Forcing data and boundary conditions

Highway 4, approximately 162 km upstream of the Gardiner Dam, was considered to be the upstream boundary of the reservoir in the model. There are no hydrometric stations close to this point in Saskatchewan; therefore, the SSR inflow was calculated based on two

stations in Alberta. The SSR inflow is the sum of two hydrometric gauge stations: “Red Deer River near Bindloss” (#05CK004) (about 150 km upstream of HWY 4) and “South Saskatchewan River at Medicine Hat” (#05AJ001) (about 300 km upstream of HWY 4) (Figure3.1b). The inflow was routed based on the Water Security Agency’s (WSA) time-lag guidelines, as described in Hudson and Vandergucht(2015).

Discharge data for Swift Current Creek, the main local tributary, were obtained from the hydrometric station “Swift Current Creek near Leinan” (#5HD039). The Qu’Appelle River dam outflow data were obtained from the hydrometric station “Elbow Diversion Canal at Drop Structure” (#05JG006). The Gardiner Dam outflow data were obtained from the hydrometric station “South Saskatchewan River at Saskatoon ”(#05HG001). Flows were lagged by one day as suggested by Pomeroy and Shook(2012). The outflow of the Gardiner dam is via the metalimnetic withdrawal supplying the hydroelectric turbines (Phillips et al.,

2015) or through the spillway. Due to lack of data, all the outflow water was considered as overflow. A sensitivity analysis showed that different withdrawal depths affect the reservoir’s heat budget only slightly, by about one degree in June and July. Water levels were obtained from the station “Lake Diefenbaker at Gardiner Dam” (#05HF003).

The inflow temperature was estimated based on correlations between the average weekly air temperature and daily stream temperature, as suggested by Morrill et al. (2005). A linear relationship was found between weekly averaged air temperatures at the “Leader Airport” meteorological station (WMO ID: 71459) and water temperatures from the monitoring station M3 (# 1 in Figure 3.3):

y= 0.85x+TIN (3.1)

where y is the stream water temperature (◦C), x is the weekly air temperature (◦C), and TIN is the inflow temperature coefficient (◦C). TIN = 4.8◦C provided a very good coefficient

of determination (R2 _{= 0.93). Measured water temperature profile data were provided by}

the Limnology Laboratory of the University of Saskatchewan. There were 16 sampling stations extending across the whole reservoir (Figure 3.3) with 57 days of sonde-profile measurements data collected in both summer and fall of 2011 and 2012 (30 days in 2011 and 27 days in 2012). A Yellow Springs Instrument (YSI) sonde, (model 6600 V2) with the

Figure 3.3: Model segmentations and locations of observation stations and meteorological stations

accuracy of ± 0.15 ◦C and the resolution of 0.01 ◦C were used for all the measurements. Details of the sampling are provided in Hudson and Vandergucht(2015).

The surface of Lake Diefenbaker freezes over completely in winter. In early spring, after the ice has melted, the reservoir water temperature profile is assumed to be homothermic since the vertical temperature gradients due to convection and turnover are small. A uniform temperature of 4 ◦_{C throughout the water column at the beginning of April provides a}

convenient point in time to initialize the water temperature simulations (Beletsky and Schwab, 2001; Hondzo and Stefan, 1993a,b; Peeters et al., 2002).

Hourly meteorological data (air temperature, dew point temperature, wind speed and wind direction) were obtained from the Lucky Lake station (WMO ID: 71455) located west of the reservoir (Figure 3.3). Since this is a land station, a wind sheltering coefficient (WSC) was applied to transfer wind speed measurements to the reservoir surface.

Solar radiation is an important component of heat flux and is required for temperature simulations of lakes (Beletsky and Schwab,2001). Global short wave radiation at the surface was calculated based on latitude and time of year, as verified by the National Oceanic and Atmospheric Administration (NOAA) “The NCEP/NCAR Reanalysis 1” solar radiation database.

The light extinction coefficient is the rate of attenuation of solar radiation as it passes vertically through the water column. It controls the amount of solar radiation penetrating the water column and has an impact on stratification and thermal mixing (Jankowski et al.,

2006). Values of the light extinction coefficient were calculated empirically from monthly measurements of the Secchi depth, the depth of water at which the Secchi disk is no longer visible from above the water surface:

= EXT

ZSD

(3.2) where is the light extinction coefficient (m−1_{), Z}

SD is the Secchi depth (m) and EXT is

the extinction coefficient (a constant). Values for EXT depend on lake water characteristics and typically range between 1.3 (e.g.,Lindenschmidt,1998) and 1.7 (e.g.,French et al.,1982).

On average, the surface of Lake Diefenbaker is covered with ice for five months of the year (December-May). Ice layers work as a barrier to solar radiation, wind effects and further cooling of the water. The ice thickness, formation and melting times are important influences on the heat budget, mixing regime and water quality processes in the lake. Physical properties such as albedo and surface temperature change the attenuation of solar radiation in ice and snow, which strongly affects the reservoir’s heat balance (Patterson and Hamblin, 1988). Ice models are necessary to calculate the heat transfer between ice and water. The current version of CE-QUAL-W2 includes an ice model; however, its application is limited due to its omission of a snow pack on the ice cover. In order to include snow effects on the ice surface, two empirical coefficients were added to the ice algorithm to mimic the reduction in heat conduction through the ice, from the water to the air. Although further development would be desirable, this simple adaptation to the algorithm greatly improved ice thickness and ice-on period calculations. Table 3.1 summarizes the governing equations

for calculating ice thickness.

For this current study, ice thickness data from Lake Diefenbaker were not sampled during the winter period of 2011 – 2012. Therefore, the ice coefficients were calibrated using an extensive data set from Blackstrap Lake, a smaller reservoir located approximately 70 km north of Lake Diefenbaker, and then transferred to the Lake Diefenbaker model. The transfer was verified for Lake Diefenbaker using data collected during 2012 – 2013. Ice data, including ice thicknesses and the extent of the ice cover period, were obtained onsite by the Global Institute for Water Security (GIWS) at the Elbow station (site # 10 in Figure 3.3). These data were used to fine-tune the calibration of the extended ice.

In the model, ice thickness has three components (Eq. 3.3): one equation for ice growth and two for melting at the air-ice and ice-water interfaces. The first step was to correct the overestimated ice thickness (θg). θg changes with ice temperature and is a function of

solar radiation, air temperature, evaporation (RE), conduction (RC) and back radiation

from black surfaces (RB) (Eq. 3.4 – 3.7 in Table 3.1). The average values of ice calculation

components in Table 3.2 reveal that RB has a significant effect on the results by yielding

very large thickness values for very low ice temperatures. Therefore, the coefficient α was added to correctRB (Eq. 3.8). αis assigned values between zero and one and was calibrated

to equal 0.63 for Lake Diefenbaker. On some days, RB becomes too small and consequently

RN becomes positive. Positive values of RN act as a melting process, which is not intended

for θg (θg is only for ice growth). Therefore, RN should have a maximum value of zero to

prevent ice melting. During times when the ice is melting (Ti > 0◦C), the model reverts

back to the original equation (Eq. 3.5).

The ice melt algorithm contains two parts as described in Equation 3.3. The values for ice melt at the air-ice interface θA and melt at the ice-water interface θW are listed in

Table 3.2. The values for θA are mainly zero during the ice-covered period; however,

at the time of melting they may have extremely high values, leading to rapid melting of the ice cover. θA is allowed to have larger values than θW, using the coefficient β

(Eq. 3.9) to allow increased melting at the air-ice interface during warmer days in spring with no snow remaining on the ice cover. β was calibrated to be equal to 50 for Lake Diefenbaker. A value of Jday (Julian day) > 90 restricts the criteria to the melt period.

Table 3.1: Ice model equations and parameters θ=θg−θA−θW (3.3) θg = ∆t×RK1×(−Ti) θg×ρIRL1 (3.4) RN =RN1−RB−RE −RC (3.5) ∆ti=RN + RK1×(−Ti) θ (3.6) Ti =Ti+ ∆ti (3.7)

RN =RN1−αRB−RE −RC; &{Ti <0 and RN <0 and 0< α <1} (3.8)

θA=max(θA, β×θW) for Jday>90 (3.9)

θ =ice thickness (m)

θg =ice growth (m)

θA = ice melt at ice and air surface (m)

θW =ice melt at water and ice surface (m)

RK1= ice conductivity (2.12 W/m/◦C)

∆t = changes in time (second)

ρIRL1= constant (305492412) Ti= ice temperature (◦C)

RN = ice heat balance (W/m2)

RN1 = heat due to solar radiation (W/m2). f (air temperature, solar radiation and ice

albedo)

RB = heat lost due to back radiation from black surface (W/m2)

RE = heat lost due to evaporation (W/m2)

RC = heat lost due to conduction (W/m2)

∆ti= changes in ice temperature (◦C)

Jday = Julian day (continuous count of days since the beginning of the year)

α = coefficient for controlling back radiation (ice thickness)

β = coefficient for controllingθA (ice melting)

Table 3.2: Simulated values for ice calculation in CE-QUAL-W2 model in 2012 – 2013 for Blackstrap Lake Ranges in 2012/13 Tice RN RN1 RB RE RC θg (m/hr) θA (m/hr) θW (m/hr) Minimum -20 -138 158 226 -76 -290 0 -2.46 -9.3 e-5 Maximum 0 426 440 361 113 100 1.6 e-3 0 0 Average -4.3 -15 218 289 14 19

Ice formation typically occurs at the end of November/ beginning of December (Jday>300).