The purpose of this note is to comment on a simple parametrization scheme for evaluation of surface fluxes in convective conditions, proposed by Stull (1994), and referred to by the author as the “convective transport theory” (CTT). My intention is to show that CTT can be evaluated based on the surface layer similarity theory, developed by Monin and Obukhov about five decades ago, and verified experimentally on many occasions since than.

*L,*called the Monin–Obukhov length (e.g., Sorbjan 1989),where

*H*

_{0}= 〈

*w*′

*θ*′〉 is the surface kinematic flux of the virtual temperature,

*β*=

*g*/Θ is the buoyancy parameter, and

*u*

_{*}is the friction velocity. Based on the definition of

*L,*the “purely” free-convection regime (no mean wind, and consequently

*u*

_{*}= 0) can be associated with the nil value of

*L.*The case that takes place when the mean wind is weak, and

*L*is small but nonzero, will be hereafter referred to as “forced” free convection. I will briefly comment on both convective regimes.

*z*is height and

*c*

_{θ}is a universal constant that is positive in value. The above formula was first obtained by Prandtl (1932) and independently by Obukhov (1946). Equation (2) indicates that near the earth’s surface the temperature lapse rate dΘ/

*dz*is very large. Near the top of the surface layer (typically at about

*z*= 100 m),

*d*Θ/

*dz*is very small (e.g., Kaimal 1966; Businger et al. 1971). For example, assuming in (2) that

*z*= 100 m,

*H*

_{0}= 0.1 K m s

^{−1},

*β*= 0.03 m s

^{−2}K

^{−1}, and

*c*

_{θ}= 0.23 (the value obtained by curve fitting the Kansas data presented by Businger et al. 1971), one can obtain

*d*Θ/

*dz*= −0.03 K/100 m.

*z*/

*T*

_{*}, where

*T*

_{*}= −

*H*

_{0}/

*u*

_{*}is the temperature scaleObviously, the above expression makes sense only during forced convection. During free convection (

*L*= 0,

*T*

_{*}= − ∞) the expression (3) is singular. It could be noted that in some previously presented empirical formulations, the exponent “−1/3” in (3) has been replaced by the exponent “−1/2” (e.g., Businger 1988). Such an approach is inconsistent with the Monin–Obukhov similarity and implies that

*d*Θ/

*dz*∼

*u*

_{*}. Therefore, Stull’s (1994) notion that the traditional bulk aerodynamic approach fails during purely free convection applies to formulation (3), and also to all empirical expressions with the exponent different than −1/3, but not to formulation (2).

*z*

_{1}<

*z*

_{2}. The above expression can be rearranged to yieldEquation (5) indicates that for the evaluation of the surface heat flux, it is sufficient to measure temperature at any two levels within the free-convection surface layer. When

*z*

_{2}is chosen near the top of the free-convection surface layer, where the temperature lapse rate

*d*Θ/

*dz*practically vanishes, Θ

_{2}is expected to be very close to the mixed layer temperature Θ

_{ml}. Because (4) cannot be extended to the earth’s surface (e.g., Brutsaert 1982), Θ

_{1}is expected to differ from the surface skin temperature Θ

_{S}.

*z*

_{i}is the mixed layer height, allows (5) to be rewritten in the form

*H*

_{0}

*b*

_{H}

*w*

_{B}

*b*

_{H}is an empirical constant, which leads to several inconsistencies discussed below.

First of all, as it follows from (6), the parameter *z*_{i} is only formally included in (7); that is, even though *w*_{B} and *b*_{H} are dependent on the mixed layer height, their product is independent of *z*_{i}. If *b*_{H} is assumed constant (as in Stull’s formulation), this consequently implies an erroneous conclusion that *H*_{0} is uniquely dependent on *z*_{i}. The surface heat flux is caused by small eddies near the surface and cannot be influenced by large eddies of the scale *z*_{i}. An additional argument demonstrating that such a relation of *H*_{0} on *z*_{i} is generally incorrect can be obtained from analyses of “large eddy simulations” (e.g., Sorbjan 1995), which exhibit an intense growth of the mixed layer in cases when the surface heating is constant with time and also almost no change of the mixed layer height *z*_{i} when the surface heating decreases with time during late afternoons. In the formulation (6)–(7) derived from the Monin–Obukhov similarity, *b*_{H} is dependent on *z*_{1}, *z*_{2}, and *z*_{i}. When we assume that (*z*_{2}/*z*_{i})^{−1/3} ≪ (*z*_{1}/*z*_{i})^{−1/3}, it will become apparent that *b*_{H}, approximately equal to 3*c*_{θ} (*z*_{1}/*z*_{i})^{1/2}, is quite sensitive to the choice of the lower observation level. For *c**θ* ∼ 0.23, and for the value *b*_{H} = 5 × 10^{−4} suggested by Stull (1994), we find that the equivalent altitude of the lower level would be *z*_{1}/*z*_{i} = (*b*_{H}/3*c*_{θ})^{2} ∼ 5.5 × 10^{−7}, which is at the bottom of the molecular sublayer. This means that the free-convection temperature profile is extended to the earth’s surface. Such an assumption is not appropriate for two reasons. First of all, (2) is not valid in the very close vicinity of the earth’s surface (e.g., Townsend 1959; Brutsaert 1982). Second, Eq. (7) with constant *b*_{H} is not general, as it does not include any direct information about thermal characteristics of the viscous (canopy) sublayer. Such information is relevant since for a given value of the surface heat flux one would expect smaller ΔΘ over rougher surfaces.

In order to include the roughness parameter for heat in the scheme, *b*_{H} has to be assumed to be a function of *z*_{1}/*z*_{i}, and the temperature Θ_{1} at *z* = *z*_{1} has to be calculated based on the temperature profile in the viscous (canopy) sublayer (e.g., Mahrt and Ek 1984). Neverthless, it should be mentioned that for complicated heterogeneous surfaces such a simple approach might not be sufficient (e.g., Sun and Mahrt 1995).

*L*very small and negative), Monin and Obukhov suggested that the turbulent Prandtl number, Pr =

*k*

_{m}/

*k*

_{h}, can be assumed constant. This assumption leads towhich together with (2) yieldswhere

*c*

_{u}is a universal constant that is positive in value. Integrating (8) with respect to height leads towhere

*z*

_{1}<

*z*

_{2}. After some rearrangements and with the help of (5), we will arrive atThe obtained result implies that for the evaluation of the surface fluxes in the case of forced convection, it is sufficient to measure temperature and wind velocity at two levels within the surface layer. When

*z*

_{2}is chosen near the top of the free-convection surface layer,

*U*

_{2}is expected to be close to the mixed layer wind speed

*U*

_{ml}.

*u*

_{*}

^{2}

*b*

_{D}

*w*

_{B}

*U*

_{2}

*U*

_{1}

*U*

_{1}= 0 [see his Eq. (21)] and also that

*b*

_{D}is an empirical constant.

It can be easily verified that even though *w*_{B} and *b*_{D} are dependent on the mixed layer height, their product is independent of *z*_{i}. If *b*_{D} is assumed to be an empirical constant, this leads to the conclusion that *u*_{*}^{2} is directly dependent on *z*_{i}. To my knowledge, such a direct dependence has not been documented empirically.

In the formulation (11)–(12), the coefficient *b*_{D} is dependent only on *z*_{1}, *z*_{2}, and *z*_{i}. When we assume that (*z*_{2}/*z*_{i})^{−1/3} ≪ (*z*_{1}/*z*_{i})^{−1/3}, then approximately *b*_{D} = (3*c*_{u})^{−1} (3*c*_{θ})^{−3/2} (*z*_{1}/*z*_{i})^{1/2}. This shows that, analogously to *b*_{H}, *b*_{D} is sensitive also to the choice of *z*_{1} and consequently cannot be considered the universal constant. For the value *c*_{u} ∼ 0.54 [the value obtained by curve fitting the Kansas data of Businger et al. (1971)], and for *b*_{D} = 1.83 × 10^{−3} suggested by Stull (1994), it can be found that the equivalent value of *z*_{1}/*z*_{i} is practically zero. This means that the free-convection wind profile is extended to the earth’s surface. Such an assumption is not appropriate because (9) is not valid in the very close vicinity of the earth’s surface. Moreover, such an assumption excludes any information on the properties of the underlying surface.

In the case of forced free convection, surface characteristics such as the roughness parameter and the surface temperature could be included in the scheme by matching the convective profiles (4) and (9) with the profiles in the logarithmic layer and the viscous sublayer below it (e.g., Zilitinkevich 1970; Kraus and Businger 1994).

Concluding, the classic Monin–Obukhov similarity theory gives constructive hints regarding the form of coefficients *b*_{H} and *b*_{D}. Knowing the form of these coefficients, as well as understanding their validity in the surface layer (and in the viscous sublayer), allows one to estimate the expected accuracy and universality of the CTT. Based on the above presented analysis one might state that the CTT functional dependencies of *H*_{0} on ΔΘ^{3/2}, and also of *u*_{*}^{2} on the product ΔΘ^{1/2}U_{ml}, agree with the Monin–Obukhov similarity. The introduced by the CTT-dependence of the surface fluxes on *z*_{i}^{1/2} (through *w*_{B}), as well as a lack of dependence on parameters characterizing the surface roughness and conduction in the viscous sublayer, should contribute to errors in situations different from conditions for which CTT has been calibrated.

## Acknowledgments

This note was sponsored by NSF Grant ATM 9217028.

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