Uncertainty in the Parameterization of Surface Fluxes under Unstable Conditions

Piyush Srivastava aCentre for Atmospheric Sciences, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India
bCentre of Excellence in Disaster Mitigation and Management, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

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Maithili Sharan aCentre for Atmospheric Sciences, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

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Abstract

In this study, an attempt has been made to analyze the possible uncertainties in the parameterization of surface fluxes associated with the form of nondimensional wind and temperature profile functions used in weather and climate models under convective conditions within the framework of Monin–Obukhov similarity theory (MOST). For this purpose, these functions, which are commonly known as similarity functions, are classified into four categories based on the resemblance in their functional behavior. The bulk flux algorithm is used for the estimation of transfer coefficients of momentum and heat using four different classes of similarity functions. Uncertainty in the estimated values of fluxes is presented in the form of deviation in the predicted values of momentum and heat transfer coefficients and their variation with the Monin–Obukhov stability parameter. The analysis suggests that a large deviation in the values of estimated fluxes might occur if different forms of similarity functions are utilized for the estimation of surface fluxes. Recommendations are made for the form of similarity function for momentum based on the analysis of 1-yr-long turbulence observations over an Indian region. The study suggests that there is a distinct need to carry out a careful analysis of turbulence data in free-convective conditions for determining a consistent functional form of the similarity functions to be utilized in the atmospheric models universally.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Maithili Sharan, mathilis@cas.iitd.ac.in

Abstract

In this study, an attempt has been made to analyze the possible uncertainties in the parameterization of surface fluxes associated with the form of nondimensional wind and temperature profile functions used in weather and climate models under convective conditions within the framework of Monin–Obukhov similarity theory (MOST). For this purpose, these functions, which are commonly known as similarity functions, are classified into four categories based on the resemblance in their functional behavior. The bulk flux algorithm is used for the estimation of transfer coefficients of momentum and heat using four different classes of similarity functions. Uncertainty in the estimated values of fluxes is presented in the form of deviation in the predicted values of momentum and heat transfer coefficients and their variation with the Monin–Obukhov stability parameter. The analysis suggests that a large deviation in the values of estimated fluxes might occur if different forms of similarity functions are utilized for the estimation of surface fluxes. Recommendations are made for the form of similarity function for momentum based on the analysis of 1-yr-long turbulence observations over an Indian region. The study suggests that there is a distinct need to carry out a careful analysis of turbulence data in free-convective conditions for determining a consistent functional form of the similarity functions to be utilized in the atmospheric models universally.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Maithili Sharan, mathilis@cas.iitd.ac.in

1. Introduction

A major source of uncertainty in the predictive capability of the atmospheric models is associated with the inadequate parameterization of turbulent fluxes in the models. Almost all the numerical models of atmosphere utilize the Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954) for parameterization of surface fluxes (Stull 1988). The bulk algorithm for parameterization of surface fluxes based on MOST requires wind, temperature, and humidity measurements at two levels in the lowest few meters above the surface with an assumption that the lowest level above the surface lies within the atmospheric surface layer where the surface fluxes remain constant. Further, the algorithm requires roughness lengths of momentum and heat, and stability correction functions to incorporate the effect of atmospheric stability. According to the MOST, the nondimensional wind and temperature profiles are universal functions of the Monin–Obukhov stability parameter ζ:
κzu*Uz=φm(ζ)
and
κzθ*θz=φh(ζ),
where κ is the von Kármán constant; z is the height above the ground; u* and θ* are, respectively, the friction velocity and temperature scales; and φm and φh are, respectively, the nondimensional similarity functions. Here, ζ is defined as
ζ=κzgθ¯θ*u*2,
in which θ¯ is the mean potential temperature in the layer and g is the acceleration due to gravity.
In near-neutral conditions (ζ → 0), both φm and φh must be equal to a constant that is equal to unity for φm, while the corresponding value for φh is determined from field observations. In the case of strongly unstable conditions, when the value of −ζ becoming large, it is assumed that buoyancy dominates the local turbulence process (Högström 1996). In such a case, friction velocity u* ceases to be the correct scaling parameter and should not be used for a nondimensional process, which suggests that φm and φh should vary as power-law functions of ζ in local free-convective conditions (Grachev et al. 2000). The dimensional analysis suggests that the exponent should be −1/3 for both φm and φh, while Businger (1966) suggested that the exponent should be −1/4 for φm and −1/2 for φh in free-convection limit. Various authors (Obukhov 1971; Kazansky and Monin 1956; Ellison 1957; Yamamoto 1959; Panofsky 1961; Sellers 1962) have proposed a formula for φm, commonly referred to as the Kazansky and Monin–Ellison–Yamamoto–Panofsky–Sellers (KEYPS) or Obukhov KEYPS (O’KEYPS) formula in the literature as
φm4(ζ)γφm3(ζ)=1,
where γ is a constant. The KEYPS formula captures the theoretical scaling for near-neutral to moderately unstable conditions and the −1/3 power law in free-convection limit as predicted by MOST. Based on the “direction dimensional analysis,” Kader (1988), Kader and Perepelkin (1989), and Kader and Yaglom (1990) presented a different conceptual structure of the atmospheric surface layer (ASL) under unstable conditions by dividing the unstable ABL into three different sublayers, namely, dynamic sublayer, dynamic–convective sublayer, and free-convective sublayer. The three sublayer model suggested by Kader and Yaglom (1990) predicts that the function φm, in theory, follows a +1/3 power law in free-convective conditions, which is neither consistent with MOST prediction nor the historical KEYPS formula. Based on the above three types of theoretical frameworks, various forms of φm and φh functions have been developed over the years. Most of these functions are derived from turbulence observations whose characteristics depend upon the accuracy of the measurements as well as the theoretical applicability of the similarity theory in that range. Making accurate measurements in strongly stratified conditions is a challenging task, which prevents us from determining the exact functional form of similarity functions in very unstable conditions.

The parameterized transfer coefficients of momentum (CD) and heat (CH) and corresponding surface fluxes in the atmospheric models depend on the behavior of φm and φh (Yang et al. 2001; Srivastava and Sharan 2017). In an earlier study, Srivastava and Sharan (2015) have pointed out the existence of a nonmonotonic behavior of CD with atmospheric stability, which is not consistent with the widely expected monotonically increasing characteristic of drag coefficient with increasing instability of the surface layer. They argue that while MOST-based parameterized form of CD depends upon the functional behavior of similarity function φm, various operational weather and climate models use one or another form of φm and φh for parameterization of fluxes using bulk flux algorithm without quantifying the uncertainties involved in the estimated values of fluxes associated with the form of those functions. For example, the way functional forms of φm and φh are prescribed in the bulk algorithm is quite different for global and regional scale models such as the Community Earth System Model (CESM; Zeng et al. 1998), Regional Climate Model (RegCM; Giorgi et al. 2012), and the Weather Research and Forecast (WRF) Model (Skamarock et al. 2008). The CESM utilizes functional form based on theoretical prediction of Kader and Yaglom (1990) in free-convective conditions, while RegCM and WRF use either classical Businger–Dyer functions (Businger et al. 1971) or an interpolation function suggested by Grachev et al. (2000) depending upon which surface layer module is selected for simulations (Jimenez et al. 2012). The level of disparity in the utilization of similarity functions among different weather and climate models points out that uncertainties in the estimated transfer coefficients and surface fluxes due to different similarity functions might be larger than the uncertainties in the other input parameters such as prescribed surface temperature and roughness lengths of momentum and heat in the bulk flux algorithm. The motivation behind this study is to find out the level of uncertainty one may expect in the estimated fluxes and transfer coefficients associated with the functional forms of similarity functions used for parameterization under unstable conditions. Thus, an attempt has been made here to classify various functional forms of φm and φh in different categories and then quantify the possible uncertainties in the estimation of surface fluxes for different classes of functions used in the weather forecast and climate models.

2. Bulk algorithm for parameterization of transfer coefficients and surface fluxes

In a homogeneous surface layer, mean wind and temperature profiles are given by
U(z)=u*κ{ln(zz0)[ψm(ζ)ψm(z0zζ)]}
and
θ(z)θ0=θ*κ{ln(zzh)[ψh(ζ)ψh(zhzζ)]},
where z0 and zh are, respectively, roughness lengths of momentum and heat; θ0 is the potential temperature at the height zh; ψm and ψh are, respectively, the integrated similarity functions corresponding to similarity functions φm and φh. The stability parameter ζ is defined as in Eq. (3). The integrated similarity functions, commonly known as stability correction functions ψm and ψh are given by
ψm,h(ζ)=0ζ1φm,h(ζ)ζdζ
in which ζ′ is a dummy variable of integration corresponding to ζ. For land surfaces, where z0 and zh are taken as constants, Eqs. (3), (5), snf (6) are solved iteratively to estimate ζ and bulk transfer coefficients CD and CH as
CD=κ2[ln(z/z0)ψm(ζ,z0zζ)]2
and
CH=κ2[ln(z/z0)ψm(ζ,z0zζ)]1[ln(z/zh)ψh(ζ,zhzζ)]1.
Then CD and CH are used to estimate the turbulent fluxes of heat (H) and momentum (−τ) from
HρCp=CHU(θθ0)
and
τρ=CDU2,
in which ρ is the dry air density and Cp is the specific heat capacity of dry air at constant pressure.
An alternative approach of estimating the transfer coefficients and surface fluxes is based on the bulk Richardson number RiB, which is defined as
RiB=gθ¯(zz0)2(zzh)(θθ0)U2.
Using Eqs. (3), (5), and (6) in Eq. (12), the expression for RiB becomes
RiB=ζ(1z0/z)2(1zh/z)ln(z/zh)[ln(z/z0)]2[1ψh(ζ)ψh(ζh)ln(z/zh)][1ψm(ζ)ψm(ζ0)ln(z/z0)]2.
For a given value of RiB, ζ is estimated by calculating the root of least magnitude of Eq. (13).

3. Commonly used functional forms of φm and φh under unstable conditions

The expression for similarity functions φm and φh was proposed by Businger (1966) and A. J. Dyer [1965, unpublished material; see Businger (1988) for details] for unstable stratification as
φm(ζ)=(1γmζ)1/4
and
φh(ζ)=Prt(1γhζ)1/2,
in which γm = 15 and γh = 9 and Prt=(φh/φm)ζ=0 is the turbulent Prandtl number. These functions are commonly known as Businger–Dyer similarity (BD) functions in the literature. Businger–Dyer similarity functions do not satisfy the classical free-convection limit. Carl et al. (1973) have suggested an expression for φm valid for the stability range −10 ≤ ζ ≤ 0 as
φm,h(ζ)=(1βm,hζ)1/3.
Different values of the constants βm and βh can be found in the literature; for example, βm = βh = 15 (Carl et al. 1973), βm and βh = 40 (Delage and Girard 1992), and βm = βh = 12.87 (Fairall et al. 1996). Högström (1988) reformulated the universal similarity functions of Businger et al. (1971) of the form
φm(ζ)=(119.3ζ)1/4
and
φh(ζ)=0.95(111.6ζ)1/2.
The validity of these formulations was limited to −2 ≤ ζ ≤ 0 (Foken 2006). Fairall et al. (1996, 2003) suggested an interpolation function valid for the full range of instability as
ψα=ψα_D+ζ2ψα_conv1+ζ2,α=m,h,
in which ψα_D is an integrated similarity function corresponding to the Dyer similarity functions (BD functions with γm = γh = 16), and ψα_conv is that given by Carl et al. (1973). Grachev et al. (2000) revisited the convective constants appearing in the expression of ψα_conv and suggested that the value of the constant for ψm_conv should be equal to 10.15 while that for ψh_conv should be 34.15 for application purposes. Akylas and Tombrou (2005) also used the interpolation between Businger–Dyer formulas and free-convection forms, but they have interpolated φm,h functions rather than the corresponding integrated functions ψm,h suggested by Fairall et al. (1996, 2003) [Eq. (19)].
The three sublayer model of Kader and Yaglom (1990) suggests the contrasting behavior of φm with ζ in the three sublayers. In the dynamic sublayer, the theory suggests that turbulence production occurs predominately in the along wind direction, (x component) and the pressure redistribution term feeds turbulence energy into vertical z component (Högström 1996). The sublayer is valid for ζ lying in the range −1/40 < ζ < 0, which corresponds to near-neutral conditions. In this condition, φm(ζ) = 1 while φh(ζ) = Prt. In the dynamic-convective sublayer, mechanical energy is produced in the x component, whereas buoyancy produces energy in the z component independently. This sublayer is valid for ζ lying in the range −0.4 < ζ < −1/40. For this sublayer, the dimension analysis suggests the functional form for similarity functions as
φm(ζ)=Au(ζ)1/3
and
φh(ζ)=AT(ζ)1/3.
Here Au and AT are constants. In this sublayer, the similarity functions behave similarly to that in the classical free-convective conditions. However, the range of values of ζ for which this sublayer is valid is limited to ζ = −0.4. In this sublayer, both the mechanical and buoyancy production terms of turbulence kinetic energy are equally important (Kader and Yaglom 1990). In the free-convective sublayer, energy production due to buoyancy dominates the mechanical production term, and the energy produced due to buoyancy in the vertical direction is fed into the horizontal direction by pressure redistribution term (Kader and Yaglom 1990; Högström 1996). Thus, in this case, the two length scales are again coupled, and there exists only one relevant velocity scale in the vertical direction. The dimensional analysis suggests
φm(ζ)=Bu(ζ)1/3
and
φh(ζ)=BT(ζ)1/3,
in which Bu and BT are constants.

The remarkable prediction of this sublayer is that unlike the case of classical free-convection limit, in which both φm and φh follow a −1/3 power law, here φm varies as the +1/3 power of ζ for sufficiently large values of −ζ. Thus, the theory of Kader and Yaglom (1990) suggests that φm has a nonmonotonic variation with respect to −ζ under unstable conditions. Based on the three sublayer model of Kader and Yaglom (1990), various expressions for φm and φh have been suggested and one of those expressions of φm and φh, which is being utilized in CESM (Zeng et al. 1998), is taken here for the analysis.

To quantify the impact of functional forms of φm and φh on the values of estimated fluxes, we have identified four classes of φm and φh functions satisfying the following:

For a given value of RiB, the value of ζ is estimated by calculating the root of least magnitude of the transcendental equation in ζ and RiB [Eq. (13)]. For computation purposes, the range of values of RiB, z/z0, and z0/zh is taken as −2 ≤ RiB < 0, −0.5 ≤ log(z0/zh) < 29.0, and 10 ≤ z/z0 < 105. The estimated value of ζ is then utilized to compute the transfer coefficients of momentum CD and heat CH [Eqs. (8) and (9)].

4. Results and discussion

a. Uncertainty in the estimation of transfer coefficients

Figures 1a–c show the variation of the stability parameter with the bulk Richardson number for the functions φm and φh of Dyer (1974), Carl et al. (1973), Fairall et al. (1996, 2003), and Zeng et al. (1998) for different values of roughness length of momentum. For the smaller magnitude of RiB the values of ζ obtained from different functional forms of φm and φh are found to be nearly identical. However, the deviations in the predicted values of ζ for a given RiB from different formulations are found to increase with decreasing values of RiB. The −1/4 and −1/2 power law for φm and φh, respectively (class I functions), and −1/3 power law for both φm and φh (class II functions) and the interpolation functions suggested by Fairall et al. (1996, 2003) (class III functions) predict relatively smaller absolute values of ζ for the values of RiB larger than 0.2. However, the absolute values of ζ predicted by the functional forms of φm and φh following +1/3 and −1/3 power laws, respectively (class IV; Kader and Yaglom 1990), are found to be very large as compared to those predicted by class I, II, and III functional forms of φm and φh. This behavior is consistent for all the values of z/z0 (Figs. 1a–c) representative of smooth, transitional, and rough surfaces. The relatively higher magnitude of ζ for a given value of RiB suggests that both the momentum and heat fluxes predicted with the class IV functional forms of φm and φh would be relatively small as compared to those predicted by class I, II, and III functional forms. The overall impact of the class IV functional form of φm and φh is a significant reduction in the estimated magnitude of surface fluxes in the moderately to strongly unstable conditions as compared to those predicted by other commonly used functional forms of φm and φh. Figures 1d–i show the variation of CD and CH with RiB for all the four types of similarity functions φm and φh for different values of roughness length of momentum. The class I, II, and III functional forms of φm and φh suggest relatively higher values of CD for a given value of RiB in moderately to strongly unstable conditions. In contrast, the values of CD predicted by class IV functional form of φm and φh are significantly smaller as compared to those predicted by the other three functional forms. One striking feature regarding the behavior of CD observed for class IV functions is that CD shows a nonmonotonic characteristic, which is in contradiction to the prediction of other forms of φm and φh (Figs. 1d–e). The values of CH are found to increase with increasing instability for all four types of similarity functions. However, the rate of increase in the values of CH with RiB is considerably small in the case of class IV functions (Figs. 1g–i). The difference in the estimated values of CH for a given RiB is relatively more pronounced in the case of higher values roughness length z0 (Fig. 1i). Notice that in the case of the rough regime, even CH shows a nonmonotonic variation with respect to RiB in case of class IV functions, which is in contradiction to that predicted by the other three classes of similarity functions. One notable feature with the class IV functions based on the three sublayer model of Kader and Yaglom (1990) is that the range of variation of values of CD and CH with stability is relatively narrow as compared to that shown by the other three classes of correction functions. The values of CD and CH are found to be bounded by twice of their near-neutral values with the class IV stability correction functions while the other classes of functions show continuously increasing values of CD and CH with increasing instability.

Fig. 1.
Fig. 1.

The relationship between stability parameter ζ, transfer coefficients CD and CH (vertical axis) calculated from bulk flux algorithm, and bulk Richardson number RiB (horizontal axis) for four classes of the similarity functions φm and φh. Three underlying surface conditions are chosen: smooth (z0 = 0.01 m), transition (z0 = 0.1 m), and rough (z0 = 1 m) for the ratio of roughness lengths of heat and momentum z0/zh = 1.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0350.1

The analysis suggests that one may expect large deviations in the estimated fluxes if stability correction functions based on Kader and Yaglom (1990) (class IV) are utilized in place of all the other types of correction functions having similar characteristics (classes I, II, and III). To quantify the deviations in the estimated values of fluxes, we have calculated the maximum possible percentage differences in the values of stability parameter and momentum and heat transfer coefficients if class IV functions are utilized in the bulk flux algorithm rather than functions belonging to the other three classes. The percentage deviation is calculated in the range −2 ≤ RiB < 0, −0.5 ≤ log(z0/zh) < 29.0, 10 ≤ z/z0 < 105 using the formula
%differenceζ=(ζpζKYζp)×100,
%differenceCD=(CDPCDKYCDP)×100,
and
%differenceCH=(CHPCHKYCHP)×100,
in which ζp, CDp, and CHp are the estimated values of stability parameter, momentum, and heat transfer coefficients for the functions belonging to the class I, II, or III and ζKY, CDKY, and CHKY are the corresponding estimated values of those parameters if the functions belonging to class IV, that is, those are based on Kader and Yaglom (1990) model are utilized for estimating the fluxes for a given value of RiB. Figure 2a shows the maximum possible difference in the estimated values of stability parameter ζ with respect to the ratio z/z0 when class IV functions are utilized in place of class I (gray line), class II (dashed line), and class III (dotted line). Notice that the differences in the estimated values of ζ, CD, and CH depend upon three parameters: RiB, z0/zh, and z/z0. In Fig. 2a, the maximum possible differences are calculated with respect to z/z0 for the values of RiB and z0/zh lying in the range −2 ≤ RiB < 0 and −0.5 ≤ log(z0/zh) < 29.0. A large difference in the estimated values of ζ is found to occur if class IV functions are utilized in the bulk parameterization of surface fluxes based on MOST. The difference is found to increase with the increasing value of z0. Similar deviations are observed for momentum and transfer coefficients (Figs. 2b,c), suggesting that the differences in the estimated values of fluxes with different classes of stability correction functions are quite large as compared to possible uncertainties in the prescription of input parameters such as wind speed and temperature required for estimation of surface fluxes from bulk flux algorithm. Since the class IV functions are found to reduce the magnitude of the estimated fluxes considerably in moderately to strongly unstable conditions, they can be utilized as an alternative to overcome the problem of overestimation of surface fluxes by the weather forecast models under unstable conditions.
Fig. 2.
Fig. 2.

Maximum percentage difference between the values of ζ, CD, and CH computed with class IV functions and the functions lying in other three classes.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0350.1

b. Nonmonotonic behavior of drag coefficient CD

Srivastava and Sharan (2015) utilized the turbulent data over a tropical region to analyze the observational behavior of CD with respect to wind speed U and stability parameter ζ under unstable conditions. They found that the observed variation of CD with ζ is nonmonotonic and bounded by a curve that shows increasing behavior with −ζ, until it reaches a peak and then, decreases further with increasing instability (Srivastava and Sharan 2015). They have argued that MOST is unable to explain the observed decreasing behavior of CD with ζ in moderate to strong unstable conditions over tropics within the framework of commonly used similarity functions belonging to classes I, II, and III. The predicted qualitative behavior of CD with ζ is found to be consistent with those observed by Srivastava and Sharan (2015) over tropics if the class IV functions are utilized for estimation of surface fluxes. In appendix B of Srivastava and Sharan (2015), a nonmonotonic form of similarity function φm [appendix B, Eq. (B1), Srivastava and Sharan 2015] was considered, and based on the mathematical analysis, it was concluded that the parameterized drag coefficient CD increases with increasing value of −ζ until it attains a maximum value at ζ = ζc and then starts decreasing with further increasing value of −ζ. Depending on the sign of the expression [Eq. (A9), appendix A, Srivastava and Sharan (2015)], it was argued that the value of ζ at which CD changes its behavior is same at which function φm changes its behavior, that is, ζ = ζc. However, in general, both the values need not be same. This error was incurred due to wrong interpretation of the expression [Eq. (A9), appendix A, Srivastava and Sharan (2015)] and here we present a correction to the analysis given in the Srivastava and Sharan (2015).

A nonmonotonic functional form of φm is taken as
φm(ζ)={φ1(ζ),ζcζ0φ2(ζ),ζζc,
in which φ1(ζ) is a continuously decreasing function of −ζ in the range ζcζ ≤ 0 and φ2(ζ) is an increasing function for ζζc.
An expression for drag coefficient CD can be written as
CD=κ2Ym2,
in which
Ym=ln(zz0)ψm(ζ,z0zζ),
where ψm is the integrated similarity function corresponding to the similarity function φm. Notice that traditionally the drag coefficient is defined as CD = (u*/U)2, where u* is the friction velocity and U is the wind speed at reference height. In case of low wind conditions when U → 0, CD tend to have very large values. However, low wind convective conditions are associated with the effect of large-scale coherent structures within the whole boundary layer (Grachev et al. 1998), which generates random gusts. Godfrey and Beljaars (1991) suggested that the gustiness velocity associated with these random gusts should be added to the vector mean wind speed U. Thus, in low wind convective boundary layer an effective wind speed Ueff should be used in place of the usual vector averaged wind speed U, which is the sum of U and the gustiness velocity Ugs. The gustiness velocity is parameterized in terms of Deardorff convective velocity scale w* (Deardorff 1970) as Ugs = βw*, where β is an empirical constant. Thus, in the case of low wind convective conditions, the updated drag coefficient can be expressed as CD = (u*/Ueff)2, where Ueff = U + βw*.
For brevity, we put η = −ζ in Eq. (27) implying that η > 0 in unstable conditions and differentiating Ym2 with respect to η, one gets
ddη(Ym2)=2YmYm.
where Ym is the derivative of Ym with respect to η, given as
Ym=[ψm(η,z0zη)].
Here, the prime denotes the derivative with respect to η.
For the function φm(ζ), Ym is given by
Ym=1η[φ1(z0zη)φ2(η)].
The behavior of CD depends on the sign of Ym, which is dependent on the exact functional forms of φ1(η) and φ2(η) as well as the value of the ratio z/z0. For clarity, a functional form suggested by Kader and Yaglom (1990) may be taken as a representative of expression (25) defined as (Fig. 3)
φm(ζ)={(116ζ)1/4,1.574ζ00.7κ2/3(ζ)1/3,ζ1.574.
In this expression, φ1 = (1 − 16 ζ)−1/4, φ2 = 0.7 κ2/3(−ζ)1/3, and ζc = −1.574.
Fig. 3.
Fig. 3.

Variation of drag coefficient (CD) and similarity function φm suggested by Kader and Yaglom (1990) with stability parameter ζ. For the computation of values of CD, z/z0 is assumed to be 10−5. The vertical dashed lines are plotted at the values of ζ at which CD and φm change their monotonic behavior.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0350.1

For this functional form of φm, an expression for Ym can be written as
Ym=1η[(1+16z0zη)1/40.7κ2/3(η)1/3].
Note that φ1=[1+16(z0/z)η]1/4 is a monotonic decreasing function of η that attains its maximum value 1 as η → 0. The other function φ2 = 0.7κ2/3(η)1/3 is a monotonic increasing that attains its minimum value 0 as η → 0 and φ2 > 1 for η>1/(0.7κ2/3)3. This suggests that there exists a point ηm, such that Ym<0 for 0 ≤ ηηm but Ym>0 for ηηm. The exact value of ηm depends upon the value of the ratio z/z0.

Thus, the value of CD increases until it attains a maximum value at η = ηm and then starts decreasing with increasing value of η. However, the points ζc and ζm = −ηm are different (Fig. 3).

c. Observational evidence and recommendation for the form of φm over the Indian region

Notice that based on the theoretical sublayer model in convective conditions, Kader and Yaglom (1990) have suggested the existence of a nonmonotonic functional form of φm, which appears to capture the qualitative nature of drag coefficient with near-surface atmospheric stability over the Indian region (Srivastava and Sharan 2015). We have further carried out data analysis to support the findings of Srivastava and Sharan (2015). The dataset used in the present study is obtained from a fast response sensor (CSAT3 sonic anemometer) installed at 10 m height in the Birla Institute of Technology Mesra, Ranchi (23.412°N, 85.440°E), India, with an average elevation 609 m above sea level (Dwivedi et al. 2015; Sharan and Srivastava 2016; Srivastava and Sharan 2017). Turbulence measurements at 10 Hz frequency for the year 2009 are used to calculate the hourly turbulent fluxes using the eddy covariance technique. Friction velocity u* is calculated from the expression
u*=[(uw¯)2+(υw¯)2]1/4
in which u′, υ′, and w′ are, respectively, the fluctuations in longitudinal, lateral, and vertical wind components.
The stability parameter ζ is calculated from the expression (3) and the drag coefficient CD is estimated using
CD=(u*/U)2.
The data corresponding to wind speed less than 2.0 m s−1 are excluded to minimize the uncertainty in values of drag coefficient in low wind conditions. Similar to Srivastava and Sharan (2015), the dataset is further classified into five unstable sublayers (Kader and Yaglom 1990; Bernardes and Dias 2010), namely, dynamic (DNS), dynamic–dynamic-convective transition (DNS–DCS transition), dynamic-convective (DCS), dynamic-convective–free-convective transition (DCS–FCS transition), and free-convective (FCS) sublayers. Notice that Srivastava and Sharan (2015) have analyzed turbulent measurements of three months (March–May 2009) corresponding to premonsoon season, while in the present study we have extended the analysis using data for the whole year. Figure 4 shows the variation of CD with ζ along with the MOST predicted forms of CD for different classes of similarity functions φm. The bin-averaged values of CD are found to increase with increasing instability in DNS and DNS–DCS transition layers and attain a peak at ζ ≈ −0.12 (Fig. 4). A decreasing trend of CD is observed from its peak value with increasing instability in DNS–FCS transition to FCS sublayers similar to that found by Srivastava and Sharan (2015). The average values of CD in different sublayers are also found to be consistent to those reported by Srivastava and Sharan (2015) for limited dataset of three months. According to MOST, CD is parameterized using Eq. (8). Equivalently one can write Eq. (9) in the form
CD=CDN{1[κψm(ζ,z0zζ)/CDN]}2,
where CDN is the neutral drag coefficient. In Fig. 4, the continuous lines, showing the predicted variation of CD for different forms of similarity functions, are plotted using CDN obtained from the observational data. Figure 4 suggests that all the formulations agree well for the values of CD in the range −0.1 < ζ < 0, that is, in near-neutral to weakly unstable condition. With class I, II, and III functions, CD shows monotonically increasing behavior in the full range of instability, while with class IV function, CD shows a decreasing behavior after attaining a peak at around ζ = −2. Notice that the observational data also suggest the existence of nonmonotonic behavior of CD with ζ. However, the point at which CD attains a peak is ζ = −0.1, which lies in DCS sublayer rather than in FCS sublayer as predicted by class IV function in its original form. To address this issue, we have empirically updated the class IV φm function in such a way that point at which CD attains its maximum value lies in the DCS sublayer as found in the observational data. The modified functional form for φm is given by
φm={(116ζ)1/4,ζmζ<03.125κ2/3(ζ)1/3,ζζm,ζm=0.1.
Notice that with the modified functional form of φm, the observed nature of CD with ζ in the unstable atmospheric conditions is well captured by MOST (Fig. 4). However, the suitability of the new form of φm for estimating both momentum and heat fluxes in the models by bulk flux algorithm requires additional analysis of neutral transfer coefficients for heat and moisture and corresponding stability correction functions. Due to the limitation of data, at present, we are unable to evaluate the impact of the new form of φm in estimating the transfer coefficients of heat and moisture from the bulk algorithm.
Fig. 4.
Fig. 4.

Variation of drag coefficient CD with stability parameter ζ observed for the whole year 2009 over Ranchi (India) is shown with red markers. The mean values of CD in each unstable sublayers are shown with black markers along with standard deviations in the form of error bars. Depending upon the data availability, two to three bins of equal width are chosen in each sublayer. The background color scheme corresponds to different unstable sublayers (Kader and Yaglom 1990; Bernardes and Dias 2010), i.e., dynamic sublayer (light gray) to free-convective sublayer (dark gray). The continuous curves represent the predicted variation of CD using different classes (I–IV) of stability correction functions. The continuous black curve shows the predicted variation of CD with the updated functional form of φm based on Kader and Yaglom (1990). For plotting each continuous curve, the value of the neutral transfer coefficient (CDN) is taken as an average value CDN in the dynamic sublayer. The dotted vertical line shows ζ = ζm = −0.1.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0350.1

5. Issues and limitations

From the application point of view, two major points of concern are 1) “equivalent” applicability of three sublayer model for heat exchange properties and 2) the treatment of both the roughness length of heat (zh) and momentum (z0) (or neutral transfer coefficients CD and CH), which has not been analyzed in the present study due to limitation of data. Here, we have assumed that apart from the uncertainty due to the form of similarity function used in the bulk flux algorithm, all the other uncertainties have the same magnitude for each of the four classes considered here for the analysis. However, while establishing the superiority of one bulk flux algorithm over the other in terms of comparison of estimated fluxes with those observed, the uncertainty associated with the way the roughness lengths are treated additionally might play a crucial role in the overall uncertainty in the estimated fluxes. Generally, the values of z0 and zh are inversely derived from observations in field experiments or empirically estimated for practical applications (Yang et al. 2008) and these values deviate from each other considerably (Beljaars and Holtslag 1991). Numerical models have rather simplified approaches to parameterize z0 and zh over different underlying surfaces. For example, depending upon the surface characteristics fixed values of z0 and zh are assigned in the numerical models (Stull 1988), while over the open water z0 is parameterized using the Charnock relation (Charnock 1955). When the surface consists of a mix of sea ice and open water, a classical mosaic approach (Vihma 1995) is utilized to estimate effective values of z0 and zh. Although significant progress has been made, accurate prescription of “neutral” transfer coefficient of heat, momentum, and moisture (equivalently roughness lengths for heat, momentum, and moisture) over various types of surfaces still presents challenges for numerical models leading to uncertainties in the estimated fluxes. In addition, uncertainty in the estimated fluxes might be due to the errors in the input parameters as well as the limitation of the bulk algorithm itself. For example, Blanc (1985) presented an alarming review of the large scheme-to-scheme differences in estimated fluxes of heat, momentum, and moisture from more than 10 bulk flux algorithms. Blanc (1983) also performed an analytical error analysis of estimated fluxes due to errors in the input parameters and found typical errors of 100% and 60% in sensible heat and momentum fluxes, respectively. Also, the current approach of analyzing the variation of CD with respect to stability parameter ζ or RiB in free-convection limit has theoretical limitation. Based on the viscous generalization of the O’KEYPS equation, Grachev (1990) has suggested that in the free-convective conditions developed over the aerodynamically smooth surface, the drag coefficient CD depends upon three additional parameters—the molecular coefficient of kinematic viscosity, the thermal diffusivity, and sensible heat flux apart from stability parameter ζ and roughness lengthz0. However, from an application point of view, CD is still parameterized as a function of ζ in the numerical models.

6. Conclusions

In the present study, a theoretical analysis is carried out to highlight the uncertainty in the estimation of surface fluxes depending upon the functional form of similarity functions φm and φh being utilized in the bulk parameterization of surface fluxes under unstable conditions. The bulk flux algorithm, in its lowest level of complexity, is selected for estimation of transfer coefficients of heat and momentum as a representative of heat and momentum fluxes using four different classes of functions φm and φh. These classes are identified based on the functional behavior of φm and φh following different power laws. The differences in the estimated values of fluxes from the first three classes of functions are found to be reasonably small as compared to the possible uncertainty in the prescription of input parameters in the bulk flux algorithm. However, the corresponding differences in the estimated fluxes with the class IV functions are very large as compared to those obtained with the other three classes of functions. The theoretical analysis presented here suggests a large deviation in the values of estimated fluxes if different forms of stability correction functions are utilized for the estimation of surface fluxes from the bulk flux algorithm.

Out of these four classes, functions lying in the first three classes satisfy the general assumption that under unstable conditions, similarity functions φm and φh are monotonically decreasing functions of the Monin–Obukhov stability parameter ζ in the whole range of values ζ. Consequently, these three functional forms predict a monotonically increasing behavior of transfer coefficients CD and CH with ζ. However, the functional form derived from the sublayer model suggested by Kader and Yaglom (1990) shows a nonmonotonic nature of similarity function for momentum and resulting in a nonmonotonic variation of drag coefficient CD with ζ. This nonmonotonic nature of CD was first reported by Srivastava and Sharan (2015) who argued that MOST, with commonly used similarity functions, is unable to capture the observed behavior of drag coefficient in moderately to very unstable conditions.

The existence of unusual nonmonotonic variation of CD versus ζ found in case of functions suggested by Kader and Yaglom (1990) was largely remained unnoticed before Srivastava and Sharan (2015). This is possibly due to the fact that term ψm(ζ0), where ζ0 = (z0/z)ζ in MOST equations, is mostly neglected in the bulk formulation of surface fluxes (Zeng et al. 1998), which leads to shifting of point of inflection (the point at which the nature of the CD vs ζ curve changes from increasing to decreasing) toward the higher magnitude of ζ beyond ζ ~ −10 for the surfaces with lower roughness lengths. However, this term becomes important in convective conditions and over the surface with high roughness lengths where the point of inflection tends to shift toward the lower magnitude of ζ with increasing surface roughness length z0.

The turbulence observations presented here suggest that functions lying in class IV are relatively more consistent with the observational behavior of drag coefficient in moderately to very unstable conditions over tropics. However, to the authors’ knowledge, the applicability of class IV functions is yet not tested in any weather forecast models. Further, a careful analysis of turbulence data in free-convective conditions is required for determining the consistent functional form of similarity functions to avoid the large uncertainties in parameterized fluxes in the atmospheric models.

Acknowledgments

We thank the anonymous reviewers for their helpful comments. This work is partially supported by J C Bose Fellowship to M.S. from Department of Science and Technology (DST)-Science and Education Research Board (SERB), Government of India (SB/S2/JCB-79/2014). P.S. was supported by Inspire Faculty Fellowship from DST, Government of India (DST/INSPIRE/04/2019/003125). The authors declare no competing interests.

Data availability statement

The raw turbulence data for Ranchi (India) site used in this study can be obtained from the Indian National Centre for Ocean Information Service (http://www.incois.gov.in/portal/datainfo/ctczdata.jsp) upon request.

REFERENCES

  • Akylas, E., and M. Tombrou, 2005: Interpolation between Businger–Dyer formulae and free convection forms: A revised approach. Bound.-Layer Meteor., 115, 381398, https://doi.org/10.1007/s10546-004-1426-3.

    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327341, https://doi.org/10.1175/1520-0450(1991)030<0327:FPOLSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bernardes, M., and N. L. Dias, 2010: The alignment of the mean wind and stress vectors in the unstable surface layer. Bound.-Layer Meteor., 134, 4159, https://doi.org/10.1007/s10546-009-9429-8.

    • Search Google Scholar
    • Export Citation
  • Blanc, T. V., 1983: An error analysis of profile flux, stability, and roughness length measurements made in the marine atmospheric surface layer. Bound.-Layer Meteor., 26, 243267, https://doi.org/10.1007/BF00121401.

    • Search Google Scholar
    • Export Citation
  • Blanc, T. V., 1985: Variation of bulk-derived surface flux, stability, and roughness results due to the use of different transfer coefficient schemes. J. Phys. Oceanogr., 15, 650669, https://doi.org/10.1175/1520-0485(1985)015<0650:VOBDSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Businger, J. A., 1966: Transfer of heat and momentum in the atmospheric boundary layer. Proceedings of the Symposium on the Arctic Heat Budget and Atmospheric Circulation, J.O. Fletcher, Ed., The Rand Corporation, 305–332.

  • Businger, J. A., 1988: A note on the Businger-Dyer profiles. Bound.-Layer Meteor., 42, 145151, https://doi.org/10.1007/BF00119880.

  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Carl, D. M., T. C. Tarbell, and H. A. Panofsky, 1973: Profiles of wind and temperature from towers over homogeneous terrain. J. Atmos. Sci., 30, 788794, https://doi.org/10.1175/1520-0469(1973)030<0788:POWATF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Charnock, H., 1955: Wind stress over a water surface. Quart. J. Roy. Meteor. Soc., 81, 639640, https://doi.org/10.1002/qj.49708135027.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1970: Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci., 27, 12111213, https://doi.org/10.1175/1520-0469(1970)027<1211:CVATSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Delage, Y., and C. Girard, 1992: Stability functions correct at the free convection limit and consistent for both the surface and Ekman layers. Bound.-Layer Meteor., 58, 1931, https://doi.org/10.1007/BF00120749.

    • Search Google Scholar
    • Export Citation
  • Dwivedi, A. K., S. Chandra, M. Kumar, S. Kumar, and N. V. P. K. Kumar, 2015: Spectral analysis of wind and temperature components during lightning in pre-monsoon season over Ranchi. Meteor. Atmos. Phys., 127, 95105, https://doi.org/10.1007/s00703-014-0346-0.

    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372, https://doi.org/10.1007/BF00240838.

  • Ellison, A. B., 1957: Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech., 2, 456466, https://doi.org/10.1017/S0022112057000269.

    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young, 1996: Bulk parameterization of air-sea fluxes for Tropical Ocean-Global Atmosphere Coupled-Ocean Atmosphere Response Experiment. J. Geophys. Res., 101, 37473764, https://doi.org/10.1029/95JC03205.

    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Foken, T., 2006: 50 years of the Monin–Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447, https://doi.org/10.1007/s10546-006-9048-6.

    • Search Google Scholar
    • Export Citation
  • Giorgi, E., and Coauthors, 2012: RegCM4: Model description and preliminary tests over multiple CORDEX domains. Climate Res., 52, 729, https://doi.org/10.3354/cr01018.

    • Search Google Scholar
    • Export Citation
  • Godfrey, J. S., and A. C. M. Beljaars, 1991: On the turbulent fluxes of buoyancy, heat, and moisture at the air-sea interface at low wind speeds. J. Geophys. Res., 96, 22 04322 048, https://doi.org/10.1029/91JC02015.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., 1990: Friction law in the free-convection limit. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 26, 837–846.

  • Grachev, A. A., C. W. Fairall, and S. E. Larsen, 1998: On the determination of the neutral drag coefficient in the convective boundary layer. Bound.-Layer Meteor., 86, 257278, https://doi.org/10.1023/A:1000617300732.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., C. W. Fairall, and E. F. Bradley, 2000: Convective profile constants revisited. Bound.-Layer Meteor., 94, 495515, https://doi.org/10.1023/A:1002452529672.

    • Search Google Scholar
    • Export Citation
  • Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation. Bound.-Layer Meteor., 42, 5558, https://doi.org/10.1007/BF00119875.

    • Search Google Scholar
    • Export Citation
  • Högström, U., 1996: Review of some basic characteristics of the atmospheric surface layer. Bound.-Layer Meteor., 78, 215246, https://doi.org/10.1007/BF00120937.

    • Search Google Scholar
    • Export Citation
  • Jimenez, P. A., J. Dudhia, J. F. Gonzalez-Rouco, J. Navarro, J. P. Montavez, and E. Gracia-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., 1988: Three-layer structure of an unstably stratified atmospheric surface layer. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 24, 12351250.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and G. Perepelkin, 1989: Effect of the unstable stratification on wind and temperature profiles in the surface layer. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 25, 787795.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J. Fluid Mech., 212, 637662, https://doi.org/10.1017/S0022112090002129.

    • Search Google Scholar
    • Export Citation
  • Kazansky, A. B., and A. S. Monin, 1956: Turbulence in the inversion layer near the surface. Izv. Akad. Nauk SSSR, Ser. Geofiz., 1, 7986.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1954: Osnovnye zakonomernosti turbulentnogo peremeshivanija v prizemnom sloe atmosfery (Basic laws of turbulent mixing in the atmosphere near the ground). Tr. Geofiz. Inst. Akad. Nauk SSSR, 24, 163187.

    • Search Google Scholar
    • Export Citation
  • Obukhov, A. M., 1971: Turbulence in an atmosphere with a non-uniform temperature. Bound.-Layer Meteor., 2, 729, https://doi.org/10.1007/BF00718085.

    • Search Google Scholar
    • Export Citation
  • Panofsky, H. A., 1961: An alternative derivation of the diabatic wind profile. Quart. J. Roy. Meteor. Soc., 87, 109110, https://doi.org/10.1002/qj.49708737113.

    • Search Google Scholar
    • Export Citation
  • Sellers, W. D., 1962: Simplified derivation of the diabatic wind profile. J. Atmos. Sci., 19, 180181, https://doi.org/10.1175/1520-0469(1962)019<0180:ASDOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sharan, M., and P. Srivastava, 2016: Characteristics of heat flux in the unstable atmospheric surface layer. J. Atmos. Sci., 73, 45194529, https://doi.org/10.1175/JAS-D-15-0291.1.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.

  • Srivastava, P., and M. Sharan, 2015: Characteristics of drag coefficient over a tropical environment in convective conditions. J. Atmos. Sci., 72, 49034913, https://doi.org/10.1175/JAS-D-14-0383.1.

    • Search Google Scholar
    • Export Citation
  • Srivastava, P., and M. Sharan, 2017: An analytical formulation of the Obukhov stability parameter in the atmospheric surface layer under unstable conditions. Bound.-Layer Meteor., 165, 371384, https://doi.org/10.1007/s10546-017-0273-y.

    • Search Google Scholar
    • Export Citation
  • Stull, R., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 666 pp.

  • Vihma, T., 1995: Subgrid parameterization of surface heat and momentum fluxes over polar oceans. J. Geophys. Res., 100, 22 62522 646, https://doi.org/10.1029/95JC02498.

    • Search Google Scholar
    • Export Citation
  • Yamamoto, G., 1959: Theory of turbulent transfer in non-neutral conditions. J. Meteor. Soc. Japan, 37, 6067, https://doi.org/10.2151/jmsj1923.37.2_60.

    • Search Google Scholar
    • Export Citation
  • Yang, K., and Coauthors, 2008: Turbulent flux transfer over bare-soil surfaces: Characteristics and parameterization. J. Appl. Meteor. Climatol., 40, 276290, https://doi.org/10.1175/2007JAMC1547.1.

    • Search Google Scholar
    • Export Citation
  • Yang, K., N. Tamai, and T. Koike, 2001: Analytical solution of surface layer similarity equations. J. Appl. Meteor., 40, 16471653, https://doi.org/10.1175/1520-0450(2001)040<1647:ASOSLS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zeng, X., M. Zhao, and R. E. Dickinson, 1998: Intercomparison of bulk aerodynamic algorithms for the computation of sea surface fluxes using TOGA COARE and TAO data. J. Climate, 11, 26282644, https://doi.org/10.1175/1520-0442(1998)011<2628:IOBAAF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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  • Akylas, E., and M. Tombrou, 2005: Interpolation between Businger–Dyer formulae and free convection forms: A revised approach. Bound.-Layer Meteor., 115, 381398, https://doi.org/10.1007/s10546-004-1426-3.

    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327341, https://doi.org/10.1175/1520-0450(1991)030<0327:FPOLSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bernardes, M., and N. L. Dias, 2010: The alignment of the mean wind and stress vectors in the unstable surface layer. Bound.-Layer Meteor., 134, 4159, https://doi.org/10.1007/s10546-009-9429-8.

    • Search Google Scholar
    • Export Citation
  • Blanc, T. V., 1983: An error analysis of profile flux, stability, and roughness length measurements made in the marine atmospheric surface layer. Bound.-Layer Meteor., 26, 243267, https://doi.org/10.1007/BF00121401.

    • Search Google Scholar
    • Export Citation
  • Blanc, T. V., 1985: Variation of bulk-derived surface flux, stability, and roughness results due to the use of different transfer coefficient schemes. J. Phys. Oceanogr., 15, 650669, https://doi.org/10.1175/1520-0485(1985)015<0650:VOBDSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Businger, J. A., 1966: Transfer of heat and momentum in the atmospheric boundary layer. Proceedings of the Symposium on the Arctic Heat Budget and Atmospheric Circulation, J.O. Fletcher, Ed., The Rand Corporation, 305–332.

  • Businger, J. A., 1988: A note on the Businger-Dyer profiles. Bound.-Layer Meteor., 42, 145151, https://doi.org/10.1007/BF00119880.

  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Carl, D. M., T. C. Tarbell, and H. A. Panofsky, 1973: Profiles of wind and temperature from towers over homogeneous terrain. J. Atmos. Sci., 30, 788794, https://doi.org/10.1175/1520-0469(1973)030<0788:POWATF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Charnock, H., 1955: Wind stress over a water surface. Quart. J. Roy. Meteor. Soc., 81, 639640, https://doi.org/10.1002/qj.49708135027.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1970: Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci., 27, 12111213, https://doi.org/10.1175/1520-0469(1970)027<1211:CVATSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Delage, Y., and C. Girard, 1992: Stability functions correct at the free convection limit and consistent for both the surface and Ekman layers. Bound.-Layer Meteor., 58, 1931, https://doi.org/10.1007/BF00120749.

    • Search Google Scholar
    • Export Citation
  • Dwivedi, A. K., S. Chandra, M. Kumar, S. Kumar, and N. V. P. K. Kumar, 2015: Spectral analysis of wind and temperature components during lightning in pre-monsoon season over Ranchi. Meteor. Atmos. Phys., 127, 95105, https://doi.org/10.1007/s00703-014-0346-0.

    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372, https://doi.org/10.1007/BF00240838.

  • Ellison, A. B., 1957: Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech., 2, 456466, https://doi.org/10.1017/S0022112057000269.

    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young, 1996: Bulk parameterization of air-sea fluxes for Tropical Ocean-Global Atmosphere Coupled-Ocean Atmosphere Response Experiment. J. Geophys. Res., 101, 37473764, https://doi.org/10.1029/95JC03205.

    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Foken, T., 2006: 50 years of the Monin–Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447, https://doi.org/10.1007/s10546-006-9048-6.

    • Search Google Scholar
    • Export Citation
  • Giorgi, E., and Coauthors, 2012: RegCM4: Model description and preliminary tests over multiple CORDEX domains. Climate Res., 52, 729, https://doi.org/10.3354/cr01018.

    • Search Google Scholar
    • Export Citation
  • Godfrey, J. S., and A. C. M. Beljaars, 1991: On the turbulent fluxes of buoyancy, heat, and moisture at the air-sea interface at low wind speeds. J. Geophys. Res., 96, 22 04322 048, https://doi.org/10.1029/91JC02015.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., 1990: Friction law in the free-convection limit. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 26, 837–846.

  • Grachev, A. A., C. W. Fairall, and S. E. Larsen, 1998: On the determination of the neutral drag coefficient in the convective boundary layer. Bound.-Layer Meteor., 86, 257278, https://doi.org/10.1023/A:1000617300732.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., C. W. Fairall, and E. F. Bradley, 2000: Convective profile constants revisited. Bound.-Layer Meteor., 94, 495515, https://doi.org/10.1023/A:1002452529672.

    • Search Google Scholar
    • Export Citation
  • Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation. Bound.-Layer Meteor., 42, 5558, https://doi.org/10.1007/BF00119875.

    • Search Google Scholar
    • Export Citation
  • Högström, U., 1996: Review of some basic characteristics of the atmospheric surface layer. Bound.-Layer Meteor., 78, 215246, https://doi.org/10.1007/BF00120937.

    • Search Google Scholar
    • Export Citation
  • Jimenez, P. A., J. Dudhia, J. F. Gonzalez-Rouco, J. Navarro, J. P. Montavez, and E. Gracia-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., 1988: Three-layer structure of an unstably stratified atmospheric surface layer. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 24, 12351250.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and G. Perepelkin, 1989: Effect of the unstable stratification on wind and temperature profiles in the surface layer. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana, 25, 787795.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J. Fluid Mech., 212, 637662, https://doi.org/10.1017/S0022112090002129.

    • Search Google Scholar
    • Export Citation
  • Kazansky, A. B., and A. S. Monin, 1956: Turbulence in the inversion layer near the surface. Izv. Akad. Nauk SSSR, Ser. Geofiz., 1, 7986.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1954: Osnovnye zakonomernosti turbulentnogo peremeshivanija v prizemnom sloe atmosfery (Basic laws of turbulent mixing in the atmosphere near the ground). Tr. Geofiz. Inst. Akad. Nauk SSSR, 24, 163187.

    • Search Google Scholar
    • Export Citation
  • Obukhov, A. M., 1971: Turbulence in an atmosphere with a non-uniform temperature. Bound.-Layer Meteor., 2, 729, https://doi.org/10.1007/BF00718085.

    • Search Google Scholar
    • Export Citation
  • Panofsky, H. A., 1961: An alternative derivation of the diabatic wind profile. Quart. J. Roy. Meteor. Soc., 87, 109110, https://doi.org/10.1002/qj.49708737113.

    • Search Google Scholar
    • Export Citation
  • Sellers, W. D., 1962: Simplified derivation of the diabatic wind profile. J. Atmos. Sci., 19, 180181, https://doi.org/10.1175/1520-0469(1962)019<0180:ASDOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sharan, M., and P. Srivastava, 2016: Characteristics of heat flux in the unstable atmospheric surface layer. J. Atmos. Sci., 73, 45194529, https://doi.org/10.1175/JAS-D-15-0291.1.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.

  • Srivastava, P., and M. Sharan, 2015: Characteristics of drag coefficient over a tropical environment in convective conditions. J. Atmos. Sci., 72, 49034913, https://doi.org/10.1175/JAS-D-14-0383.1.

    • Search Google Scholar
    • Export Citation
  • Srivastava, P., and M. Sharan, 2017: An analytical formulation of the Obukhov stability parameter in the atmospheric surface layer under unstable conditions. Bound.-Layer Meteor., 165, 371384, https://doi.org/10.1007/s10546-017-0273-y.

    • Search Google Scholar
    • Export Citation
  • Stull, R., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 666 pp.

  • Vihma, T., 1995: Subgrid parameterization of surface heat and momentum fluxes over polar oceans. J. Geophys. Res., 100, 22 62522 646, https://doi.org/10.1029/95JC02498.

    • Search Google Scholar
    • Export Citation
  • Yamamoto, G., 1959: Theory of turbulent transfer in non-neutral conditions. J. Meteor. Soc. Japan, 37, 6067, https://doi.org/10.2151/jmsj1923.37.2_60.

    • Search Google Scholar
    • Export Citation
  • Yang, K., and Coauthors, 2008: Turbulent flux transfer over bare-soil surfaces: Characteristics and parameterization. J. Appl. Meteor. Climatol., 40, 276290, https://doi.org/10.1175/2007JAMC1547.1.

    • Search Google Scholar
    • Export Citation
  • Yang, K., N. Tamai, and T. Koike, 2001: Analytical solution of surface layer similarity equations. J. Appl. Meteor., 40, 16471653, https://doi.org/10.1175/1520-0450(2001)040<1647:ASOSLS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zeng, X., M. Zhao, and R. E. Dickinson, 1998: Intercomparison of bulk aerodynamic algorithms for the computation of sea surface fluxes using TOGA COARE and TAO data. J. Climate, 11, 26282644, https://doi.org/10.1175/1520-0442(1998)011<2628:IOBAAF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    The relationship between stability parameter ζ, transfer coefficients CD and CH (vertical axis) calculated from bulk flux algorithm, and bulk Richardson number RiB (horizontal axis) for four classes of the similarity functions φm and φh. Three underlying surface conditions are chosen: smooth (z0 = 0.01 m), transition (z0 = 0.1 m), and rough (z0 = 1 m) for the ratio of roughness lengths of heat and momentum z0/zh = 1.

  • Fig. 2.

    Maximum percentage difference between the values of ζ, CD, and CH computed with class IV functions and the functions lying in other three classes.

  • Fig. 3.

    Variation of drag coefficient (CD) and similarity function φm suggested by Kader and Yaglom (1990) with stability parameter ζ. For the computation of values of CD, z/z0 is assumed to be 10−5. The vertical dashed lines are plotted at the values of ζ at which CD and φm change their monotonic behavior.

  • Fig. 4.

    Variation of drag coefficient CD with stability parameter ζ observed for the whole year 2009 over Ranchi (India) is shown with red markers. The mean values of CD in each unstable sublayers are shown with black markers along with standard deviations in the form of error bars. Depending upon the data availability, two to three bins of equal width are chosen in each sublayer. The background color scheme corresponds to different unstable sublayers (Kader and Yaglom 1990; Bernardes and Dias 2010), i.e., dynamic sublayer (light gray) to free-convective sublayer (dark gray). The continuous curves represent the predicted variation of CD using different classes (I–IV) of stability correction functions. The continuous black curve shows the predicted variation of CD with the updated functional form of φm based on Kader and Yaglom (1990). For plotting each continuous curve, the value of the neutral transfer coefficient (CDN) is taken as an average value CDN in the dynamic sublayer. The dotted vertical line shows ζ = ζm = −0.1.

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