Inland water bodies such as reservoirs and lakes are very important parts of the continental land surface [1
]. Reservoirs are typically built to store water for water supply, hydropower or flood control [2
]. Knowledge of the mixing characteristics and temperature profile of a lake are important for its operation and management [3
]. The thermal structure of water bodies, temperature stratification dynamics and changes in temperature values have a direct impact on the heat storage of lakes and their water quality [4
]. Understanding the heat budget of lakes and reservoirs is crucial for estimating evaporation in the energy budget methods that are widely used [7
]. However, measurements of heat exchange between the water surface and atmosphere are scarce. In most cases carrying out measurements for shallow and small inland water bodies is difficult and expensive and needs high level of expertise to obtain reliable measurements over the water surface even for measuring conventional meteorological parameters such as air temperature, wind velocity and so on. Although experimental temperature profiles in lakes are commonly available, the vertical resolutions are often not sufficient for assessing small-scale turbulence effects or investigating variations of water temperature induced by radiative forcing, air temperature as well as wind velocity in shallow waters [7
]. As small shallow lakes and reservoirs respond to atmospheric conditions very quickly, precise estimation of the heat transfer between the atmosphere and their surface is extremely important to model the temperature dynamics and stratification in these water bodies [11
]. In these water bodies, the near-surface water temperature commonly follows the radiative forcing (solar radiation) trend with an increase during the day and a decrease during the night. The gradient temperature can transport vertically into the water column by (effective) thermal diffusivity, which can be enhanced by the atmospheric parameters, water surface waves and the dynamics of the flow in the water body. Eddy diffusivity and thermal conductivity are important parameters in simulating the diurnal evolution of the temperature in the water bodies. Wind over the water surface affects lake currents, sensible and latent heat fluxes and turbulence as well as surface waves. The time-dependent effects of wind shear stress over the surface can change the flow pattern and thermodynamics of the lake. Therefore, considering the effects of heat transfer and wind-induced flow in small water bodies is so complicated and needs the use of high-resolution simulation to determine the flow parameters.
In the case of shallow and small inland water bodies, which have been used in this study, simulating the flow field requires an additional degree of complexity beyond simulation of a deep and large water body. Including the effects of the time-varying driving forces such as short-wave radiation, air temperature, wind speed and its direction, precipitation, cloud cover, water surface temperature, and variation in water composition (such as salinity and density) in a shallow water body simulation is difficult. In addition, implementing an appropriate approach to compute the heat fluxes through the water surface, the evaporative flux and the source heat due to the penetration of the incident short-wave radiation, which comes with a high degree of uncertainty, needs to be handled carefully. As these complexities introduce approximations and consequently modeling errors, developing a model which be able to simulate the flow variation in a small water body considering the aforementioned uncertainties is very promising. In this paper, we have developed a fully three-dimensional and unsteady hydrothermal model that is capable of simulating the effects of wind and atmospheric conditions over a complex bathymetry to predict the circulation patterns as well as the temperature distribution in the water body. In this model, the atmospheric conditions, with particular attention to heat fluxes over the water surface (sensible and latent heat fluxes), are applied dynamically to reduce the model uncertainties. To verify the capabilities of the model developed in this study, it was applied to a small shallow reservoir in the Upper East Region of Ghana. To evaluate the performance of the model against the observed values of temperature, some quantitative metrics, include root mean square error (RMSE), the mean absolute error (MAE), the relative mean error (RME) and mean error (ME) were applied. From the metrics of model performance evaluation, the results show that the simulated temperature values are in good agreement with the observed values.
2. Water Bodies Modeling
In the last two decades, an increasing interest in predicting the temperature profiles in reservoirs and lakes has been high due to the correlation between temperature, water quantity and water quality [12
]. Transport processes in water bodies are inherently three-dimensional, driven by wind, surface thermodynamics, and the topography of the lake. Hence, assessing the water temperature as well as water circulation, inherently requires transport modelling [13
]. Mathematical modelling of water temperature in lakes and reservoirs have been carried out over the years to investigate thermal dynamics in water bodies [5
]. However, in many real-world cases, it is not always possible to solve the water temperature equations analytically due to the non-linearity of some parameters at the air–water interface [5
] even though water temperature has been simulated in these models at various levels of complexity [16
One-dimensional models (1-D) are extensively applied to estimate vertical temperature profiles in lakes in time. In 1-D models, variations in the lateral directions are assumed to be small with respect to variations in the vertical direction [4
]. In terms of a global or regional coupled atmosphere-lake modeling system for water bodies, 1-D models are the best choices since they require low computational resources and are sufficiently fast for long-term simulations [17
]. Generally, one-dimensional models are not able to consider horizontal advection terms and this seems to be one of the disadvantages.
In the early 1980s, two-dimensional (2-D) laterally averaged models were used extensively to predict the flow field and temperature distribution in water bodies [16
]. Although these 2-D models are computationally efficient and easily implemented, they are not appropriate for simulating flow fields in shallow lakes because these 2-D models are not able to describe the fully three-dimensional flow field in shallow water bodies [12
Due to the inabilities of 2-D models in capturing mechanisms influencing mixing and temperature dynamics precisely, especially in morphometrically complex lakes and reservoirs, a number of three-dimensional models have recently been presented [11
]. Flow field prediction and consequently the temperature dynamics determination in water bodies are accomplishable only through fully 3-D models [16
]. Liu et al.
] developed and applied a three-dimensional finite element model to the subtropical alpine Yuan-Yang Lake (YYL) in northeastern region of Taiwan. Leon et al.
] evaluated the capability of the 3-D model (ELCOM) for coupling it with the Canadian Regional Climate Model (CRCM) on Great Slave Lake, Canada. Politano et al.
] solved a fully three-dimensional model to predict the temperature distribution at McNary Dam using the commercial code Fluent; and a later study by Wang et al.
] developed a 3-D numerical model extending the approach of Politano et al.
] using the open-source code OpenFOAM.
While numerous 3-D models have been described to characterize thermal dynamics in lakes, they have usually been utilized for large and deep lakes where the representation of the boundary geometry is less important than for shallow small lakes [3
]. Only a limited number of CFD simulations for temperature distribution in shallow and small inland water bodies can be found [29
3. Description of Study Site and Data Collection
The Upper East Region of Ghana (UER) has more than 160 small and shallow reservoirs which have different surface areas ranging from 1 to 100 hectares [30
]. These small reservoirs are operationally efficient with their flexibility, closeness to the point of use, and requirement for few parties for management [31
]. The studied lake is a small and shallow reservoir located in this region. Lake Binaba (10°53′20″ N, 00°26′20″ W) is a man-made lake mainly used for irrigation, fishing, livestock watering, construction, domestic uses and recreation. As shown in Figure 1
a natural stream has been dammed, storing and supplying water for all these uses in Binaba, a small town in the sub-humid region of Ghana [32
]. The average lake surface area is estimated around 31 ha with an average and maximum depth of 1.1 m and 4.0 m, respectively. To monitor the meteorological parameters, a floating measurement station was installed over the water surface. Measurements taken included atmospheric parameters (air temperature, wind speed at 2.0 m above the water surface, wind direction and relative humidity), incoming short-wave radiation, sensible heat flux using an Eddy Covariance (EC) System and water temperature profile. These parameters were used to validate the model. Atmospheric measurements and a water thermistor string were situated near the dam body, where the lake depth is around 4.0 m. The water temperature profile was measured with an Onset HOBO Tidbit v2 water temperature data logger with nominal accuracy of ±0.21 °C [33
] and in the following depths: 0.100, 0.200, 0.500, 1.100, 1.550, 1.850, 2.150, 2.800, and 3.465 m.
The air temperature fluctuated from 18.0 to 40.0 °C with an average of 28.7 °C while the water surface temperature varied between 24.0 °C and 32.5 °C with an average of 27.5 °C during the measurement period. Measurements were done between 23 November 2012 and 22 December 2012. A four-day period was selected from the observations to simulate the lake temperature and to validate the model as well. Figure 2
a shows the diurnal changes of air temperature, with daily variations of approximately 16.0 °C. Incoming short-wave (solar) radiation measurements from the atmosphere to the water surface are shown in Figure 2
b. The daily maximum value was recorded around 1:00 p.m. with a value above 800 Wm−2
for most of days. The measured values of relative humidity (RH) over the water surface are shown in Figure 2
c. The wind speed and directions, are shown in Figure 2
d with south-western direction being the most dominant direction with a maximum speed of 4.0 m/s. Since the wind speed values have been averaged over 30-min intervals (as for the other parameters), instantaneous wind speed may be larger. The variation of atmospheric pressure during the study period was very small and could be ignored. Therefore, the pressure was taken to be a constant 102 kPa for all of the simulations.
7. Numerical Results and Discussion
A large number of simulations were run during the model development. The simulation was run for four days (345,600 s) where the starting time of calculations was at 12:00:00 a.m. on 24 November 2012. The simulated flow field in the water body shows the existence of an unsteady and three-dimensional flow for most times due to the effects of the reservoir geometry, dynamic atmospheric conditions and the coupling of energy (temperature) changes with the flow field.
To validate the model, the distribution of simulated temperature in the water body was compared with observations as shown in Figure 4
. For each depth where the temporal temperature profile is depicted in Figure 4
, the mean error (
are measured and simulated temperature values, respectively) or relative mean error (
) are provided as a measure of the bias of the simulated values. The calculated values of ME and RME for each depth are presented in Table 2
. As shown in Table 2
, the maximum difference between the simulated and observed values is −1.60 where the minus sign means the model overestimated the temperature. Alongside the ME and RME, the root mean square error (
) and the mean absolute error (
) are provided as the measures of the overall goodness-of-fit of the simulations to the observations. The values of RMSE between the simulated and observed values of temporal temperature profiles at different depths range from 0.11 to 0.44 °C with an average value of 0.33 °C. Similarly, the calculated MAE ranges from 0.03 to 0.31 °C with an average of 0.21 °C. As was expected considering the depicted temperature profiles in Figure 4
, the RMSE as well as the MAE are increasing in greater depths. A good agreement between simulated and observed temperatures demonstrate the capability of the model to represent temperature dynamics in the small and shallow inland water bodies. These results clearly indicate the variability in the temperature and velocity distributions and the daily thermal cycle predicted by the model for the studied meteorological conditions. Although deviations between modeled and observed temperature profiles at some depths, especially at greater depths were relatively large, general trends and daily temperature fluctuations due to heat transfer are reasonably reproduced by the model. These large deviations between the simulated and observed values are mainly due to the existing uncertainties in thermal boundary condition assigned to the bottom and sides of lake considering the available data or applicable measurements. As shown by Suarez et al.
], in shallow water bodies the thermal interaction between the reservoir bottom, which includes both the bottom and the sides, and the sediment beneath the reservoir significantly affects the reservoir thermal structure. In addition, ignoring the variations of turbidity in the water column and changes of the extinction coefficient of water in this simulation can be considered as the error sources, especially at greater depths [41
]. To evaluate the effects of upper thermal boundary condition (on the water surface) on the temperature profiles, both boundary conditions for temperature described above were considered. In both simulations, the same differences were found between the simulated and observed temperature values at greater depths.
In Figure 4
, the simulated water temperature (S.) and observed values (M.) at different depths are depicted. These temporal profiles of temperature were generated by using the heat flux as the temperature boundary condition (Section 6.1
) on the water surface. These profiles show that in each day of simulation, two different time periods can be detected. In the first time period, which commonly (in the four-day simulated period) ranges from 12:00:00 till 6:00:00 a.m., the simulated temperature matched the observed ones. In this period, due to the good agreement between the modeled and observed temperatures the model can predict the heat budget of the lake precisely. In the second time period, which commonly expands from 6:00:00 a.m. till 12:00:00 a.m., the model overestimated the temperature and consequently the heat content of the lake. As these time periods occur periodically in the simulated period, it seems that the model’s excess heat during the second time period (from 6:00:00 a.m. till 12:00:00 a.m.) is matched by excess cooling at the first period. Therefore, in spite of the uncertainties and errors discussed above, the model could be applied precisely for estimating heat budget of lakes through a one-day time step. This aspect of the model can be promising in energy budget method for evaporation estimation where ignoring the heat content of the lake usually makes significant error in the estimated evaporation from the water surfaces [7
To analyze the simulated temperature profiles with respect to the incoming short-wave radiation, by following the proposed approach by Vercauteren et al.
], the amplitude and the phase shift of the observed and the simulated daily temperature variations (24 November 2012) as a function of the depth are plotted in Figure 5
. As Figure 5
shows the model overestimated both the amplitude and the phase shift in all depths and the differences between the simulated and measured values are increased with depth. Although the applied model in this study is fully 3-D and considers the horizontal flows, analyzing the amplitude and phase shift of the temperature signals helps to find the order of the importance of radiation as well as the turbulent diffusivity (or heat conductivity) on temperature profiles.
As on the selected day (24 November 2012), the wind speed was low (Figure 2
d), it is expected that incoming short-wave radiation has the dominant effect (in comparison with the turbulent diffusivity) on the temperature profiles. The overestimated amplitude and phase shift of temperature signals show that the model overestimated the radiation effects especially on the calm days and consequently radiation can lead to an overestimation of the turbulent heat transfer conductivity [7
]. It can be concluded that the optical properties of the water bodies should be considered carefully in shallow lake models to enable one predict the temperature signals due to the significant effects of radiation on both the amplitude of the temperature oscillations as well as the phase shift.
The vertical simulated temperature distribution through the water body, in seven distinctive time frames were plotted in Figure 6
. These time frames were chosen in a way that they cover both the heating and cooling phases in the lake. To show the performance of the model with respect to the vertical temperature profiles, the measured vertical temperature profiles are plotted as well in Figure 6
with dotted lines for the same time frames. At the beginning part of the simulation (from 12:00:00 a.m. to 7:00:00 a.m.) the water surface is cooling and the value of temperature source (
) is equal to zero. During the cooling time, the wind speed over the water surface is low (less than 1.0 m/s). Looking at the simulated temperature profile, during the cooling phase (at t
= 7 h) there are very small differences at different depths and the lake could be considered as a well-mixed water body (Figure 6
). This condition could be useful in making some simplifications in calculating the heat budget of the lake to calculate evaporation from the water surface in energy budget methods. As the radiative heating intensifies, the water temperature in the top layers near the water surface increase as a consequence of the penetrating short-wave radiation from the water surface. It should be mentioned that during the heating phase (from 7:00:00 a.m. to 6:00:00 p.m.) the values of
over the water surface remains negative. Due to the effects of absorbed radiation, applied as source term in temperature equation, the temperature increases especially in the top layers near the water surface from 7:00:00 a.m. until 2:00:00 p.m. where the incoming short-wave radiation is increasing. As the incoming short-wave radiation decreases on the water surface from 2:00:00 p.m. to 12:00:00 a.m., the temperature decreases in the top layers to reach to the well-mixed condition (at t
= 24 h). The behavior of the water body in the heating phase is completely different from the cooling phase. In the heating phase the water body is not well-mixed; hence, to estimate the heat budget applicable in the evaporation calculation, the non-uniform simulated distribution of temperature is used.
According to Equation (2) in the governing equations of the model, the flow was coupled with energy in the lake. Therefore, changes in temperature impact the flow field. The velocity distribution at different depths and stream lines in the lake show the transient boundary conditions over the water surface and complex bathymetry of lake which make the flow in the lake unsteady and fully 3-D. Assuming that the top grid cells near the free water surface represent the temperature and velocity at the free surface, the horizontal distributions of velocity field at the water surface are presented in Figure 7
and Figure 8
present the velocity fields at two different depths, at the water surface and at 1 m beneath the water surface at 1:00 p.m. As expected, there were return currents at this depth. The simulated vertical velocity profiles show non-uniform distributions, where flow near the bottom and sides tend to follow the bathymetry.
As shown in Figure 9
and Figure 10
, the higher wind speeds caused more mixing in the water column in the vertical direction and consequently lead to higher return flows, which were generated between the surface and deeper layers. Wind induced circulation mainly affects the region near the free water surface and its effects are negligible near the bottom of the lake. In deep regions, this process consequently separates the bottom layer from the top mixed layer and leads to stratification. However, in shallow regions, winds at the water surface can generate circulation throughout the whole depth, from the surface to the bottom of the lake, and therefore in the shallow parts there was no significant stratification most of the time (Figure 13
). In general, as can be seen in Figure 8
and Figure 11
, the velocity distributions in the horizontal section are greatly dependent on wind speed and its direction at the water surface. Higher wind velocities induce strong horizontal circulation as corroborated by Lee [53
shows the simulated temperature values in a horizontal section at 1 m beneath the water surface. As can be seen, the temperature distribution is not uniform and the temperature difference between the points at similar depth is around 1.4 °C. The vertical distribution of temperature in a vertical section is illustrated in Figure 13
, which shows that the behavior of shallow and deep parts are different, and shallow parts respond faster to air heating. Since surface temperature is a complex function of several parameters, such as wind speed, incoming short-wave (solar) radiation, wind direction, humidity, air temperature, etc.
, it is difficult to detect a general clear pattern in water temperature. However, generally, the simulated results indicate that heating during the day is normally related to incoming solar radiation, while cooling at night is more complicated and is more a function of wind speed and its direction. Water in the surface layer starts to warm after sunrise as incoming solar radiation increases (around 7:00:00 a.m.), and this increase continues until short-wave radiation reduces (at 3:00 p.m.), after which surface water temperatures reduce gradually.
Apart from the wind effects on flow field in water bodies, the coupling energy and momentum equations drive the circulation. As solar radiation increases, the temperature in the top layers increases. This increase extends vertically by effective thermal conductivity to the bottom of the lake. With an increase in wind during the day, the velocities at the surface also increases, but no significant vertical circulation is seen throughout the water because of the existence of stable stratification. Water temperature in the top layer is more sensitive to the meteorological conditions. Steeper temperature gradients in the top layers near the water surface are correctly predicted by the model due to the high heat fluxes at the water-air interface.
The main sources of error in the results could be related to the following:
Estimating heat fluxes over the water surface as boundary condition is very uncertain especially for latent heat flux. The location, climate, shape, depth, bathymetry, atmospheric stability conditions, etc. make it difficult to estimate evaporation accurately from the water surface.
There are no measurements for some important parameters that can affect the flow field and temperature in the water body, such as turbidity, and heat fluxes at the bottom and side walls where using simplified temperature boundary conditions could be considered as a source of error.
The measurements were taken only at one point. This means that the distribution of parameters over the water surface was assumed homogeneous. For shallow and small lakes with limited fetch, this assumption could produce a large error in the results.
Coupling the turbulent flow and heat transfer in a shallow water body is complex and computational issues such as numerical errors, mesh dependency and residuals control should be considered.
Errors in field measurements on the water surface especially for water surface temperature or heat fluxes.
Due to the limitation of computational resources, it is not possible to use a finer mesh or very small time steps. In this study different settings for numerical schemes and mesh sizes as well as the time steps were considered to find the optimum situation to make a balance between the needed computational resources and the desired accuracy according to the aims of simulations. For the computational grid used alongside the implemented adaptive time-stepping technique (Section 5.2
), different time step values (
seconds) were used in this simulation to prevent numerical stability issues. Four days of simulations, as described in this paper, took about 20 h on the HPC Cloud-based virtual machine with 12 Intel processors at 2.7 GHz and 96 GB RAM [72