The Priestley–Taylor (PT) coefficient (

Evaporation from wet surfaces, including oceans, lakes, and reservoirs, is relevant to global hydrological cycles and water availability. There is a long history of developing theories and methods to estimate wet-surface evaporation (Bowen, 1926; Penman, 1948; Priestley and Taylor, 1972; Thornthwaite and Holzman, 1939; Yang and Roderick, 2019). Among the existing models, the Priestley–Taylor (PT) model or equation is known for its transparent structure and low input requirement (Priestley and Taylor, 1972). The PT equation is widely used in evaporation estimation across varied scales and is the basis for various hydrologic and land surface models. Specifically, the PT equation comes from the equilibrium evaporation (

Atmospheric boundary layer box model describing the energy and water fluxes at the saturated surface and in the atmosphere above. The dotted line represents the removable upper boundary of the box.

In this case, the PT equation introduced a parameter,

A general method to quantify the changes in

Based on a recent study by Liu and Yang (2021), here, we aim to derive a physically clear, transparent, and calibration-free equation for estimating

Here, we use a model based on the atmospheric boundary layer (ABL) as the basis for the Bowen ratio (Bo, defined as the ratio of sensible heat fluxes to latent heat fluxes,

According to Eqs. (3) and (4), we can obtain a formula to calculate the rate of VPD (

When air is not saturated, we can rewrite Eq. (6) as follows:

Under a relatively long-term period (monthly and/or longer), there is a potential VPD budget

The surface energy balance is expressed as follows:

Comparison between observations and

According to the above derivations, we can know that

Firstly, we decompose

For practical use, we simplified Eqs. (20) and (21) as follows:

Values of

We select data from eddy covariance measurements on several water surfaces (Han and Guo, 2023): (i) Lake Taihu, located in the Yangtze River Delta, China, with an area of

Location and date period of each waterbody.

Observations from global flux sites (FluxNet2015 database) are also selected. We first examine days without water stress based on the following steps (Maes et al., 2019). At each site, the evaporative fraction (i.e., EF, latent heat flux over the sum of latent and sensible fluxes) is first calculated, and the days with an EF exceeding the 95th-percentile EF and with an EF larger than 0.8 remain. Secondly, the days with soil moisture lower than 50 % of the maximum soil moisture (taken as the 98th percentile of the soil moisture series) are removed. Days having rainfall and negative values of latent and sensible heat fluxes are also not included. As a result, a total of

We also collect ocean surface data from 11 CMIP6 models (under scenario SSP585, Table 2) from 2021 to 2100 to see the temporal changes in

CMIP6 models used in this study.

CSIRO is the Commonwealth Scientific and Industrial Research Organization, CCCma is the Canadian Centre for Climate Modelling and Analysis, NCAR is the National Center for Atmospheric Research, CMCC is the Euro-Mediterranean Center on Climate Change, CAS is the Chinese Academy of Sciences, MPI-M is the Max Planck Institute for Meteorology, and NCC is the Norwegian Climate Centre.

We used yearly and monthly (from January to December) climatology data from Lake Taihu to investigate the temporal variation in

Temporal and spatial relationships of

Sensitivity of

Spatial relationships of

Based on Eqs. (20) to (22),

Contributions of changes in temperature (

Note that, since

Derived

Based on CMIP6 ocean surface data, we also detected significant negative relationships of

Temporal relationships of

Ocean surface temperature, specific humidity, and heat fluxes for the first 10 years (2021–2030) and for the end of the 21st century (2091–2100).

Stylized diagram showing the average changes in heat fluxes over global ocean surfaces.

In this study, we employed an open boundary layer model with a governing potential VPD budget (Raupach, 2001, 2000), originally integrated by Liu and Yang (2021), to formulate an expression for the Priestley–Taylor coefficient,

It should be noted that, except for the PT model, the PM-based model can also be used to estimate wet-surface evaporation (Penman, 1948; Shuttleworth, 1993). While PM-based equations encapsulate all processes that possibly affect evaporation, the PT model, taking evaporation as a simple function of radiation and temperature, takes more account of the feedback and/or balance between the surface and near-atmosphere (Fig. 1). Besides, it has been noted that the PM-based models may fail at certain points and cannot capture the sensitivity of evaporation to temperature changes (Liu et al., 2022; McColl, 2020). So, in this case, also considering the fact that the PT model is currently one of the most popular equations due to its low input requirements, revisiting this classic model can greatly promote its adaption under the changing climate. Meanwhile, some revised PT equations can also be used to estimate the parameter

In Sect. 2.1, it was suggested that

We recommend utilizing the derived model under warm conditions, for example, when the air temperature exceeds zero, to account for the prerequisite of a well-mixed boundary layer. In extremely cold regions or seasons, the water surface temperature can be lower than the air temperature, resulting in a downward sensible heat flux (De Bruin, 1982). Under such circumstances, the boundary layers exhibit relative stability and may not reach a well-mixed state. Additionally, we advise adopting a temporal scale ranging from weekly to monthly when applying the derived model. This is because the potential VPD budget (the governing equation) may not be rapidly achieved, such as on a diurnal or daily basis. Furthermore, over a longer term, the sensible heat flux typically manifests as upward in the majority of scenarios compared to a fine temporal scale.

The derived formula for

Data of Lake Taihu can be obtained from the Harvard Dataverse,

Conceptualization: ZL, HY. Data curation: ZL. Formal analysis: ZL. Funding acquisition: HY. Methodology: ZL, HY. Software: ZL. Supervision: HY. Writing – original draft: ZL. Writing – review and editing: CL, TW, HY.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the National Natural Science Foundation of China (grant nos. 51979140 and 42041004).

This paper was edited by Bob Su and reviewed by two anonymous referees.