A model of evapotranspiration based on the theory of maximum entropy production



[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

[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.

2. Model Formulation

[5] The development of the MEP model boils down to formulating the dissipation or entropy production function D [see Wang and Bras, 2009, equation (9)] to include the latent heat flux term. Then extremization of D under the constraint of conservation of energy leads to a unique partition of net radiation into latent, sensible, and ground heat fluxes. We start from the case of bare soil.

2.1. Nonvegetated Land Surface

[6] By analogy to the case of dry bare soil, D including evaporation (latent heat flux) E, sensible heat flux H, and ground heat flux G over a nonvegetated land surface may be written as

equation image

where Is, Ia, and Ie are the thermal inertia parameters (W m−2K−1 s1/2) associated with the corresponding fluxes. Is characterizes a thermal property of the soil varying with moisture content [e.g., Verhoef, 2004]. A convenient method for estimating Is is given by Wang et al. [2010], which is based on an analytical solution of the diffusion equation. For the idealized case of sinusoidal function of surface (skin) soil temperature Ts as a first-order approximation of actual Ts, there is a linear relationship between the amplitude of diurnal variation of G and that of Ts with a coefficient proportional to Is. Then Is may be estimated as the regression coefficient using observed diurnal variations of G and Ts. The method has been used in producing a global map of Is [Bennett et al., 2008]. Ia has been parametrized [Wang and Bras, 2009] using an extremum solution of the Monin-Obukhov similarity equations [Wang and Bras, 2010]. Ie is the “thermal inertia” for the transport of “latent heat,” a new parameter to be defined next.

2.1.1. Formulation of Ie

[7] Theoretically, Ie ought to be formulated to take into account the turbulent diffusion of water vapor and the movement of liquid water because evaporation requires transport of water vapor in the atmospheric boundary layer (ABL) and flow of liquid water in the soil. There are several leads suggestive about how to formulate Ie.

[8] 1. The turbulent mixing responsible for the transport of heat in the ABL is also responsible for the transport of water vapor. This view would imply a functional dependence of Ie on Ia.

[9] 2. Because ET may be expressed in terms of surface soil temperature and surface soil moisture according to the maximum principle of evaporation [Wang et al., 2004], Ie is expected to be expressed in terms of these two surface variables as well. Wang et al. [2004] have shown that the physics of evaporation allows a diagnostic relationship relating E to the thermal and water condition of the land surface characterized by soil temperature and soil moisture as well as the intensity of turbulence characterized by fluxes of sensible heat or momentum. (The link between the maximum principle of evaporation and the proposed MEP model is further discussed in section 4 below.)

[10] 3. Water vapor within an infinitesimal layer next to the evaporating surface is presumably in equilibrium with the liquid water at the soil surface. It follows that surface (skin) specific humidity is determined by the surface soil temperature and surface water potential (or soil water content through a retention curve).

[11] Based on these heuristic arguments, an expression of Ie is postulated, satisfying the above conditions:

equation image

where Ia is, according to Wang and Bras [2009],

equation image

where equation image is the density of air, cp the specific heat of the air under constant pressure, κ the von Kármán constant (∼0.4), z the distance above the surface, g the gravitational acceleration, Tr a reference temperature (∼ 300 K), and C1 and C2 two empirical constants characterizing the stability of the potential temperature and wind velocity profiles [see Wang and Bras, 2009, p. 4]. I0 is the concise expression of the H-independent coefficient.

[12] equation image in equation (2), a dimensionless parameter characterizing the phase-change related state of the evaporating surface, is postulated as follows:

equation image

where Ts is the surface (skin) soil temperature, qs the surface specific humidity, and all other variables are defined in Appendix A, in which some heuristic arguments leading to this postulation are corroborated.

[13] When not directly measured, qs in equation (4) may be estimated from other meteorological and/or hydrological variables. The most convenient method is to compute qs from relative humidity and temperature using the Clausius-Clapeyron equation. Alternatively, qs may be derived from surface soil water potential equation image or surface soil moisture if the retention curve is known according to equation (A2).

[14] The postulated equation image function in equation (4) may be justified by the limiting cases of dry and saturated soil. For the case of dry soil, equation image as qs = 0. For the case of saturated soil, equation image becomes

equation image

where equation image is the slope of the saturation water vapor pressure curve at Ts and equation image the psychrometric constant [e.g., Brutsaert, 1982, p. 215]. As shown below, the MEP model predicted fluxes for the cases of equation image (dry) and equation image (saturated) agree with previously published studies.

[15] It is important to re-emphasize that the general expression of equation image in terms of Ts and qs in equation (4) is postulated (inferred or guessed) instead of “derived.” Its usefulness can only be confirmed through the correct prediction of ET and the heat fluxes based on it. Clues for guessing the functional form of equation image come from concrete example(s). For instance, the well-known Edelfsen-Anderson equation of qs in equation (A2) reveals what the general equation image function may look like (see Appendix A). Yet, postulating equation (4) does not necessarily imply that qs must follow equation (A2). That is, equation image prescribed in equation (4) as a priori does not uniquely determine qs as a function of Ts and equation image (or soil moisture), but is consistent with specific models such as equation (A2). The case study below does support the postulated equation image function. We hope that further independent tests will follow.

2.1.2. MEP solution of E, H and G

[16] Using the parametrization of Ia and Ie according to equations (2)(4), D defined in equation (1) is expressed as a nonlinear (and nonquadratic) function of E, H, and G with Ts and qs as the external parameters. Following the MEP formalism, extremizing D over all possible combinations of E, H, and G under the constraint of conservation of energy for a given net radiation Rn,

equation image

which leads to

equation image
equation image

where I0 is defined in equation (3) with

equation image

where equation image is given in equation (4). equation image and equation image functions are displayed in Figure 1.

Figure 1.

equation image is defined in equation (4) and equation image in equation (8).

[17] The nonlinear algebraic equations (6)(8) have a unique solution of E, H, and G given Rn, Ts, and qs, referred to as the “MEP model of evaporation over nonvegetated land surface.” equation image in equation (9) is recognized as the reciprocal Bowen ratio in terms of equation image as a function of equation image.

2.1.3. Two limiting cases

[18] For the limiting case of saturated soil, the Bowen ratio equation image predicted by the MEP theory agrees closely with the well-known classical equation [e.g., Priestley, 1959, p. 116, (8.21)] as shown in Figure 2.

Figure 2.

The MEP model of Bowen ratio for a saturated surface. Equation (8) versus the classical model of Priestley [1959, p.116, (8.21)].

[19] For the limiting case of dry soil, equations (6)(8) with equation image reduce to the MEP solution of H and G over a dry soil reported previously [Wang and Bras, 2009] because of the following properties of B (see Figure 1):

equation image
equation image


equation image

for a fixed Ts. Other properties of the MEP solution include equation image (as well as equation image) increasing with Ts when equation image for a given equation image or qs, and equation image (≤11/12) decreasing with equation image.

2.2. Vegetated Land Surface

[20] Consider a vegetated land surface covered with a closed canopy. The MEP model of transpiration Ev may be viewed as another limiting case of the MEP model formulated in section 2.1 when Is = 0. In this study, the energy balance is defined at the “material surface” (i.e., ground or canopy top) instead of for a finite control volume. Hence, G should be understood as the heat flux at the surface of the “leaf matrix” continuum conceived as a “porous media of leaf materials.” This concept is an analogy to the “soil matrix” continuum treated as a “porous media of soil materials.” Consequently, ground heat flux (at the soil surface) does not enter the MEP formalism for the case of vegetated land surface. “Is” is the thermal inertia of the leaf matrix, depending on its effective bulk density, specific heat, and conductivity, which is two to three orders of magnitude smaller than that of a soil matrix. As a result, G, the heat flux through the leaf matrix, is negligible compared to latent and sensible heat fluxes, and the energy balance equation over a canopy can be well approximated by Ev + H = Rn. Nonetheless, there is no theoretical difficulty to include the “G” term in the MEP model of transpiration when it is needed.

[21] It is straightforward to show that equations (6)(8) when Is = 0 lead to explicit expressions of Ev and H,

equation image
equation image

where equation image is given in equation (9) and equation image in equation (4) except that Ts and qs represent leaf temperature and specific humidity at the leaf surface, respectively. Equations (13) and (14) are referred to as the “MEP model of transpiration.”

[22] qs is not always equal to the specific humidity within the substomatal cavity, often assumed to be in equilibrium with leaf liquid water at leaf temperature Ts and leaf water potential equation image, because of the stomatal control on the movement of water vapor. Thus, qs may be written as a function of Ts, equation image, and the openness of stomatal apertures,

equation image

where qsat, the saturation specific humidity within the substomatal cavity, may be obtained from equation (A2). equation image, referred to as the “stomatal function” herein, characterizes the effect of the openness of stomatal apertures on the transport of water vapor satisfying the condition

equation image

where equation image corresponds to complete closure of stomatal pores and equation image to stomatal pores being fully open so that rapid exchange of water vapor through the stomatal pores leads to specific humidity at the leaf surface being equal to that within the intercellular air space.

[23] The “stomatal function” equation image (a dimensionless quantity) introduced in equation (15) plays a similar role as the familiar concept of “stomatal conductance” [e.g., Nobel, 1991, p.399], as a plant property, to characterize the effect of geometrical structures of leaf stomates (e.g., their sizes and shapes) on the exchange of water vapor and CO2 between the stomatal cavity and the atmosphere. The main difference is that stomatal conductance is formulated in terms of the flux-gradient equation, while equation image is not. equation image may be estimated, e.g., through equation (15), independent of water vapor flux. Note that lack of local gradient of water vapor between the intercellular air space and leaf surface does not necessarily imply vanishing water vapor flux. Instead, it indicates that the water vapor in and out of the stomatal pores is in a quasi-equilibrium state when stomatal pores are fully open. It is important to emphasize that qs in equation (15) represents the water vapor resulting from transpiration, i.e., the phase change of liquid water of the leaf tissue, driven by the radiative energy supply. When all stomatal pores are closed (i.e., equation image), qs vanishes and so does Ev according to equation (13). Developing models of equation image is beyond the scope of this study. Nonetheless, equation (15) offers an explicit expression linking the leaf variables to qs that might be used in modeling equation image.

3. Model Validation

3.1. Evaporation

3.1.1. SGP97 site

[24] The proposed MEP model was first tested against field observations from the Southern Great Plains 1997 (SGP97) field experiment (data is available online http://disc.gsfc.nasa.gov/fieldexp/SGP97/). (See Famiglietti et al. [1999] for more information about the SGP97 field experiment.) The experiment site (central facility CF01) is located in an area partially covered by grazed pasture with significant bare soil exposed. There were a number of rain events through the observation period. Surface soil moisture was assumed to be saturated after rainfall events as the continuous record of surface soil moisture is not available. Near surface air temperature (at ∼ 2m height) is used as a surrogate of surface (soil) temperature. The near-surface air temperature is expected to be close to the surface soil temperature over saturated soils. In the test, qs in equation (4) is assumed to be saturated at the measured near-surface air temperature.

[25] Figure 3 compares the time series of the MEP model predicted E, H, and G with the observed fluxes. The close agreement between the MEP solution of E and the observed E is evident (see also the scatter plot in Figure 4). The MEP solution of H also agrees well with the observed values. The MEP solution of G appears to be somewhat overestimated relative to the observed G. This may be because of the fact that the soil heat flux measured at 5 cm below the surface at this site tends to underestimate soil heat flux at the ground level.

Figure 3.

The MEP model predicted evaporation E, sensible heat flux H, and ground heat flux G (broken red), according to equations (6)(8), versus the observed fluxes (solid blue) from SGP97 experiment at the site of CF01ARM during 31 May–30 June 1997.

Figure 4.

The scatter plots of Figure 3.

3.1.2. Lucky Hills site

[26] Long-term (20 years) meteorological and hydrological data at 20 min resolutions have been collected at Lucky Hills site located in the Walnut Gulch Experimental Watershed in southern Arizona. Although not a bare soil site, Lucky Hills site is covered with open shrubs of 1 m height that transpire little outside of the monsoon seasons lasting about three months starting in July so that soil evaporation dominates during the dry seasons. Turbulent heat fluxes were measured using the Bowen-ratio method. Relative humidity was measured at the 2.4 m level. Soil moisture was measured at multiple depths between 5 and 200 cm. Soil heat flux was measured at 8 cm depth. Net radiation and surface soil temperature were also measured and used in this study. More details about the data products are described in Emmerich and Verdugo [2008]. The datasets are open to the public, http://www.tucson.ars.ag.gov/dap/.

[27] To show the capability of the MEP model in capturing the dependence of the surface heat fluxes on soil moisture, data collected during 16 November – 26 December 2007 are used in the test. Three wetting-drying cycles occurred during this period as the records of soil moisture at 5 cm depth and rainfall show in Figure 5. Transpiration from plants was insignificant indicated by the carbon dioxide flux (not shown). Thermal inertia of the soil Is may be expressed as

equation image

where equation image is the volumetric soil moisture, equation image the thermal inertia of (still) liquid water with equation image kg m−3 the density, cw = 4.18 × 10−3 J kg−1 K−1 the specific heat, kw = 0.58 W m−1 K−1 the heat conductivity, and Ids (≃0.8 × 103 W m−2 K−1 s1/2) the thermal inertia of dry soil at the Lucky Hills site [Wang et al., 2010]. equation image used in the test is shown in Figure 5a.

Figure 5.

(a) Volumetric soil moisture equation image sampled at 5 cm depth, and (b) 20-min cumulative rainfall measured at Lucky Hills site of the Walnut Gulch Experimental Watershed 16 November–26 December 2007.

[28] Figure 6 compares the MEP model predicted E, H, and G with the observed fluxes with the corresponding scatterplot in Figure 7. The MEP model predicted H agrees closely with the observed H. The agreement between modeled and observed G is good considering that soil heat flux measured at 8 cm depth is expected to be lower than that at the ground level. The agreement between the modeled and observed E is reasonable with larger scatters. Two reasons might be responsible for the increased scatter. First, the specific humidity at the 2.4 m level is expected to be different from the surface value because of the mixing process in the boundary layer. Second, evaporation measured by the Bowen ratio system appears to be overestimated, indicated by a number of spurious spikes in the records with unrealistically large values as shown in Figure 6a where the vertical axis is cut off at 200 W m−2. Nonetheless, the wetting-drying cycles of soil moisture reflected in the surface heat fluxes are captured by the MEP model. A new field experiment has been planned to measure turbulent fluxes using the eddy-covariance method and air humidity very close (e.g., within ≤ 5 cm) to soil surface, which would lead to a more definite test of the MEP model for unsaturated soils.

Figure 6.

The MEP model predicted evaporation E, sensible heat flux H, and ground heat flux G (broken red), according to equations (6)(8), versus the observed fluxes (solid blue) at Lucky Hills site of the Walnut Gulch Experimental Watershed 16 November–26 December 2007.

Figure 7.

The scatter plots of Figure 6.

3.2. Transpiration

[29] Harvard Forest site. The MEP model of transpiration was tested using measurements of fluxes and meteorological variables during the period of 19 August – 8 September 1994 at Harvard Forest. A detailed description of the site and experiment is available online http://www-as.harvard.edu/chemistry/hf/index.html. An observation tower, where an eddy-covariance device and other instruments are installed, is located in a densely wooded area. The height of the canopy top is 24 m. Air temperature sampled at 30, 28, and 22.6 m, used as a surrogate of leaf temperature (not directly measured at the site), is nearly identical, suggesting that the leaf temperature is in equilibrium with the air temperature. There is little difference between the relative humidity measured at 28 and 22.6 m with a mean value of ∼80% over this period of time, while the relative humidity at 30 m (i.e., 6 m above the canopy top) remains at the 10% − 20% level. This condition suggests that the high humidity at the canopy surface results from transpiration from the trees. In addition, the high relative humidity at the canopy surface indicates that stomatal pores were wide open as the plants were not under water stress. Therefore, the measured air temperature at the lower levels is used to compute the leaf surface specific humidity, assumed to be saturated at the leaf temperature, with equation image. The measured water vapor flux is assumed to be predominantly from transpiration as the site is covered with dense trees and ground covered with a layer of falling leaves. The soil evaporation is expected to be insignificant relative to transpiration during growing seasons.

[30] Figure 8 shows a close agreement between the MEP model prediction of Ev and H and the observed fluxes. The corresponding scatter plot in Figure 9 indicates that the MEP model slightly overestimates Ev and H relative to the observed Ev and H. This is consistent with the fact that the observed Rn is consistently higher than observed total flux Ev + H (figure not shown). Like the bare soil case, further test of the MEP model, in particular under the condition of equation image when the plants are under substantial water stress, needs direct measurement of leaf surface specific humidity and leaf temperature in new field experiment(s).

Figure 8.

The MEP model predicted transpiration Ev and sensible heat flux H (broken red), according to equations (13) and (14), versus the observed fluxes (solid blue) from the Harvard Forest experiment during 19 August–8 September 1994.

Figure 9.

The scatter plots of Figure 8.

4. On the MEP Model of ET

[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 Ts, qs, and Rn. The reverse is not true. That is, Ts and qs 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 equation image or equation image. 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 Ts and qs in the MEP model predicted fluxes: E, H, and G are explicitly dependent on Ts only when qs ≠ 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 Ts 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 Ie (associated with E) instead of Ia (associated with H). The role of temperature gradient in H has been represented by the H -dependent Ia 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 qs. That is, soil moisture (or soil water potential) and the retention curve are redundant when qs is measured. For the case of transpiration, leaf water potential and the stomatal function are redundant when qs is known. When qs is not directly measured, all that the MEP model needs are those for retrieving qs.

[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 Ia and Ie 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.

5. Conclusions

[40] The proposed MEP model of ET is an effective tool in modeling the energy budget over land surface because of its unique features: (1) the model is built on the state-of-the-art nonequilibrium thermodynamics, (2) the model only needs input of surface variables (i.e., temperature, humidity, and net radiation), (3) all model parameters are either physical properties of the system or universal empirical coefficients, and (4) the model formulation allows a unique solution of the fluxes with reduced sensitivity to the uncertainties in the model input and parameters. Upon further tests, the MEP model would potentially lead to the desired parametrization of land surface hydrology that meets the demands of climatological and environmental studies.

Appendix A:: Postulation of σ

[41] Is in equation (1) is a well-defined parameter because soil heat flux is expressed as the product of temperature gradient and molecular diffusivity according to Fourier's law. Parametrization of Ia in equation (1) [Wang and Bras, 2009] used an analogy of heat conduction to formally express sensible heat flux as the product of temperature gradient and turbulent diffusivity. Yet Ie introduced in equation (1) is a very different concept from Is or Ia because what is transferred in the boundary layer is water vapor (a mass flux) instead of heat (an energy flux). Therefore, a key step in the parametrization of equation image is to formally express the water vapor flux as the product of temperature gradient and turbulent diffusivity so that E in equation (1) can be treated as a “heat flux.”

[42] In the boundary layer turbulence models, there is an analogy between the flux of water vapor versus the local gradient of specific humidity and heat flux versus the local gradient of temperature [see Wang and Bras, 2009, equation (B10)]. Using this analogy, E may be formally expressed as

equation image

where qs is the surface specific humidity within an infinitesimal layer next to the evaporating surface, equation image the vaporization heat of liquid water, and z the vertical coordinate. KH is the eddy diffusivity for heat according to the assumption of the same transport mechanism underlying transfer of heat and water vapor. Note that equation (A1) is only assumed within the infinitesimal layer regardless of its validity at finite distances above the surface.

[43] It is commonly assumed that the air right above the evaporating surface remains in equilibrium with the liquid water in the soil with temperature Ts and soil water potential equation image. A specific model of qs is given by Edelfsen and Anderson [1943, p.141],

equation image

where T0 is a reference temperature taken to be 0°C, e0 the saturation vapor pressure at T0, P0 the ambient pressure, and Rv gas constant for water vapor. Equation (A2) allows E in equation (A1) to be expressed in terms of a (local) temperature gradient,

equation image

where equation image represents the quantity in the parentheses. Equation (A3) is formally similar to that for H [see Wang and Bras, 2009, equation (B10)]. Therefore, Ie may be defined in the form of equation (2) analogous to the definition of Ia.

[44] Because Ie is associated with the turbulent transport of water vapor, equation image ought to be expressed in terms of the surface specific humidity qs (or its equivalents) determined by the thermal and moisture condition of the soil. That is, we seek a general expression of equation image. The functional form of equation image is revealed by

equation image

where qs is given in equation (A2) and the difference between the last two terms is no more than 2% for realistic values of Ts and the entire range of equation image.

[45] These heuristic arguments lead to a postulation of equation image,

equation image

[46] The above thought processes leading to equation (A5) should not be interpreted as a “derivation” since the reasoning is built on analogy instead of more fundamental physical laws.


[47] This work was supported by ARO grants W911NF-07−1−0126, W911NF-10−1−0236, and NSF grant EAR-0943356. Data sets for Lucky Hills site were provided by the USDA-ARS Southwest Watershed Research Center. Funding for these data sets was provided by the United States Department of Agriculture, Agricultural Research Service. Part of the work was conducted at the MA Institute of Technology. We are grateful to the computer support provided by Soroosh Sorooshian and Dan Braithwaite of University of CA at Irvine. The expert comments by three anonymous reviewers have markedly improved the quality of this manuscript and we thank them. We also acknowledge the support of the editor Praveen Kumar and the associate editor during the revision process.