mg/L) as a function of surface temperature and lake elevation. Equation (16) represents plant respiration as a first-order kinetic process where YCHO2 is a yield coefficient representing the ratio of mg chlorophyll-a to mg oxygen (0.008), k

_{r}is the respiration rate coefficient (day

^{−1}), and θ

_{r}is the temperature adjustment coefficient. Equation (17) represents biochemical oxygen demand (BOD) as a function of detrital mass expressed in oxygen equivalents. Here k

_{b}is the first order decay coefficient (day

^{−1}), θ

_{b}is the temperature adjustment coefficient, and BOD is detritus as oxygen equivalent (mg/L). In the regional DO model and MINLAKE2018, k

_{r}= k

_{b}= 0.1 day

^{−1}and θ

_{r}= θ

_{b}= 1.047 are used [8]. Sedimentary oxygen demand (SOD) from the lake bottom sediments is a boundary condition (sediment surface flux), but since it occurs for each layer it is treated as a sink term in the one-dimensional (vertical) transport equation [26]. Sedimentary oxygen flux per control volume (A*dz) is given by Equation (18), where S

_{b}is the sedimentary oxygen demand coefficient (g O

_{2}/m

^{2}-day), which is directly related to lake trophic status. There are several factors that are commonly considered responsible for SOD variation in a water body. The most important factor responsible for SOD variation is the temperature near the sediment-water interface [27], the velocity of the water overlying the sediment [28], the organic content of the sediment, and the oxygen concentration of the overlying water [3].

#### 2.3. Sediment Temperature Simulation

_{s}(°C) is the sediment temperature and K

_{s}(m

^{2}/day) is the sediment thermal diffusivity. The boundary conditions for the sediment temperature model are given by the following equations:

_{w}

_{(i)}is the simulated water temperature in the water layer (i) at the previous time step. The first boundary condition assures the continuity of temperature at the water-sediment interface. The second boundary condition implies an adiabatic boundary (there is no heat transfer) at 10 m below the sediment surface, where seasonal temperature fluctuations are damped out. The initial sediment temperature at the sediment-water interface is set equal to the initial water temperature at the sediment surface. Sediment temperature increases/decreases exponentially with sediment depth until it approaches a constant value at 10 m below the sediment/water interface. Sediment temperatures at 10 m below the sediment-water interface (T

_{S}

_{10}) at different water depths are very important input data (depending on the geographic location of the lake) that have been studied by Fang and Stefan [30].

#### 2.4. Modifications to the Daily Model

_{2}(mg Chl-a)

^{−1}h

^{−1}[24,31] so that no change is needed. In the daily MINLAKE2012 model, daily solar radiation was redistributed as a sinusoidal function over the photo period and used to compute hourly oxygen production by photosynthesis that was summed as daily oxygen production [8]. In the hourly MINLAKE2018 model, hourly solar radiation from the weather input file was used directly to compute the hourly photosynthetic oxygen production. The vertical heat diffusion coefficient K

_{z}(m

^{2}/h) was calculated using Equation (21) [21,32] in epilimnion and Equation (2) in metalimnion and hypolimnion:

_{z}in Equation (2) for the daily model varies with lake surface area but does not change with time during a simulation, but hourly K

_{z}in Equation (21) is a function of hourly wind speed W in mph (mile/h). Wang et al. [33] reported that the wind speed showed significant correlation to the half-hourly turbulent heat flux and energy budget over a small lake in open water season. The wind speed is the most significant factor governing physical processes (evaporation, heat flux, and energy budget) in lakes for time periods shorter than daily [19,33]. Moreover, the wind speed changes throughout the day and causes the short-term mixing in the lakes. Since the short-term mixing was not of interest in daily model MINLAKE2012, wind speed was not used in turbulent diffusion coefficient calculation but in computing the wind (kinetic) energy applied on the lake surface, which is balanced with the potential energy of lifting colder/denser water at lower depths to determine the mixed layer depth in each day [8]. Equation (21) was compared with Equation (2) and another equation (${K}_{z}=28\times dt\times {W}^{1.3}$) used by Riley and Stefan (1987); Equation (21) seemed to perform the best for hourly model, though additional study on specifying K

_{z}is needed. The unsteady heat transfer Equation (1) and dissolved oxygen transport Equation (10) were then used to simulate hourly temperature and dissolved oxygen through all layers of the lake.

_{s}in Equation (19) was multiplied by dt to account for the hourly time step. To accurately simulate the sediment temperature gradient, the 10-m sediment below the lake bottom was divided into 20 layers each of 0.5 m thickness, while it was 10 layers for the daily model. In shallow lakes, solar radiation is likely to penetrate the whole water depth and directly heat the bottom sediment below the water. In the daily model, the heat absorbed by any water layer was quantified as a subtraction of the heat reached on the top and bottom surfaces of the layer as shown in Equation (9), which means all the heat reaching the area A(i) − A(i + 1) was used to warm up water only (not directly heat the bottom sediments). Therefore, direct solar heating to sediment was ignored even though heat exchange between the water and sediment was considered [12] when sediment temperature is simulated. The heat absorbed by a water layer due to radiation attenuation (k) was modified as:

_{(i)}and r

_{(i+1)}are the radius of the top and bottom surface areas of the water layer i, respectively; and tanα is equal to [r

_{(i)}− r

_{(i+1)}]/$\text{}\Delta {z}_{i}$ and approximates the slope of the lake bottom for layer i. These changes subsequently may change the sediment temperature profile and sediment heat flux of the lake.

#### 2.5. Modeled Shallow Lakes

_{s}

^{0.25}/H

_{max}, where A

_{s}in m

^{2}and H

_{max}in m are the surface area and the maximum depth of the lake, respectively) is a very important characteristic parameter of a lake that affects stratification, lake habitat, etc. Since all the study lakes have a geometry ratio between 3 and 10, they are weakly stratified. The lower the geometry ratio, the higher the stratification of a lake. From Table 1, Carrie Lake is relatively more stratified (geometry ratio 3.12 m

^{0.5}) and Red Sand Lake is the least stratified (geometry ratio 8.34 m

^{0.5}). Based on chlorophyll-a concentration, Belle, Pearl and Portage lakes are eutrophic (mean Chl-a > 10 µg/L, [34]) and Carrie and Red Sand lakes are mesotrophic (mean Chl-a between 4 and 10 µg/L, [34])

## 3. Modeling Results

#### 3.1. Model Calibration

#### 3.2. Diurnal Variations

#### 3.3. Profile Comparison

#### 3.4. Comparison of Long-Term Surface Temperature Simulation

#### 3.5. Comparison of Heat Flux, DO Production and Reaeration

_{sn}and H

_{a}in Equation (3)) has a clear diurnal variation for most of the days while the hourly flux out (sum of H

_{br}, H

_{c}, and H

_{e}in Equation (3)) has almost no diurnal variation but some fluctuations in each day. At night when the solar radiation H

_{sn}is absent, water loses heat to the atmosphere as the flux out is greater than the flux in, while during the day, due to the increase of shortwave solar radiation, the water body gains heat to increase the water temperature in epilimnion (Pearl Lake in Figure 5). The daily model has a constant heat flux in and flux out over a 24-h period whereas the hourly model considers the heat flux variations hour by hour (24 values for each day). As a result, the hourly model can represent daily variations more accurately than the daily model, which is evident in Figure 7b. Results from the daily model illustrate that heat could transfer in one direction (cooling or warming) for several consecutive days while the results from the hourly model show that heat transfer from and to the waterbody is a more dynamic process that occurs within the day and depends on the time of day.

_{s}in Equation (15)) is less than the surface DO concentration. On 5 June, the hourly reaeration rate is still negative whereas the daily reaeration rate is positive. From 6 June to 10 June, hourly reaeration becomes positive which means that saturated DO is higher than surface DO. On 8 June after midnight, the hourly reaeration rate suddenly increases and becomes 4.32 mg/L/h at 4 a.m. because of a strong wind speed of 19 mph (the average wind speed was 6.18 mph for the first 10 days of June at Saint Cloud). The daily model cannot account for hourly variations and has a constant reaeration rate of 0.57 mg/L/h. In the daily model, hourly photosynthesis was calculated by redistributing daily solar irradiance as a sine function over the photoperiod (14 h in June) and added together to get a daily oxygen production by photosynthesis for solving the daily DO balance equation [10], while the hourly model uses hourly solar radiation from the weather data file to get the hourly photosynthetic oxygen production and solve the hourly DO balance equation. In Figure 7c daily photosynthesis was averaged over the photoperiod starting from 6:00 a.m. to compare with hourly photosynthesis. In the presence of light, hourly photosynthetic oxygen production increases and then becomes zero during the night when there is no oxygen production. Figure 7c shows that both the hourly and daily models have similar estimates on photosynthetic oxygen production but differ in surface aeration.

#### 3.6. Impact of Direct Solar Radiation Heating on Sediment Bed

## 4. Discussion

#### 4.1. Short-Term Mixing Prediction

_{max}= 7.6 m), Carrie (H

_{max}= 7.9 m), and Portage (H

_{max}= 4.6 m) lakes. In 2009 Belle Lake was completely mixed from 6 June to 9 June and on 30 June from both the hourly and daily model simulations. On 28–29 June, the daily model results show that the lake is well mixed whereas the hourly model predicts weak stratification. This occurs due to the sudden increase in daily wind speed on those days. The hourly model simulates water temperature hour by hour using hourly wind speed which increased gradually and hence, no complete mixing was simulated by the hourly model. Carrie Lake is more stratified than Belle Lake (Figure 10b), which is related to a smaller geometry ratio (Table 1). Observed half-hourly water temperature data in these lakes were collected and converted into hourly observed data for comparison. At the surface, the simulated water temperatures match well with the observed data with an RMSE of 1.2 °C, 1.7 °C, and 1.4 °C for Belle (2008–2011), Carrie (2008–2015), and Portage (2008–2015), respectively. Since Portage is a very shallow lake with a maximum depth of 4.3 m, there are small differences between the surface and bottom water temperatures. For most of the days, the daily model predicts essentially the same temperatures at the lake surface and bottom whereas the hourly model predicts the temporal variations of water temperature more correctly. Owing to the lack of necessary details associated with the daily weather data, the daily model underpredicts the water temperature in well-mixed conditions at Portage Lake.

#### 4.2. Stratification Prediction

#### 4.3. Application in Lake Management

#### 4.4. Future Studies

## 5. Conclusions

- MINLAKE2018 was calibrated against measured profiles in five shallow Minnesota lakes (Table 4) with an average standard error of 1.48 °C for temperature and 2.02 mg/L for DO. With the help of available surface water temperature hourly data, the average RMSE of long-term water temperature simulation was 1.50 °C with a standard deviation of 0.32 °C. For Pearl Lake, the average RMSE for water temperature simulation at six different depths is 1.30 °C with a standard deviation of 0.15 °C.
- When compared with the daily MINLAKE2012 model, for Pearl Lake (H
_{max}= 5.6 m), the hourly model calculated 12% and 13% more temperature stratification for ice cover period and open water season, respectively (Table 6). Similarly, for DO, stratification increases were 14% and 20% for ice cover period and open water season, respectively. For other lakes, hourly model simulation also resulted in increased stratification percentages for water temperature and DO. The hourly model can capture diurnal changes and mixing events that lasted a few hours within a day, which the daily model ignores. Moreover, it was observed that the daily model could not predict most of the weak stratifications of shallow lakes in the fall season (Figure 12). As a result, to ensure desired water quality for aquatic organisms and fish habitat, the hourly model is suitable for shallow lakes all year round. - The hourly model MINLAKE2018 performs better than the daily model MINLAKE2012 in water temperature and DO profile simulation (Figure 2). The RMSEs of temperature and DO from MINLAKE2018 decreased by 17.3% and 18.2%, respectively, and Nash-Sutcliffe efficiency increased by 10.3% and 66.7%, respectively, in comparison to MINLAKE2012.
- Sediment heating subroutine was modified to include direct heating of sediment from solar radiation for all sediment layers. After modification, the sediment heat flux pattern became coincident with the solar radiation pattern eliminating the lag time between the change in solar radiation and the change in heat flux to appear. The magnitude of sediment heat flux was reduced for both cases (heat flux going from water to sediment or sediment to water) after the sediment subroutine was modified.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Effect of wind sheltering coefficient (increasing from 0.67 to 1) on simulated (

**a**) water temperature and (

**b**) dissolved oxygen profiles on 7 July 2010; (

**c**) water temperature and (

**d**) dissolved oxygen profiles on 22 September 2008 in Belle Lake including observed profiles.

**Figure 2.**Measured versus simulated water temperature and DO in Portage Lake and Carrie Lake for simulation over several years.

**Figure 3.**Time series of simulated and observed maximum and minimum hourly surface water temperatures each day at (

**a**) Portage Lake and (

**b**) Belle Lake in 2009 including simulated daily temperatures.

**Figure 4.**Time series of simulated daily, simulated and observed hourly surface (1 m at Belle and 1.5 m at Portage Lake) temperatures at 4 PM at Belle Lake (2008–2012) and Portage Lake (2009–2013).

**Figure 5.**Simulated water temperature profiles in Pearl Lake and DO profiles in Belle Lake at five different hours comparing with observed (green squares) and simulated daily profiles.

**Figure 6.**Time series of observed (orange) and simulated (blue) hourly surface water temperatures in five study lakes over 4–8 years of simulation.

**Figure 7.**Time series of (

**a**) air temperature and solar radiation, (

**b**) calculated daily and hourly heat fluxes, and (

**c**) two DO source terms through the water surface at Carrie Lake on 1–10 June 2009.

**Figure 8.**Comparison of heat flux calculated by the modified and previous sediment models on an hourly basis on (

**a**) day with ice and snow cover, (

**b**) day with ice cover but no snow cover, (

**c**) high solar radiation day in June, (

**d**) low solar radiation day in June. The y-axis scales on (

**a**,

**b**) are different from on (

**c**,

**d**).

**Figure 10.**Time series of simulated daily and hourly water temperatures at two depths (1 m, 7.5 or 4 m) at (

**a**) Belle Lake in June 2009, (

**b**) Carrie Lake in June 2009, and (

**c**) Portage Lake in June 2015 including observed hourly surface temperatures. Daily values were plotted at 4:00 p.m. each day.

**Figure 11.**Time series of simulated daily and hourly DO at two depths (1 m, 7.5 or 4 m) at (

**a**) Belle Lake in June 2009, (

**b**) Carrie Lake in June 2009, and (

**c**) Portage Lake in June 2015. Daily values were plotted at 4:00 p.m. each day.

**Figure 12.**(

**a**) Time series of observed and simulated hourly water temperature at 1.2 m and 5 m in Pearl Lake in open water season in 2015, (

**b**) differences of simulated and observed hourly temperatures between 1.2 m (surface) and 5 m (near the bottom) during open water season in 2015 in Pearl Lake including simulated daily temperature differences and 1 °C difference as stratification criterion.

Lake | Surface Area, A_{s}_{,} (km^{2}) | Max. Depth H_{max}_{,} (m) | Geometry Ratio ^{1}(m) ^{0.5} | Mean Chl-a (μg/L) | Trophic Status | Simulation Years | Number of Days with Profile Data |
---|---|---|---|---|---|---|---|

Carrie | 0.37 | 7.90 | 3.12 | 6.71 | Mesotrophic | 2008–2012 | 50 |

Belle | 3.71 | 7.60 | 5.77 | 27.10 | Eutrophic | 2008–2012 | 73 |

Pearl | 3.05 | 5.55 | 7.53 | 16.91 | Eutrophic | 2008–2012 | 36 |

Portage | 1.54 | 4.57 | 7.71 | 15.98 | Eutrophic | 2008–2015 | 86 |

Red Sand | 2.11 | 4.57 | 8.34 | 4.43 | Mesotrophic | 2008–2015 | 87 |

^{1}Geometry ratio of a lake is A

_{s}

^{0.25}/H

_{max}where A

_{s}in m

^{2}and H

_{max}in m.

**Table 2.**Calibration parameters for hourly and daily MINLAKE models [35].

Calibration Parameter | Effect on Model Results | Description of the Parameter |
---|---|---|

Wstr | Temperature and DO profiles | Wind sheltering coefficient |

BOD | DO Profiles | Biochemical oxygen demand depending on lake trophic status |

Sb20 | DO Profiles | Sediment oxygen demand, lake tropic dependent |

EMCOE(1) | Temperature and DO Profiles | Multiplier for diffusion coefficient in the metalimnion |

EMCOE(2) | DO Profiles | Multiplier for SOD below the mixed layer |

Pmax | DO Profiles | Maximum photosynthesis rate for oxygen production |

Parameter/Lakes | Red Sand Lake | Portage Lake | Carrie Lake | Pearl Lake | Belle Lake |
---|---|---|---|---|---|

Wstr | 0.47 (0.47) | 0.37 (0.37) | 1 (1.0) | 0.6 (0.4) | 0.67 (1.0) |

BOD | 0.75 (0.75) | 1.5 (1.5) | 1 (0.75) | 0.75 (1.5) | 1.5 (1.0) |

Sb20 | 0.75 (0.75) | 1.5 (1.5) | 1.5 (0.75) | 0.75 (1.5) | 1.5 (1.8) |

EMCOE(1) | 1 (7) | 1 (3) | 1 (3) | 1 (0.8) | 1 (4) |

EMCOE(2) | 1 (3) | 1.1 (1) | 0.82 (1.2) | 1 (0.7) | 1 (0.5) |

Pmax | 9.6 (16.8) | 9.6 (8.5) | 9.6 (9.6) | 9.6 (8.5) | 9.6 (7.7) |

**Table 4.**Statistical error parameters for the hourly and daily MINLAKE models against observed profile data in five study lakes.

Lake Name | Hourly Model (MINLAKE2018) | |||||
---|---|---|---|---|---|---|

Water Temperature | Dissolved Oxygen | |||||

RMSE ^{1} (°C) | NSE ^{2} | Slope ^{3} | RMSE (mg/L) | NSE | Slope | |

Carrie Lake | 2.21 | 0.85 | 1.04 | 1.69 | 0.78 | 0.96 |

Pearl Lake | 1.03 | 0.98 | 0.98 | 2.23 | 0.35 | 1.00 |

Belle Lake | 1.03 | 0.96 | 1.03 | 1.53 | 0.69 | 1.00 |

Red Sand Lake | 1.86 | 0.94 | 0.97 | 2.77 | 0.36 | 0.99 |

Portage Lake | 1.41 | 0.97 | 0.98 | 1.91 | 0.31 | 0.99 |

Average ± STD ^{4} | 1.48 ± 0.32 | 0.96 ± 0.02 | 0.98 ± 0.01 | 2.02 ± 0.49 | 0.50 ± 0.22 | 0.99 ± 0.02 |

Lake Name | Daily Model (MINLAKE2012) | |||||

Carrie Lake | 2.47 | 0.77 | 1.08 | 2.76 | 0.42 | 0.92 |

Pearl Lake | 1.04 | 0.97 | 0.98 | 2.58 | 0.13 | 0.98 |

Belle Lake | 1.14 | 0.96 | 1.01 | 2.09 | 0.43 | 0.94 |

Red Sand Lake | 2.48 | 0.79 | 0.97 | 2.90 | 0.29 | 0.98 |

Portage Lake | 1.82 | 0.86 | 1.03 | 2.03 | 0.22 | 0.96 |

Average ± STD | 1.79 ± 0.69 | 0.87 ± 0.09 | 1 ± 0.03 | 2.47 ± 0.39 | 0.30 ± 0.13 | 0.96 ± 0.03 |

^{1}RMSE stands for Root Mean Square Error,

^{2}NSE for Nash-Sutcliffe Efficiency [36],

^{3}Slope of linear regression between simulated and observed,

^{4}STD for Standard Deviation.

**Table 5.**Statistical error parameters for the hourly MINLAKE model against observed time-series data.

Surface Depths | Carrie Lake | Pearl Lake | Belle Lake | Red Sand Lake | Portage |
---|---|---|---|---|---|

RMSE | 1.82 | 1.22 | 1.19 | 1.95 | 1.33 |

NSE | 0.95 | 0.98 | 0.98 | 0.94 | 0.99 |

Slope | 0.98 | 0.98 | 0.99 | 0.97 | 0.99 |

Pearl Lake | 1.7 m | 2.4 m | 3.4 m | 4.4 m | 5.0 m |

RMSE | 1.08 | 1.18 | 1.47 | 1.47 | 1.42 |

NSE | 0.98 | 0.98 | 0.97 | 0.96 | 0.97 |

Slope | 0.98 | 0.99 | 0.98 | 0.97 | 0.98 |

**Table 6.**Water temperature and dissolved oxygen stratification in study lakes presented as % hours of stratification (hourly model) or % days of stratification (daily model) in 2009–2011.

Lake Name | Geometry Ratio (Secchi Depth) | % Hours or Days of Temperature Stratification | |||

Ice Cover Period | Open Water Season | ||||

Hourly Model | Daily Model | Hourly Model | Daily Model | ||

Carrie | 3.12 (1.48 m) | 89 | 89 | 65 | 64 |

Belle | 5.77 (1.46 m) | 86 | 80 | 37 | 35 |

Pearl | 7.53 (1.85 m) | 93 | 81 | 80 | 67 |

Portage | 7.71 (2.00 m) | 89 | 83 | 26 | 25 |

Red Sand | 8.34 (3.04 m) | 90 | 89 | 32 | 16 |

Lake Name | Geometry Ratio (Secchi Depth) | % Hours or Days of DO Stratification | |||

Ice Cover Period | Open Water Season | ||||

Hourly Model | Daily Model | Hourly Model | Daily Model | ||

Carrie | 3.12 (1.48 m) | 75 | 66 | 58 | 71 |

Belle | 5.77 (1.46 m) | 88 | 76 | 56 | 52 |

Pearl | 7.53 (1.85 m) | 93 | 79 | 67 | 47 |

Portage | 7.71 (2.00 m) | 89 | 67 | 48 | 38 |

Red Sand | 8.34 (3.04 m) | 90 | 87 | 37 | 42 |

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## Share and Cite

**MDPI and ACS Style**

Tasnim, B.; Jamily, J.A.; Fang, X.; Zhou, Y.; Hayworth, J.S.
Simulating Diurnal Variations of Water Temperature and Dissolved Oxygen in Shallow Minnesota Lakes. *Water* **2021**, *13*, 1980.
https://doi.org/10.3390/w13141980

**AMA Style**

Tasnim B, Jamily JA, Fang X, Zhou Y, Hayworth JS.
Simulating Diurnal Variations of Water Temperature and Dissolved Oxygen in Shallow Minnesota Lakes. *Water*. 2021; 13(14):1980.
https://doi.org/10.3390/w13141980

**Chicago/Turabian Style**

Tasnim, Bushra, Jalil A. Jamily, Xing Fang, Yangen Zhou, and Joel S. Hayworth.
2021. "Simulating Diurnal Variations of Water Temperature and Dissolved Oxygen in Shallow Minnesota Lakes" *Water* 13, no. 14: 1980.
https://doi.org/10.3390/w13141980