Uncertainty in the Parameterization of Surface Fluxes under Unstable Conditions

Piyush Srivastava; Maithili Sharan

doi:10.1175/JAS-D-20-0350.1

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Bulk algorithm for parameterization of transfer coefficients and surface fluxes
3. Commonly used functional forms of φm and φh under unstable conditions
4. Results and discussion
5. Issues and limitations
6. Conclusions
Acknowledgments
Data availability statement
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Fig. 1.

The relationship between stability parameter ζ, transfer coefficients C_D and C_H (vertical axis) calculated from bulk flux algorithm, and bulk Richardson number Ri_B (horizontal axis) for four classes of the similarity functions φ_m and φ_h. 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/z_h = 1.

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Fig. 2.

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

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Fig. 3.

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

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Fig. 4.

Variation of drag coefficient C_D with stability parameter ζ observed for the whole year 2009 over Ranchi (India) is shown with red markers. The mean values of C_D 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 C_D using different classes (I–IV) of stability correction functions. The continuous black curve shows the predicted variation of C_D with the updated functional form of φ_m based on Kader and Yaglom (1990). For plotting each continuous curve, the value of the neutral transfer coefficient (C_DN) is taken as an average value C_DN in the dynamic sublayer. The dotted vertical line shows ζ = ζ_m = −0.1.

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Editorial Type:: Article

Article Type:: Research Article

Uncertainty in the Parameterization of Surface Fluxes under Unstable Conditions

Piyush Srivastava

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

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

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Online Publication:: 30 Jun 2021

Print Publication:: 01 Jul 2021

DOI:: https://doi.org/10.1175/JAS-D-20-0350.1

Page(s):: 2237–2247

Received:: 20 Nov 2020

Accepted:: 31 Mar 2021

Published Online:: 30 Jun 2021

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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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.

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

Abstract

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

Keywords: Fluxes; Surface fluxes; Numerical weather prediction/forecasting; Parameterization

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:

\frac{κ z}{u_{*}} \frac{\partial U}{\partial z} = φ_{m} (ζ)

(1)

and

\frac{κ z}{θ_{*}} \frac{\partial θ}{\partial z} = φ_{h} (ζ),

(2)

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 φ_m and φ_h are, respectively, the nondimensional similarity functions. Here, ζ is defined as

ζ = \frac{κ z g}{\bar{θ}} \frac{θ_{*}}{u_{*}^{2}},

(3)

in which

\bar{θ}

is the mean potential temperature in the layer and g is the acceleration due to gravity.

In near-neutral conditions (ζ → 0), both φ_m and φ_h must be equal to a constant that is equal to unity for φ_m, while the corresponding value for φ_h 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 φ_m and φ_h 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 φ_m and φ_h, while Businger (1966) suggested that the exponent should be −1/4 for φ_m and −1/2 for φ_h 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 φ_m, commonly referred to as the Kazansky and Monin–Ellison–Yamamoto–Panofsky–Sellers (KEYPS) or Obukhov KEYPS (O’KEYPS) formula in the literature as

φ_{m}^{4} (ζ) - γ φ_{m}^{3} (ζ) = 1,

(4)

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 φ_m, 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 φ_m and φ_h 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 (C_D) and heat (C_H) and corresponding surface fluxes in the atmospheric models depend on the behavior of φ_m and φ_h (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 C_D 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 C_D depends upon the functional behavior of similarity function φ_m, various operational weather and climate models use one or another form of φ_m and φ_h 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 φ_m and φ_h 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 φ_m and φ_h 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) = \frac{u_{*}}{κ} {\ln (\frac{z}{z_{0}}) - [ψ_{m} (ζ) - ψ_{m} (\frac{z_{0}}{z} ζ)]}

(5)

and

θ (z) - θ_{0} = \frac{θ_{*}}{κ} {\ln (\frac{z}{z_{h}}) - [ψ_{h} (ζ) - ψ_{h} (\frac{z_{h}}{z} ζ)]},

(6)

where z₀ and z_h are, respectively, roughness lengths of momentum and heat; θ₀ is the potential temperature at the height z_h; ψ_m and ψ_h are, respectively, the integrated similarity functions corresponding to similarity functions φ_m and φ_h. The stability parameter ζ is defined as in Eq. (3). The integrated similarity functions, commonly known as stability correction functions ψ_m and ψ_h are given by

ψ_{m, h} (ζ) = \int_{0}^{ζ} \frac{1 - φ_{m, h} (ζ^{'})}{ζ^{'}} d ζ^{'}

(7)

in which ζ′ is a dummy variable of integration corresponding to ζ. For land surfaces, where z₀ and z_h are taken as constants, Eqs. (3), (5), snf (6) are solved iteratively to estimate ζ and bulk transfer coefficients C_D and C_H as

C_{D} = κ^{2} {[\ln (z / z_{0}) - ψ_{m} (ζ, \frac{z_{0}}{z} ζ)]}^{- 2}

(8)

and

C_{H} = κ^{2} {[\ln (z / z_{0}) - ψ_{m} (ζ, \frac{z_{0}}{z} ζ)]}^{- 1} {[\ln (z / z_{h}) - ψ_{h} (ζ, \frac{z_{h}}{z} ζ)]}^{- 1} .

(9)

Then C_D and C_H are used to estimate the turbulent fluxes of heat (H) and momentum (−τ) from

- \frac{H}{ρ C_{p}} = C_{H} U (θ - θ_{0})

(10)

and

\frac{τ}{ρ} = C_{D} U^{2},

(11)

in which ρ is the dry air density and C_p 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 Ri_B, which is defined as

{Ri}_{B} = \frac{g}{\bar{θ}} \frac{{(z - z_{0})}^{2}}{(z - z_{h})} \frac{(θ - θ_{0})}{U^{2}} .

(12)

Using Eqs. (3), (5), and (6) in Eq. (12), the expression for Ri_B becomes

{Ri}_{B} = ζ \frac{{(1 - z_{0} / z)}^{2}}{(1 - z_{h} / z)} \frac{\ln (z / z_{h})}{{[\ln (z / z_{0})]}^{2}} \frac{[1 - \frac{ψ_{h} (ζ) - ψ_{h} (ζ_{h})}{\ln (z / z_{h})}]}{{[1 - \frac{ψ_{m} (ζ) - ψ_{m} (ζ_{0})}{\ln (z / z_{0})}]}^{2}} .

(13)

For a given value of Ri_B, ζ is estimated by calculating the root of least magnitude of Eq. (13).

3. Commonly used functional forms of φ_m and φ_h under unstable conditions

The expression for similarity functions φ_m and φ_h was proposed by Businger (1966) and A. J. Dyer [1965, unpublished material; see Businger (1988) for details] for unstable stratification as

φ_{m} (ζ) = {(1 - γ_{m} ζ)}^{- 1 / 4}

(14)

and

φ_{h} (ζ) = \Pr_{t} {(1 - γ_{h} ζ)}^{- 1 / 2},

(15)

in which γ_m = 15 and γ_h = 9 and

{Pr}_{t} = {(φ_{h} / φ_{m})}_{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 φ_m valid for the stability range −10 ≤ ζ ≤ 0 as

φ_{m, h} (ζ) = {(1 - β_{m, h} ζ)}^{- 1 / 3} .

(16)

Different values of the constants β_m and β_h can be found in the literature; for example, β_m = β_h = 15 (Carl et al. 1973), β_m and β_h = 40 (Delage and Girard 1992), and β_m = β_h = 12.87 (Fairall et al. 1996). Högström (1988) reformulated the universal similarity functions of Businger et al. (1971) of the form

φ_{m} (ζ) = {(1 - 19.3 ζ)}^{- 1 / 4}

(17)

and

φ_{h} (ζ) = 0.95 {(1 - 11.6 ζ)}^{- 1 / 2} .

(18)

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

ψ_{α} = \frac{ψ_{α_D} + ζ^{2} ψ_{α_conv}}{1 + ζ^{2}}, α = m, h,

(19)

in which

ψ_{α_D}

is an integrated similarity function corresponding to the Dyer similarity functions (BD functions with γ_m = γ_h = 16), and

ψ_{α_conv}

is that given by Carl et al. (1973). Grachev et al. (2000) revisited the convective constants appearing in the expression of

ψ_{α_conv}

and suggested that the value of the constant for

ψ_{m_conv}

should be equal to 10.15 while that for

ψ_{h_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 φ_m,h functions rather than the corresponding integrated functions ψ_m,h suggested by Fairall et al. (1996, 2003) [Eq. (19)].

The three sublayer model of Kader and Yaglom (1990) suggests the contrasting behavior of φ_m 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, φ_m(ζ) = 1 while φ_h(ζ) = Pr_t. 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

φ_{m} (ζ) = A_{u} {(- ζ)}^{- 1 / 3}

(20)

and

φ_{h} (ζ) = A_{T} {(- ζ)}^{- 1 / 3} .

(21)

Here A_u and A_T 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

φ_{m} (ζ) = B_{u} {(- ζ)}^{1 / 3}

(22)

and

φ_{h} (ζ) = B_{T} {(- ζ)}^{- 1 / 3},

(23)

in which B_u and B_T are constants.

The remarkable prediction of this sublayer is that unlike the case of classical free-convection limit, in which both φ_m and φ_h follow a −1/3 power law, here φ_m varies as the +1/3 power of ζ for sufficiently large values of −ζ. Thus, the theory of Kader and Yaglom (1990) suggests that φ_m 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 φ_m and φ_h have been suggested and one of those expressions of φ_m and φ_h, which is being utilized in CESM (Zeng et al. 1998), is taken here for the analysis.

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

−1/4 and −1/2 power law for φ_m and φ_h, respectively (Businger et al. 1971; Högström 1996), for the full range of instability [Eqs. (14) and (15); class I functions].
−1/3 power law for both φ_m and φ_h (Carl et al. 1973) for the full range of instability [Eq. (16); class II functions].
−1/4 and −1/2 power law in near neutral for φ_m and φ_h, respectively, and −1/3 power law in very unstable conditions for both φ_m and φ_h (Fairall et al. 1996; Grachev et al. 2000; Fairall et al. 2003) [Eq. (19); class III functions].
−1/4 and −1/2 power law in near-neutral for φ_m and φ_h, respectively, with a −1/3 power law in very unstable conditions for φ_h and +1/3 power law in very unstable conditions for φ_m (Kader and Yaglom 1990; Zeng et al. 1998; class IV functions).

For a given value of Ri_B, the value of ζ is estimated by calculating the root of least magnitude of the transcendental equation in ζ and Ri_B [Eq. (13)]. For computation purposes, the range of values of Ri_B, z/z₀, and z₀/z_h is taken as −2 ≤ Ri_B < 0, −0.5 ≤ log(z₀/z_h) < 29.0, and 10 ≤ z/z₀ < 10⁵. The estimated value of ζ is then utilized to compute the transfer coefficients of momentum C_D and heat C_H [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 φ_m and φ_h 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 Ri_B the values of ζ obtained from different functional forms of φ_m and φ_h are found to be nearly identical. However, the deviations in the predicted values of ζ for a given Ri_B from different formulations are found to increase with decreasing values of Ri_B. The −1/4 and −1/2 power law for φ_m and φ_h, respectively (class I functions), and −1/3 power law for both φ_m and φ_h (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 Ri_B larger than 0.2. However, the absolute values of ζ predicted by the functional forms of φ_m and φ_h 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 φ_m and φ_h. 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 Ri_B suggests that both the momentum and heat fluxes predicted with the class IV functional forms of φ_m and φ_h 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 φ_m and φ_h 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 φ_m and φ_h. Figures 1d–i show the variation of C_D and C_H with Ri_B for all the four types of similarity functions φ_m and φ_h for different values of roughness length of momentum. The class I, II, and III functional forms of φ_m and φ_h suggest relatively higher values of C_D for a given value of Ri_B in moderately to strongly unstable conditions. In contrast, the values of C_D predicted by class IV functional form of φ_m and φ_h are significantly smaller as compared to those predicted by the other three functional forms. One striking feature regarding the behavior of C_D observed for class IV functions is that C_D shows a nonmonotonic characteristic, which is in contradiction to the prediction of other forms of φ_m and φ_h (Figs. 1d–e). The values of C_H are found to increase with increasing instability for all four types of similarity functions. However, the rate of increase in the values of C_H with Ri_B is considerably small in the case of class IV functions (Figs. 1g–i). The difference in the estimated values of C_H for a given Ri_B 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 C_H shows a nonmonotonic variation with respect to Ri_B 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 C_D and C_H with stability is relatively narrow as compared to that shown by the other three classes of correction functions. The values of C_D and C_H 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 C_D and C_H with increasing instability.

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 ≤ Ri_B < 0, −0.5 ≤ log(z₀/z_h) < 29.0, 10 ≤ z/z₀ < 10⁵ using the formula

% difference ζ = (\frac{ζ_{p} - ζ_{KY}}{ζ_{p}}) \times 100,

(24a)

% difference C_{D} = (\frac{C_{D_{P}} - C_{D_{KY}}}{C_{D_{P}}}) \times 100,

(24b)

and

% difference C_{H} = (\frac{C_{H_{P}} - C_{H_{KY}}}{C_{H_{P}}}) \times 100,

(24c)

in which ζ_p,

C_{D_{p}}

, and

C_{H_{p}}

are the estimated values of stability parameter, momentum, and heat transfer coefficients for the functions belonging to the class I, II, or III and ζ_KY,

C_{D_{K Y}}

, and

C_{H_{K Y}}

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 Ri_B. 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, C_D, and C_H depend upon three parameters: Ri_B, z₀/z_h, and z/z₀. In Fig. 2a, the maximum possible differences are calculated with respect to z/z₀ for the values of Ri_B and z₀/z_h lying in the range −2 ≤ Ri_B < 0 and −0.5 ≤ log(z₀/z_h) < 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.

b. Nonmonotonic behavior of drag coefficient C_D

Srivastava and Sharan (2015) utilized the turbulent data over a tropical region to analyze the observational behavior of C_D with respect to wind speed U and stability parameter ζ under unstable conditions. They found that the observed variation of C_D 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 C_D 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 C_D 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 φ_m [appendix B, Eq. (B1), Srivastava and Sharan 2015] was considered, and based on the mathematical analysis, it was concluded that the parameterized drag coefficient C_D increases with increasing value of −ζ until it attains a maximum value at ζ = ζ_c 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 C_D changes its behavior is same at which function φ_m changes its behavior, that is, ζ = ζ_c. 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 φ_m is taken as

φ_{m} (ζ) = {\begin{matrix} φ_{1} (ζ), & ζ_{c} \leq ζ \leq 0 \\ φ_{2} (ζ), & ζ \leq ζ_{c} \end{matrix},

(25)

in which φ₁(ζ) is a continuously decreasing function of −ζ in the range ζ_c ≤ ζ ≤ 0 and φ₂(ζ) is an increasing function for ζ ≤ ζ_c.

An expression for drag coefficient C_D can be written as

C_{D} = \frac{κ^{2}}{Y_{m}^{2}},

(26)

in which

Y_{m} = \ln (\frac{z}{z_{0}}) - ψ_{m} (ζ, \frac{z_{0}}{z} ζ),

(27)

where ψ_m is the integrated similarity function corresponding to the similarity function φ_m. Notice that traditionally the drag coefficient is defined as C_D = (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, C_D 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 U_eff should be used in place of the usual vector averaged wind speed U, which is the sum of U and the gustiness velocity U_gs. The gustiness velocity is parameterized in terms of Deardorff convective velocity scale w_* (Deardorff 1970) as U_gs = βw_*, where β is an empirical constant. Thus, in the case of low wind convective conditions, the updated drag coefficient can be expressed as C_D = (u_*/U_eff)², where U_eff = U + βw_*.

For brevity, we put η = −ζ in Eq. (27) implying that η > 0 in unstable conditions and differentiating

Y_{m}^{2}

with respect to η, one gets

\frac{d}{d η} (Y_{m}^{2}) = 2 Y_{m} Y_{m}^{'} .

(28)

where

Y_{m}^{'}

is the derivative of Y_m with respect to η, given as

Y_{m}^{'} = - [ψ_{m}^{'} (η, \frac{z_{0}}{z} η)] .

(29)

Here, the prime denotes the derivative with respect to η.

For the function φ_m(ζ),

Y_{m}^{'}

is given by

Y_{m}^{'} = - \frac{1}{η} [φ_{1} (\frac{z_{0}}{z} η) - φ_{2} (η)] .

(30)

The behavior of C_D depends on the sign of

Y_{m}^{'}

, 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)

φ_{m} (ζ) = {\begin{matrix} {(1 - 16 ζ)}^{- 1 / 4}, & - 1.574 \leq ζ \leq 0 \\ 0.7 κ^{2 / 3} {(- ζ)}^{1 / 3}, & ζ \leq - 1.574 \end{matrix} .

(31)

In this expression, φ₁ = (1 − 16 ζ)^−1/4, φ₂ = 0.7 κ^2/3(−ζ)^1/3, and ζ_c = −1.574.

For this functional form of φ_m, an expression for

Y_{m}^{'}

can be written as

Y_{m}^{'} = - \frac{1}{η} [{(1 + 16 \frac{z_{0}}{z} η)}^{- 1 / 4} - 0.7 κ^{2 / 3} {(η)}^{1 / 3}] .

(32)

Note that

φ_{1} = {[1 + 16 (z_{0} / z) η]}^{- 1 / 4}

is a monotonic decreasing function of η that attains its maximum value 1 as η → 0. The other function φ₂ = 0.7κ^2/3(η)^1/3 is a monotonic increasing that attains its minimum value 0 as η → 0 and φ₂ > 1 for

η > {1 / (0.7 κ^{2 / 3})}^{3}

. This suggests that there exists a point η_m, such that

Y_{m}^{'} < 0

for 0 ≤ η ≤ η_m but

Y_{m}^{'} > 0

for η ≥ η_m. The exact value of η_m depends upon the value of the ratio z/z₀.

Thus, the value of C_D increases until it attains a maximum value at η = η_m and then starts decreasing with increasing value of η. However, the points ζ_c and ζ_m = −η_m are different (Fig. 3).

c. Observational evidence and recommendation for the form of φ_m 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 φ_m, 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_{*} = {[{(\bar{u^{'} w^{'}})}^{2} + {(\bar{υ^{'} w^{'}})}^{2}]}^{1 / 4}

(33)

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 C_D is estimated using

C_{D} = {(u_{*} / U)}^{2} .

(34)

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 C_D with ζ along with the MOST predicted forms of C_D for different classes of similarity functions φ_m. The bin-averaged values of C_D 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 C_D 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 C_D 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, C_D is parameterized using Eq. (8). Equivalently one can write Eq. (9) in the form

C_{D} = \frac{C_{D N}}{{1 - [κ ψ_{m} (ζ, \frac{z_{0}}{z} ζ) / \sqrt{C_{D N}}]}^{2}},

(35)

where C_DN is the neutral drag coefficient. In Fig. 4, the continuous lines, showing the predicted variation of C_D for different forms of similarity functions, are plotted using C_DN obtained from the observational data. Figure 4 suggests that all the formulations agree well for the values of C_D in the range −0.1 < ζ < 0, that is, in near-neutral to weakly unstable condition. With class I, II, and III functions, C_D shows monotonically increasing behavior in the full range of instability, while with class IV function, C_D 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 C_D with ζ. However, the point at which C_D 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 φ_m function in such a way that point at which C_D attains its maximum value lies in the DCS sublayer as found in the observational data. The modified functional form for φ_m is given by

φ_{m} = {\begin{matrix} {(1 - 16 ζ)}^{- 1 / 4}, & ζ_{m} \leq ζ < 0 \\ 3.125 κ^{2 / 3} {(- ζ)}^{1 / 3}, & ζ \leq ζ_{m} \end{matrix}, ζ_{m} = - 0.1 .

(36)

Notice that with the modified functional form of φ_m, the observed nature of C_D with ζ in the unstable atmospheric conditions is well captured by MOST (Fig. 4). However, the suitability of the new form of φ_m 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 φ_m in estimating the transfer coefficients of heat and moisture from the bulk algorithm.

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 (z_h) and momentum (z₀) (or neutral transfer coefficients C_D and C_H), 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 z_h 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 z_h over different underlying surfaces. For example, depending upon the surface characteristics fixed values of z₀ and z_h 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 z_h. 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 C_D with respect to stability parameter ζ or Ri_B 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 C_D 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, C_D 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 φ_m and φ_h 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 φ_m and φ_h. These classes are identified based on the functional behavior of φ_m and φ_h 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 φ_m and φ_h 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 C_D and C_H 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 C_D with ζ. This nonmonotonic nature of C_D 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 C_D 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 ψ_m(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 C_D 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.

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Bulk algorithm for parameterization of transfer coefficients and surface fluxes
3. Commonly used functional forms of φm and φh under unstable conditions
4. Results and discussion
5. Issues and limitations
6. Conclusions
Acknowledgments
Data availability statement
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Fig. 1.

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Fig. 2.

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

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Uncertainty in the Parameterization of Surface Fluxes under Unstable Conditions

Abstract

Abstract

1. Introduction

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

3. Commonly used functional forms of φ_m and φ_h under unstable conditions

4. Results and discussion

a. Uncertainty in the estimation of transfer coefficients

b. Nonmonotonic behavior of drag coefficient C_D

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

5. Issues and limitations

6. Conclusions

Acknowledgments

Data availability statement

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Uncertainty in the Parameterization of Surface Fluxes under Unstable Conditions

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Abstract

1. Introduction

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

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

4. Results and discussion

a. Uncertainty in the estimation of transfer coefficients

b. Nonmonotonic behavior of drag coefficient CD

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

5. Issues and limitations

6. Conclusions

Acknowledgments

Data availability statement

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3. Commonly used functional forms of φ_m and φ_h under unstable conditions

b. Nonmonotonic behavior of drag coefficient C_D

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