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Keywords:

  • surface heat fluxes;
  • maximum entropy production

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[1] A model of heat fluxes over a dry land surface is proposed based on the theory of maximum entropy production (MEP) as a special case of the maximum entropy principle (MaxEnt). When the turbulent heat transfer in the atmospheric boundary layer is parameterized using a Monin-Obukhov similarity model, a dissipation function or entropy production function may be expressed in terms of the heat fluxes following the MEP formalism. A solution of the heat fluxes can be obtained by finding the extreme of the dissipation function under the constraint of conservation of energy for a given energy input (i.e., net radiation) at the surface. The MEP solution of the surface heat fluxes is tested using observations from fields experiments.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[2] Modeling heat fluxes remains a major challenge in the study of energy and water balance at the land surface. Recent studies have concluded that no single 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]. One major problem is identified as nonclosure of the surface energy balance, indicating that existing models of surface heat fluxes still have room to improve. The difficulty is, to a large extent, due to lack of a mature theory for nonequilibrium systems. The classical nonequilibrium thermodynamics [de Groot and Mazur, 1984; Kondepudi and Prigogine, 1998] is most successful for the near-equilibrium systems, but inadequate for the far-from-equilibrium processes [Jaynes, 1980] including turbulence in the atmospheric boundary layer. Even though the classical treatment of turbulent transfer in the atmosphere [Priestley, 1959] works reasonably well in practice, further improvement in modeling the surface energy balance needs a better tool than offered by the classical nonequilibrium thermodynamics.

[3] The recent advances in the nonequilibrium thermodynamics including the fluctuation theorem and the hypothesis of maximum entropy production (MEP) [e.g., Dewar, 2005] offer fresh approaches to modeling surface fluxes. The MEP theory is a derivative of the principle of maximum entropy (MaxEnt) first formulated as a general method to assign probability distributions in statistical mechanics [Jaynes, 1957]. The theoretical foundation of MaxEnt is Bayesian probability theory [Jaynes and Bretthorst, 2003] where the concept of entropy is defined as a quantitative measure of information [Shannon, 1948] for any systems that need to be described probabilistically for making statistical inferences. The MaxEnt states that “Out of all the possible probability distributions which agree with the given constraint information, select the one that is maximally noncommittal with regard to missing information” [Gregory, 2005, p. 185]. The MaxEnt distribution is interpreted as the most probable and macroscopically reproducible state among all physically possible states. The generality and power of the MaxEnt are rooted in the Bayesian interpretation of probability that allows the concept of probability to be applicable to any situation where statistical inference based on incomplete information is sought. In that sense, the MaxEnt is much more than a physical law. Yet, it carries physical significance. For example, the (information) entropy, a central concept in the MaxEnt, reduces to the familiar thermodynamic entropy for thermodynamic systems at equilibrium [Tribus, 1961]. The link between the information entropy and the thermodynamic entropy becomes even more intimate and intuitive in the MEP theory [Dewar, 2003]. Nonetheless, it is important to emphasize that the entropy (production) in the MEP theory is not necessarily related to the thermodynamic entropy (production).

[4] Since Paltridge [1975] used the MEP argument to explain the basic patterns of global climate in terms of the maximum rate of thermodynamic dissipation toward a steady state, there have been a number of applications of the MEP theory in earth and planetary climatology and hydrodynamics [Ozawa et al., 2003]. During the last decade, the MEP theory has been tested as an organizing principle governing biological and ecological systems. For example, Kleidon and Fraedrich [2004] have shown that global biotic productivity corresponds to the states of MEP. Juretic and Zupanovic [2003] found that the MEP may govern the photosynthetic processes. Evidence accumulated so far suggests that the MEP theory could explain transport phenomena in the far from equilibrium systems over a wide range of space and time scales [Kleidon and Schymanski, 2008]. We intend to show that the MEP not only explains the observed behavior of nonequilibrium thermodynamic processes, but also can provide a predictive tool in modeling the transport processes quantitatively. This paper investigates the case of heat fluxes over a dry land surface as a proof-of-concept study, which ultimately could lead to a MEP model of surface energy balance including latent heat flux.

[5] Traditionally, ground heat flux is derived from the gradient of soil temperature across a small depth near the surface given the thermal conductivity of the soil materials [e.g., Sellers, 1965]. Ground heat flux may also be derived from the time history of skin temperature as the heat transfer in the soil is described by a diffusion equation given the thermal inertia of the soil materials [Wang and Bras, 1999]. Turbulent sensible heat is often derived from the gradient of air temperature according to the well-known Monin-Obukhov similarity theory [Monin and Obukhov, 1954]. Also based on the Monin-Obukhov similarity theory is the flux variance model [Tillman, 1972; Katul et al., 1995] where sensible heat flux is derived from the standard deviation of turbulent fluctuations of air temperature (at a single level). In these classical models, ground and sensible heat fluxes are estimated using either single-level or multiple-level temperature records (with other necessary input parameters). As shown later in the paper, the proposed MEP model of heat fluxes over a dry soil is different from the existing models in that the MEP model of ground and sensible heat fluxes does not use soil and air temperature records if net radiation is available.

[6] The paper is organized as follows. We first review the MEP formalism following Dewar [2005] in section 2. The formulation and validation of the MEP solution of ground and sensible heat fluxes is presented in section 3. Further understanding of the partition between ground and sensible heat fluxes from the perspective of an optimality principle is the subject of section 4, followed by concluding remarks (section 5).

2. MEP Formalism

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[7] The details of the MEP formalism are described in the original work of Dewar [2005]. Here we summarize the key results that are most relevant to the proposed MEP model below. The MEP is a derivative of the MaxEnt. In the MaxEnt formalism, the probabilities pi of variables xi (1 ≤ in where n is a positive integer) are derived by maximizing the Shannon information entropy, SI

  • equation image

subject to the constraints

  • equation image

where fk are functions of xi, Fk are given parameters (as the constraints) representing available information about xi, and m (≪ n) is a given integer. The brackets stand for mathematical expectation. The MaxEnt distributions pi can be derived by maximizing SI in equation (1) subject to the constraints in equation (2), leading to

  • equation image

where λk (1 ≤ km) are the Lagrange multipliers associated with the constraints Fk, which are related through the well known Legendre transform.

[8] The MEP theory results from the particular situation of antisymmetric functions fk in equation (2) satisfying fk(xi+) = −fk(xi) when all potential outcomes, xi, can be grouped in pairs (i+, i−). For anti-symmetric functions (whose physical meaning becomes clear later in the paper), the corresponding MaxEnt distribution satisfies the generic “fluctuation theorem” (FT),

  • equation image

The FT implies that some microscopic states or phase space trajectories are overwhelmingly favored over the others, leading to macroscopic transport of heat and matter, unless the exponent of the exponential function in equation (4) is always close to zero.

[9] The behavior of the MaxEnt distribution for the antisymmetric functions is determined by the properties of the exponent of the exponential function in equation (4), whose mathematical expectation

  • equation image

is referred to herein as the “dissipation function” or “entropy production function” for a reason explained by Dewar [2005].

[10] D in equation (5) satisfies orthogonality conditions [see Dewar, 2005, equations (18) and (19)], which imply that D reaches an extremum given certain constraint on the Fks. D is maximum when the constraint is a certain nonlinear (e.g., quadratic) function of Fk, and minimum when that constraint is a linear function. Therefore, “maximum entropy production” is in fact a misnomer in the sense that D could reach either a maximum or a minimum or a saddle point, theoretically, depending on the functional form of the constraint on Fks. The well-known Prigogine's minimum entropy production theorem can be viewed as a special case of the general linear constraint, leading to a minimum D. Hopefully, MEP will be replaced by a more suitable term in the future to avoid confusion. In the development of MEP model below, we deal with the situation of a linear constraint on the unknown surface heat fluxes through the surface energy balance equation.

[11] We emphasize that the orthogonality conditions of D may not be exact since the derivation invokes a quadratic approximation of the exponent function in equation (4). Grinstein and Linsker [2007] argued that this approximation implies that the orthogonality conditions are valid only for near-equilibrium systems where Fks are close to zero [see Grinstein and Linsker, 2007, equation (2)]. A careful examination of Grinstein and Linsker's derivation indicates that Grinstein and Linsker's conclusion is in part due to a mathematical artifact. (It is incorrect to differentiate λk with respect to Fn as in equation (2) of Grinstein and Linsker [2007] because equation (15) of Dewar [2005] is only an expression of λk evaluated at given equation image. ∂λk/∂Fn must be obtained from equation (10) or equation (9) where S(equation image) is given in equation (8) of Dewar [2005]). Grinstein and Linsker are correct that the quadratic approximation is less accurate than hoped by Dewar [2005]. Nonetheless, the formulation of MEP for the case of Fk not close to zero put forward by Dewar [2005] does not invalidate the MEP hypothesis for far from equilibrium systems. A case study of land surface energy balance presented below provides evidence that the MEP hypothesis may hold for a far from equilibrium system where constitutive relations such as turbulent heat transfer are highly nonlinear. This is a topic for ongoing and future research.

[12] The key to apply the MEP method is to obtain an expression of D where λk must be expressed as explicit functions of Fk according to the Legendre transformation once fks in equation (2) are identified. For the case of heat exchange processes at the land surface, fk may include the kinetic energy of molecules or turbulent eddies, etc. Identification of fk requires deep understanding of the physical processes at a microscopic level and characterized in mathematically tractable forms. Yet most of the microscopic details of the physical processes, difficult to fully capture, are irrelevant to the macroscopic observable properties. Therefore, it is possible to find D without knowing fk exactly. The situation is analogous to the classical equilibrium thermodynamics where the states of individual molecules play no role in the ideal gas law. Finding D without specification of fk indeed can be done for the case of land surface energy balance, of interest in this paper. Below we show how to use an idealized case (described as the toy model below), where the analytical function D is known, to formulate D involving turbulent heat flux through analogy.

3. Application of MEP in Land Surface Energy Balance

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[13] The surface heat fluxes predicted by the MEP theory are referred to as the MEP solution of surface heat fluxes herein. Using the MEP framework outlined above, it would be more realistic to formulate D without getting into microscopic details of molecular conduction in the soil and turbulent diffusion in the air. Anticipating some commonality of transport processes in nonequilibrium systems, we start by building a toy model of the surface energy balance where heat transfer is assumed to be through conduction. The MEP solution of the toy model not only reveals the functional form of D, but also elucidates the physical significance of the Lagrange multipliers when the constraints are imposed on fluxes.

3.1. A Toy Model

[14] Consider one-dimensional heat conduction in one (vertical) semi-infinite column (labeled 1) on top of the other (labeled 2) with different thermal properties. Figure 1 illustrates the configuration of the system. A source of heat is located at the interface z = 0 with a prescribed input heat flux F0 varying with time. Conservation of energy requires F0 to be distributed between two heat fluxes into the two columns, F1 and F2, which are defined as positive when heat flows away from the interface. F1 and F2 can be determined by solving the corresponding diffusion equations governing the heat transfer within the media as described in Appendix A. An analytical solution of F1 and F2 is given in equations (A25) and (A26),

  • equation image

where I1 and I2 are the thermal inertia of the two media defined in equation (A19), respectively.

image

Figure 1. Graphical illustration of the toy model configuration. The top graph refers to z > 0, and the bottom graph refers to z < 0. T1(z, t) and T2(z, t) are temperature at location z and time t within the top graph and the bottom graph, respectively. F1 and F2 are heat fluxes at the boundary of the top graph and the bottom graph, respectively, which are defined as positive going toward the interior of the two media. The thermal properties of the corresponding medium include bulk density ρ, specific heat c, thermal conductivity μ, thermal diffusivity κ, and thermal inertia I.

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[15] The solution of Fk (k = 1, 2) can be obtained without solving the differential equations. It is straightforward to verify that the exactly the same solution of Fk results from minimizing D defined as

  • equation image

under the constraint equation (A9), i.e., F1 + F2 = F0, with λk in equation (5) expressed as

  • equation image

which is recognized as the orthogonality conditions given by Dewar [2005, equations (18) and (19)].

[16] The physical meaning of λk becomes clear once Fk is expressed in terms of temperature at the interface [Wang and Bras, 1999]

  • equation image

where Tk(t, 0) is the surface temperature and τ is the integration variable. Comparing equation (8) with equation (7) reveals that λk is the half-order time derivative of the temperature at the interface. The validity of the MEP argument is confirmed by the physical reality of a continuous distribution of temperature throughout the two columns at all times. The straightforward application of the MEP demonstrates its power in deriving highly nontrivial results from seemingly trivial propositions.

[17] The toy model is an example where the concept of “entropy production” is not related to the production of thermodynamic entropy expressed as the ratio of flux to temperature that the term “MEP” may allude to.

3.2. Ground and Sensible Heat Fluxes Over Dry Soil

[18] Consider the energy budget over a dry nonvegetated land surface. The major difference between the above toy model and the energy budget over a land surface is due to the transport mechanism, i.e., heat transfer in the atmosphere results from turbulent diffusion instead of conduction in solid media. Yet it remains true that the temperature is continuous at the land-atmosphere interface. Since this property plays a key role in the MEP solution of the heat fluxes in the toy model, the energy budget over a dry land surface is expected to be solved in the same way using the MEP framework. By analogy, the dissipation function of ground and sensible heat fluxes, G and H, is defined as

  • equation image

where Is is the thermal inertia for heat conduction in the soil as a composition of density, specific heat and thermal diffusivity. Ia is the “thermal inertia” for turbulent heat transfer in the air. Is is a well-defined physical property of the soil material, while Ia is yet to be defined.

[19] Assuming Ia has the same composition as Is defined in equation (A19), an eddy diffusivity must be parameterized using a turbulent transfer model where an eddy diffusivity appears as the coefficient formally relating the turbulent heat flux to the gradient of mean (potential) temperature. The most successful models of turbulent transport in the atmospheric boundary layer are those based on Monin-Obukhov similarity theory (MOST) [e.g., Arya, 1988]. In the original MOST, two dimensionless equations are established to relate the gradients of mean temperature and wind velocity to the fluxes of heat and momentum according to the Buckingham π theorem [Buckingham, 1914]. Consequently, the eddy diffusivity must be expressed in terms of two of the four state variables (i.e., the gradients of mean wind velocity and temperature and the fluxes of heat and momentum). Our goal is to reduce the degree of freedom from two to one so that Ia can be formulated as a function of the heat flux alone. This can be done by using an extremum solution based on the MOST briefly described in Table B1 in Appendix B. The extremum solutions were derived by removing the nonuniqueness in the relationships between wind shear/temperature gradient and momentum/heat flux described by the similarity equations, equations (B1) and (B2). The extremum hypothesis leads to a third equation linking wind shear, temperature gradient, momentum flux and heat flux, allowing any three of the four parameters to be expressed in terms of the other [Wang and Bras, 2009].

[20] Following the eddy diffusivity defined in equation (B12) and the proposed extremum solution based on the MOST summarized in Table B1, we obtain an expression of Ia,

  • equation image

where ρ, Cp, κ, g and z carry their usual meanings, and T0 is a reference temperature. C1 and C2 are coefficients related to the universal constants in the empirical functions (α, β, γ1 and γ2 in Table B1) representing the effect of the stability on the mean profiles of wind speed and (potential) temperature within the surface layer [Businger et al., 1971],

  • equation image

I0 defined in equation (10),

  • equation image

is referred to as the “apparent thermal inertia of the air,” only depends on external parameters such as z and T0. Note that I0 does not have the same unit as Is due to the factor of ∣Hequation image in the expression of Ia that does have the same unit as that of Is. It is important to point out that parameterization of Ia based on equation (B11) by no mean implies linearity between heat flux and temperature gradient, which is true only for conduction. Turbulent heat flux in general is a nonlinear function of the corresponding temperature gradient, leading to a flux-dependent thermal inertia parameter.

[21] The dissipation function D with Ia parameterized in terms of equation (10) is

  • equation image

which is not quadratic in H. Minimizing D under the constraint of conservation of energy at the land surface,

  • equation image

for a given net radiation Rn requires

  • equation image

Substituting equation (14) into equation (13) leads to a nonlinear algebraic equation for H,

  • equation image

G and H according to equations (14) and (15) are the MEP solution of the energy budget over a dry land surface.

[22] The solution of H, hence G, is unique since equation (15) has only one real root for realistic values of the parameters (i.e., Is, I0 and Rn). G and H have the same sign according to equation (14) following the usual sign convention for G and H. As a special case, H = 0 and G = 0 when Rn = 0.

[23] Equation (14) would be formally identical to equation (7) were not for the 11/12 factor. The difference is due to the fact that D defined in equation (12) is not a quadratic function of G and H, while D in equation (6) for the toy model is a quadratic function of Fk. This numerical example suggests that the orthogonality conditions given by Dewar [2005], which hold exactly for near-equilibrium systems, are indeed a good approximation for far from equilibrium systems. Equation (14) without the 11/12 coefficient is recognized as the formula obtained by Priestley [1959, p. 105, equation (8.10)] based on the classical theory,

  • equation image

The MEP theory justifies Priestley's result based on a more fundamental principle except that Priestley's equation is difficult to apply without the parameterization of KH based on the newly derived extremum solution of MOST summarized in Table B1. The new parameterization of KH based on the equations given in Table B1 is briefly described in the end of Appendix B. To our knowledge, there has been no application of Priestley's formula in land surface models.

3.3. Validation of the MEP Solution

[24] The MEP solution of G and H in equations (14) and (15) will be compared with observations from two field experiments at Owens Lake, California in 1993 and at Lucky Hill near Tombstone, Arizona in 2008. The measured energy balance at the two sites is shown in Figure 2.

image

Figure 2. Observed energy balance Rn = G + H for (a) Owens Lake and (b) Lucky Hill.

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3.3.1. Owens Lake

[25] The data at this bare soil site was collected during 20 June to 2 July 1993. Sensible heat flux was measured using an eddy covariance device, including a sonic anemometer mounted at z = 2.5 m above the ground, colocated with other sensors to measure radiative fluxes, ground heat flux, soil skin temperature among other variables. Twenty minute averaged variables are used in this study. More details about the experiment site, instrument setup, sampling schemes, etc. have been reported in an earlier publication [Katul, 1994]. Thermal inertia of the soil material at this location was estimated as the (regression) coefficient in equation (8), Is ≃ 0.83 × 103 J m−2 K−1 s−1/2, using the independently measured skin temperature and ground heat flux. I0, defined in equation (10), is computed as

  • equation image

where T0 = 300 K is used. It turns out that I0 is insensitive to T0 due to the 1/6 power dependence. For example, the extreme diurnal variation of temperature T0 ∼ 280–330 K at the two sites causes no more than 3% change in I0, which is small enough to be negligible. The observed energy balance at this site is shown in Figure 2a.

[26] Figures 3 and 4 compare the MEP solutions of G and H with the observed fluxes. The agreement is excellent. In Figure 3, the MEP G and H are computed with the observed Rn as input to equation (15). In Figure 4, the MEP G and H are computed with the observed total fluxes, G + H, to replace Rn in equations (15). Since the observed energy balance is closed without significant biases (see Figure 2a), the agreement between the two MEP solutions (under different inputs) are comparable to that between observed Rn and G + H. In addition, the agreement between the MEP predictions and the observed fluxes is as good during the nighttime as during the daytime. Notice the observed G and H (as well as Rn not shown here) cross zeroes at the same times predicted by the MEP. The second test using the Lucky Hill data demonstrates the behavior of the MEP solution when the observed energy balance has some biases.

image

Figure 3. (a) Time series of the MEP solution of G and H (dashed lines) versus the observed fluxes (solid lines) with the observed Rn as the input to equation (15); MEP solution of (b) G and (c) H versus observations. Twenty minute average data collected at Owens Lake, California, 20 June to 2 July 1993.

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image

Figure 4. (a) Time series of the MEP solution of G and H (dashed lines) versus the observed fluxes (solid lines) with the observed total fluxes G + H as the input to equation (15) (to replace Rn); MEP solution of (b) G and (c) H versus observations. Same data set as that of Figure 3.

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3.3.2. Lucky Hill

[27] The site (31°44.564′N, 110°3.251′W) is located in the Walnut Gulch watershed, Arizona. The field experiment was carried out 2–17 June 2008 during the premonsoon season with strong insolation and occasional clouds. An instrument tower was set up at a flat spot with sparse dry shrubs. The topsoil layer of tens of centimeters was completely dry after an extended rain-free period. Ground heat flux was measured using heat flux plates placed ∼ 1 cm below the surface. The observed energy balance at this site is shown in Figure 2b. Skin temperature was measured using an infrared thermometer mounted on the tower ∼ 1 m above the ground. Sensible heat fluxes were derived from 10 Hz samples of a RM Young sonic anemometer and a temperature sensor mounted at z = 4.3 m above the ground. Four components of radiative fluxes were measured using Epply Precision Spectral Pyranometers and Precision Infrared Radiometers (Pyrgeometers). All variables were sampled at 0.1 Hz except those measured by the eddy covariance device. Twenty minute averaged variables are used below. The thermal inertia of the soil at this site has been estimated as Is ≃ 0.83 × 103 J m−2 K−1 s−1/2, in the same way as that at Owens Lake. I0 is computed as

  • equation image

[28] Figure 5 compares the MEP predictions with the observed fluxes corresponding to the case of Figure 3. They agree qualitatively but have visible differences. The MEP G has reduced diurnal amplitudes relative to, but in phase with the observed G (Figures 5a and 5b). According to Figure 2b, the observed Rn is underestimated at high Rn, and overestimated at low Rn relative to the observed G + H. Figure 5b shows that these biases in the observed energy budget lead to the same biases in the MEP G relative to the observed G due to the energy balance constraint. Figure 5b confirms G in phase with Rn predicted by the MEP (the scatterplot of the observed G versus Rn not shown). Figure 5c shows the biases as well as phase differences between the MEP H and the observed H. The biases in the MEP H are due to the same reason for those in the MEP G. The phase differences, which do not exist between the MEP and observed G, are caused by the phase differences between the observed Rn and H (figure not shown) since the MEP predicts H in phase with Rn (e.g., peaking at the same times). This test demonstrates that the MEP is able to use the available information effectively to pick up true signals and highlight the inconsistency in the inputs.

image

Figure 5. (a) Time series of the MEP solution of G and H (dashed lines) versus the observed fluxes (solid lines) with the observed Rn as the input to equation (15); MEP solution of (b) G and (c) H versus observations. Twenty minute average data collected at Lucky Hill in Walnut Gulch watershed, Arizona, 2–17 June 2008.

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[29] Using the observed total flux G + H input, the MEP G and H are in much closer agreement with the observed G and H as shown in Figure 6. The bias in G and phase difference in H are reduced according to Figures 6b and 6c, respectively. It appears that the MEP G has slight phase shift relative to the observed G. This is due to the phase difference between the observed G and H (figure not shown). The signal of relative phase in the input is preserved but divided between the MEP G and H.

image

Figure 6. (a) Time series of the MEP solution of G and H (dashed lines) versus the observed fluxes (solid lines) with the observed total fluxes G + H as the input to equation (15) (to replace Rn); MEP solution of (b) G and (c) H versus observations. Same data set as that of Figure 5.

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[30] The MEP solution of heat fluxes appears to be a function of z even though the physical process of the partition of radiative energy occurs at the land-atmosphere interface. This is due to the parameterization of Ia (and H) using the MOST model. In reality, H has always been measured at a certain distance above the ground in field experiments. Since z has appreciable effect on the MEP solution due to the 2/3 power dependence of I0 on z, the value of z must be appropriately chosen for the locations where no field measurements of H (e.g., using eddy covariance device) are available to compare with. Note that the MOST model for the surface layer is only valid for a limited range of z. z cannot be too small due to the assumption of the MOST that mean wind speed and temperature profiles are not affected by the characteristics of the underlying surface such as roughness. z cannot be too large either because H at large z could be substantially different from H at the surface [e.g., Wang and Bras, 2001]. It would be ideal if z is set at the lower bound of the surface layer, zs, where H is expected to be the closest to H at the surface (which has never been measured directly). The theoretical value of zs is rather difficult to determine. Nonetheless, the good agreement between the MEP H with the observed H indicates that zs at the two sites is close to the measurement height of H. Our results suggest that, depending on the surface roughness, zs ∼ 2–3 m for flat nonvegetated surfaces, while zs ∼ 4–5 m for surfaces covered with sparse vegetation of ∼1 m in height.

[31] The only meteorological input to the MEP model is net radiation. The model parameters are either physical (i.e., thermal inertia of the soil) or universal constants. The model does not need calibration (i.e., no tuning parameters). No classical theories would give a unique solution of G and H (over a dry soil) based on conservation of energy only (i.e., G + H = Rn for given Rn) since there are more unknowns than governing equations. The problem is underdetermined using conventional methods. This is a typical situation where an inference is sought based on incomplete information. In that sense, the MEP solution is a “guess,” arguably the best one, instead of a guaranteed prediction. Our confidence on the MEP solution may be measured by the FT (see equation (4)). The odds that the MEP prediction fails is estimated as 1 in 1017 for common situations, e.g., D ∼ 40 for G, H ∼ 100 and Is, Ia ∼ 103. Certainly, the performance of the MEP method depends on the input information. The power of the MEP is reflected in its capability, as an inference tool, of extracting the relevant information about G and H given the net energy, Rn.

[32] An intriguing aspect of the analysis is that the surface temperature seems to be irrelevant since neither the dissipation function nor the test of the MEP solution needs the information of surface temperature. Yet, surface temperature is expected to be an important diagnostic for the surface energy balance as the emitted long-wave radiation, a component of Rn, strongly depends on the skin temperature. In fact, the effect of the surface temperature on the MEP solution of G and H is included through Rn even though a given Rn does not uniquely determine the surface temperature. In addition, the surface heat fluxes are directly related to space-time variations in the surface temperature instead of temperature itself. Nonetheless, the surface temperature does contain essential information about the surface energy budget that we turn our attention to next.

4. A Stationary Hypothesis of Energy Balance

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[33] Many nonequilibrium systems can be understood and described by extremum principles [Weinstock, 1952; Sieniutycz and Salamon, 1990]. The idea that nature takes an optimal path (or fastest approach) to reach an equilibrium state under certain constraints has led the hypotheses of maximum evaporation and transpiration [Wang et al., 2004, 2007] governing the energy budget over the land surface. It has also been suggested that turbulent sensible heat flux is the next most effective (to evaporation) heat removal mechanism [Kim and Entekhabi, 1998]. If that is true, which as we show below it is not, H is expected to be maximized among all possible values allowed by the available energy for the case of dry soil. The hypothesis of maximum H, or equivalently minimum G due to conservation of energy H + G = Rn for a given Rn, can be tested using the observations of heat fluxes and skin temperature.

[34] In the following analysis, G = G(Ts) is a functional of the skin temperature time history [Wang and Bras, 1999], where Ts is understood as the skin temperature at current time while the skin temperature at earlier times is treated as given parameters. The necessary condition for H to be extremum is that all partial derivatives of the Lagrangian function,

  • equation image

with respect to the state variables Ts, H and the Lagrangian multiplier λh vanish,

  • equation image

for a given Rn. The choice of H as a state variable explicitly leads to a partition of net radiation as a result of the extremization procedure without specification of arbitrary models for turbulent heat transfer. Heat fluxes are commonly used as independent state variables in the nonequilibrium thermodynamics [Sieniutycz and Salamon, 1990, p. 3]. Equations (17) and (18) yield the necessary condition of a potential extremum H,

  • equation image
  • equation image

The superscript Rn in equation (19) emphasizes that the derivative is evaluated under the condition of fixed Rn. The corresponding Hessian matrix ℋfh (see Appendix C) has three (nonzero) eigenvalues,

  • equation image

where the superscript “0” indicates the derivatives are defined at the stationary point of fh under the constraint G(Ts) + H = Rn. The eigenvalues do not have the same sign since the curvature of G function is not zero in general. Then the stationary point of H is a saddle point instead of an extremum due to the indefiniteness of ℋfh. If equation (19) holds, neither is H maximum, nor is G minimum due to the constraint G + H = Rn. Validation of equation (19) implies that Rn is partitioned in such as way that both fluxes reach a stationary saddle point.

[35] The estimated ∂G/∂Ts terms using the observed Ts, G and Rn for the Owens Lake and the Lucky Hill site are shown in Figures 7 and 8, respectively. They are close to zero, meaning that the values are small compared to the ratio of the amplitude of variation in G to that of Ts, ∼10 W m−2 K−1. At both sites, the observed G ranges from −100 to 500 W m−2 and Ts from 10 to 60 C (Ts data not shown). The estimated derivatives only significantly depart from zero value at high and low levels of Rn. There are two plausible explanations for this departure. First, there are fewer data points at high and low values of Rn that may cause large numerical errors in estimating the partial derivatives. Second, the nonzero derivatives may result from the violation of energy conservation as the observed energy fluxes tend to be more out of balance, especially at the Lucky Hill site (see Figure 2), at very high and very low Rn. Hence we view the results as solid evidence supporting the hypothesis that H and G reach stationary saddle point in balancing net radiation.

image

Figure 7. Validation of equation (19) using the observed Ts, G and Rn measured at the Owens Lake site.

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image

Figure 8. Validation of equation (19) using the observed Ts, G and Rn measured at the Lucky Hill site.

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[36] The hypothesis, if further confirmed to be true, is a general governing principle for the energy budget over a dry soil since the test of the hypothesis is (turbulent transport) model independent. Note that equation (19) holds for the case considered in section 3.1 where an analytical solution of the heat fluxes exists (see Appendix A). Temperature independent F1 and F2 in equations (A25) and (A26) guarantees vanishing derivatives for prescribed F0. But neither can be maximum or minimum because F1 and F2 are symmetric. Then the only possibility is that the stationary point is a saddle. It would not be surprising that the conclusion for F1 and F2 would be also valid for G and H if one thinks of heat conduction as a limiting case of turbulent diffusion.

[37] The above analysis indicates that the energy budget over a dry soil depends on change of temperature instead of absolute level of temperature. This is not a new finding as all existing models of G and H [e.g., Sellers, 1965; Wang and Bras, 1999; Monin and Obukhov, 1954; Tillman, 1972; Katul et al., 1995] use either spatial or temporal variations of temperature. The MEP solution of heat fluxes is consistent with this property of energy budget over dry soil: the surface heat fluxes are not informative about the surface temperature, at least not directly, which gives a physical meaning to equation (19).

[38] The stationary hypothesis and the MEP model are intended to tell a full story about the partition of net radiation into surface heat fluxes over a dry soil seen from complementary point of view. The stationary hypothesis describes a certain aspect of the process that the MEP model does not, and vice versa. The MEP model of surface heat fluxes is formulated using an epistemological argument based on the Bayesian probability theory. The result is a useful algorithm, as a statistical inference, for computing the surface heat fluxes using the available information. The stationary hypothesis of surface heat flux is developed using an ontological argument based on our experience about the physical world. The result is a rationale, as a fundamental principle, for explaining the underlying physical mechanisms leading to the observed the surface heat fluxes. Together they not only shed new light on the process of surface energy balance but also predict the energy budget with information considered to be incomplete from traditional viewpoint. The MEP hypothesis has little direct bearing on the stationary hypothesis except that the latter may explain why the MEP model of heat fluxes does not need temperature input. Logically, the stationary hypothesis is completely independent of the MEP model of heat fluxes in the sense that (1) testing the stationary hypothesis does not require models of sensible and ground heat flux, which makes the hypothesis most general; (2) the stationary hypothesis does not lead to models of sensible and ground heat flux either (at least not yet). Nonetheless, the stationary hypothesis supported by the observational evidence (i.e., ∂G/∂Ts = 0) conditioned on conservation of energy (i.e., G + H = Rn) is consistent with the MEP model that the solution of surface heat fluxes does not depend on Ts, at least not directly.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[39] The proposed model of the surface heat fluxes based on the principle of maximum entropy production and the stationary hypothesis of the energy balance are two complementary components of a new view of the heat exchange at the land-atmosphere interface. The stationary hypothesis adds another example of natural phenomena understood from the perspective of an optimality argument. The hypothesis was tested under the most general conditions even though it does not lead to a model of the heat fluxes. That task is fulfilled by the MEP theory that allows the heat fluxes to be parameterized once transport models are selected. The MEP method for modeling and estimating surface heat fluxes complements the existing methods as it only needs net radiation input without requiring other meteorological variables such as temperature and wind speed. The MEP method will not replace the existing ones, instead it offers an alternative method that would enhance the models of land surface energy balance.

[40] The significance of the proposed MEP model is that the concept of entropy production is not limited to the thermodynamic entropy production. In general, the MaxEnt and MEP formalisms provide a tool through extremization of “entropy production” to make predictions (as statistical inference). Depending on the quantities of interest, the “entropy (production)” defined in the MaxEnt and MEP formalisms is not always related to the thermodynamic entropy. This study presents an example where the “entropy production” is different from both thermodynamic entropy production and (thermal) energy dissipation that are often used in the models of nonequilibrium systems. We expect the MEP method to be applicable to the energy balance over a wet or a vegetated land surface where the challenge is to formulate the entropy production or dissipation function including the latent heat flux term; an ongoing research subject. This study offers new possibilities of improving hydrology models using the MEP theory. The encouraging results offer genuine hope that the next generation of land surface models will be capable of capturing all features of the surface energy balance under all conditions.

Appendix A:: Analytical Solution of Heat Conduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[41] The heat transfer in the two semi-infinite columns in contact at z = 0 are described by

  • equation image
  • equation image

where T is temperature and ρ, c and μ are constant density, specific heat and heat conductivity associated with self-explanatory indices. t and z carry their conventional meaning. The prescribed initial and boundary conditions are

  • equation image
  • equation image
  • equation image
  • equation image

where T0 is a constant. A second boundary condition at z = 0 must be specified for solving T1 and T2. Assume a source of heat is located at z = 0 (the contact point of the two columns) with a given heat flux input, F0, partitioned into F1 and F2,

  • equation image
  • equation image

satisfying conservation of energy,

  • equation image

where appropriate sign is understood. F1 and F2 are unknowns to be determined from equations (A1)(A9).

[42] Solutions of equations (A1) and (A2) under initial and boundary conditions equations (A3)(A9) may be deduced by means of Laplace transform (ℒ transform) with respect to time t. Denote

  • equation image
  • equation image

where ℒ is the Laplace transform operator,

  • equation image

[43] equation image transform of equation (A1) with the initial condition equation (A3) leads to

  • equation image

where equation image1 = μ1/(ρ1c1). Equation (A13) has a general solution,

  • equation image

where C1(s) and D1(s) are two arbitrary functions. Zero flux boundary condition equation (A4) requires C1(s) = 0. Therefore,

  • equation image

where D1(s) is yet to be determined. Similarly, equation image transform of equation (A2) with the initial condition equation (A5) leads to

  • equation image

where κ1 = μ2/(ρ2c2) and C2(s) is an arbitrary function to be determined.

[44] Continuous temperature at the interface z = 0 through equation (A6) requires

  • equation image

Substituting equations (A15)(A17) into ℒ-transformed equations (A9), we obtain

  • equation image

where equation image0 is the ℒ transform of F0 with thermal inertia parameters defined as

  • equation image

Then,

  • equation image
  • equation image

[45] Inverse ℒ transform of equations (A20) and (A21) gives the solution of temperature at the interface z = 0,

  • equation image

[46] According to equations (A7) and (A8), ℒ transform of F1 and F2 are

  • equation image
  • equation image

Then, inverse ℒ transform of equations (A23) and (A24) where ∂equation image1(t, 0)/∂z and ∂equation image2(t, 0)/∂z are obtained using equations (A20) and (A21) by differentiating with respect to z first and then letting z = 0 leads to the desired solution,

  • equation image
  • equation image

Appendix B:: An Extremum Solution of Surface Layer Turbulence

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[47] According to Monin and Obukhov [1954], mean wind shear Uz and temperature gradient Θz within a stationary and homogeneous surface layer can be described by the following equations:

  • equation image
  • equation image

where z is the vertical coordinate (i.e., distance from the surface), κ is the von Karman constant. Velocity scale u* (friction velocity) and temperature scale θ* are defined through shear stress τ and heat flux H as

  • equation image
  • equation image

where ρ is the (constant) air density, Cp is the heat capacity of the air at constant pressure. L is the Obukhov length,

  • equation image

where g is the gravitational acceleration, and T0 is a representative temperature.

[48] ϕm and ϕh in equations (B1) and (B2) are empirical functions introduced by Monin and Obukhov to represent the effect of the stability on the mean profiles of wind speed and temperature. The most popular functional forms of ϕm and ϕh are proposed by Businger et al. [1971]. Alternative forms of ϕm and equation imageh have been suggested by other authors [e.g., Beljaars and Holtslag, 1991], but they do not differ qualitatively from those of Businger et al. According to Businger et al.,

  • equation image
  • equation image

under stable conditions, L > 0; and

  • equation image
  • equation image

under unstable conditions, L < 0. The empirical constants in the above expressions are estimated as

  • equation image

[49] It can be shown that equations (B1) and (B2) do not always lead to unique relationships between Uz, Θz, u* and H. To remove the nonuniqueness, we seek those among all possible solutions allowed by equations (B1) and (B2) that satisfy the requirement that “momentum flux always reaches such values that heat flux and wind shear are minimized under stable condition; and that heat flux and temperature gradient are minimized under unstable condition.” Then we obtain expressions of Θz and H in terms of u* shown in Table B1, called the extremum solution based on Monin-Obukhov similarity theory (MOST).

Table B1. Summary of the Extremum Solution Based on Monin-Obukhov Similarity Theorya
 StableUnstable
  • a

    The constants are taken as α ∼ 0.75 or 1, β ∼ 4.7, γ1 ∼ 15, γ2 ∼ 9.

Θz(α + equation image)equation image2equation image
equation imageequation imageequation image

[50] By combining the extremum solutions of Θz and H in terms of u* given in Table B1, H can be formally expressed in the form

  • equation image

with the eddy-diffusivity KH defined as

  • equation image

where C1 is defined in section 3.2. The “thermal inertia” for the air, Ia, may be defined formally,

  • equation image

Appendix C:: Sufficient Conditions for Extremum

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[51] Positivity of the Hessian matrix of a multivariate function f(x1, x2,…, xn) evaluated at the stationary point equation image0 = (x10, x20,…, xn0) determines whether f(equation image0) is an extremum [Dineen, 2001]. The Hessian matrix of f is defined as

  • equation image

and the stationary point of f, equation image0, is determined by

  • equation image

where ∇ is the gradient operator. The criteria for extremum f(equation image0) are

[52] 1. f(equation image0) is a local minimum (maximum) if ℋf is positive (negative) definite at equation image0,

[53] 2. ℋf is positive (negative) definite if and only if all eigen values of ℋf are positive (negative),

[54] 3. f(equation image0) is a saddle point if ℋf is indefinite, i.e., its eigen values change signs.

[55] The Hessian matrix of fh in equation (17) is

  • equation image

At the stationary point,

  • equation image

then ℋfh has three eigenvalues,

  • equation image

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information

[56] This work was supported by ARO under project W911NF-07-1-0126 and NSF under grant EAR-0309594. We are grateful to Gabriel Katul of Duke University for providing the Owens Lake data used in this study. We thank David Goodrich and John Smith of USDA-ARS for their support during the field experiment involving graduate students at MIT, Ryan Knox and Gajan Sivandran.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. MEP Formalism
  5. 3. Application of MEP in Land Surface Energy Balance
  6. 4. A Stationary Hypothesis of Energy Balance
  7. 5. Conclusions
  8. Appendix A:: Analytical Solution of Heat Conduction
  9. Appendix B:: An Extremum Solution of Surface Layer Turbulence
  10. Appendix C:: Sufficient Conditions for Extremum
  11. Acknowledgments
  12. References
  13. Supporting Information
FilenameFormatSizeDescription
wrcr12238-sup-0001-taB01.txtplain text document1KTab-delimited Table B1.

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