### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Model Validation
- 4. On the MEP Model of ET
- 5. Conclusions
- Appendix A:: Postulation of σ
- Acknowledgments
- References

[1] Building on a proof-of-concept study of energy balance over dry soil, a model of evapotranspiration is proposed based on the theory of maximum entropy production (MEP). The MEP formalism leads to an analytical solution of evaporation rate (latent heat flux), together with sensible and ground heat fluxes, as a function of surface soil temperature, surface humidity, and net radiation. The model covers the entire range of soil wetness from dry to saturation. The MEP model of transpiration is formulated as a special case of bare soil evaporation. Test of the MEP model using field observations indicates that the model performs well over bare soil and canopy.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Model Validation
- 4. On the MEP Model of ET
- 5. Conclusions
- Appendix A:: Postulation of σ
- Acknowledgments
- References

[2] Evapotranspiration (ET) is arguably the most challenging hydrological process to predict. Even though the basic physics of ET is well understood [e.g., *Shuttleworth*, 1993; *Brutsaert*, 1982], we are still facing some difficulties in modeling ET, a crucial component of the land surface water and energy balance [*Desborough et al.*, 1996; *Henderson-Sellers et al.*, 2003]. Efforts to improve ET simulation models, too numerous to be summarized here, have primarily focused on improving the parametrization of physical processes, in particular the turbulence in the atmospheric boundary layer [e.g., *Tillman*, 1972; *Katul et al.*, 1996], and on incorporating more field and remote sensing observations [e.g., *Kalma et al.*, 2008]. Toward the same goal, in this study we propose a different kind of ET model taking advantage of the emerging theory of maximum entropy production (MEP) [*Dewar*, 2005] as a derivative of the maximum entropy (MaxEnt) theory [*Jaynes and Bretthorst*, 2003].

[3] The MaxEnt theory was developed as a general inference tool for any systems that need to be described probabilistically. The MEP is derived from applying the MaxEnt to nonequilibrium thermodynamic systems. An excellent overview of the MEP theory and its applications to a range of subjects is given by *Kleidon and Lorenz* [2005]. More insightful views about the potential applications of the MEP theory in land surface hydrology are reported by *Kleidon and Schymanski* [2008]. The MEP method differs conceptually from traditional “physically based” approaches [e.g., *Sellers et al.*, 1997]. Basically, the MEP theory addresses the question “what is the best prediction based on the available information?,” while the classical physical theories deal with the question “what are the fundamental laws governing the physical world?.” A proof-of-concept MEP model of surface heat fluxes over a dry soil [*Wang and Bras*, 2009] has demonstrated the usefulness and potential of the MEP theory in modeling the land surface energy balance. The MEP theory offers a possibility of a new approach to predicting ET (and heat fluxes). We attempt to realize that possibility by formulating an MEP model of ET guided by the case study of heat fluxes over a dry soil.

[4] A description of the MEP formalism is given by *Wang and Bras* [2009], and hence not repeated here. Section 2 focuses on model formulation starting from bare soil evaporation followed by transpiration from a canopy. The MEP solution of evaporation and transpiration together with sensible and ground heat fluxes are expressed as implicit or explicit analytical functions of surface temperature, humidity, and net radiation. In particular, no gradient variables are used as model input. Section 3 presents model validation using observations from several field experiments. Section 4 discusses some important properties of the MEP model. Section 5 gives a brief summary and our view on the potential applications of the MEP model.

### 4. On the MEP Model of ET

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Model Validation
- 4. On the MEP Model of ET
- 5. Conclusions
- Appendix A:: Postulation of σ
- Acknowledgments
- References

[31] The proposed MEP model of ET demonstrates the potential of the MEP theory in modeling land surface energy balance. For example, transpiration is often considered more difficult to model than (bare soil) evaporation because of the complexity associated with plant physiology. Yet, the MEP model of transpiration turns out to be a trivial limiting case of the MEP model of bare soil evaporation. The MEP model of ET confirms the findings of the earlier theoretical studies on evapotranspiration processes [*Wang et al.*, 2004, 2007] and translates them into predictive capability. It also sheds more light on the fundamental physics of ET from the perspective of optimality principle. As argued by *Wang et al.* [2004, 2007], the thermodynamic system of the land surface at macroscopic level evolves toward a potential equilibrium as quickly as possible by maximizing evapotranspiration under the constraint of conservation of energy. The MEP theory further reveals that the maximum evapotranspiration corresponds to a macroscopic state associated with the largest number of microscopic configurations of the system, i.e., maximum evapotranspiration is the macroscopically most probable phenomenon. More importantly, the MEP theory is able to make efficient use of information provided by a small number of observables to predict macroscopic fluxes.

[32] Note that the MEP model of ET is not a dynamic model for the surface state variables in the sense that it does not predict how temperature and moisture evolve with time. A dynamic model of surface soil temperature and soil moisture requires more information than that required by the MEP model to predict the surface fluxes. For example, predicting the states at the current time will need initial conditions of the states at a certain previous time. Instead, the MEP model predicts the surface fluxes using instantaneous temperature, humidity, and net radiation independent of their time histories. This feature makes the MEP model parsimonious in model input when applied to modeling surface energy balance.

[33] Note also that the MEP model provides a unique solution of *E*, *H*, and *G* given *T*_{s}, *q*_{s}, and *R*_{n}. The reverse is not true. That is, *T*_{s} and *q*_{s} cannot be uniquely determined from given *E*, *H*, and *G* because of the fact that the MEP solution of *E*, *H*, and *G* depends on or . This property makes physical sense since the surface state in terms of temperature and humidity is not expected to be fully specified from the surface fluxes alone. An interesting feature of the MEP model is reflected in the unequal role of *T*_{s} and *q*_{s} in the MEP model predicted fluxes: *E*, *H*, and *G* are explicitly dependent on *T*_{s} only when *q*_{s} ≠ 0 (e.g., over nondry soils). Therefore, the MEP model of ET is consistent with the MEP model of heat fluxes over a dry land surface [*Wang and Bras*, 2009] that does not need *T*_{s} input to predict *H* and *G*. This property may be viewed as an indicator of the dominant role of soil moisture in the land surface energy balance.

[34] The MEP model is not only parsimonious in model input, but also less sensitive to the uncertainties of model input than the bulk transfer based models because no temperature and humidity gradients appear in equations (6)–(9). Temperature only enters the model formulation through *I*_{e} (associated with *E*) instead of *I*_{a} (associated with *H*). The role of temperature gradient in *H* has been represented by the *H* -dependent *I*_{a} through the framework of the Monin-Obukhov similarity theory. Again, this property is consistent with the MEP model for the dry case [*Wang and Bras*, 2009] where temperature is not even a model input.

[35] The MEP model and the classical Penman's model have several features in common [*Penman*, 1948]. Both models are energy based and assume identical eddy diffusivity for heat and water vapor transfer in the ABL. Yet, the MEP model distinguishes itself from Penman's model in several major ways. First of all, the MEP model is built on a theory of nonequilibrium thermodynamics, while the Penman's model relies on empirical equations of turbulent transport. Second, the MEP model is formulated over the entire range of soil wetness from dry to saturation, while the Penman's model is applicable only to saturated surfaces. Generalization of the Penman's equation to unsaturated land surfaces often introduces empirical functions/parameters to characterize the effect of soil moisture on evaporation. The MEP method is most attractive for modeling transpiration, which turns out to be simpler than the bare soil case, as the eddy-diffusivity parameters for heat and water vapor cancel in the equations (see equations (13) and (14)). As a result, the MEP model of transpiration uses even fewer parameters, hence is less sensitive to the turbulent transport models than its bare soil counterpart. On the contrary, with additional parameters such as stomatal and aerodynamic conductance (or resistance) the Penman-Monteith model is more vulnerable to the modeling errors of turbulent transport. Third, the MEP model is more parsimonious in input parameters. Only three input variables are needed: net radiation, surface temperature, and surface specific humidity. The Penman and Penman-Monteith model require more input variables including temperature measured at two levels, air humidity, and those needed to parametrize stomatal and aerodynamic resistance. Fourth, Penman's model requires ground heat flux as input, while the MEP model predicts ground heat flux.

[36] The effectiveness of the MEP model is rooted in the power of the MEP theory, which allows efficient use of information in the sense that redundant information relevant to the surface fluxes will be automatically filtered out. This is possible because the Bayesian probability theory behind the MEP formalism automatically takes the redundancy, if any, into account in relating the macroscopic fluxes to measurable quantities such as temperature and humidity. For example in the MEP model of bare soil evaporation (equations (7) and (8)), the effect of soil moisture or soil water potential and soil property through a retention curve has been represented by a single variable *q*_{s}. That is, soil moisture (or soil water potential) and the retention curve are redundant when *q*_{s} is measured. For the case of transpiration, leaf water potential and the stomatal function are redundant when *q*_{s} is known. When *q*_{s} is not directly measured, all that the MEP model needs are those for retrieving *q*_{s}.

[37] It is important to emphasize that the MEP model is not derived from more fundamental physical laws; rather it is inferred using the (Bayesian) probability rules. The MEP model predicting the surface fluxes without using some parameters such as water vapor deficit should not be interpreted as implying that these parameters are not related to the evapotranspiration process. The MEP model does not answer the question “what are the fundamental physical processes behind evapotranspiration?.” Instead, it answers the question “what would be the best estimate of evapotranspiration based on the information of net radiation, surface temperature, and humidity?.” An answer to the former question has been given in terms of the maximum principles [*Wang et al.*, 2004, 2007]. The connection between the MEP model of ET and the principles of maximum evaporation/transpiration is analogous to that between the MEP model of surface heat fluxes and the corresponding stationary hypothesis of energy balance over dry soil (a view elaborated in the last paragraph of section 4 of *Wang and Bras* [2009]). There is no contradiction that the MEP model is different from other formulations such as Penman's equation, viewed as a “physical law,” since they (1) answer different questions, and (2) use different information. Yet, the MEP theory, as an inference tool, could give the same results as predicted by “physical laws.” *Wang and Bras* [2009, section 3.1] presents an example where the MEP theory “guesses” the same result as the “derived” one.

[38] A theoretical limitation of the proposed MEP model is the assumption that the atmospheric boundary layer turbulence can be described by the Monin-Obukhov similarity equations. For the cases where Monin-Obukhov model is not suitable, the formulation of the MEP model may take different forms since the parametrization of *I*_{a} and *I*_{e} is turbulence-model dependent. This limitation is not too severe for practical purposes as the Monin-Obukhov model has been shown to be adequate for describing turbulent flow in the surface layer under a majority of natural conditions. In fact, the turbulence-model dependent MEP solution of surface fluxes is an advantage in the sense that incorrect prediction of the fluxes by equations (6)–(9) based on the Monin-Obukhov similarity theory could indicate the need to develop an alternative turbulence model suitable for the specific situations. Failure of the MEP model may also be caused by other potentially flawed assumptions underlying the model development including (1) the same transport mechanism of heat and water vapor in the ABL, (2) determination of ET on surface soil temperature and moisture, and (3) equilibrium between liquid water and water vapor at the evaporation surface. Therefore, this MEP model of ET is not the end of story. Instead, it is the beginning of a new framework that opens more possibilities of ET models suitable for diverse meteorological and ecohydrological environments.

[39] The MEP method offers a potential solution to the problem of “no single (existing) land surface model is capable of capturing all features of the surface energy balance under all conditions” [*Desborough et al.*, 1996; *Henderson-Sellers et al.*, 2003]. The proposed MEP model indicates that the most relevant information about the surface heat fluxes are net radiation, surface specific humidity, and surface temperature. The MEP formalism is able to use this information effectively. Because the MEP model is built on a sound theoretical foundation and includes no location-dependent or species-specific (empirical) tuning parameters, it has the potential to perform satisfactorily regardless of the environmental conditions. Even though not a prognostic model itself, the MEP model may be used as a component of a land surface model to describe dynamic processes, a topic of future research.