Persistence of Logarithmic Temperature Profile in Unstably Stratified Atmospheric Boundary Layers with and Without Sand

Abstract

Monin-Obukhov similarity theory (MOST) is a widely used framework in atmospheric boundary layer (ASL) research. MOST predicts that the mean potential temperature profile deviates from the logarithmic law in buoyancy-dominated turbulence. However, recent studies show that the logarithmic profile remains intact in unstably stratified atmospheric boundary layers. Using data from the Qingtu Lake Observation Array(QLOA) with Reynolds number up to O(106), we investigate the similarity functions and mean temperature profile with different stability parameters and sand-bearing conditions. By assessing the accuracy of the logarithmic profile and the MOST-based empirical expressions obtained by Hogstrom and Wilson, we discover that the logarithmic law is a better fit than the MOST expressions. The Von Kármán constant of the potential temperature profile has a power function dependences on the stability parameter. In sand-laden ASLs, it is remarkable to find that the logarithmic law still holds, and the error of the MOST expressions amplifies. The von Kármán constant of potential temperature increases in sandiness conditions. Still, a quantitative theory that describes the sand effect on the mean temperature profile remains to be studied.

1 Introduction

The logarithmic mean velocity profile in the near-wall region of boundary-layer turbulence has been observed in laboratory and field observations and numerical simulations [16, 19, 20]. Similarly, the logarithmic profile for mean temperature was also observed [15, 25]. Kader [13] pointed out that the average temperature profile in wall turbulence satisfies the law of the wall. However, when buoyancy affects the flow field, the universal log law is insufficient for accurately describing the mean velocity [11, 26] and temperature profiles [7, 17, 24, 27].

Monin-Obukhov similarity theory (MOST) [21, 23] is frequently used to account for buoyancy effects. MOST suggests that four parameters govern the quasi-steady turbulence structure in the Atmospheric Surface Layer (ASL): the distance z from the surface, the friction velocity $u_\tau$ (which is the square root of the magnitude of the mean kinematic surface stress), the mean temperature flux at the surface $Q_0$, and the buoyancy parameter $g/\theta _0$, where $\theta _0$ is a characteristic potential temperature. Combining the dependent variables yields dimensionless parameters. The Monin-Obukhov length is defined as $L=-\left( u_\tau \right) ^3\theta _0/\kappa g Q_0$ [22], where $\kappa$ is the von Kármán constant. Thus, z/L is defined as the stability parameter. MOST implies that the mean velocity and potential temperature gradients are expressed as:

$$\begin{aligned} \frac{\kappa z}{u_\tau } \frac{\partial U}{\partial z}&=\phi _m\left( \frac{z}{L}\right) ,\end{aligned}$$

(1a)

$$\begin{aligned} -\frac{\kappa z u_\tau }{Q_0} \frac{\partial \varTheta }{\partial z}&=\frac{\kappa z }{\theta _\tau } \frac{\partial \varTheta }{\partial z}=\phi _h\left( \frac{z}{L}\right) , \end{aligned}$$

(1b)

where U is the mean velocity and $\varTheta$ is the mean potential temperature at the same height. $\phi _m$ and $\phi _h$ are Monin-Obukhov similarity functions. Many different forms of similarity functions have been proposed [3, 4, 9, 14, 31], and they depart from the logarithmic profiles. Particularly, Hogstrom [9] suggested:

$$\begin{aligned} \phi _h= {\left\{ \begin{array}{ll} 1 + 7.8\dfrac{z}{L}, &{} \dfrac{z}{L}>0, \\ \left( 1 - 12\dfrac{z}{L}\right) ^{-1/2}, &{} \dfrac{z}{L}<0. \end{array}\right. } \end{aligned}$$

(2)

Wilson [31] proposed

$$\begin{aligned} \phi _h=\left( 1+7.9 \left| \frac{z}{L}\right| ^{2/3}\right) ^{-1/2} \end{aligned}$$

(3)

for unstable conditions. The expression of law of the wall [28] is

$$\begin{aligned} \frac{u}{u_\tau }=\frac{1}{\kappa } \ln {\frac{z}{\delta _\nu }} +C, \end{aligned}$$

(4)

where $\delta _\nu =\nu /u_\tau$ is the viscous length , which defines the Kármán constant.

However, recent studies find the logarithmic profile remains intact in unstably stratified atmospheric boundary layers [1, 2, 8, 10]. It has been shown that in turbulent flows driven by both shear and buoyancy, the mean temperature profile may deviate from the traditional power law observed in Rayleigh-Bénard Convection [1, 2] and natural convection [10]. Instead, a logarithmic temperature profile with varying slopes has been reported to be more prevalent than the velocity log law, and they are observed even in extremely buoyant flows [1]. These findings motivate further examination of the existence of a temperature log law in such turbulent flows. Johansson et al. [12] found that the similarity functions also depend on $z_i/L$, where $z_i$ is the convective boundary layer, by performing large eddy simulations and studying field measurements. Cheng et al. [5] investigates the existence and slope of the logarithmic temperature profile in turbulent flows driven by both shear and buoyancy, such as the unstably stratified atmospheric boundary layer. Their results suggest that the buoyancy force does not modify the logarithmic nature of the mean potential temperature profile but modulates its slope instead.

Inspired by this work, we verified this conclusion under different Reynolds numbers and sediment conditions compared to theirs, examining the variables that contribute to the expression of temperature profiles. We compared the accuracy of the logarithmic profile and the MOST-based empirical expressions obtained by Hogstrom and Wilson based on the data of QLOA [18, 29, 30].

This paper is arranged as follows. Section 2 introduces the observation facility, data selection and pretreatment. Section 3 contains the similarity functions, mean velocity and potential temperature profiles, comparison of logarithmic profile and MOST expressions, and the slopes of the profiles vary with different abscissas. Section 4 gives the conclusions.

2 Experimental Facility and Data Details

This study utilizes data collected from the QLOA site situated on the flat and arid Qingtu Lake in western China. The observation array comprises of 21 towers, with the main tower standing at a height of 32 m and 20 smaller towers at 5 m height, all capable of measuring the three components of wind speed and dust concentration simultaneously. To measure wind speed and temperature, eleven acoustic anemometers (Campbell scientific, CSAT-3B) were mounted on the main tower between 0.9 and 30 m. The temperature measurement range is from $-40^{\circ }$C to 60$^{\circ }$C, with a minimum resolution of 1$^{\circ }$C and less than 1$^{\circ }$C absolute error.

Due to the complexity and unpredictability of environmental conditions during field observations, specific selection and preprocessing of original data were performed to ensure accurate representation of the turbulent boundary layer at high Reynolds numbers. The pretreatment involved stratification stability judgment and classification, wind direction adjustment, and long-term trend filtering. Additionally, quality verification was conducted to ensure that the wind data were statistically stable. The non-stationary index was calculated based on the study by Foken et al. [6] to identify high-quality data with $\textrm{IST}_u<30\%$ and $\textrm{IST}_\theta <30\%$. The definition of $\textrm{IST}_u$ and $\textrm{IST}_\theta$ is

$$\begin{aligned} \textrm{IST}_u=\left| \left( \textrm{CV}_{um}-\textrm{CV}_{u1 h}\right) / \textrm{CV}_{u1 h}\right| \times 100 \%, \end{aligned}$$

(5)

where $\textrm{CV}_{um}=\sum _{i=1}^{12} \textrm{CV}_{ui} / 12, \textrm{CV}_{u1}, \textrm{CV}_{u2}, \ldots , \textrm{CV}_{u12}$ are the streamwise velocity variances every 5 min in 1 h, and $\textrm{CV}_{u1 h}$ is the total streamwise velocity variance for the hour.

$$\begin{aligned} \textrm{IST}_\theta =\left| \left( \textrm{CV}_{\theta m}-\textrm{CV}_{\theta 1\,h}\right) / \textrm{CV}_{\theta 1\,h}\right| \times 100 \%, \end{aligned}$$

(6)

with $\textrm{CV}_{\theta m}=\sum _{i=1}^{12} \textrm{CV}_{\theta i} / 12, \textrm{CV}_{\theta 1}, \textrm{CV}_{\theta 2}, \ldots , \textrm{CV}_{\theta 12}$ are the potential temperature variances every 5 min in 1 h, and $\textrm{CV}_{u1 h}$ is the total potential temperature variance for the hour.

Table 1. Main parameters of the selected representative wall flow data intervals. The $Re_\tau$ of the flow is $O(10^6)$. $\nu$ is the kinematic viscosity coefficient and $\alpha$ is the thermal diffusion coefficient. The z/L here is calculated with the minimum experimental height 0.9m. The PM10 measured at 5m is used to reflect the amount of sediment concentration.

High-quality data with $\textrm{IST}_u<30\%,\textrm{IST}_\theta <30\%$ are selected in the work. Unstable stratified data is mainly studied. The used data were measured from 2015 to 2017 and can be classified into two types: sand-laden flow data observed during sandstorm weather and clean air data measured in clean air with little sand.

As shown in Table 1, several intervals are selected to analyze the data. No.1 is a clean air data and N0.2–N0.6 are data measured in sandstorm. $\theta _\tau =Q_0/u_\tau$ is the friction temperature. Negative z/L and $\theta _\tau$ corresponds to an unstable situation. The kinetic viscosity coefficient and thermal diffusion coefficient of them have little difference. The key distinction is the stability parameter and sediment concentration. The measured data of 0.9m and 10.24m is applied in the calculation of z/L and $\textrm{PM10}$ separately.

Fig. 1.

$\left\langle wu \right\rangle$ and $\left\langle w\theta \right\rangle$ of clean air and sand-laden flow. $Q_0$ equals $\left\langle w\theta \right\rangle$ on the surface. (a) The kinetic surface flux and potential temperature flux of different heights in clean air with $u_\tau$ and $\theta _\tau$ on the top right corner The blue line is $\left\langle wu \right\rangle$ while the red line is $\left\langle w\theta \right\rangle$. (b) The flux in a sand-laden flow.

From Fig. 1, the kinetic flux and potential temperature flux are relatively steady with elevation change, which indicates the defined characteristic parameters $u_\tau$ and $\theta _\tau$ can be applicable to each height for both sets of data. No significant change is observed due to the presence of sediment.

3 Results

3.1 Similarity Functions

Figure 2 shows the similarity functions of the measured data and verifies the expressions of Businger [3] and Hogstrom [9]. Businger suggests the approximate expression is:

$$\begin{aligned} \phi _h= {\left\{ \begin{array}{ll} 0.74 + 4.7 \dfrac{z}{L}, &{} \dfrac{z}{L}>0, \\ 0.74 \left( 1 - 9\dfrac{z}{L}\right) ^{-1/2}. &{} \dfrac{z}{L}<0. \end{array}\right. } \end{aligned}$$

(7)

This work focus on the unstable condition, so only the negative z/L data is plotted. In Fig. 2, measured value of $\phi _h$ at several heights and different time intervals are plotted. $\phi _h$ of different heights at the same time is close to each other. Hogstrom’s formula exhibits better performance than Businger’s, but discrepancies with our experimental data still exists. The predicted values are widely underestimated but the general trend is consistent with the empirical formula. The similarity functions for the sand-laden and clean air cases are essentially equivalent.

Fig. 2.

Similarity function $\phi _h$ of clean air and sand-laden flow. The circles are the clean air data and the asterisks are the sand-laden data. The dotted line is Businger’s (1971) expression and the dot dash line is Hogstrom’s(1988) expression. The figure has a logarithmic ordinate.

3.2 The Potential Temperature Profile

Similar to Eq. (4), the potential temperature profile expression is:

$$\begin{aligned} \frac{\varTheta -\varTheta _0}{\theta _\tau }=\frac{1}{\kappa _\theta } \ln {\frac{z}{z_r}}, \end{aligned}$$

(8)

where $\varTheta$ is the mean potential temperature, $\varTheta _0$ is the mean potential temperature at the lowest height, $\kappa _\theta$ is the potential temperature Kármán constant, and $z_r$ is the reference height.

Figure 3 is the potential temperature profile. The horizontal axis is $z^+=z/\delta _\nu =z u_\tau /\nu$ and the vertical axis is $\varTheta ^+=(\varTheta -\varTheta _0)/{\theta _\tau }$. The predicted profile of Hogstrom and Wilson deviates more from the measured data than the clean air situation.

In unstable conditions, the mean temperature grows with height. It can be seen from the left figure that both the three estimates of the profile under the clear wind flow are in good agreement with the actual measured data. Under the sand-laden flow, the measured data has a larger fluctuation and the two empirical expressions deviate more from the logarithmic profile. The potential temperature profiles in sand-laden flow have larger fluctuation than clean air, but they still satisfy the logarithmic profile in general.

3.3 The Root-Mean-Square Error (RMSE) Comparison of Log Law and MOST-Based Empirical Equations

To assess the accuracy of the logarithmic law equation and the MOST-based empirical equations, we computed the root mean square error (RMSE) between the measured temperature ($\varTheta _i$) and the estimated temperature ($\hat{\varTheta }_i$). The RMSE is defined as:

$$\begin{aligned} \sigma _{l,H,W} = \sqrt{\frac{1}{n}\sum _{i=1}^{n}(\varTheta _i - \hat{\varTheta }_i)^2}, \end{aligned}$$

(9)

where n represents the total number of data points. The subscripts l, H, and W correspond to the logarithmic law expression, Hogstrom’s expression, and Wilson’s expression, respectively. A smaller value of $\sigma$ indicates a lower error, indicating a better fit between the estimated and measured temperatures.

Fig. 3.

The potential temperature profiles of clean air and sand-laden flow. The titles represent the time of data measurement. The triangular scatters are the measured potential temperature. The black dashed line is the log law line. The red and green lines are the MOST-based empirical results. Subfigure (a) is the clean air while (b) is the sand-laden flow.

From Fig. 4, the logarithmic coordinate plot, it can be seen that error in unstable conditions is unexpectedly smaller than in near-neutral conditions. $\sigma$ has a straight line tendency in the logarithmic graph and the log law equation always has the minimum error.

The presence of sand particles does not significantly affect the error, as there is no significant difference of RSME between the sand-laden flow and clean air scenarios. The error of two empirical formulas is consistently higher than the log law equation.

The conclusion remains in the sand-laden flow. The log-law expression has the least error in all cases. Strong unstable conditions have smaller errors than weak unstable conditions. The PM10 concentration’s influence is not remarkable (Fig. 5).

Fig. 4.

RMSE of different expressions. Different color and different marker styles represent different equations’ RMSE. The straight lines denotes the trends.

Fig. 5.

RMSE of sand-laden flow. The color and size represent the value of RMSE. A larger size indicates that RMSE is larger.

3.4 Slope of the Potential Temperature Profile

This section shows the dependents of $\kappa _\theta$ on different parameters.

We discover that the logarithmic profile remains in even strong unstable conditions when the buoyancy term plays an important role in turbulence. Different stability conditions mainly impact the slope of the mean temperature profile. The slope of the profile equals $\theta _\tau /\kappa _\theta$. The Kármán constants are power functions of $-\delta _\tau /L$, $-u_\tau ^2/gL$, $-(\nu ^2/g)^{1/3}/L$ and $-(\alpha ^2/g)^{1/3}/L$ from Fig. 6. The abscissa employed distinct dimensionless characteristic lengths from the boundary layer flow.

Based on the fitting results of Fig. 6, we have:

$$\begin{aligned} \kappa _\theta =&C_1 \left( -\frac{\nu }{u_\tau L} \right) ^{0.251}\approx C_1 \left( -\frac{\nu }{u_\tau L} \right) ^{4/15} , \end{aligned}$$

(10a)

$$\begin{aligned} \kappa _\theta =& C_2 \left( -\frac{u_\tau ^2}{g L} \right) ^{0.642}\approx C_2 \left( -\frac{u_\tau ^2}{g L} \right) ^{2/3}, \end{aligned}$$

(10b)

$$\begin{aligned} \kappa _\theta =& C_3 \left( -\frac{\nu ^{2/3}}{g^{1/3} L} \right) ^{0.347} \approx C_3 \left( -\frac{\nu ^{2/3}}{g^{1/3} L} \right) ^{1/3},\end{aligned}$$

(10c)

$$\begin{aligned} \kappa _\theta =& C_4 \left( -\frac{\alpha ^{2/3}}{g^{1/3} L} \right) ^{0.338} \approx C_4 \left( -\frac{\alpha ^{2/3}}{g^{1/3} L} \right) ^{1/3} , \end{aligned}$$

(10d)

where $C_{1,2,3,4}$ are constants. As Table 1 shows, $\nu$ and $\alpha$ vary slightly at different time intervals and can be approximated as constants.

Equation (10b) and Eq. (10c) can be rewritten as:

$$\begin{aligned} u_\tau =g^{1/2} (-L)^{1/2} \left( \frac{\kappa _\theta }{C_2}\right) ^{3/4}, \quad \nu =g^{1/2} (-L)^{3/2} \left( \frac{\kappa _\theta }{C_3}\right) ^{9/2}. \end{aligned}$$

(11)

Substituting them in Eq.(10a), we get:

$$\begin{aligned} C_1=\frac{C_3^{6/5}}{C_2^{1/5}}, \end{aligned}$$

(12)

which shows that the scalings are consistent. The Kármán constant in the sand-laden situation has little difference with clean air conditions. A larger PM10 give rise to a larger bias from the trend line. The sand-laden data has same slope but larger $\kappa _\theta$ in the logarithmic plots than clean air data. A rough estimate gives that $\kappa _\theta$ in sand-laden data is 20–40% higher compared to clean air. However, the number of sandstorm data point is quite small. To verify the conclusion, more data from different experiment condition should be used for verification in the future.

Fig. 6.

The velocity and potential temperature Kármán constant of clean air and sand-laden flow. The clean air data markers are blue and the sand-laden data markers are red. Larger red circles represent larger PM10 concentrations. The dotted lines are the trend lines calculated by logarithmic polynomial fitting. Different horizontal coordinates are used in the subfigures.

4 Conclusion and Discussion

We verified the logarithmic law remains valid even in the presence of unstably stratified atmospheric conditions by investigating QLQA data with and without sand. By studying the similarity functions, potential temperature profile, root-mean-square error and slope of the profiles, we find that when buoyancy effects become significant, the empirical equations derived from the Monin-Obukhov similarity theory do not exhibit superior performance compared to the logarithmic law. We focus on the impact of stability parameters on the Kármán constant of the potential temperature profile, $\kappa _\theta$. A power expression of $\kappa _\theta$ about $u_\tau$ and Obukhov length L is found. $\kappa _\theta =C_1 \left( -\frac{\nu }{u_\tau L} \right) ^{4/15}$ , $\kappa _\theta =C_2 \left( -\frac{u_\tau ^2}{g L} \right) ^{2/3}$ and $\kappa _\theta =C_3 \left( -\frac{\nu ^{2/3}}{g^{1/3} L} \right) ^{1/3}$. Flows containing sand have approximately $20-40\%$ higher $\kappa _\theta$ values compared to sand-free flows.

Due to the limited data, we do only provide qualitative impact of sand on the mean temperature profile, and the quantitative effect should be studied in the future. Also, we need theoretical explanations to the scalings for the Kármán constant of the potential temperature profile.