Derivation of the Penman-Monteith equation with the thermodynamic approach. I: A review and theoretical development

Authors: ZERIHUN, DAWIT - University Of Arizona; SANCHEZ, CHARLES - University Of Arizona; French, Andrew

Submitted to: American Society of Civil Engineers Journal of Irrigation and Drainage
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 10/13/2022
Publication Date: 3/10/2023
Citation: Zerihun, D., Sanchez, C.A., French, A.N. 2023. Derivation of the Penman-Monteith equation with the thermodynamic approach. I: A review and theoretical development. American Society of Civil Engineers Journal of Irrigation and Drainage. 149(5).Article 04023007. https://doi.org/10.1061/JIDEDH.IRENG-9887.
DOI: https://doi.org/10.1061/JIDEDH.IRENG-9887

Interpretive Summary: The Penman-Monteith Equation (PM) models steady-state evapotranspiration (ET) from soil and plants into the overlying air and is considered a foundation for ET under well-watered extensive landscapes. Its creation in 1948, and extension in 1965, has provided a physically based general method for estimating ET anywhere over the Earth. However, literature describing the equation’s derivation has explanatory gaps that make full understanding of PM difficult. Filling those gaps can be done by reviewing the PM derivations for calm and windy conditions. PM is derived as a flux momentum process, and then again as a thermodynamic process. The flux approach is simpler to derive while the thermodynamic one reveals physical assumptions in a clear way. Results from this study will be useful for scientists, engineers and students to gain a more complete understanding of the PM derivation and how it represents ET.

Technical Abstract: A review of the thermodynamic approach to the derivation of the Penman-Monteith equation, proposed by Monteith, is presented. Evaporation is described here as a process consisting of a pair of formal thermodynamic subprocesses (namely, adiabatic cooling and diabatic heating) that leads to changes in the energy states of the ambient air in a manner that is readily quantifiable. The Penman-Monteith equation is derived in two steps. As an initial approximation, first a form of the equation that models evaporation into a stationary ambient air, from a source/sink surface, is developed based on thermodynamic equations of state applied to a suitably defined system. In a subsequent step, transfer coefficients (resistance parameters) are introduced into the basic equations accounting for the dynamic effects of wind-surface interactions (and the effects of crop physiological response to atmospheric conditions) on evaporation, leading to the Penman-Monteith equation. Although less compact than the conventional approach, the thermodynamic approach to the derivation of the Penman-Monteith equation has the advantage of being revealing of some key assumptions and concepts that are generally implicit in the conventional approach. The initial step of the thermodynamic formulation accentuates the notion that the Penman-Monteith equation is fundamentally a description of the process of vapor and heat transfer between a source/sink surface and a stationary ambient air. The subsequent step, on the other hand, emphasizes the fact that wind and crop effects on evaporation are taken into account in an approximate sense. The thermodynamic approach also shows that evaporation is essentially a process driven by energy supply and as such each term of the Penman-Monteith equation represents a separate heat source for evaporation. Furthermore, the approach readily reveals some useful interpretations of the mathematical/physical attributes of key parameters of the Penman-Monteith equation. Although the manuscript essentially reviews the approach proposed by Monteith, the derivation here places emphasis on basic assumptions and attempts to fill gaps, mainly in terms of providing extended discussion in places where the original discussion was concise. It also focusses on physical interpretations of the terms and parameters, of the Penman-Monteith equation, accruing from the thermodynamic formulation. Remarkably, the resultant equations for latent heat flux, sensible heat flux, and the final air temperature cannot be evaluated directly. Thus, numerical solutions to this system of equations are presented and evaluated in the companion manuscript.