Seasonal and Diurnal Variations in the Priestley– Taylor Coefficient for a Large Ephemeral Lake

: The Priestley–Taylor equation (PTE) is widely used with its sole parameter ( α ) set as 1.26 for estimating the evapotranspiration (ET) of water bodies. However, variations in α may be large for ephemeral lakes. Poyang Lake, which is the largest freshwater lake in China, is water ‐ covered and wetland ‐ covered during its high ‐ water and low ‐ water periods, respectively, over a year. This paper examines the seasonal and diurnal variations in α using eddy covariance observation data for Poyang Lake. The results show that α = 1.26 is overall feasible for both periods at daily and subdaily scales. No obvious seasonal trend was observed, although the standard deviation in α for the wetland was larger than that for the water surface. The mean bias in evaporation estimations using the PTE was less than 5 W ∙ m − 2 during both periods, and the root mean square errors were much smaller than the average evaporation measurements at daily scale. U ‐ shaped diurnal patterns of α were found during both periods, due partly to the negative correlation between α and the available energy (A). Compared to the vapor pressure deficit (VPD), wind speed (u) exerts a larger contribution to these variations. In addition, u is positively correlated with α during both periods, however, VPD was positively and negatively correlated with α during the high ‐ water and low ‐ water periods, respectively. Subdaily α exhibited contrasting clusters in the (u, VPD) plane under the same available energy ranges. Our study highlights the seasonal and diurnal course of α and suggests the careful use of PTE at subdaily scales. and wetland ‐ covered periods. Such results indicate that the PT equation is applicable for LE estimation in our study area if the available energy can be


Introduction
Inland water bodies cover less than 4% of the total terrestrial land [1]. However, they exert huge impacts on the terrestrial water cycle, energy cycle, and climate system [2][3][4] because of their higher heat capacity and lower albedo compared to the land surface [5,6]. Understanding the evaporative process of inland waters is therefore important for the application of numerical atmospheric models [7][8][9], especially in regions with large lakes [4]. There are 122 large lakes (>1000 km 2 ) across the globe [10], 10 of which are located in China. Poyang Lake is the largest freshwater lake (3680 km 2 ) in China. The surrounding area of Poyang Lake serves as an important food production base for approximately 10 million people [11]. However, due to hydrological droughts, Poyang Lake's surface area has decreased in the past three decades [12], which may have great negative impacts on the regional economy and natural ecosystem.

Background and General Definitions
In a closed system, evaporation from the water's surface will eventually make the air above the surface saturated (the vapor pressure deficit equals zero). Evaporation is considered equilibrium evaporation (Eeq) under such saturated conditions. Eeq can be theoretically determined as the available energy flux (A) times Δ/(Δ+γ) because the Bowen ratio (B = H/LE) equals γ/Δ in this case, where Δ is the slope of the saturated vapor pressure to the air temperature and γ is the psychrometric constant. Δ and γ are the functions of air temperature and air pressure, respectively. The actual evaporation from wide water surfaces and wetlands is estimated using the concept of equilibrium evaporation and the coefficient α to account for the effect of the drying power of the air on evaporation (Equation (1)). Because A equals Rn − G, α is, therefore, written as Equation (2), where EF is the evaporative fraction (EF = LE/(H + LE)). EF is assumed to be insensitive to wind speed in equilibrium conditions due to the similarity of evaporation and heat conduction [29,30]. The surface heat flux (G) is the heat flux that is conducted from the surface into the water (or from water body to the surface). Due to the high heat capacity of the water body [5,6], the G of lake systems usually takes up a large proportion of the surface energy balance [31][32][33].
Note that Penman [34] derived the evaporation equation (Equation (3)) for water bodies by combining the considerations of the surface energy balance and the heat transfer process (Equation 3). The first term of Equation (3) is theoretically equal to equilibrium evaporation, and the second term accounts for the effect of the drying power of the air (Ea), which is estimated by the product of the vapor pressure deficit and a function of wind speed. γ/(Δ+γ) represents the fraction of the drying power that transfers into latent heat flux. When the vapor pressure deficit tends to be zero, Equation (3) tends to be Eeq.
Therefore, theoretically, the α of water bodies is correlated with meteorological factors. For example, α increases with an increase in wind speed or the vapor pressure deficit due to the increase of Ea if the available energy flux remains constant. Similarly, α decreases with an increase of the available energy flux if the other factors remain constant.

Site Description
Poyang Lake (28°22′-29°45′ N, 115°47′-116°45′ E) is located on the south bank of the Yangtze River. The Poyang Lake basin has a humid subtropical climate with an annual mean air temperature of 17.5 °C and a multiyear mean precipitation of 1635.9 mm for 1960-2010. Five rivers (Xiushui, Ganjiang, Fuhe, Xinjiang, and Raohe) are the main water suppliers to the lake [35], and the lake discharges to the Yangtze River at Hukou (Figure 1). Poyang Lake has an average depth of 8 m with a high degree of seasonal variation [11]. The inundated area varies remarkably from more than 3000 km 2 in summer to less than 1000 km 2 in winter [35,36]. The major river routes are covered by water throughout the year, whereas most of the lake basin is only covered by water during the high-water period. The lake periodically reveals its bottom surfaces (i.e., mudflat, grassland, etc.), during the low-water period. The measuring system in this study was set in a periodically inundated zone of Poyang Lake, which changes from a water surface to a wetland periodically within a year.
The high-water period of Poyang Lake normally extends from April to October. Xingzi station is the best site for representing the overall water level (based on the Wusong elevation datum) status of Poyang Lake [37]. The high-water and low-water periods (See Figure in Section 3) for the EC (eddy covariance) site used in this study reveal the times when the water levels at Xingzi station were higher than 14 m, and lower than 12 m, respectively, according to the study that quantified the relationship between the water surface areas of the Poyang Lake and the water level at the Xingzi station [38]. The times when the water levels are within the range of 12-14 m are defined as the transition period. The water level data of Xingzi station were acquired from the Hydrological Bureau of Jiangxi Province (http://www.jxssw.gov.cn/). The sensible and latent heat fluxes of the lake were measured by eddy covariance devices from a 38 m tower (29.09° N, 116.38° E) on Sheshan Island, which is located at the center of Poyang Lake ( Figure 1). The EC system measures fluxes from the effective upwind source, which can be determined by a footprint analysis of the relative contributions of the flux at each point (x,y) to the EC site (x0,y0). A footprint analysis using the Kljun model [39,40] reveals that the major source area (85%) was within 1~2 km of the wind direction under unstable conditions. Compared to unstable conditions, the extent of the 85% source area was larger under stable conditions. However, the major source area was still within 6 km in most cases except when the surface layer was extremely stable (Zm/L0 > 1, Zm is the measuring height and L0 is the Monin-Obukhov length). Data were discarded when Zm/L0 > 1. Due to the relatively small area of the island (less than 1 km 2 ), the flux contribution from Sheshan Island can be ignored because it is usually small, less than 5% from the major northwest winds ( Figure 2B). The water coverage fractions within the EC footprints were generally larger than 90% and smaller than 20% during the high-water and low-water periods, respectively [27]. The data used in this paper are deposited in a public domain repository (https://figshare.com/articles/data_water-717367/11968551).

Data Processing
Turbulences in the air temperature, humidity, and three-dimensional wind velocity were measured by a CO2/H2O analyzer and a 3-D Sonic Anemometer (EC150, Campbell Scientific Inc., Logan, UT, USA). The eddy covariance technique and subsequent corrections were then used to estimate the 30 min scale H and LE using the high-frequency measurements of departures from the mean values. H and LE were calculated using the following equations (Equation (4) and Equation (5)): where ρ is the dry air density (kg•m −3 ), cp is the specific heat of dry air (J•g −1 •K −1 ), and λ is the latent heat of vaporization (J•g −1 ), which is dependent on the air temperature. w′, T′, and q′ are the deviations from the time-averaged vertical wind velocity (m•s −1 ), air temperature (K), and water vapor density (g•m −3 ), respectively. In addition, the meteorological forcings were measured at a 30 min scale. The surface radiation components, including the downward/upward short-wave and long-wave radiations, were measured by the pyrgeometers/pyranometers (CNR4, Kipp & Zonen B.V., Delft, The Netherlands). The air temperature and relative humidity were measured by an HMP155A (Vaisala, Helsinki, Finland). We handled the half-hourly flux measurements using the spike detection technique [41]. The half-hourly data at the timings when the downward shortwave radiation was larger than zero were chosen for diurnal analysis. We also averaged the chosen 30 min scale data for each day to analyze the seasonal variations in α. Overall, we analyzed the variations in α and their controlling factors at two temporal scales-subdaily (diurnal variations) and daily (seasonal variations)-in both the high-water and low-water periods. Because of the large uncertainty in determining G for the water surface, the available energy was surrogated by the sum of the measured H and LE. Because evaporation is mainly controlled by the energy input and aerodynamic transport of water vapor, we chose two categories of controlling factor candidates of α in this study: first, the available energy that drives the land-atmosphere energy transfer process and second, the factors that reflect the evaporative demand of the air and local advection, that is, the wind speed (u) and water vapor pressure deficit of the air (VPD).
We used the stepwise regression method to explore the first-order controlling factors of α. Stepwise regression is a systematic method for constructing an explanatory multilinear model. It adds or removes one of the predictive terms (x1, x2, …, xi, …, xn) from the model at each step based on the statistical significance of the term in the regression. Pcoeff is the p-value of the estimated coefficient of a candidate term xi. A potential predicted term is accepted in the final regression model (status set as ′′In′′) if Pcoeff and PF are smaller than the threshold of 0.05. The stepwise analysis was conducted using MATLAB 2015b.

Environmental Conditions and Energy Fluxes
The  Over-lake observations of energy fluxes in 2015 are shown in Figure 4. LE displayed a unimodal seasonal pattern, which generally matches those of the driving forces (e.g., Rs↓, Rl↓, and Ta). LE peaked in the summer (376.1 W•m −2 ), and the minimum value (−26.9 W•m −2 ) occurred in the winter. Compared to LE, H displayed an opposite trend with a smaller mean (11.3 W•m −2 ) in the high-water period than that in the low-water period (16.9 W•m −2 ), although the available energy was larger when the water level was higher (92.2 W•m −2 vs. 62.7 W•m −2 on average) ( Table 1). One reason for the smaller H in the summer is that a larger portion of the available energy was consumed by LE (EFmean = 0.86, EF = LE/(H + LE)) in the high-water period than that (EFmean = 0.71) in the low-water period.  Compared to those in the daily scale scenario, both energy fluxes and the evaporative fraction EF exhibited much larger variations at subdaily scales ( Figure 5). However, the diurnal cycle patterns for each variable can also be seen from the averages of the values at the same timings of each day ( Figure 5). During the daytime, H exhibited a weak increase over time from early morning and reached its maximum at noon (around 11:30) and then decreased slowly afterward during the high-water period ( Figure 5A). Similar, but more obvious, trends for H and LE were found in the low-water period, whose peaks occurred just after noon (i.e., at 13:30) ( Figure 5B). In contrast, the LE in the high-water period increased after 9:30, maintained relatively high values from 13:00 until 16:00, and then decreased. These results indicate that, compared to the mudflat, the existence of the water surface smooths the variations in H, delays the time of the peak value of LE, and prolongs the duration of the high values.

Seasonal and Diurnal Variations in α
The seasonal and diurnal variations in α are explored in this section. α showed no obvious seasonal trend as Poyang Lake experienced water-rising and water-falling ( Figure 6). The mean α was 1.25 during the high-water period, which is close to the widely accepted value of 1.26. A similar but slightly larger mean α (1.29) was found during the low-water period ( Figure 6 and Table 2). This result indicates that the PT equation with α = 1.26 can yield reasonable ET estimates on average over the lake and wetland surfaces in Poyang Lake. However, α exhibits large variations around 1.26 during both periods. The standard deviation (sd) in α during the high-water period was 0.19, which is 15.2% of the mean value. In contrast, α during the low-water period exhibited a larger variation (sd = 0.31) than that during the high-water period.  At subdaily scales, α was 1.27 and 1.30 on average during the high-water and low-water periods, respectively, indicating that α = 1.26 is also reasonable at subdaily scales. However, unlike seasonal variations, α generally showed U-shaped diurnal variations during both periods, especially during the high-water period. During the high-water period, α generally decreased from 8:30 to its minimum value at around 11:30, and then increased to relatively high values in the evening (17:00) (Figure 7). The parameter α exhibited large variations during the daytime for the low-water period.

The Controlling Factors of Parameter α
We performed a stepwise regression analysis to determine the first-order controlling factors of α. We selected the factors that are related to the advection (VPD and u) and the factor that represents the energy supply for sensible and latent heat fluxes (i.e., the available energy) for analysis. Statistics of the regression results are shown in Table 3. RMSE (root mean square error) of the daily predictions were 0.146 and 0.256 during the high-water and low-water periods, respectively, which were relatively small (~11.6%-20.3%) compared to the mean value of α. In addition, although the RMSE of the subdaily predictions (0.214 and 0.329) were larger than those in the daily cases, they were still smaller than 30% of the mean value of α, indicating the capability of the regression equations in explaining the variations α.
We found that the available energy is negatively correlated with α during both the high-water and low-water periods at subdaily scales ( Table 3). Note that the available energy exhibits unimodal patterns at subdaily scales ( Figure 8). These results partly explain the U-shaped diurnal patterns in α. VPD exerts positive and negative impacts on subdaily α during the high-water and low-water periods, respectively. In contrast, u exerts positive impacts during both periods. Note that the VPD and wind speed exhibit increasing and decreasing trends, respectively, from 8:00 to 17:00 during both periods (Figure 8). VPD and u tend to cancel out one another's impacts on α during the high-water period, but amplify one another's impacts on α during the low-water period.  Table 3. Stepwise regression analysis for α during both the high-water and low-water periods at daily and subdaily scales. RMSE represents the root mean squared error of the prediction from the regression equation. r represents the correlation coefficient between the predictions and "ground-truth" α. The distributions of α with VPD and u are further analyzed under different available energy ranges (Figure 9 and Figure 10). We found that extreme (large) values generally occur at the lowest available energy range ( Figure 9A and Figure 10A). The role of wind speed is significant in most of the available energy ranges at both time scales. Compared to wind speed, VPD exerts less obvious impacts on α. However, α tends to be highest when VPD is close to 0 in most of the available energy ranges. As the available energy increases, α seems to be clustered in different (u, VPD) regions, which can be separated by lines of VPD = a × u + b ( Figure 9E, F). For example, the points above and below the line y = x (where x = u and y = VPD) were generally smaller and larger than 1.26, respectively, when the available energy was larger than 310 W•m −2 during the high-water period (e.g., Figure 9E).

Contributions of Local Advection and Energy Control to the α Variations
According to Penman's equation [34], α increases with an increase of wind speed or VPD due to the increase of Ea if the available energy remains constant under water-surface conditions. We estimated the contributions of VPD and u to the α variations by multiplying the standard deviations in VPD and u with their respective coefficients of VPD and u from Table 3. The results are shown in Table 4. Compared to wind speed, VPD exerted much smaller effects on α during the high-water period (i.e., 0.06 vs. 0.11 at a daily scale) ( Table 4). This result may be because evaporation from the lake exerted feedbacks on the ambient air humidity. Therefore, a low VPD (small atmospheric demand) does not necessarily indicate that the evaporation is small. Instead, a small VPD may be the result of a large latent heat flux from the lake.
Compared to the high-water period, VPD was found to exert a larger and more negative impact on daily α during the low-water period, indicating that LE decreases with Ea if the air temperature and available energy remain constant. As Granger and Gray [42] noted, VPD reflects the moisture conditions of an unsaturated surface to some extent (i.e., a higher VPD indicates a drier surface). Therefore, this complementary effect may be responsible for the negative correlation between α and VPD during the low-water period. Our results show that α exhibited no obvious seasonal trend throughout the year, whereas diurnal variation was characterized as U-shaped under wetland surface conditions. This result is consistent with the diurnal U-shaped EF patterns found for vegetative surfaces [43,44]. The stepwise regression analysis shows that a negative correlation exists between the available energy and α, which is partly responsible for the U-shaped diurnal variations over both periods.
Worth noting, compared to daily scales, clusters of subdaily α in different (u, VPD) regions were found across a range of available energy values (Figure 9). The natural speculation is that the two clusters of points that were above and below the y = ax + b line occurred at distant timings. However, this speculation is not true because the DOYs above and below the y = ax + b line exhibited no obvious contrast at each available energy range. A more probable explanation is that the contrasting clusters of points in Figure 9 occurred at the timings in adjacent DOYs or in the same DOY and that the diurnal variations in α caused the emergence of the clusters. For example, the available energy was similar at 8:00, 11:00, and 11:30 in DOY 155 (i.e., 209.7, 206.8, and 194.2 W•m −2 , respectively). The corresponding α at those timings was 1.32, 1.09, and 1.07, respectively. In the morning, u was relatively large and VPD was relatively small; thus, the point at 8:00 is located below the y = ax + b line. Further, u decreases and VPD increases with time ( Figure 8). Therefore, points at 11:00 and 11:30 are located above the y = ax + b line, and α remained relatively low (1.09 and 1.07) at noon. In summary, because u decreases and VPD increases with time during the day (Figure 8), the points in Figure 9 and Figure 10 tend to rotate in a counterclockwise direction in the u-VPD plane. Because wind speed makes the most (positive) contribution to diurnal α variations (Table 4), clusters of subdaily α in different (u, VPD) regions were found.

Use of the PT Equation in Changing Surface Conditions
The Priestley-Taylor equation [18] has been widely used as a stand-alone model [45] or as a core part in many ET models [46][47][48][49][50]. The remarkable changes in the water level of Poyang Lake provide a unique opportunity for comparatively studying the variations in α and their controlling factors over water-covered and non-water-covered surfaces. Although the meteorological conditions were quite different between the high-water and low-water periods at our study site, we can still deduce that the existence of the water surface exerted a significant impact on the evaporative process. For example, although the available energy was higher in the high-water period, the H in the high-water period was smaller on average than in the low-water period. In addition, the existence of the water surface remarkably smoothed the variation in LE.
Compared with energy fluxes, no obvious seasonal trend was observed for α. In addition, α = 1.26 is overall feasible for both water-covered and wetland-covered periods. Such results indicate that the PT equation is applicable for LE estimation in our study area if the available energy can be determined. The accuracy of the LE estimation using the PT equation (with α as 1.26) is shown in Table 5. During both the water-covered and wetland-covered periods, the mean biases were less than 5 W•m −2 , indicating that the PT equation is feasible in estimating the overall latent heat flux of Poyang Lake at an annual scale. In addition, the root mean square error (RMSE) of the LE predictions at a daily scale was much smaller than the mean measurements in LE, indicating that the PT equation (with α as 1.26) is capable of modeling seasonal ET variations. Worth noting, the RMSE of the LE predictions at a 30 min scale was still less than 50% of the mean ET measurements during the water-covered period. Considerations of the diurnal courses of α and the impacts of wind speed are needed to better model LE at a 30 min scale, especially during the wetland-covered period.

Conclusions
The Priestley-Taylor equation has been proven to be robust in estimating the long-term ET of water bodies. However, its sole parameter α may deviate from the widely accepted value of 1.26, especially under ephemeral lake conditions. In this study, we provided a detailed examination of the seasonal and diurnal variability of α over a large ephemeral lake. The results showed that α = 1.26 is overall feasible for not only the water-covered but also the wetland-covered periods. α exhibits no obvious seasonal trend, whereas U-shaped diurnal patterns were found during both periods. Previous studies showed that local advection causes a deviation in α. However, stepwise regression analysis indicates that such U-shaped variations are also correlated with variations in the available energy. Our study indicates that the PT equation is still applicable at subdaily scales if the diurnal variations of α are properly accounted for.