We hypothesize that the system of liquid water in leaf tissues and the water vapor in the atmosphere tends to evolve towards a potential equilibrium as quickly as possible by maximization of the transpiration rate. We make two assumptions in formulating the transpiration rate: (1) stomatal aperture is directly controlled by guard cell turgor (or leaf water potential); (2) CO2 flux can be used as a nonparametric equivalent of stomatal conductance for a given stomatal function (not necessarily optimal in terms of the water use efficiency for photosynthesis). Transpiration is then expressed as a function of leaf temperature, CO2 flux (as a surrogate of stomatal conductance), and sensible heat flux characterizing the transport mechanism at a given level of radiative energy input. Maximization of transpiration constrained by the energy balance equation leads to vanishing derivatives of transpiration with respect to leaf temperature and CO2 flux. We have obtained observational evidence in support of the proposed hypothesis.
 More than 70% of the earth's land surface is covered with seasonal or perennial vegetation ranging from tropical forest to arctic tundra [Friedl et al., 2002]. Numerous investigations have revealed that the vegetation cover has a tremendous impact on the earth's atmosphere and vice versa through the exchange of water, heat, momentum, and CO2 [Shukla and Mintz, 1982]. Gas exchange, particularly water vapor and CO2, plays a crucial role in the global hydrological cycle and climate change, which in turn influences the state and evolution of the vegetation [Gash et al., 1996]. Despite the significance of these fluxes, physical and physiological laws of vegetation-atmosphere dynamics are still not adequately understood to model mechanistically the exchange of water vapor and CO2 at the land-atmosphere interface [Henderson-Sellers et al., 2003]. Over the years, workers have relied on empirical models based on Fick's law (i.e., linear flux-gradient relationship) and analogies to Ohm's law of electrical resistances to parameterize the energy and mass fluxes. The phenomenological nature of the empirical models leaves little room for their improvement and limits their utility across spatial and temporal scales. Here we seek a general physical principle, governing the energy balance at the canopy surface, independent of the nature of transport processes. Such a principle should lead to constraints on the partition of solar radiative energy into various energy and mass fluxes at the canopy-atmosphere interface. We believe that a general and fundamental law of transpiration may exist to uniquely determine the energy partition and CO2 flux, and to offer opportunities to improve the models of land-vegetation-atmosphere interaction dynamics. The goal of this paper is to layout a mathematical/physical framework and develop some initial tests of this framework against canopy-scale data.
 Transpiration, or evaporation of water from plants, is a complex phenomenon involving numerous physical, chemical and biological processes. Physiologists emphasize the coupling of transpiration and photosynthesis at the leaf scale, while hydrologists mostly focus on transport of water vapor at the canopy scale. In this study, transpiration is viewed as the process of phase change of liquid water at the leaf scale and turbulent transport of the water vapor at the canopy scale. Since transpiration is species and environment-dependent and environment-reactive in general, a quantitative description of transpiration is far more difficult than evaporation from a nonvegetated land surface due to the complex and diverse features of plant physiology. Among many factors, transpiration is strongly affected by the size and shape of stomatal aperture, referred to here as the stomatal conductance and often expressed empirically in terms of several plant and meteorological variables [e.g., Jarvis, 1976; Tuzet et al., 2003]. No definitive mechanistic theory is available, to our knowledge, to quantitatively relate the geometry of stomatal aperture to stomatal conductance. A major challenge we are facing is to establish an equation linking the stomatal conductance to observable parameters that holds under the most general conditions independent of the turbulent transport models.
 A recent study of Wang et al.  suggested that there exists a maximum principle of evaporation over nonvegetated land surfaces. This development motivated us to hypothesize that transpiration may follow a similar principle based on the thermodynamics of liquid-vapor equilibrium. Theory based on extremum principles has a long history of application in plant physiology and ecology [e.g., Cohen, 1966; Cowan and Farquhar, 1977; Givnish, 1983; Sellers et al., 1992, 1997; Eagleson, 2002]. Although not a modeling study per se, this work parallels Cowan and Farquhar's school of thought with some unique features. First, no plant form and function and pathways of carbon fixation, optimal or otherwise, are assumed a priori. Second, the formulation is nonparametric and (transfer) model-independent. It is important to point out that the proposed hypothesis of maximum transpiration does not imply optimal stomatal functions in terms of water-use efficiency as proposed by Cowan and Farquhar or vice versa. This point will be discussed further in section 4.
 We are putting forward a hypothesis with ecological significance that is tested using a mechanistic approach. Although the hypothesis as presented here does not directly lead to predictive models of transpiration, it reveals an intrinsic energetic constraint on the biological responses to the environment for all plant functions. The hypothesis, if proven true, could serve as a first principle in developing new models of transpiration to predict leaf temperature, heat and CO2 exchange rates between the canopy and the atmosphere. We focus here on initial tests of the hypothesis using existing field observations, with the hope that future studies will follow. We do not address the explicit development of physiological transpiration models; this paper is, rather, a first step toward understanding the constraints on how plants organize themselves through their structure-function relationships.
2.1. Formulation of Transpiration
2.1.1. Physics of Transpiration
 Transpiration is coupled to a suite of complex species-dependent physical, chemical and biological processes and is strongly influenced by the natural environment in which the plants live [Salisbury and Ross, 1991; Nobel, 1991]. Nevertheless, the fundamental mechanism behind transpiration, from the perspective of the phase change of liquid water, is the same as that of evaporation over bare soil, a view shared by some scientists [e.g., Shuttleworth, 1993]. Table 1 compares the two processes. Evaporation from either plant tissues or bare soil surfaces requires four necessary conditions: energy input, presence of liquid water, fugacity, and a transport mechanism. The most important source of energy for transpiration comes from sunlight (photosynthetically active radiation). Availability of liquid water is characterized by water vapor pressure inside the substomatal cavity, assumed to be saturated, determined by leaf temperature and leaf water potential according to the Clausius-Clapeyron equation. Turbulent mixing is the dominant mechanism for transferring the water vapor leaving leaf stomates into the ambient atmosphere. Unlike the case of bare soil where molecular diffusion of water vapor within the soil layer does not impose a rate-limit to evaporation [e.g., Saravanapavan and Salvucci, 2000], diffusion of water vapor out of leaf stomatal cavity is strongly affected by the openness of stomates. Atmospheric water vapor deficit is associated with the transport of vapor away from the leaf surface. Several recent studies have indicated that transpiration is not directly responsive to this deficit. Later in this paper we will argue that very near the leaf surface this deficit is in the diagnostic result of the thermodynamics occurring at the leave surface. Although knowledge of this deficit can, and is, used in successful predictive models it is not the cause of the phase change that occurs at the leave surface. State variables at the surface, are sufficient to largely determine the transpiration or phase change. We will offer observational evidence of this interpretation.
Table 1. Fundamental Elements in the Processes of Bare Soil Evaporation and Plant Transpiration
Bare Soil Evaporation E
Plant Transpiration Ev
Supply of energy
Supply of water
soil moisture/water potential
leaf water content/potential
θs or Ψs
turbulence in PBL
turbulence in PBL
Pathway of water vapor
into the atmosphere
2.1.2. Expression of Transpiration
 Based on the above argument, the transpiration, Ev, may be formulated in a general form,
where Tl is the leaf temperature, Ψl the leaf water potential, gs the stomatal conductance, H the sensible heat flux, and R the radiative energy input. It is important to emphasize that R is an external parameter rather than an independent variable in the Ev function. The functional form (or model) of Ev is not assumed in this work. Several points are worth mentioning,
 1. H is treated as an independent variable in equation (1). It represents one of the essential elements of transpiration process, a surrogate for the transport mechanism. H cannot be substituted by the other independent variables such as Tl. H characterizes the turbulent transport capacity due to the combined effect of forced and free convection in the surface layer. The choice of H as one of the independent variables is not unique. For example, friction velocity may be selected instead of H to represent the transport mechanism. Nonetheless, the choice of H as a state variable explicitly leads to partition of radiative input as a result of the maximization that avoids selection of arbitrary models of sensible heat flux.
 2. Environmental variables such as air temperature, atmospheric humidity, CO2 concentration, soil moisture, etc. are not explicitly included as independent variables in the expression of Ev. Although it is a common practice to parameterize Ev using vapor pressure deficit, experimental investigations [e.g., Feild and Holbrook, 2000] suggest the lack of a causal link between Ev and vapor pressure deficit. We believe it is possible to establish a diagnostic function of Ev without using these atmospheric and soil related parameters since the leaf temperature and water potential reflect the atmospheric and soil conditions as a result of strong interaction in the soil-canopy-atmosphere system. Justification of equation (1) will be discussed later in the paper.
 3. R in equation (1) is an external parameter since the effect of R on Ev is through the energy balance equation (a mathematical constraint in the formulation of the maximization). That is, Ev is uniquely determined by Tl, Ψl, gs and H assuming that liquid water is in equilibrium with the water vapor in the substomatal cavity. It will become clear later in the paper that the proposed hypothesis allows a solution of Tl, Ψl, gs and H as functions of given R for which Ev is maximum.
2.2. A Maximum Hypothesis
 Transpiration occurs when the liquid water in the leaf tissues tends to reach equilibrium with the water vapor in the substomatal cavities. We hypothesize that the evolution towards a liquid-vapor equilibrium proceeds at the fastest rate limited by the five identified factors of Table 1. The idea was suggested by the fact that the processes of phase change of liquid water in nature share some common features as Table 1 indicates. A theoretical justification of the idea may come from the emerging theory of the maximum entropy production principle (MEPP) [Dewar, 2005]. Some view MEPP as an analogy to the second law of equilibrium thermodynamics that describes the general behavior of a nonequilibrium thermodynamic system. One implication of MEPP is that maximum transpiration rate (and the corresponding heat fluxes) under the constraint of given energy input is the most probable and macroscopically reproducible thermodynamic process among all possible partitions of surface energy fluxes. The theory of MEPP has been applied to a number of scientific and engineering problems (see Ozawa et al.  and Kleidon and Lorenz  for summary). As shown in Dewar , MEPP reduces to various principles of maximum fluxes under certain conditions, e.g., MEPP is equivalent to maximum evaporation over nonvegetated soil (Dewar and Wang, private communication). There is growing evidence indicating that this conclusion may be generalized to vegetated surfaces. For instance, Juretić and Županović  found that photosynthetic proton pumps operate close to the maximum entropy production mode. Kleidon  has shown that an optimal stomatal conductance exists at which the rate of photosynthesis, and therefore vegetation productivity or transpiration, is at a maximum.
 The hypothesis of maximum transpiration is defined as follows: The dynamics of water and heat exchange at the leaf-atmosphere interface lead to a state of leaf temperature, leaf water potential, and sensible heat flux into the atmosphere that maximizes latent heat expenditure for transpiration under a given energy input, meteorological environment, and a given physical and physiological form and function of a plant, i.e.,
where Rn is the net radiation at the canopy surface. gs in equation (2) is assumed to be a given species-dependent function of Ψl. ∀ is understood as “for all possible combinations that satisfy the energy balance equation”. gs is not included under ∀ in equation (2) since this study does not deal with the optimality of gs in terms of efficiency of water use versus CO2 fixation. Formulation of gs is discussed next. Note that R in equation (2) is not necessarily equal to Rn, for example, R may represent incoming solar radiation.
 It is worth reiterating the meaning of equation (2). The energy balance equation, Ev + H = Rn, as a constraint in equation (2), does not uniquely determine the partition between Ev and H. There are countless number of ways of the partition that satisfy the energy balance through various combinations of Tl, Ψl and H. It is important to re-emphasize that gs is not a “free” variable when gs is controlled by Ψl (see discussion in section 2.3). Whether the stomatal function in terms of water-use efficiency is optimal, a subject of future research, may be determined by including gs as an independent variable like Tl in equation (2). The partition between Ev and H (for a certain stomatal function) is uniquely determined when Ev is maximized among all possible combinations of Tl, Ψl and H that balance the energy budget. This maximum Ev under a given meteorological condition (e.g., wind, cloudiness, etc.) is achieved for a unique combination of Tl, Ψl and H corresponding to the given stomatal function.
2.3. Formulation of Stomatal Conductance
 The effect of stomatal structure on the exchange of gases is very difficult to quantify analytically as there is no unique way to define stomatal conductance. Theoretically, stomatal conductance gs is a function of the size and shape of stomatal aperture. Although the physiological mechanism is still not fully understood, gs is shown to be correlated with several leaf and environmental variables including leaf temperature, leaf water potential, sunlight intensity, leaf-air vapor pressure deficit, and ambient CO2 concentration [e.g., Willmer and Fricker, 1996, p. 126]. The correlation relationships have led to an empirical model of stomatal conductance [Jarvis, 1976] as well as mechanistic models at smaller scales [e.g., Tuzet et al., 2003]. In bio-hydrological applications, stomatal conductance is commonly defined as the water vapor flux divided by the difference in water vapor pressure between the leaf intercellular spaces and the leaf surface using the Fick's law analogy. We do not adopt this definition in this study.
 A popular view is that stomatal conductance responds to the vapor pressure deficit. This view has been questioned by some researchers. For example, Lynn and Carlson  doubt the effect of water vapor deficit on stomatal conductance. Mott and Parkhurst  provided empirical evidence suggesting that stomatal conductance does not directly respond to atmospheric humidity. Mott and Parkhurst's work distinguishes itself in two aspects from most of the correlation analyses. First, they used an objective definition of stomatal conductance following Cowan's [1977, p. 214]. Here “objective” means that the definition of stomatal conductance is independent of transport models such as the Fick's law analogy of the gradient-flux relation. Second, stomatal behaviors were investigated by independently varying water vapor deficit and transpiration rate. A number of other studies [Monteith, 1995; Bunce, 1996; Lhomme et al., 1998] have supported Mott and Parkhurst's conclusion.
 Stomates are movable pores whose aperture are regulated by surrounding cells, known as guard cells. When turgor pressure increases, the guard cells swell causing an expansion of the aperture of the stomatal pore so that the conductance increases [Sharpe et al., 1987]. Guard cells regulate their turgor through active transport of ions; import of potassium ions lowers the osmotic potential of the cells and causes an influx of water, which raises the turgor pressure and leads to stomatal opening [Kramer and Boyer, 1995]. The myriad factors that affect stomatal conductance, e.g., CO2 concentration, soil nutrients, and leaf water status, act through their effects on guard cell turgor. Empirical and mechanistic equations have been developed to describe these relations [Comstock and Mencuccini, 1998; Frank and Farquhar, 2001]. As a result of these studies, a well defined relationship is expected to exist between the leaf water potential (hydrostatic pressure) and the guard cell turgor under given internal and environmental conditions. Therefore, a general equation may be written as,
where the external environmental variables pertinent to chemical and biological processes are omitted for brevity. Equation (3) is either a function or functional relationship that does not need to be specified for the purpose of this study.
 A mechanistic model of stomatal conductance in terms of equation (3) is not available at present, to our knowledge. Identification of the gs function in equation (3) would be difficult without assuming a transfer model. An added difficulty in using equation (3) is the fact that Ψl is not a routinely measured parameter in field experiments. gs defined using the Fick's law analogy is a derived parameter not directly observable. We need to find a substitute of Ψl or gs in equation (3) that is a measurable parameter. Note that CO2 diffuses in and out of the substomatal cavities through the same stomatal aperture as water vapor. Therefore, CO2 flux is a logical candidate as the surrogate of stomatal conductance. This idea is not new. The seminal work by Cowan and Farquhar  on an econometric model for optimal stomatal behavior was developed assuming that both transpiration and photosynthesis rate are functions of stomatal conductance, which implies that, arithmetically if not biologically, stomatal conductance is a function of CO2 flux. Modeling studies [e.g., Lynn and Carlson, 1990; Gunderson et al., 2002] also support the concept initiated by Cowan and Farquhar. Hence, we may write,
where A is the CO2 flux. Again, the unknown function or functional relationship in equation (4) need not be specified here.
 The governing equations for the maximum Ev can be obtained using the Lagrangian multiplier method. The Lagrangian function, f, is defined as,
where λ is the Lagrangian multiplier, and the external parameter R is omitted for clarity. The Lagrange multiplier method will select, among all possible combinations of Tl, A and H that satisfy the energy balance constraint in equation (7), the solution of Tl, A and H that maximizes Ev.
 The derivation of the governing equations below is for the case of given net radiation Rn. Other cases can be handled in the same manner. Differentiating f in equation (7) with respect to Tl, A, H and λ and setting the corresponding derivatives to zero while keeping Rn constant lead to,
where the superscripts indicate the given constraint (i.e., Rn for this case) and the subscripts stand for the independent variables held constant in computing a particular partial differential. The above equations yield,
where the given constraint Rn in the superscript is omitted for conciseness. Keep in mind that all derivatives in equations (12)–(15) are evaluated under the condition of given Rn. The constrained derivative is referred to as “conditional derivative” herein. Equations (12)–(15) indicate that the conditional derivatives vanish for all values of Rn if the maximum hypothesis holds. The solution not only offers a complete description of the energy budget over a vegetated land surface, but also contains important information about the CO2 exchange between canopy and the environment. For example, a legitimate Ev model has to agree with equations (12)–(13) if the maximum hypothesis is true. Once an Ev model is selected, equations (12)–(14) can constrain or even fully determine the model parameters.
 In the next section, we will use field observations to compute the derivatives for different values of Rn. The evaluation of equations (12)–(13) does not require Ev models in the form of equation (1) or equation (5) once Ev, Tl, A, H, and Rn are measured independently. Hence, the test of the hypothesis through validation of equations (12)–(13) is transpiration-model independent. If the Ev function (or a transpiration model) is specified, equations (12)–(14) can be used to predict Tl, A (or Ψl) and H given Rn input. It is now clear that R in equation (5) is an external parameter rather than an independent variable.
 The validity of the maximum hypothesis of transpiration requires that: (1) Ev can be formulated in terms of three state variables; (2) Ev is maximized over all possible combinations of the state variables under the constraint of surface energy balance. The test of the hypothesis includes two tasks: (1) validating equations (12)–(13) (the necessary conditions for Ev to be maximum), (2) justifying equation (5), i.e., Ev is adequately described by three state variables. The second task is actually independent of the first one. A nonparametric method is desirable for testing the hypothesis without assuming transport models. Since equations (12)–(13) are necessary conditions for a maximum (or, more rigorously, an extremum) Ev, lack of validation of equations (12)–(13) would constitute a falsification of the hypothesis.
3.1. Computation of the Derivatives of Ev
 Field experiments gather measurements sequentially, not stratified by given radiation. The first step in calculating the derivatives is to transform the time series measurements into new sequences by re-arranging the time series in an ascending (or descending) order of the constraint parameter, say Rn. Then the transformed sequences are divided into a number of subsequences with equal number of data points. The resulting subsequences are conditioned within a narrow range of Rn values. The next step is to write the Taylor's expansion of Ev in terms of Tl, A, and H, up to the first order, with the expansion coefficients being the corresponding first-order partial derivatives. Then, the desired conditional derivatives of Ev can be obtained as the (linear) regression coefficients relating Ev to Tl, A, and H within the sorted sequences of observations. Rn is not a regressor.
 According to equation (5), the Taylor's expansion of Ev up to the first order is,
where “0” stands for a reference point of a subsequence at which the derivatives are defined, i is the ith point of a certain subsequence, and m is the number of data points in the subsequence. Since the first order expansion of Ev is linear in the (unknown) derivative terms, the reference points in equation (16) need not be specified in solving the regression coefficients according to equation (16). Equation (16) does not require specification of a physiological model for Ev.
3.2. Data Sets
 The data used in this study are AmeriFlux products (http://public.ornl.gov/ameriflux/). A description of the observation network is given by Baldocchi et al. . Observations from three experimental sites with different types of vegetation are selected: an area covered with arctic tundra in Happy Valley of Alaska during 1995, a mountainous region with high elevation covered with forest in Black Hills of South Dakota during 2001, and a temperate forest area in Harvard Forest of Massachusetts during 2000. (This data set was prepared specifically for us by S. Wofsy and his research group at Harvard University. Data product and a description of the site and experiment are available online (http://www-as.harvard.edu/chemistry/hf/index.html).) We selected these data sets because the measured energy budgets have relatively small biases. Equations (12)–(13) hold when the surface energy budget is balanced. Large biases in the surface energy balance imply large errors in the flux data, making the estimates of the derivatives less accurate. Only daytime (upward or positive) latent heat flux and (downward or negative) CO2 flux data are used in estimating the derivatives.
 We first present the analysis using 30 min resolution measurements of surface energy fluxes, CO2 flux, air temperature at 20 cm height (used as a surrogate for leaf temperature, which was not available) during the period of June 14 to August 28, 1995 at Happy Valley, Alaska. Vegetation height at this location is 25 cm. The estimated derivatives in equations (12)–(13) are shown in Figure 1. Although somewhat noisy, it is evident that ∂Ev/∂Tl and ∂Ev/∂A are fluctuating around zero, and there is little (linear) trend in them over the range of Rn values. It is not clear whether the wave-like patterns in the derivatives are real or numerical artifact. Nonetheless, the graph suggests that equations (12)–(13) hold and the data do not lead to a rejection of the maximum transpiration hypothesis.
 In the second example, 30 min resolution measurements of surface energy fluxes, CO2 flux, and air temperature (used as a surrogate for leaf temperature, which was not available) during the period of August 18–September 27, 2001 at Black Hills, South Dakota, were analyzed. As in the first case, the computed ∂Ev/∂Tl and ∂Ev/∂A in Figure 2 are fluctuating around zero with little (linear) trend over the range of Rn values up to 400 W m−2, suggesting the validity of equations (12)–(13). What causes the deviation of the derivatives from the theoretical value of zero for relatively large Rn, i.e., Rn > 400 W m−2, is not clear. One possible reason is that the error in surface energy balance becomes large so that the estimated derivatives are less accurate (see the discussion in section 4).
 The same analysis was repeated using the fluxes and air temperature time series collected during the period of May 24–June 23, 2000, at Harvard Forest, MA. The air temperature is again used as a surrogate for the missing leaf temperature. Average value of air temperature measured at 28 m and 22.6 m should be fairly close to the leaf temperature at the 24 m canopy top. Evidently, the estimated derivatives shown in Figure 3 are in good agreement with equations (12)–(13), an affirming evidence in support of the maximum hypothesis.
 The above analysis leads to promising, albeit imperfect, evidence that supports a maximum hypothesis of transpiration. More tests are needed because of the difficulty in accurately estimating the derivative variables from noisy data. We have found (results not shown) that the calculated derivatives are sensitive to the bias in the surface energy budget (i.e., Rn = Ev + H). Independent measurement of the individual heat flux components often has significant errors (∼20%) in the surface fluxes. Our analysis shows that as the surface energy budget becomes more imbalanced, the estimated derivatives depart further away from what predicted by the hypothesis. Section 4 offers a detailed discussion of the possible causes of the estimation error in the derivatives using existing data products. Our analysis also shows that the vanishing derivatives according to equations (12)–(13) are nontrivial solutions. We have tested equations (12)–(13) using simulated data obtained from various land surface models, and found that equations (12)–(13) do not always hold. Detailed analysis of the model simulations is beyond the scope of this study since it calls for understanding whether the failure can be attributed to model error, parameter estimation or error in conceptualization of the process. Additional work along these lines is on-going. Nonetheless, our preliminary analysis does indicate the discrimination power of equations (12)–(13). It is revealing that no large deviations of the hypothesis from the data have been observed with the discriminating test.
 We acknowledge that equations (12)–(13) are only necessary conditions for extremum Ev. Although we cannot prove the potential extremum Ev is a maximum, the most probable situation is that Ev reaches maximum when equations (12)–(13) hold, given what is known about transpiration. Theoretically, the two other possible scenarios are: Ev reaches minima or stationary saddle-points. Definite determination can be made by evaluating the second or higher order derivatives of Ev, which seems much more difficult, if not impossible, using the time series observations without specifying a Ev model. This is a subject of future research.
 The formalism and test of the maximum hypothesis through equations (12)–(15) is founded on the validity of equation (5). As discussed in section 2.1, equation (5) builds on the analogy between the transpiration and evaporation processes. In practice, however, Ev is often parameterized as a function of atmospheric and soil variables such as water vapor deficit, air temperature, root zone soil moisture, etc. It is logical to ask: how many parameters are adequate in determining Ev, and must the selected parameters include certain atmospheric and/or soil variables?
 If one accepts that the five basic elements (i.e., state variables Ψl, Tl, H and gs and constraint R) fully characterize the transpiration processes (see Table 1), then Ev can be formulated with no more than four nonredundant state variables. Ev would be determined once the four state variables are specified. Among them, Ψl and Tl characterize the energetics of liquid water, while H and gs quantify the movement of water vapor. Liquid water is presumably at equilibrium with water vapor within substomatal cavities, while the exchange rate of gases with the ambient air is controlled by stomatal aperture and turbulent mixing. R affects Ev indirectly by changing Ψl and Tl through photosynthesis and the energy balance constraint. When a mechanistic link between Ψl and gs exists, Ev only has three independent variables.
 The choice of independent variables is not unique. Due to the strong coupling in the soil-canopy-atmosphere system, the three variables do not have to include either atmospheric or soil parameters. Air temperature is not needed when Tl is used and vice versa. Friction velocity may replace H as well. For the purpose of this study, we select Tl, A and H as the independent variables (see more discussion in section 2.1 and section 2.2.3 of Wang et al. ). The absence of vapor pressure deficit (VPD) in equation (5) seems to contradict the conviction that transpiration depends on VPD among other things. The nature of such correspondence, i.e., causal or correlation, may be debatable. Nonetheless, it is sufficient to argue for equation (5) that VPD does not affect transpiration through impacts on stomatal conductance. Recent data from the Amazon [Hutyra et al., 2007] also show that evapotranspiration follows closely the measured energy inputs but not the vapor pressure deficit. Below we use atmospheric humidity qa to demonstrate that qa is a redundant variable when Ev is formulated as a function of Tl, A and H.
 A statistical justification of equation (5) may be given using a mathematical tool called parameter restriction method combined with meta-analysis technique [e.g., Schulze et al., 2003]. The methodology is outlined in Appendix A. The statistical analysis will tell us whether including qa as an additional independent variable of Ev,
leads to significant improvement of its predictability compared to that of Ev in equation (5). The null hypothesis being tested is as follows: qa does not significantly increase the predictability of Ev when Tl, A, and H are used as the independent variables of Ev.
 The hypothesis is tested with the same data sets as described in section 3.2. The atmospheric humidity is represented by the water vapor deficit, relative humidity, and specific (absolute) humidity, respectively. The first two parameters are (air) temperature dependent, and the third is not. In the analysis, the observed sequences are divided into N = 30 subsequences sorted in the order of ascending values of Ev. The idea is to find out whether including qa as an extra independent variable in Ev can explain better the observed variabilities in Ev. A statistic is constructed from the squared errors in the restricted model equation (5) and the unrestricted model equation (17). If the hypothesis is true, this statistic will follow an F-distribution. A p-value is obtained associated with the statistic computed for a particular subsequence. Then a new statistic can be constructed from the N individual p-values. The new statistic will follow a χ2 distribution associated with a overall p-value, , which is used to determine whether the hypothesis should be accepted. We obtained for the three cases: = 0.32 using Happy Valley data, = 0.93 using Black Hill data, = 0.55 using Harvard Forest data. These numbers are sufficiently large so that the hypothesis cannot be rejected. The result is robust in terms of N, i.e., the overall p-values tend to increase with smaller N.
 The results presented above support the hypothesis that “the dynamics of water and heat exchange at the leaf-atmosphere interface lead to a state of leaf temperature, leaf water potential, and sensible heat flux into the atmosphere that maximizes latent heat expenditure for transpiration under a given energy input, meteorological environment, and physical and physiological form and function of a plant”. At the very least the available data are consistent with the predicted behavior and the hypothesis cannot be disproved based on that information. Furthermore, the data support the assertion that the transpiration is well defined by the three state variables measured at the leaf surface. We argue this is due to the close and fast mechanical and thermodynamic links between the surface and the boundary layer.
4.1. Estimation Errors in the Derivatives
 We do observe noise in the results and it begs a discussion of the possible sources of error in the data analysis. We believe that the relatively large “random errors” in the estimates of the derivatives are caused by several reasons. One major problem with the data sets used in this study is the measurement errors in the surface fluxes of water vapor, heat, and CO2. Uncertainties from the surface and atmospheric characteristics and the limitations of the eddy covariance method often preclude balancing more than 80% of the surface energy budget [Baldocchi and Vogel, 1996; Wilson et al., 2002]. This 20% measurement error will be translated into the uncertainties on the calculated derivatives. A second problem is caused by the mixing of the plant and soil fluxes of water vapor and CO2. The measured water vapor flux reflects both transpired water from the leaf surfaces and evaporated water from the soil since the water vapor sensor sits above the canopy. In closed canopy systems, i.e., most ecosystems except deserts, the vast majority of water flux comes from the plants and not from the soil, so the problem with respect to water vapor flux should not be too severe as far as energy balance is concerned.
 On the other hand, the multiple CO2 sources and sinks in ecosystems add substantial complexity to the interpretation of measured CO2 exchange. In addition to photosynthesis by leaves, which provides a significant sink for CO2, stems, branches, roots, and soil flora constitute a large source of CO2, usually smaller than the photosynthetic flux during the day. This combination of downward (into the leaf) and upward (from plant and soil respiration) fluxes complicates the assumption that the stomatal conductance derived from CO2 is a surrogate of stomatal conductance to water vapor. Nonetheless, this difficulty is unlikely to cause a major concern to this study since photosynthesis is considered the dominant component of CO2 exchange over heavily vegetated regions under study. Indeed, the test results would be more conclusive if data of separate CO2 sources are available in the future. In addition, the fluctuation of ambient CO2 concentration is not included the formulation of the hypothesis. We do not expect it to be important for the purpose of this study since our test are all done over a small range of ambient CO2 concentration. The effect of CO2 concentration may become important if the maximum hypothesis can be generalized over a much longer time scale than that considered here.
 The use of air temperature as the surrogate for leaf temperature in our analysis also contributes to the uncertainties in the estimated derivatives. The issue is complicated by the canopy structure and vertical leaf density distribution. Nonetheless, when the air temperature, Ta, is measured at a point very close (∼20 cm) to canopy, we expect Ta and Tl to be well correlated through a linear regression equation of the form Tl = ATa + B with the coefficient A on the order of unity. Then equation (12) still holds when Tl is replaced by Ta and the corresponding derivatives would differ only by a scale factor (i.e., A ∼ 1) without biases. Even though the leaf and air temperature are correlated, the discrepancies may still introduce significant error (or scatter) in the estimated derivatives, which is rather sensitive to the noise level of the signals. We are not able to quantify this effect at the moment.
 An issue that likely has little quantitative impact on the estimation of the derivatives is the variability in stomatal behavior across species in the canopy. Measurements over the canopy reflect an average of different stomatal response of the individuals composing the canopy. These differences are nevertheless far smaller than those caused by variability in water and energy over the area.
 The link between transpiration and atmospheric humidity has long been under debate. Our statistical justification of equation (5) suggests that the possibility of transpiration not directly responding to atmospheric humidity cannot be ruled out. This is not surprising considering the similarity between transpiration and (soil) evaporation. The phase change must follow the same basic physical laws and share some common features no matter where it occurs, within stomatal pores or over the soil surface. The lack of sensitivity or direct correspondence of transpiration to dryness of the air by no means implies decoupling of canopy from the environment. On the contrary, it indicates strong interaction in the soil-canopy-atmosphere system. As a result, the soil and atmospheric conditions are well reflected in the thermal and water status of the plants, leading to interdependence of mass fluxes, heat fluxes and leaf temperature in a diagnostic manner.
4.2. Maximum Transpiration and Optimal Stomatal Function
 Cowan and Farquhar hypothesized an optimal stomatal function that maximizes water use efficiency in terms of CO2 assimilation. Their theory states that transpiration is minimized for a fixed amount of CO2 assimilation, or equivalently CO2 assimilation is maximized for a fixed amount of water loss through transpiration. There is empirical evidence supporting the theory based on the fact that transpiration is a monotonic function of CO2 assimilation rate with a positive curvature. According to Cowan and Farquhar, Ev and A are related in the following way,
where λ is a nonzero constant (a proposition that is under debate) Lagrangian multiplier. Although equation (18) is formally similar to equation (13), the two equations are not related. The derivative in equation (18) is defined under a given A (or cumulative A over an arbitrary period of time), while that in equation (13) is defined under a given radiative energy input assuming Ev is expressed in the form of equation (5). Equation (18) represents the necessary condition for a minimum Ev, while equation (13) corresponds to the necessary condition for a (presumably) maximum Ev. The apparent contradiction between the minimum transpiration of Cowan and Farquhar and the maximum transpiration of our hypothesis may be reconciled with the help of the Ev − A relationship. Note that equation (18) is also the necessary condition for maximum A given Ev. This maximum A in turn corresponds to a maximum Ev, according to the Ev − A curve, limited by the energy input and the partition of sensible and latent heat flux described by our proposed hypothesis. Together the two equations can be interpreted to indicate that an optimal CO2 gain is always associated with a maximum transpiration water loss for a given radiative energy input. The maximum efficiency and maximum transpiration hypothesis address different aspects of plant physical/physiological processes. Cowan and Farquhar's theory would imply that natural selection favors those plants with optimal stomatal function in terms of water use efficiency. Our hypothesis suggests that the partition of radiative energy favors transpiration for any given stomatal function, optimal or otherwise. The fast establishment of the liquid-vapor equilibrium always leads to a maximum latent heat flux by selecting a particular combination of leaf temperature and leaf water potential among all possible states of leaf temperature and leaf water potential that satisfy the energy balance.
 Numerous studies have attempted to use optimality approaches to understand the regulation of leaf water flux by plants. Since the seminal work of Cowan and Farquhar on this topic, the value of these optimality models has been seen at the scales ranging from the leaf to the whole plant and the uniform stand [e.g., Monteith, 2000]. Recent theoretical and empirical studies have demonstrated, however, that such optimality approaches are not valid at scales larger than the genotype [Anten and Hirose, 2001]. Our formulation is independent of this restriction, however, because it develops a fundamental physical law of maximum transpiration driven by the tendency to quickly approach a liquid-vapor equilibrium. It is important to re-emphasize that this maximum transpiration is associated with a particular stomatal function, which is not necessarily optimal in terms of water use efficiency. The implications of this maximization rule across ontogeny and across biological scales, from organism to ecosystem and from physiological to evolutionary time scales, are not yet understood but stand as a crucial area for future research. By developing a first-principles framework we hope to build a system from small-scale observations that can be used at the canopy (and eventually landscape) scale. This cross-scale capability is one potential advantage of a first-principles approach such as the proposed hypothesis of maximum transpiration.
4.3. Further Test of the Hypothesis
 The proposed hypothesis may be further tested in two other ways. One is to follow the same approach described in section 3 while using alternative sources of data. This approach includes using a different set of state variables to characterize the basic elements of the transpiration process listed in Table 1, for example, representing transport mechanism through friction velocity instead of sensible heat flux, or considering other surrogate(s) of stomatal conductance than CO2 flux. The purpose remains to validate the mathematical conditions under which Ev is maximized. The second one is to investigate the consequences if the hypothesis holds. This may be done by (1) comparing energy budgets of vegetation versus bare soil to show that the presence of vegetation affects (likely enhances) energy dissipation through latent heat transfer; (2) show that as canopies develop from bare soil, latent heat transfer tends toward the upper limit allowed by all possible stomatal functions under unlimited water supply. Although the second approach addresses an issue beyond the scope of this study, it highlights the ecological significance of the hypothesis.
 We have obtained some evidence suggesting that the partition of latent and sensible heat fluxes over vegetated land surface leads to such leaf temperature and leaf water potential that transpiration is maximized for any given plant form and stomatal function. A distinctive feature of this work is the nonparametric formulation of the hypothesis. A model-independent definition of stomatal conductance is less restrictive than the common models of stomatal conductance based on either correlation analysis or Fick's law analogy of water vapor transport. Our results support some earlier studies suggesting that vapor pressure deficit does not regulate stomatal behavior. We believe that this is due to the strong coupling of the atmospheric boundary layer and the surfaces of the canopy. The proposed hypothesis provides a general framework to solve the heat flux partition over the vegetated land surface once a model of Ev is specified although modeling Ev in the form of equation (1) or equation (5) is beyond the scope of this paper. The proposed hypothesis is not a “model” per se; rather it may serve as a general guideline in developing vegetation models and assessing existing models. One elementary application of the hypothesis would be to model calibration, i.e., when empirical parameters are introduced in a transpiration model, equations (12)–(15) could be used to calibrate the model parameters from observations. Future work in this field should aim at a mechanistic model of transpiration. A framework such as is presented in this paper is the first step toward that goal.
 All the potential modeling development hinges on the proof of the hypothesis. What does it imply if further studies disprove the hypothesis? Were it the case, we must conclude that the phase change of water (e.g., liquid to vapor) is not the most effective way to dissipate the energy input, and the living systems would not organize themselves through maximally dissipating energy gradients. This hypothesis does not speak to the question of whether stomatal function is optimal in terms photosynthetic water use efficiency because the hypothesis itself is independent of physiological function. If our hypothesis turns out to be false, then we must look elsewhere to understand the thermodynamic laws underlying the structure and function of water use by plants.
 The parameter restriction analysis includes these steps:
 1. Divide the entire multidimensional domain of observed variables, Tl, A, H, and qa into N subdomains assuming that Ev can be approximated by piece-wise linear functions over the subdomains. Note that R is not an independent variable of Ev.
 2. Estimate a linear regression function Ev,i of the unrestricted model in equation (17) for all subdomains i = 1, 2, …N.
 3. Calculate the unrestricted residual sum of squared errors,
where Ev,io is the observed value of Ev.
 4. Estimate a linear regression function Ev,i of the restricted model in equation (5) for the same subdomains. This is called restricted since the regression coefficient of qa is restricted to be zero.
 5. Calculate the restricted residual sum of squared errors,
 6. Form a test statistic
where T is the number of observations used in calculating the regression coefficients, k the number of regressors in the unrestricted model (i.e., 4 in equation (17)), and s the number of regressors restricted to zero in the restricted model (i.e., 1 in equation (5)) for each of the N subdomains.
 It can be shown [e.g., Pindyck and Rubinfeld, 1998, p. 134] that ω follows the F-distribution, Fs,T−k, under the hypothesis (i.e., qa does not significantly increase the predictability of Ev when Tl, A, and H are used as the independent variables of Ev). Large ω indicates a large decrease in the predictive capability of the restricted model relative to the unrestricted model. A p-value associated with ω can be defined as,
which is between 0 and 1. Hence, the hypothesis that the unrestricted model does not increase predictability cannot be rejected at the significance level α when p > α for an individual subdomain where the linear regression is performed.
 Now we use a standard meta-analysis procedure to create a overall p-value based on the individual p-values, pi, for each of the N subdomains. To do so, a new statistic, Γ, is introduced,
 It turns out that Γ follows χ2N2 distribution under the hypothesis. The overall p-value, , can be defined as,
so that the hypothesis cannot be rejected at the significance level α when > α for the entire domain of the observed independent variables.
 This work was supported by NSF under grant EAR-0309594. Manuel Lerdau is grateful for the support of a Bullard Fellowship and a HRDY Visiting Professorship from Harvard University, and the support of the Mellon Foundation. AmeriFlux data are made available by NASA Oak Ridge National Laboratory (ORNL) Distributed Active Archive Center (DAAC). We are grateful to Steve Wofsy of Harvard University for providing observations from the Harvard Forest site.