The complementary principle is an important methodology for estimating actual evaporation by using routinely observed meteorological variables. This review summaries its 56-year development, focusing on how related studies have shifted from adopting a symmetric linear complementary relationship (CR) to employing generalized nonlinear functions. The original CR denotes that the actual evaporation (E) and “apparent” potential evaporation ($E_{p_{\mathrm{a}}}$ ) depart from the potential evaporation ($E_{p_{\mathrm{0}}}$ ) complementarily when the land surface dries from a completely wet environment with constant available energy. The CR was then extended to an asymmetric linear relationship, and the linear nature was retained through properly formulating $E_{p_{\mathrm{a}}}$ and/or $E_{p_{\mathrm{0}}}$ . Recently, the linear CR was generalized to a sigmoid function and a polynomial function. The sigmoid function does not involve the formulations of $E_{p_{\mathrm{a}}}$ and $E_{p_{\mathrm{0}}}$ but uses the Penman (1948) potential evaporation and its radiation component as inputs, whereas the polynomial function inherits $E_{p_{\mathrm{0}}}$ and $E_{p_{\mathrm{a}}}$ as inputs and requires proper formulations for application. The generalized complementary principle has a more rigorous physical base and offers a great potential in advancing evaporation estimation. Future studies may cover several topics, including the boundary conditions in wet environments, the parameterization and application over different regions of the world, and integration with other approaches for further development.

The generalized complementary function to estimate actual evaporation, namely, the ratio of actual evaporation to Penman potential evaporation (E/E_Pen) as a function of the proportion of the radiation term in E_Pen (E_rad/E_Pen), has been increasingly recognized. Existing analytical forms of this generalized function present deficiencies because they are limited by improper boundary conditions resulting from inadequate understanding of physical constraints. In this study, its zero‐ and first‐order boundary conditions were rigorously derived by adopting the physical constraints for E in Penman's combination theory, and a sigmoid feature of relationship between E/E_Pen and E_rad/E_Pen was derived. Minimum and maximum limits of E_rad/E_Pen were introduced based on the derived boundary conditions, and accordingly, a new sigmoid function was developed. By restricting it to be approximately equivalent to the linear advection‐aridity (Brutsaert & Stricker, 1979, https://doi.org/10.1029/WR015i002p00443) function under normal environments, the new sigmoid function satisfies the upper limits of Penman's open water evaporation and Priestley‐Taylor's minimal advection evaporation in parallel. The sigmoid feature and the new sigmoid function were validated by tower‐based data from FLUXNET. This work improves our understanding of the three‐stage pattern of the complementary behavior, and the new function demonstrates a favorable potential for use in evaporation estimation.

The influences of agricultural irrigation on trends in surface air temperature from 1959 to 2006 over Xinjiang, Northwest China are evaluated using data from 90 meteorological stations. The 90 stations are located in landscapes with markedly different cultivated land uses. The increasing trends in daily average temperature (T a), maximum temperature (T max), and minimum temperature (T min) for May-September (the main growing season) are negatively correlated with cultivated land proportions within 4 km of the meteorological stations, as indicated by year 2000 land use data. The correlations between the trends in T max and cultivated land proportions are the most significant. The trends in T a , T max , and T min for May-September are expected to decrease by-0.018,-0.014, and-0.016 • C per decade, respectively, along with a 10% increase in cultivated land proportion. As irrigated cultivated land occupies over 90% of total cultivated land, the dependence of temperature trends on cultivated area is attributed to irrigation. The cooling effects on stations with cultivated land proportion larger than 50% are compared to temperature trends in a reference group with cultivated land proportion smaller than 10%. The irrigation expansion from 1959 to 2006 over Xinjiang is found to be associated with cooling of May-September T a , T max , and T min by around-0.15 • C to-0.10 • C/decade in the station group with extensive irrigation. Short periods of rapid irrigation expansion co-occurred with the significant cooling of the May-September temperature.

Understanding whether a positive or negative correlation between actual (E) and potential (E_p) evaporation is of great importance in detecting changes of E from E_p. In this study, such correlation was evaluated via the linear regression slope (k) between E and E_p derived from a nonlinear complementary relationship model. k varies with the relative degree of variability in the radiation term (E_rad) and aerodynamic term (E_aero) of E_p, and is further affected by water availability. The sign of k is always positive under conditions with high E_rad variability but at the same time with low E_aero variability (commonly true on an hourly basis). Under conditions with high E_aero variability but at the same time with low E_rad variability, the sign of k tends to change from negative to positive with more readily water availability. On a daily or annual basis, the sign of k would be related with water availability. Under water-limited conditions, the E_aero variability is more significant than the E_rad variability, and negative k is expected. Under energy-limited conditions, the E_rad variability is compared to or even much obvious than the E_aero variability, and positive k is expected. This implies a negative correlation between E and E_p under water-limited conditions but a positive correlation under energy-limited conditions on the daily or annual basis. The above analysis is confirmed in a grassland site in Northeast China at daily and half-hourly time scales. The correlation between annual E and E_p over China was evaluated using data of 690 meteorological stations from 1956 to 2005. The k values change from negative in arid regions to positive in humid regions. According to k and the decreasing trends in annual E_p, it is deduced that E increased for most stations in arid regions but decreased for most stations in humid regions.