Zhe Zhng , , Yu Shi , Hijion Sun , , Lei Liu , Fei Hu , ,

a State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

b University of Chinese Academy of Sciences, Beijing, China

Keywords:Turbulent heat diffusivity Counter gradient transportation Atmospheric boundary layer Urban pollution

ABSTRACT Turbulent diffusion efficiently transports momentum, heat, and matter and affects their transfers between the atmosphere and the surface. As a key parameter in describing turbulent diffusion, the turbulent heat diffusivity K H has rarely been studied in the context of frequent urban pollution in recent years. In this study, K H under urban pollution conditions was directly calculated based on K-theory. The authors found an obvious diurnal variation in K H , with variations also in the vertical distributions between each case and over time. Interestingly,the height corresponding to the high occurrence frequency of negative K H rises gradually after sunrise, peaks at noon, falls near sunset, and concentrates around 140 m during most of the night. The K H magnitude and fluctuation are smaller in the pollutant accumulation stage (CS) at all levels than in the pollutant transport stage and pollutant removal stage. Turbulent diffusion may greatly affect PM 2.5 concentrations at the CS because of the negative correlation between PM 2.5 concentrations and the absolute value of K H at the CS accompanied by weak wind speeds. The applicability of K-theory is not very good during either day or at night. These problems are inherent in K-theory when characterizing complex systems, such as turbulent diffusion, and require new frameworks or parameterization schemes. These findings may provide valuable insight for establishing a new turbulence diffusion parameterization scheme for K H and promote the study of turbulent diffusion, air quality forecasting, and weather and climate modeling.

1. Introduction

Turbulent diffusion plays a crucial role in the atmospheric boundary layer. It not only efficiently transports momentum, heat, and matter and affects their transfers between the atmosphere and land or ocean at scales ranging from local to global ( Stull, 1988 ; Baklanov et al., 2011 ;Holtslag et al., 2013 ) but also plays an increasingly prominent role in pollution events ( Li et al., 2017 ). As an important parameter used to describe turbulent diffusion, the distribution pattern of turbulent heat diffusivity Kis not clear. Few studies have investigated Kin a systematic way in the context of frequent urban heavy pollution in recent years. K-theory can be used to determine K. Specifically, a turbulent scalar flux, such as potential temperature, is proportional to its local average gradient, where the proportional coefficient is K, which is called gradient diffusion theory or small-eddy closure because of the gradient transport of small eddies (Stull, 1988) . However, large eddies can make this approach fail in many cases. In this or other complicated situations, negative Koccurs ( Stull, 1988 ; Garratt, 1992 ), indicating the emergence of counter gradient transportation (CGT), which is extremely challenging and has not been fully understood ( Zhou et al., 2018 ).

Johnson and Heywood (1938) calculated values of Kby assuming that it was constant over a thin layer, as specified by two successive instrumental heights on a wireless mast up to approximately 88 m. The distributions of Kwith height and time in the lowest 100 m were determined by assuming a relationship with the time-variant heat flux at the ground ( Jehn and Gerhardt, 1950 ). Wu (1965) studied

K

in the lowest 400 m of the atmosphere based on some assumptions. Estimates of

K

have been made by Wong and Brundidge (1966) , who found that the negative values around sunset are concentrated above 100 m, whereas those around sunrise extended over all levels. However, the study also made some assumptions and did not give a clear physical explanation.To eliminate the occurrence of negative

K

, many researchers have derived the counter gradient term based on different turbulent physics processes to correct K-theory ( Deardorff, 1966 , 1972 ; Holtslag and Moeng,1991 ; Zilitinkevich et al., 1999 ), which has become the most representative modification of K-theory. The physical meanings of the gradient and counter gradient terms in atmospheric boundary layer schemes have been studied ( Zhou et al., 2018 ). Analysis and verification have been carried out either by the indirect derivation of basic meteorological elements under certain assumptions or by large eddy simulation, which is usually based on a homogeneous underlying surface. The extreme lack of turbulent flux data limits the direct calculation and analysis of

K

.The purpose of this letter is to analyze the characteristics of

K

and its vertical distribution and temporal variation under urban pollution conditions, where

K

is directly calculated based on gradient diffusion theory and data collected from a 325-m meteorological tower. The conclusions of this paper will provide some insight for improving or establishing a new parameterization scheme for

K

and pollutant diffusion( Li et al., 2016 ) in the urban boundary layer and even for climate prediction models ( Holtslag et al., 2013 ). Finally, due to the universality of turbulent diffusion, this paper will also provide a reference for the study of diffusion problems in other fields. The rest of the study is organized as follows. The data and methods used to calculate

K

are described in Section 2 . Section 3 provides results including

K

and the characteristics of its vertical distribution and time variation. Conclusions and future work can be found in Section 4 .

2. Data and methods

2.1. Data

The data were collected from a 325-m meteorological tower built by the Institute of Atmospheric Physics (IAP), Chinese Academy of Sciences, in northcentral Beijing (39.97°N, 116.37°E). The tower is equipped with three-dimensional sonic anemometers at 7 levels (8, 16,47, 80, 140, 200, and 280 m) and anemometers and thermometers at 15 levels (8, 16, 32, 47, 65, 80, 100, 120, 140, 160, 180, 200, 240, 280, and 320 m). A more detailed description of the tower and instruments can be found in Liu et al. (2018) , Chen et al. (2018) , and Wang et al. (2019) and the quality control processing of the data and the special processing of turbulence are outlined in Vickers and Mahrt (1997) , Kaimal and Finnigan (1994) , and Lyu et al. (2018) . The Olympic Sports Center,an air quality station approximately 2.5 km from the tower, provides hourly mass concentrations of PM. More detailed information can be found on the website of China National Environmental Monitoring Center ( http://www.cnemc.cn ). The four heavy pollution episodes selected in this paper occurred during 3-8 November, 29 November-4 December, 26-31 December 2017, and 10-15 January 2018, during which the turbulence data were not affected due to all sunny or cloudy days.

2.2. Methods

To close the atmospheric turbulence equations, the simplest turbulent flux parameterization assumes that a turbulent scalar flux, such as the potential temperature

θ

, is proportional to its local mean gradient in an idealized environment of dryness, horizontal homogeneity, and quasi-stationarity:

Fig. 1. Diurnal variation in the average turbulent heat diffusivity K H at 8, 16,47, 80, 140, 200, and 280 m on the 325-m meteorological tower in Beijing,China.

Vertical gradients in the mean potential temperature can be calculated by differentiation using the following function of height:

where

d,

e

, and

f

are obtained by fitting vertical potential temperature data via the least squares method (Sorbjan, 1987) . The fitting effect of Eq. (2) on the potential temperature is better than that of the various fitting and interpolation methods we have tested because this equation is very good at describing global and local variations in the potential temperature profile. Sometimes sharp local trends in the potential temperature profile may not be captured, although they are rare or may be associated with fluctuations in the data. Moreover, this method may not work well for tens of meters above the ground, especially at night,where the complex structure and thermal characteristics of the urban surface result in an abnormal potential temperature profile. To address this issue this study, first, the moving average of five values of potential temperature data was calculated to eliminate small-scale fluctuations.Then,

K

was obtained by substituting the heat flux and potential temperature gradient into Eq. (1) . In addition, meaningless

K

values caused by a potential temperature gradient approaching 0 were removed.

3. Results and discussion

Fig. 1 depicts the diurnal variation in the average

K

at all levels during the observation period. The absolute values of

K

present an obvious diurnal variation, reaching a maximum in the daytime and a minimum at night. Wong and Brundidge (1966) also obtained

K

curves with the same trend for six cases in winter. Specifically, these values at all levels rise slowly around sunrise (0700 LST), then peak at approximately 1330 LST, and finally drop sharply at approximately 1500 LST ( Fig. 1 ).This pattern reflects the periodic response of turbulent activity in the atmospheric boundary layer to changes in high and low surface temperatures. Surprisingly, negative

K

occurs often near the ground, i.e., at the 140 m level at night and near the 280 m level during the day. The negative

K

indicates counter gradient diffusion according to Eq. (1) .This phenomenon will be further described and discussed later in this article. The absolute values of

K

in each case are often higher than those in Fig. 1 because the negative values offset the positive values in the averaging process. In addition to the above changes, the vertical distribution of

K

at different times and cases is shown in more detail in Fig. 2 .

Fig. 2. Vertical distribution of the average turbulent heat diffusivity K H for 23 cases under four heavy pollution episodes. The legend includes the local time.The colors of the vertical lines and the length of the horizontal lines represent the local time and the standard deviation of all cases, respectively.

In Fig. 2 , 12 selected instances in the day are divided into four parts(a, b, c, and d) to clearly show the average

K

vertical distributions.The vertical distributions at different instances (0100-0900 and 1700-2300 LST) during the night and early morning are similar and cluster around 0 but vary from during the day (0900-1500 LST; Fig. 2 (a-d)).This is caused by different turbulence intensities between the convective boundary layer in the daytime and the nocturnal boundary layer at night. The absolute

K

values at the same time increase with height during the day because turbulent diffusion is hindered more by a complex surface at lower levels than at higher levels ( Fig. 2 (b, c)). Significant differences in the distributions in different cases occur in the morning and increase with height (0900-1100 LST in Fig. 2 (b, c)). Variations in stratification in the nocturnal boundary layer may lead to divergence in the distribution and unclear patterns ( Fig. 2 (a, d)). Wong and Brundidge (1966) found similar results in near-surface layers over flat surfaces and attributed them to fluctuations in the lapse rate dependent on case and time. However, we argue that the more fundamental causes are the interactions among the randomness of turbulence itself, complex urban topography, and human activities. The presence of negative

K

also causes a large standard deviation in

K

in different cases ( Fig. 2 (a-d)),and we discuss the distribution of negative

K

in Fig. 3 .

Fig. 3. (a) Frequency distributions of negative turbulent heat diffusivity K H and(b) the average heat flux for 23 cases under four heavy pollution episodes.

Fig. 3 depicts the frequency distribution of negative

K

and the average daily heat flux variations for 23 cases. The height of negative

K

rises gradually after sunrise, reaches its peak at noon, and then falls near sunset. In particular, the frequency of negative

K

is unusually significant near sunrise and sunset ( Fig. 3 (a)). Jehn and Gerhardt (1950) and Staley (1956) also observed negative

K

at sunrise and sunset, and Wong and Brundidge (1966) also found that the negative values around sunset were concentrated above 100 m, whereas those around sunrise extended over all levels. However, that study did not provide a clear physical explanation. The high frequency of negative

K

is concentrated at approximately 140 m during most of the night ( Fig. 3 (a)).

In addition, the occurrence frequency of negative

K

below 50 m at night reaches approximately 0.9, which is different from the results obtained by Wong and Brundidge (1966) for a relatively flat and lower surface ( Fig. 3 (a)). This is probably related to the turbulence caused by the complex structure and thermal properties of a city surface. The stochastic volatility of the temperature data near the upper and lower boundaries eventually affects the slope of the fitted temperature, which is also an important reason for negative

K

near the lowest and highest levels.Many studies have studied only the correction of negative

K

without giving a clear physical explanation ( Zhou et al., 2018 ). We give only some tentative explanations for the occurrence of negative

K

or counter gradient diffusion. The applicability of K-theory is not very good during the day or at night. These problems are inherent in K-theory when describing complex systems, such as turbulent diffusion, and require new frameworks or parameterization schemes.

Fig. 4. Temporal variations in PM 2.5 (black lines) and the turbulent heat diffusivity K H (dots) during (a) 3-8 November 2017, (b) 29 November-4 December 2017,(c) 26-31 December 2017, and (d) 10-15 January 2018. The red dots represent K H at 8, 16, 47, and 80 m, and the blue dots represent K H at 140, 200, and 280 m.

Fig. 4 shows the temporal variations in the PMconcentrations and

K

at all levels during four heavy pollution events. Based on the characteristics shown in Fig. 4 and the pollution stages described in Zhong et al. (2017) , each six-day pollution event is divided into three stages: the pollutant transport stage (TS), pollutant accumulation stage(CS), and pollutant removal stage (RS). As described in Fig. 1 , the

K

values in Fig. 4 all exhibit significant diurnal changes. From the perspective of each of the four pollution events, the similar characteristics of PMand

K

changes suggest that the two may be closely related.Compared with the large variation range and fluctuation in

K

at all levels during the TS and the rapid increase in

K

with the rapid decline in PMduring the RS, the magnitude and fluctuation in

K

during the CS are significantly smaller. The average wind speed during the CS is weaker and the vertical variation is smaller than those at all levels during the TS and RS, indicating the weak clearing effect of the horizontal wind. The PMconcentration is negatively correlated with the absolute values of

K

during the CS in Fig. 4 . In fact,

K

and PMare related with a time lag because the influence of turbulence diffusion on the PMconcentration is not immediate but requires a certain response time. For example, when turbulence diffusion increases or decreases gradually, PMalso decreases or accumulates gradually. The correlation between PMand

K

is not significant because the PMconcentration can still increase due to emissions or transmission when

K

is small and nearly unchanged in the stable boundary layer at night.In addition, the diffusion of heat can reflect the diffusion of substances to a large extent because heat in the boundary layer is mainly transmitted by material mixing ( Lutgens and Tarbuck, 1991 ). The pollutant concentration during the CS may largely be affected by turbulent diffusion when other influencing factors remain unchanged.The atmospheric boundary layer schemes in models have difficulty distinguishing between diffusion of haze and clean days in the complex terrain region in China ( Li et al., 2016 ). Thus, the pattern of

K

may provide a direction for improvement in parameterization schemes of the atmospheric boundary layer and PMsimulations.

4. Conclusions

In this paper,

K

values under urban pollution conditions were directly calculated based on K-theory and data measured from the 325-m meteorological tower in Beijing, China, before analyzing the characteristics of

K

as well as its vertical distribution and temporal variation.

K

in the urban near-surface boundary layer shows obvious diurnal variations;

K

usually increases with height in the daytime and concentrates around 0 at night. The vertical distribution of

K

varies with case and time. Moreover,

K

varies with the stage of pollution. In particular, the height corresponding to the high occurrence frequency of negative

K

rises gradually after sunrise, peaks at noon, falls near sunset, and concentrates around 140 m during most of the night. The

K

magnitude and fluctuation in the CS are significantly smaller than those in the TS and RS at all levels. Turbulent diffusion may greatly affect pollutant concentrations in the CS because of the negative correlation between PMconcentrations and the absolute value of

K

during the CS when accompanied by a weak wind speed.The applicability of K-theory is not very good during the day or at night. These problems are inherent in K-theory when describing complex systems, such as turbulent diffusion, and require new frameworks or parameterization schemes. The findings of this study may provide some insight for improving or establishing new parameterization schemes for

K

and pollutant diffusion in the urban boundary layer and promote the research of turbulent diffusion, air quality forecasting, and weather and climate modeling. In the future, the mechanism of negative

K

and the distribution of

K

need to be further studied and explored under broader conditions and long-term observations. For example, we hope to study the pattern of the turbulent structure in the boundary layer and the relationship between

K

and other parameters, such as the Richardson number, to provide a better representation of turbulent processes in the atmospheric boundary layer.

Funding

This work was jointly supported by the National Natural Science Foundation of China [grant numbers 41975018 and 41675012 ] and the National Key Research and Development Program of China [grant number 2017YFC0209605 ].

Acknowledgments

We acknowledge the State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Science, for providing the turbulence data.We also thank Dr. Rui Lyu for his helpful advice.