The complementary relationship (CR) between actual evapotranspiration (ET) and apparent potential evapotranspiration (PETa) is widely adopted as a simple yet powerful approach for ET estimation over land. However, most existing CR formulations remain empirical, largely due to a lack of clear physical interpretation of its key parameter. In this study, we show that the CR naturally emerges from the surface energy balance and clarify the physical meaning of its parameter: the wet Bowen ratio, defined as the Bowen ratio when the surface becomes saturated. Fundamentally, the CR originates from the partitioning of available energy: ET is directly linked to the latent heat flux, while PETa is proportional to the sensible heat flux. Additionally, the CR can be interpreted as the atmospheric response (encapsulated by PETa) to ET dynamics across wet and dry conditions. In contrast, ET exhibits a positive relationship with the energy-based potential evapotranspiration (PETe), which controls and drives ET over land. This physically grounded relationship among ET, PETa, and PETe advances our understanding of the spatial and temporal variations in ET, as well as its critical role in land-atmosphere interactions, thereby facilitating the practical application of the CR for ET estimation across diverse environments.
- Open Access
- Article
Physical Interpretation of the Complementary Relationship for Evapotranspiration
- Sha Zhou 1,2,*,
- Bofu Yu 3
Author Information
Received: 16 Nov 2025 | Revised: 08 Jan 2026 | Accepted: 19 Jan 2026 | Published: 02 Feb 2026
Abstract
References
- 1.
Katul, G.G.; Oren, R.; Manzoni, S.; et al. Evapotranspiration: A Process Driving Mass Transport and Energy Exchange in the Soil-Plant-Atmosphere-Climate System. Rev. Geophys. 2012, 50. https://doi.org/10.1029/2011RG000366.
- 2.
Fisher, J.B.; Melton, F.; Middleton, E.; et al. The Future of Evapotranspiration: Global Requirements for Ecosystem Functioning, Carbon and Climate Feedbacks, Agricultural Management, and Water Resources. Water Resour. Res. 2017, 53, 2618–2626. https://doi.org/10.1002/2016WR020175.
- 3.
Yang, Y.; Roderick, M.L.; Guo, H.; et al. Evapotranspiration on a Greening Earth. Nat. Rev. Earth Environ. 2023, 4, 626–641. https://doi.org/10.1038/s43017-023-00464-3.
- 4.
Gu, B.; Zhou, S.; Yu, B.; et al. Multifaceted Changes in Water Availability with a Warmer Climate. npj Clim. Atmos. Sci. 2025, 8, 31. https://doi.org/10.1038/s41612-025-00913-4.
- 5.
Wang, K.; Dickinson, R.E. A Review of Global Terrestrial Evapotranspiration: Observation, Modeling, Climatology, and Climatic Variability. Rev. Geophys. 2012, 50. https://doi.org/10.1029/2011RG000373.
- 6.
Zhang, Y.; Kong, D.; Gan, R.; et al. Coupled Estimation of 500 m and 8-Day Resolution Global Evapotranspiration and Gross Primary Production in 2002–2017. Remote Sens. Environ. 2019, 222, 165–182. https://doi.org/10.1016/j.rse.2018.12.031.
- 7.
Miralles, D.G.; Bonte, O.; Koppa, A.; et al. GLEAM4: Global Land Evaporation and Soil Moisture Dataset at 0.1 Resolution from 1980 to near Present. Sci Data 2025, 12, 416. https://doi.org/10.1038/s41597-025-04610-y.
- 8.
Zhang, K.; Kimball, J.S.; Running, S.W. A Review of Remote Sensing Based Actual Evapotranspiration Estimation. WIREs Water 2016, 3, 834–853. https://doi.org/10.1002/wat2.1168.
- 9.
Zhou, S.; Keenan, T.F.; Williams, A.P.; et al. Large Divergence in Tropical Hydrological Projections Caused by Model Spread in Vegetation Responses to Elevated CO2. Earth’s Future 2022, 10, e2021EF002457. https://doi.org/10.1029/2021EF002457.
- 10.
Raoult, N.; Douglas, N.; MacBean, N.; et al. Parameter Estimation in Land Surface Models: Challenges and Opportunities With Data Assimilation and Machine Learning. J. Adv. Model. Earth Syst. 2025, 17, e2024MS004733. https://doi.org/10.1029/2024MS004733.
- 11.
Bouchet, R.J. Evapotranspiration Reelle, Evapotranspiration Potentielle, et Production Agricole. Ann. Agron. 1963, 14, 743–824.
- 12.
Morton, F.I. Operational Estimates of Areal Evapotranspiration and Their Significance to the Science and Practice of Hydrology. J. Hydrol. 1983, 66, 1–76. https://doi.org/10.1016/0022-1694(83)90177-4.
- 13.
Han, S.; Tian, F. A Review of the Complementary Principle of Evaporation: From the Original Linear Relationship to Generalized Nonlinear Functions. Hydrol. Earth Syst. Sci. 2020, 24, 2269–2285. https://doi.org/10.5194/hess-24-2269-2020.
- 14.
Zhang, L.; Brutsaert, W. Blending the Evaporation Precipitation Ratio With the Complementary Principle Function for the Prediction of Evaporation. Water Resour. Res. 2021, 57, e2021WR029729. https://doi.org/10.1029/2021WR029729.
- 15.
Ma, N.; Szilagyi, J.; Zhang, Y. Calibration‐Free Complementary Relationship Estimates Terrestrial Evapotranspiration Globally. Water Resour. Res. 2021, 57, e2021WR029691. https://doi.org/10.1029/2021WR029691.
- 16.
Ma, N.; Szilagyi, J.; Zhang, Y. Hydrological Responses to Warming: Insights From Centennial-Scale Terrestrial Evapotranspiration Estimates. Water Resour. Res. 2025, 61, e2025WR041001. https://doi.org/10.1029/2025WR041001.
- 17.
Brutsaert, W. A Generalized Complementary Principle with Physical Constraints for Land‐surface Evaporation. Water Resour. Res. 2015, 51, 8087–8093. https://doi.org/10.1002/2015WR017720.
- 18.
Zhou, S.; Yu, B. Physical Basis of the Potential Evapotranspiration and Its Estimation over Land. J. Hydrol. 2024, 641, 131825. https://doi.org/10.1016/j.jhydrol.2024.131825.
- 19.
Kahler, D.M.; Brutsaert, W. Complementary Relationship between Daily Evaporation in the Environment and Pan Evaporation: Daily and Pan Evaporation. Water Resour. Res. 2006, 42. https://doi.org/10.1029/2005WR004541.
- 20.
Brutsaert, W.; Parlange, M.B. Hydrologic Cycle Explains the Evaporation Paradox. Nature 1998, 396, 30. https://doi.org/10.1038/23845.
- 21.
Granger, R.J. A Complementary Relationship Approach for Evaporation from Nonsaturated Surfaces. J. Hydrol. 1989, 111, 31–38. https://doi.org/10.1016/0022-1694(89)90250-3.
- 22.
Szilagyi, J. On the Inherent Asymmetric Nature of the Complementary Relationship of Evaporation. Geophys. Res. Lett. 2007, 34, L02405. https://doi.org/10.1029/2006GL028708.
- 23.
Crago, R.; Szilagyi, J.; Qualls, R.; et al. Rescaling the Complementary Relationship for Land Surface Evaporation. Water Resour. Res. 2016, 52, 8461–8471. https://doi.org/10.1002/2016WR019753.
- 24.
Aminzadeh, M.; Roderick, M.L.; Or, D. A Generalized Complementary Relationship between Actual and Potential Evaporation Defined by a Reference Surface Temperature. Water Resour. Res. 2016, 52, 385–406. https://doi.org/10.1002/2015WR017969.
- 25.
Szilagyi, J.; Crago, R.; Qualls, R. A Calibration-Free Formulation of the Complementary Relationship of Evaporation for Continental-Scale Hydrology. J. Geophys. Res. Atmos. 2017, 122, 264–278. https://doi.org/10.1002/2016JD025611.
- 26.
Szilagyi, J.; Ma, N.; Crago, R.D.; et al. Power-Function Expansion of the Polynomial Complementary Relationship of Evaporation. Water Resour. Res. 2022, 58, e2022WR033095. https://doi.org/10.1029/2022WR033095.
- 27.
Tu, Z.; Yang, Y.; Roderick, M.L.; et al. Potential Evaporation and the Complementary Relationship. Water Resour. Res. 2023, 59, e2022WR033763. https://doi.org/10.1029/2022WR033763.
- 28.
Zhang, L.; Cheng, L.; Brutsaert, W. Estimation of Land Surface Evaporation Using a Generalized Nonlinear Complementary Relationship. J. Geophys. Res. Atmos. 2017, 122, 1475–1487. https://doi.org/10.1002/2016JD025936.
- 29.
Brutsaert, W.; Cheng, L.; Zhang, L. Spatial Distribution of Global Landscape Evaporation in the Early Twenty-First Century by Means of a Generalized Complementary Approach. J. Hydrometeorol. 2020, 21, 287–298. https://doi.org/10.1175/JHM-D-19-0208.1.
- 30.
Tu, Z.; Yang, Y.; Ruan, F.; et al. Global Terrestrial Evaporation from Physically-Based, Calibration-Free Complementary Relationship. J. Hydrol. 2025, 660, 133382. https://doi.org/10.1016/j.jhydrol.2025.133382.
- 31.
Priestley, C.H.B.; Taylor, R.J. On the Assessment of Surface Heat Flux and Evaporation Using Large-Scale Parameters. Mon. Wea. Rev. 1972, 100, 81–92.
- 32.
Penman, H.L. Natural Evaporation from Open Water, Bare Soil and Grass. Proc. R. Soc. London. Ser. A Math. Phys. Sci. 1948, 192, 120–145.
- 33.
Zhou, S.; Yu, B. Neglecting Land–Atmosphere Feedbacks Overestimates Climate-Driven Increases in Evapotranspiration. Nat. Clim. Chang. 2025, 15, 1099–1106. https://doi.org/10.1038/s41558-025-02428-5.
- 34.
Szilagyi, J. On the Thermodynamic Foundations of the Complementary Relationship of Evaporation. J. Hydrol. 2021, 593, 125916. https://doi.org/10.1016/j.jhydrol.2020.125916.
- 35.
Yang, Y.; Roderick, M.L. Radiation, Surface Temperature and Evaporation over Wet Surfaces. Q. J. R. Meteorol. Soc. 2019, 145, 1118–1129. https://doi.org/10.1002/qj.3481.
- 36.
Kim, Y.; Garcia, M.; Johnson, M.S. Land-Atmosphere Coupling Constrains Increases to Potential Evaporation in a Warming Climate: Implications at Local and Global Scales. Earth’s Future 2023, 11, e2022EF002886. https://doi.org/10.1029/2022EF002886.
- 37.
Zhou, S.; Yu, B. Reconciling the Discrepancy in Projected Global Dryland Expansion in a Warming World. Glob. Chang. Biol. 2025, 31, e70102. https://doi.org/10.1111/gcb.70102.
- 38.
Pastorello, G. The FLUXNET2015 Dataset and the ONEFlux Processing Pipeline for Eddy Covariance Data. Sci. Data 2020, 7, 225.
- 39.
Zhou, S.; Williams, A.P.; Berg, A.M.; et al. Land–Atmosphere Feedbacks Exacerbate Concurrent Soil Drought and Atmospheric Aridity. Proc. Natl. Acad. Sci. USA 2019, 116, 18848–18853. https://doi.org/10.1073/pnas.1904955116.
- 40.
Milly, P.C.D.; Dunne, K.A. A Hydrologic Drying Bias in Water-Resource Impact Analyses of Anthropogenic Climate Change. JAWRA J. Am. Water Resour. Assoc. 2017, 53, 822–838. https://doi.org/10.1111/1752-1688.12538.
- 41.
Milly, P.C.D.; Dunne, K.A. Potential Evapotranspiration and Continental Drying. Nat. Clim. Chang. 2016, 6, 946–949. https://doi.org/10.1038/nclimate3046.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Suite 4002 Level 4, 447 Collins Street, Melbourne, Victoria 3000, Australia
General Inquiries: info@sciltp.com