the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Mar 2025
New derivation and interpretation of the complementary relationship for evapotranspiration
Abstract. The complementary relationship (CR) between actual evapotranspiration (ET) and apparent potential evapotranspiration (PETa) is widely used as a simple yet effective method for ET estimation. However, most existing CR formulations are empirical, lacking rigorous derivation based on physics. In this study, the complementary relationship was derived analytically with a physically meaningful parameter: the wet Bowen ratio, defined as the Bowen ratio when the surface becomes saturated. This parameter can be computed from observations without calibration. Fundamentally, the CR is shown to originate from partitioning of the net radiation, with ET directly linked to the latent heat and PETa proportional to the sensible heat. Additionally, ET is linearly related to and constrained by the energy-based potential evapotranspiration (PETe). The physically-based relationship among ET, PETa, and PETe has important implications for our understanding of the spatial and temporal variations in ET and would promote practical application of the complementary relationship for ET estimation across different environments.
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Status: open (until 01 May 2025)
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RC1: 'Comment on egusphere-2025-1124', Anonymous Referee #1, 21 Mar 2025
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The authors claim to have derived a new physically-based CR via ‘rigorous derivation based on physics’, unlike previous versions, which they deem only empirical.
In fact, what the authors achieve is making use of several hypothetical and highly speculative assumptions (lines 210-215):
i) The surface temperature of a small, freely evaporating water body is always the same as that of the surrounding drying land (this would require a heat conduction as effective as evaporative cooling, which is highly unlikely under realistic conditions, thus the corresponding potential evaporation rate remains speculative only);
ii) The Bowen ratio (βw) written for such a small water body does not change during drying of the environment (contradicting the constant surface net radiation assumption stated).
None of the above assumptions are valid in general and none been ever confirmed rigorously by any study.
They proceed further and claim that neither the Penman nor the Priestley-Taylor equation is appropriate for estimating the corresponding apparent potential evaporation rate or the evaporation rate of the wet environment, even though that these equations are the backbone of practically all existing CR methods. Yet, when they decide to discuss the practical applications of their version of the CR they turn to a modified version of the Penman equation with an empirical coefficient (k’) to be determined from measurements (eqs. 25 & 26). Note that the original Penman equation does not have this additional coefficient. Also, as the land surface temperature is typically unknown in practical applications, they introduce another empirical coefficient (α) to convert the Bowen ratio of equilibrium evaporation into βw in eq. (24).
One would expect that when a new method is introduced then its practical predictive superiority is showcased over existing similar methods it is supposed to replace. Such a validation is completely missing here.
The authors’ main equation (eq. 17), when combined with eqs. 12 and 15 yields simply: ET = Rn – H, which is a rather trivial formulation of the energy balance equation. All the authors do is combine this energy balance equation with the definition of the Bowen ratio and express them in a way that looks like a CR equation, i.e., their eq. 17. For βw they use the actual land surface and air temperature plus vapor pressure values (i.e., eq. 11) by capitalizing on assumption ii). An additional problem is that they still need to know H unless they employ the above mentioned modified Penman equation.
So what is the new insight from the authors’ ‘theoretically sound’ CR? I am not sure.
Based on these observations I can only recommend rejection of the manuscript. A thoroughly revised version of the manuscript that is not based on highly questionable assumptions [i.e., i) and ii)] could only be publishable if the authors demonstrate its practical predictive superiority (i.e., that it indeed leads to better ET estimates when differences in the number of parameters to calibrate and input requirements are properly accounted for) over existing CR models and drops any claim that it is a ‘theoretically sound’ and ‘rigorously derived’ CR version (in opposition to other existing CR versions) as all versions of the CR today are empirical to varying degrees, if not else then for the Penman equation (with its empirically derived wind-function) they employ.Citation: https://doi.org/10.5194/egusphere-2025-1124-RC1 -
AC1: 'Reply on RC1', Sha Zhou, 24 Mar 2025
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Please see the supplement for our response letter
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AC1: 'Reply on RC1', Sha Zhou, 24 Mar 2025
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