## 1. Introduction

Our response to the comments by van de Wiel et al. (2011) on our paper (Wang and Bras 2010, hereafter WB10) follows as closely as possible the order and format of van de Wiel et al. After addressing the issues raised in van de Wiel et al., we will make further remarks on the extremum solution (ES) of the Monin–Obukhov similarity equations (MOSE).

Van de Wiel et al. misinterpreted the meaning of the extremum hypothesis (EH) in their statement about the stable surface layer and thermal equilibrium. The ES to the EH is much like Newton’s law of motion to the principle of stationary (or least) action (e.g., Morin 2007). The EH selecting the flux–gradient relationships among those allowed by the MOSE is analogous to the principle of stationary (or least) action selecting the actual motion of an object among those with the same endpoint conditions (through extremizing the action functional). Saying “no observational evidence of a preferred stability state in the stable surface layer (SSL) exists in the literature” misses the points. Thermal equilibrium is irrelevant to the EH as turbulent transport is necessarily associated with nonequilibrium systems.

Van de Wiel et al. are concerned about the stability functions. Although in theory the empirical stability correction functions *ϕ _{m}* and

*ϕ*in the MOSE are

_{h}*ζ*dependent, there is nothing fundamental in the similarity formalism that rules out the possibility of constant

*ϕ*and

_{m}*ϕ*. Indeed, the idea of constant rather than variable

_{h}*ϕ*and

_{m}*ϕ*seems to be far-fetched given the relatively large spread of

_{h}*ζ*data from field experiments. Yet the possibility is indicated by the histogram of some

*ζ*data. Figure 1 illustrates that data points of

*ζ*cluster around two well-defined peaks whose values are consistent with those predicted by the ES (i.e.,

*ζ*

^{ES}~ 0.1 for the stable condition and

*ζ*

^{ES}~ −0.2 for the unstable condition). Furthermore,

*ϕ*(

_{m}*ζ*

^{ES}) and

*ϕ*(

_{h}*ζ*

^{ES}) having much reduced variability (see Table 1 and discussion below) suggests that the possibility of

*ϕ*and

_{m}*ϕ*reducing to constants should not be ruled out.

_{h}Several previously proposed models of *ϕ _{m}* and

*ϕ*functions (Högström 1996) evaluated at

_{h}*ζ*

^{ES}(~0.2 for unstable conditions and 0.1 for stable conditions) according to the ES of the MOSE (WB10). Model A, B, C, D, and E of

*ϕ*and

_{m}*ϕ*for unstable and stable conditions are described in Tables II, III, IV, and V of (Högström 1996), respectively.

_{h}A major weakness of the existing models of *ζ*-dependent *ϕ _{m}* and

*ϕ*is lack of universality since the similarity theory requires

_{h}*ϕ*and

_{m}*ϕ*(variable or constant) in the MOSE to be location independent. However, specific forms of

_{h}*ϕ*and

_{m}*ϕ*have to be used to fit the data collected at different sites (e.g., Högström 1996). Note that the differences between various proposed functions of

_{h}*ϕ*and

_{m}*ϕ*tend to be greater at larger

_{h}*ζ*[see Tables II, III, IV, and V of Högström (1996)], suggesting that

*ζ*-dependent

*ϕ*and

_{m}*ϕ*lose universality when

_{h}*ζ*significantly deviates from

*ζ*

^{ES}. Yet, the same models of

*ϕ*and

_{m}*ϕ*considered in Högström (1996) agree closely when evaluated at

_{h}*ζ*

^{ES}(see Table 1). We interpret the close match between various models of

*ϕ*and

_{m}*ϕ*near

_{h}*ζ*

^{ES}and the clustering of

*ζ*points around

*ζ*

^{ES}as a hint of constant

*ϕ*and

_{m}*ϕ*. This view is confirmed by the maximum entropy production (MEP) model of surface heat fluxes (Wang and Bras 2009, 2011). After all, validity and usefulness of the empirical

_{h}*ϕ*and

_{m}*ϕ*in the MOSE must be tested in terms of its predictive capability.

_{h}Van de Wiel et al. object to our suggestion that EH has a solid foundation built on modern nonequilibrium thermodynamics while at the same time saying that ultimate proof is not yet available. We see no contradiction: indeed there is more work to do but the consistency and quality of the results are certainly encouraging and tantalizing. We believe in scientific inquiry and search for new knowledge and approaches, and hope that the field keeps an open and inquisitive mind.

It is not clear what are the fundamental flaws in our interpretation of the nonuniqueness. Van de Wiel et al.’s explanation of the maximum downward heat flux is not physically consistent because heat flux, however small (or large), does not vanish under stable conditions. Van de Wiel et al.’s interpretation of the maximum downward sensible heat flux reveals that they misinterpreted the EH as explained above. Note that the nonquantitative concepts of weak and strong stable regimes are not used in the formalism of the ES. At any rate, sensible heat flux will not vanish as long as the system does not reach thermal equilibrium. The ES captures sensible heat flux under all stratification conditions in the framework of MOSE: stable or unstable, weak or strong, however defined.

Indeed, we do not consider the nonconvergence issue to be the biggest weakness of the classical MOSE-based atmospheric surface layer (ASL) models. More severe issues include assuming height-invariant momentum and heat fluxes and use of ancillary parameters such as aerodynamic surface temperature and various roughness lengths not belonging to the MOSE. The MOSE will have different forms if these parameters are included in the similarity formalism. In addition, there are theoretical difficulties in defining aerodynamic surface temperature (e.g., Mahrt 1998). The ES does not use these parameters, does not assume constant profiles, and does not include nuisance variables; hence it does not suffer from the problems of nonconvergence.

Van de Wiel et al. question the value of our contributions and cite many past and good papers, including several of their own contributions. We are not trying to take merit away from any previous work; let us try to highlight some important differences. The analyses of Taylor (1971), De Bruin (1994), Malhi (1995), van de Wiel et al. (2007), and Basu et al. (2008) assumed height-invariant fluxes and used a roughness length not appearing in the MOSE. Therefore, their findings only hold under restrictive conditions. The analysis of WB10 makes no assumption about flux profiles and uses no ancillary parameters such as roughness length that do not belong to the MOSE. It is important to point out that the nonuniqueness referred to in WB10 is in terms of friction velocity *u*_{*} versus wind shear (i.e., gradient of wind speed), whereas that referred to in Taylor (1971), De Bruin (1994), Malhi (1995), van de Wiel et al. (2007), and Basu et al. (2008) is *u*_{*} versus wind speed. In fact, the issue of nonuniqueness studied by WB10 is much broader than that dealt with by Taylor, De Bruin, Malhi, van de Wiel et al., and Basu et al., including *u*_{*} versus wind shear and sensible heat flux under a stable regime and *u*_{*} versus temperature gradient and sensible heat flux under an unstable regime treated in a mathematically and physically consistent and unified manner. We are not aware of previously published comprehensive analysis of the nonuniqueness of the MOSE as reported in WB10. More on the nonuniqueness (or sensible heat flux duality, as it is called by van de Wiel et al.) appears below.

## 2. Surface sensible heat flux duality

The analyses of the stationary Richardson number *R _{i}* [a term used by Monin and Obukhov (1954)] of Malhi (1995), van de Wiel et al. (2007), and Basu et al. (2008), very different from the derivation presented in section 2 of van de Wiel et al., is limited to a special case of log-linear profile (height-invariant fluxes) as Malhi (1995, p. 394) wrote: “The above analysis is reliant on the validity of the log-linear profile law, and is thus likely to break down when applied to a strongly stratified surface layer.” Note that the maximum sensible heat flux obtained by Malhi (1995), van de Wiel et al. (2007), and Basu et al. (2008) is expressed in terms of wind speed instead of wind shear as in van de Wiel et al. Again, Eq. (11b) of Basu et al. (2008) was derived under the condition of log-linear profile whereas no assumption was made in WB10.

Contrary to van de Wiel et al.’s claim, the evidence of the dual nature of sensible heat flux *H* provided by Malhi (1995), Mahrt (1998), and Basu et al. (2006) is not conclusive. For example, Fig. 2 of Malhi (1995) displayed all data points of *H* versus *ζ.* It is not clear whether the pattern will remain when the data are sorted according to wind shear (or wind speed for the case of log-linear profile) as the *H* duality is conditioned on fixed wind shear (or wind speed). The scattering pattern of Fig. 2 of Malhi (1995) might be interpreted as an indicator of *H/u*^{3} depending on one or more variables other than *ζ*. Even though the pattern remains with the sorted data, the possibility of scattering due to sampling errors of turbulence quantities, especially under stable condition, cannot be ruled out easily. It is well known that accurate measurement of turbulent fluxes at nighttime is difficult using eddy-covariance methods.

As demonstrated in WB10, the duality (or more generally nonuniqueness) is not limited to sensible heat flux under stable conditions. The nonuniqueness of sensible heat flux also exists, theoretically, under unstable conditions (see Fig. 4 of WB10). The nonuniqueness is an inherent property of the MOSE as long as *ϕ _{m}* and

*ϕ*are

_{h}*ζ*dependent. The nonuniqueness disappears when

*ϕ*and

_{m}*ϕ*reduce to constants. The evidence presented above and in WB10 opens such a possibility.

_{h}## 3. Is the extremum solution the “only” physically realistic state?

This question is misleading. No such claim was made in WB10. Rather our statement is very specific: “it [the ES] is the only mathematically consistent and physically realistic *solution* of the Monin–Obukhov similarity equations.” To say the ES is the only physically realistic *state*, one must show that all physically realistic states can be described by the MOSE, a task beyond the scope of WB10. Nonetheless, the ES is the only MOSE-based turbulence model of ASL not reliant on the assumption of height-invariant fluxes, without using ancillary parameters such as roughness lengths that are not appropriate for the similarity formalism leading to the MOSE, with consistent asymptotic properties, universal (site-independent), and simple. More importantly, it has led to a new (MEP) model for predicting surface heat fluxes requiring minimum input, which no other models can do.

## 4. Implications of surface sensible heat flux duality for numerical modeling

The problems that Basu et al. (2008) and Holtslag et al. (2007) dealt with are not necessarily representative. It does not make sense to use a fixed surface sensible heat boundary condition in numerical simulations of the ASL processes in the first place. A more common issue in modeling and monitoring land–atmosphere interactions is how to estimate and/or predict surface heat fluxes with the input of wind, temperature, humidity, etc. using the MOSE method. As a matter of fact, the theoretical analysis of WB10 was completed several years ago, but we held back the results until the MEP model of surface energy balance was formulated and tested (Wang and Bras 2009). The MEP model is most suitable for the kinds of numerical simulations of Basu et al. (2008) and Holtslag et al. (2007). We encourage van de Wiel et al. to try it. Results will speak for themselves.

## 5. Closing remarks

### a. More on the extremum solution of WB10

We would like to highlight two features of the ES vis-à-vis the classical treatment of the MOSE. First, the ES applies to any vertical distributions of fluxes, whereas a majority of the existing MOSE-based models must assume uniform flux profiles to integrate the MOSE analytically. Uniform flux profiles rarely, if ever, exist in reality, while the phenomenon of height-varying fluxes has been well documented (e.g., Nieuwstadt 1984; Malhi 1995; Mahrt 1998; Derbyshire 2005; Cheng and Brutsaert 2005; Basu et al. 2006, to name a few). The solution of the MOSE under this restrictive assumption is useful, but not as general as the ES.

Second, the ES is universal, at least much more so than those using *ζ*-dependent stability correction functions. Van de Wiel et al. acknowledged “there is no consensus in the boundary layer meteorology field in terms of the exact form of the empirical stability correction functions.” We can think of the following reasons for no consensus: 1) universal functions do exist but have not been identified because (a) they cannot be formulated analytically and (b) they cannot be determined using available datasets; and 2) universal functions do not exist. A number of models of *ϕ _{m}* and

*ϕ*have been proposed based on observations from various field experiments (e.g., Högström 1996; Cheng and Brutsaert 2005; Grachev et al. 2007). These proposed functions tend to agree at small

_{h}*ζ*and differ at large

*ζ*. One could argue that this is because the nondecreasing

*ϕ*and

_{m}*ϕ*must converge to unity as

_{h}*ζ*→ 0 and are more uncertain as

*ζ*→ ∞. Whatever the reason, universal

*ϕ*and

_{m}*ϕ*functions defined over the entire range of

_{h}*ζ*remain elusive. To meet the requirement of universality, it makes sense to use

*ϕ*(

_{m}*ζ*

^{ES}) and

*ϕ*(

_{h}*ζ*

^{ES}) (which are rather uniform across different sites according to Table 1, with

*ζ*

^{ES}being the cluster center of

*ζ*illustrated in Fig. 1) in the MOSE for practical purposes. Note that the ES (i.e., the three expressions in Table 1 of WB10 as well as

*ζ*

^{ES}) cannot be derived by directly fixing

*ζ*in the MOSE. As a result, observational confirmation of the ES ought to be viewed as strong evidence, if not a definite proof, in support of the EH (see Figs. 5 and 6 of WB10).

### b. Is any work in the study of ASL turbulence “absolute”?

Van de Wiel et al. claimed that the solution of the sensible heat flux duality in numerical modeling offered by Holtslag et al. (2007) and Basu et al. (2008) is absolute so that the EH must be rejected (as an alternative). Is there such a thing as “absolute” in the study of ASL turbulence, or in any field of scientific research for that matter? We think not. Modeling turbulence is a wide open area with many unresolved issues. Fortunately, others agree with us. For example, Mahrt (1998, p. 276) wrote “… formulation of turbulence in the very stable boundary layer is uncertain and the stable boundary layer contains a number not present in existing models, … even small future advances justify more work.” To advance the study of ASL turbulence, we must venture into new territory. WB10 is a step in that direction.

## Acknowledgments

This work was supported by ARO Grant W911NF-10-1-0236 and NSF Grant EAR-0943356.

## REFERENCES

Arya, S. P., 1999:

*Air Pollution Meteorology and Dispersion*. Oxford University Press, 310 pp.Basu, S., F. Porté-Agel, E. Foufoula-Georgiou, J.-F. Vinuesa, and M. Pahlow, 2006: Revisiting the local scaling hypothesis in stably stratified atmospheric boundary-layer turbulence: An integration of field and laboratory measurements with large-eddy simulations.

,*Bound.-Layer Meteor.***119**, 473–500.Basu, S., A. A. M. Holtslag, B. J. H. van de Wiel, A. F. Moene, and S.-J. Steeneveld, 2008: An inconvenient “truth” about using sensible heat flux as a surface boundary condition in models under stably stratified regimes.

,*Acta Geophys.***56**, 88–99.Cheng, Y., and W. Brutsaert, 2005: Flux-profile relationships for wind speed and temperature in the stable atmospheric boundary layer.

,*Bound.-Layer Meteor.***114**, 519–538.De Bruin, H. A. R., 1994: Analytical solutions of the equations governing the temperature fluctuation method.

,*Bound.-Layer Meteor.***68**, 427–433.Derbyshire, S. H., 2005: Boundary-layer decoupling over cold surfaces as a physical boundary instability.

,*Bound.-Layer Meteor.***90**, 297–325.Grachev, A. A., E. L Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, 2007: SHEBA flux-profile relationships in the stable atmospheric boundary layer.

,*Bound.-Layer Meteor.***124**, 315–333.Högström, U., 1996: Review of some basic characteristics of the atmospheric surface layer.

,*Bound.-Layer Meteor.***78**, 215–246.Holtslag, A. A. M., G.-J. Steeneveld, and B. J. H. van de Wiel, 2007: Role of land-surface temperature feedback on model performance for the stable boundary layer.

,*Bound.-Layer Meteor.***125**, 361–376.Mahrt, L., 1998: Stratified atmospheric boundary layers and breakdown of models.

,*Theor. Comput. Fluid Dyn.***11**, 263–279.Malhi, Y. S., 1995: The significance of the dual solutions for heat fluxes measured by the temperature fluctuations method in stable conditions.

,*Bound.-Layer Meteor.***74**, 389–396.Monin, A. S., and A. M. Obukhov, 1954: Basic turbulence mixing laws in the atmospheric surface layer.

,*Tr. Inst. Teor. Geofiz. Akad. Nauk. SSSR***24**, 163–187.Morin, D., 2007:

*Introduction to Classical Mechanics*. Cambridge University Press, 719 pp.Nieuwstadt, F. T. M., 1984: The turbulent structure of the stable, nocturnal boundary layer.

,*J. Atmos. Sci.***41**, 2202–2216.Taylor, R. J., 1971: A note on the log-linear velocity profile in stable conditions.

,*Quart. J. Roy. Meteor. Soc.***97**, 326–329.van de Wiel, B. J. H., A. F. Moene, G.-J. Steeneveld, O. K. Hartogensis, and A. A. M. Holtslag, 2007: Predicting the collapse of turbulence in stably stratified boundary layers.

,*Flow Turbul. Combust.***79**, 251–274.van de Wiel, B. J. H., S. Basu, A. F. Moene, H. J. J. Jonker, G.-J. Steeneveld, and A. A. M. Holtslag, 2011: Comments on “An extremum solution of the Monin–Obukhov similarity equations.”

,*J. Atmos. Sci.***68**, 1405–1408.Wang, J., and R. L. Bras, 2009: A model of surface heat fluxes based on the theory of maximum entropy production.

,*Water Resour. Res.***45**, W11422, doi:10.1029/2009WR007900.Wang, J., and R. L. Bras, 2010: An extremum solution of the Monin–Obukhov similarity equations.

,*J. Atmos. Sci.***67**, 485–499.Wang, J., and R. L. Bras, 2011: A model of evapotranspiration based on the theory of maximum entropy production.

,*Water Resour. Res.***47**, W03521, doi:10.1029/2010WR009392.