[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 *F*_{0} varying with time. Conservation of energy requires *F*_{0} to be distributed between two heat fluxes into the two columns, *F*_{1} and *F*_{2}, which are defined as positive when heat flows away from the interface. *F*_{1} and *F*_{2} 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 *F*_{1} and *F*_{2} is given in equations (A25) and (A26),

where *I*_{1} and *I*_{2} are the thermal inertia of the two media defined in equation (A19), respectively.

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

under the constraint equation (A9), i.e., *F*_{1} + *F*_{2} = *F*_{0}, with *λ*_{k} in equation (5) expressed as

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

[16] The physical meaning of *λ*_{k} becomes clear once *F*_{k} is expressed in terms of temperature at the interface [*Wang and Bras*, 1999]

where *T*_{k}(*t*, 0) is the surface temperature and *τ* is the integration variable. Comparing equation (8) with equation (7) reveals that *λ*_{k} 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

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

[19] Assuming *I*_{a} has the same composition as *I*_{s} 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 *I*_{a} 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 *I*_{a},

where *ρ*, *C*_{p}, κ, *g* and *z* carry their usual meanings, and *T*_{0} is a reference temperature. *C*_{1} and *C*_{2} are coefficients related to the universal constants in the empirical functions (*α*, *β*, *γ*_{1} and *γ*_{2} 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],

*I*_{0} defined in equation (10),

is referred to as the “apparent thermal inertia of the air,” only depends on external parameters such as *z* and *T*_{0}. Note that *I*_{0} does not have the same unit as *I*_{s} due to the factor of ∣*H*∣ in the expression of *I*_{a} that does have the same unit as that of *I*_{s}. It is important to point out that parameterization of *I*_{a} 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 *I*_{a} parameterized in terms of equation (10) is

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

for a given net radiation *R*_{n} requires

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

*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., *I*_{s}, *I*_{0} and *R*_{n}). *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 *R*_{n} = 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 *F*_{k}. 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,

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 *K*_{H} based on the newly derived extremum solution of MOST summarized in Table B1. The new parameterization of *K*_{H} 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.