Introduction

Despite the pollutant emissions that have been witnessed significant reduction in recent years, there are still heavy fine particulate matter (PM2.5) and ozone (O3) pollution events in some regions of China (Chen et al., 2022; Qin et al., 2021). Qin et al. (2021) reported that the occurrence of PM2.5-O3 double high pollution (DHP) events in China’s major city clusters were driven by a combination of the emissions of pollution sources and regional meteorological factors. It clearly implies that an enhanced understanding of the cross-correlation between high PM2.5 and O3 in a city area has a primary significance to formulating regional targeted PM2.5-O3 DHP coordinated control strategies.

PM2.5 and O3 not only have common precursors nitrogen dioxides (NOx) and volatile organic compounds (VOCs), but also interact with each other through a variety of atmospheric photochemical pathways, causing extremely complex non-linear responses (Chen et al., 2019a; Ding et al., 2013; Liu et al., 2023; Wu et al., 2021b). Firstly, PM2.5 can influence the formation and thickness of cloud by acting as a cloud condensation nuclei and enhance atmospheric extinction capacity to alter the photolysis rate, thus indirectly affect the formation of O3. For instance, Zhu et al. (2019) discovered the strong negative correlation between PM2.5 and O3 in northern China in winter, due to the influence of PM2.5 on photolysis rate. Chu et al. (2020) also reported that the increase in actinic flux owing to the reduction of air particles, leads to the rising of O3 levels in China. Secondly, O3 can enhance the atmospheric oxidation capacity and thus produce photochemical oxidants such as hydroxyl (·OH) radicals, hydrogen peroxide (H2O2) and aldehyde (R-CHO). These oxidants can result in the rapid nucleation of secondary aerosol particles and boost the explosive growth of PM2.5. For instance, Jia et al. (2017) analyzed the interaction between PM2.5 and O3 in Nanjing in different seasons and found that high O3 levels in hot season could accelerate the formation of secondary particles. Fu et al. (2020) revealed that atmospheric oxidizing capacity enhanced by high O3 levels, played a key role in the conversion efficiency of NOx into nitrate (the main component of PM2.5 in North China Plain). At last, some other studies highlighted that the heterogeneous reactions of chemicals on particle surface also affect the formation and consumption of O3 (Qu et al., 2018; Xu et al., 2012). Thus, the complexity of the interaction mechanism between PM2.5 and O3 and the effectiveness of PM2.5-O3 DHP coordinated control in city clusters are closely linked.

Due to the temporal variability of meteorology and pollution emissions, the time scale dependence can be observed in the dominant mechanism of the cross-correlation between PM2.5 and O3. The photochemical mechanism by which photochemical oxidants such as O3 contribute to the rapid nucleation of secondary aerosol particles occurs mainly on the time scale of seconds to hours. For example, Wang et al. (2014) found that the explosive growth of PM2.5 levels in Beijing-Tianjin-Heibei mainly occurred at the time scale of several hours. At the same time, the photochemical mechanism by which particulate matter affects O3 formation by scattering solar radiation occurs on daily, weekly, monthly and even longer time scales. For example, Xing et al. (2017) studied the impact of aerosol direct effects (ADEs) on tropospheric O3 in China and found that ADEs led to a decrease in the average O3 concentrations at time scale up to 1 month. In addition, the correlation between PM2.5 and O3 is also affected by the selection of research period. For example, Zhao et al. (2018) pointed out that the PM2.5 was positively correlated with the daily maximum of O3 at a 3-month scale in summer (from June to August 2016) and negatively correlated with it at a 3-month scale in winter (from December 2016 to February 2017). Conversely, Chou et al. (2011) discovered an inverse correlation between PM2.5 and O3 at the time scale of nearly 1 month during the Beijing Olympic Games in August 2008. During the city-lockdown period from January 23 to February 13, 2020 in northern China, a positive correlation between O3 and PM2.5 at the time scale of 22 days was observed by Le et al. (2020). These ambiguous conclusions, which show both positive and inverse correlation between PM2.5 and O3, are largely related to the sampling periods. Finally, several studies have showed that the correlation between PM2.5 and O3 is also affected by meteorological factors (Chen et al., 2019b; Liu & Shi, 2021). For example, Chen et al. (2017) showed that the correlation coefficients of O3 and five other conventional pollutants (including PM2.5) were less than 0.4 in warm seasons, while they were weak or negatively correlated in cold seasons. Therefore, it is essential to investigate the differences in the correlation between PM2.5 and O3 and evolution over time scales in different climate zones.

It is generally believed that the differences in regional pollution emission will interfere with the assessment of the impact of meteorology on air pollution. In 2020, with the outbreak of the COVID-19, the Chinese government adopted a series of nationwide compulsory measures such as city closure, travel restriction and plant shutdown to control the spread of the epidemic. As a result, primary pollutants were greatly reduced, which led to a similar initial atmospheric condition in different regions (Bherwani et al., 2021; Gautam, 2020; Gautam et al., 2021). Kumar et al. (2022) compared the air pollutant levels during the Christmas and New Year celebration in 2019, 2020 and 2021 and found lower total concentration of air pollutants than the previous year when there was no pandemic situation in India. Such confinement served as a natural experiment for scholars to accurately evaluate the impacts on the formation of regional pollution. For example, Chelani and Gautam (2022) studied the correlation between PM2.5 and NO2 concentrations and indicated that the inherent temporal characteristics of the short-term air pollutant concentrations (APCs) time series do not change even after withholding the emissions. However, current studies have mainly considered the variations in pollutant concentrations during COVID-19. The cross-correlation between secondary pollutants PM2.5 and O3 has rarely been studied.

Due to the non-linearity and complexity of evolution of secondary pollutants, existing atmospheric chemistry models are hard to meet the requests of high accuracy for research data and may result in deviations in the simulation of air quality under extreme conditions (Liu et al., 2020). In order to eliminate the misleading or non-robust effects caused by non-stationarity of nonlinear series, Peng et al. (1995) first proposed detrended fluctuation analysis (DFA), which is suitable for investigating the long-term auto-correlation of non-stationary time series. Further, Podobnik and Stanley (2008) proposed a detrended cross-correlation analysis (DCCA) method suitable for studying the long-term persistence of cross-correlations between two non-stationary time series. This method can effectively avoid the pseudo-correlation between the series due to non-stationary data, and analyze the time scale characteristics of the interaction between two groups. DCCA has been efficiently applied in the analysis of the interaction between pollutants at multiple time scales. For example, Shi (2014) used the DCCA method to reveal that the relationship between ambient dioxins and precipitation (or PM10) manifested a long-term cross-correlation at the time scale ranging from 1 month to 1 year. Du et al. (2021) applied DCCA method to report that the high-frequency mode of NO2 between urban area and scenic spots in Zhangjiajie at different time scales showed long-term cross-correlation at 24-h time scale. He (2016) used DCCA method to find that there was a long-term cross-correlation between pollutants and meteorology at a 10-year time scale, and it was more obvious in rural areas. DCCA method has been widely used in some other fields (Chen et al., 2018a; Liang et al., 2017; Rohit & Mitra, 2018; Yuan & Fu, 2014), and its scientificity and applicability have been proved by many scholars.

Taking Beijing-Tianjin-Heibei (BTH) and Pearl River Delta (PRD) as the study areas, this study establishes a new VM-DCCA method based on DCCA method to extract high pollutants, thus to compare time scale characteristics of the cross-correlation between high PM2.5 and O3. The primary objective is to consider the spatiotemporal evolution of the interaction between these two pollutants and attempt to provide a quantitative basis of devising sustainable coordinated control strategies of high PM2.5 and O3 in city clusters in different climate zones.

Materials and methods

Study areas

Both the Beijing-Tianjin-Hebei (BTH) and the Pearl River Delta (PRD) are the major city clusters in China. The BTH (39°28′N-41°05′N, 115°20′E-117°30′E) is located in the north of North China Plain, including two municipalities (Beijing and Tianjin) and 11 prefecture level cities (Baoding, Tangshan, Langfang, Shijiazhuang, Qinhuangdao, Handan, Zhangjiakou, Chengde, Cangzhou, Xingtai and Hengshui). This region belongs to temperate continental monsoon climate, with high temperature and rainy in summer due to the impact of the ocean water vapor and cold and dry in winter due to the impact of the cold inland air (Sun et al., 2021). The PRD (21°31′N-23°10′N, 112°45′E-113°50′E) is located in the south central part of Guangdong Province, China, including 9 cities (Dongguan, Foshan, Guangzhou, Huizhou, Jiangmen, Shenzhen, Zhaoqing, Zhongshan and Zhuhai). This region belongs to subtropical monsoon climate, with abundant rainfall and heat throughout the year. The rain and heat are in the same season, and the four seasons are relatively evenly distributed (Li et al., 2018). The total GDP of BTH and PRD were 8.69 and 8.95 trillion, accounting for about 8.54% and 8.77% of the national GDP respectively by 2020. There are a large number of advanced manufacturing bases and factories with global influence and developed industries in the two regions, which lead to intensive energy consumption and pollutant emission in the two regions, bringing serious atmospheric pollution to the local (Chen et al., 2018b; Hu et al., 2021).

Data source

According to the National Emergency Response Plan for Public Health Emergencies, control measures were implemented throughout the country on around January 24, 2020, and all work and production activities resumed in early May. In this study, January 24 to May 31, 2020 are set as the COVID-19 period and the same period in 2019 is set as the non-COVID-19 period for comparison. Hourly observations of PM2.5 and O3 are obtained from the China National Environmental Monitoring Centre (http://www.cnemc.cn/). Due to instrument calibration, power failure and some other factors, it is inevitable to cause the lack of individual data, accounting about 0.1%. These lacking data have been replaced by the average of the two adjacent data. Besides, daily meteorological data including minimum temperature (T-Min), maximum temperature (T-Max), precipitation (Pre), relative humidity (HR), wind speed (WS) and atmospheric pressure (P) are obtained from the China Meteorological Administration (http://cma.gjzwfw.gov.cn/). The statistical analyses of these meteorological factors are shown in Fig. 1 (For each box, the upper and lower boundaries of the box represent the 75th and 25th percentile respectively, the horizontal line in the box represents the median, the beard represents the minimum and maximum respectively, and the square represents the average, which has been marked). According to Fig. 1, there are no significant differences between BTH and PRD in WS and P, while the other four meteorological factors are completely different. During the non-COVID-19 period, the T-min, T-max, Pre and HR in PRD are about 14.59℃, 8.75℃, 6.39 mm, 38.3% higher than BTH respectively, and about 13.15℃, 8.24℃, 4.39 mm, 25.9% higher respectively during the COVID-19 period. In this way, the great differences in meteorological factors between the BTH and PRD are conducive to the comparative analysis of the impacts of different meteorological factors on compound atmospheric pollution.

Fig. 1
figure 1

The statistics of meteorological factors during the non-COVID-19 period and COVID-19 period in BTH and PRD

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Methods

Detrended cross-correlation analysis based on maximum time series of variable time scales

Compared with traditional method for correlation analysis, the advantage of DCCA related methods is that the scales characteristic of correlations can be quantified. Piao and Fu (2016) compared the Pearson method with DCCA method and reported that DCCA related methods can quantify scale-dependent correlations and are robust to contaminated noises and amplitude ratio, but not Pearson method. The calculation process is as follows.

Firstly, the series \(\left\{ {x_{{\text{k}}} } \right\}\) and \(\left\{ {y_{{\text{k}}} } \right\}\) are calculated as follows:

$$x_{k} = \sum\limits_{t = 1}^{k} {\left( {x_{{\text{t}}} - \overline{x}} \right)} \;{\text{and}}\;y_{k} = \sum\limits_{t = 1}^{k} {\left( {y_{t} - \overline{y}} \right)} ,\;k = 1,2, \cdots ,S$$
(1)

where \(\left\{ {x_{{\text{t}}} } \right.,t = 1,2, \cdots ,\left. S \right\}\) and \(\left\{ {{\text{y}}_{{\text{t}}} } \right.,t = 1,2, \cdots ,\left. S \right\}\) are the original time series, \(\overline{x}\) and \(\overline{y}\) are the averages of them, respectively.

Secondly, the above series \(\left\{ {x_{k} } \right\}\) and \(\left\{ {y_{k} } \right\}\) are divided into equal segments with length \(s\) (\(s\) represents the length of time scale), the least square method is used for linear fitting of each segment, thus, the local trend of data fluctuation at a specific time scale \(s\) is obtained. Then the trend of each data segment is combined as local trend \(x_{k,t}\), \(y_{k,t}\).

Thirdly, the trend signal is subtracted from the cumulative signal to obtain the residual signal, and the covariance of each residual signal is calculated as follows:

$$\mathop f\nolimits_{DCCA}^{2} (s,t) = \frac{1}{s - 1}\sum\limits_{k = t}^{t + s} {(x_{t} - \tilde{x}_{k,t} )} (y_{t} - \tilde{y}_{k,t} )$$
(2)

The covariance of the whole time series is

$$F(s) \equiv \sqrt {\sum\limits_{t = 1}^{N - n} {[\frac{1}{s - 1}\sum\limits_{k = t}^{t + s} {(x_{k} - \tilde{x}_{k,t} )} (y_{k} - \tilde{y}_{k,t} )]} }$$
(3)

Finally, repeat the above steps for each time scale \(s\) to obtain logarithmic relationship curves of \(F(s)\) and \(s\). If \(\ln s\sim \ln F(s)\) satisfy a linear relationship, a power-law relationship \(F(s) \propto s^{\alpha }\) exists. A linear fit is performed at this point, and the slope is the DCCA exponent \(\alpha\).

For a specific time scale, \(\alpha\) quantitatively represents the cross-correlation between the two groups of non-stationary time series. \(\alpha > 0.5\) represents that there is a lasting positive cross-correlation in a power-law form between the two groups of time series, that is, there is a trend of increase (or decrease) in a group of time series that will be followed by the same trend in the other group of time series at a certain time scale. The larger the \(\alpha\), the stronger the positive cross-correlation between the two groups of time series. \(\alpha < 0.5\) indicates that there is a continuous negative cross-correlation between the two groups of non-stationary time series variables, and the specific meaning is opposite to that when \(\alpha > 0.5\). \(\alpha = 0.5\) indicates that there is no cross-correlation between the two groups of time series.

Due to the adverse effects of extreme pollution (Kumar et al., 2023), the correlation between high PM2.5 and O3 should be further explored. The formation of high pollution strongly relies on the time scale, but the traditional DCCA method cannot identify the correlation between high pollutants at different time scales. So the simultaneous series of high PM2.5 and O3 at different time scales should be obtained first.

This paper applies the method proposed by Muchnik et al. (2009) to obtain the series of high pollutants. Specifically, Fig. 2a shows typical hourly average concentrations series of PM2.5 and O3 for 48 sequential hours. They are divided into several segments with interval length of 4 h, and the maxima of each interval are marked. Thus, Fig. 2b represents the maximum sequences of PM2.5 and O3 at a 4-h interval. Similarly, by changing the length of the period, maximum sequences of PM2.5 and O3 at different interval length can be obtained. At last, the VM-DCCA exponent \(\alpha_{VM}\) can be further obtained by the same steps of DCCA, and the meaning of \(\alpha_{VM}\) is similar to \(\alpha\). \(\alpha_{VM}\) varies with the interval length, which reflects the cross-correlation between high PM2.5 and O3 at different time scales.

Fig. 2
figure 2

Maximum sequences of PM2.5 and O3 obtained from original sequence with interval length of 4 h

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Self-organized criticality

Self-organized criticality (SOC) theory was proposed by Bak et al. (1987), which explained the dynamic mechanism of the power-law distribution characteristics of complex systems. In general, a complex system consists of many cells that have short-term correlation. Under certain conditions, the system spontaneously evolves and reaches the critical state. At this state, a slight perturbation would lead to large-scale events through spatial association between each cell, and the “frequency-size” distribution of these events obeys a power-law relation.

The “sandpile model” is used to illustrate the physical significance of SOC theory. Individual grains are dropped and gradually form a small pile, and then simulate and calculate how many grains will slide when a grain falls. In the initial stage, the falling grains just add to the growing pile, but as the grains accumulate to a certain extent, it stops growing and reaches a critical slope in a statistically stationary state. At this moment, the result of adding grains is unpredictable. The addition of grains may cause either a small perturbation or even trigger a large avalanche. Statistical analysis suggests that the frequency of avalanches is a power-law function with the size of the pile in the long-term grain adding process.

Results

Concentrations variations of PM2.5 and O3 during the COVID-19 period

Figure 3 shows the variations of PM2.5 and O3 during the COVID-19 period compared with the non-COVID-19 period. For BTH, PM2.5 in most cities shows a downward trend, ranging from 4.72 to 32.34%, and shows a slight increase in a small number of cities. By contrast, O3 in most cities shows an upward trend, ranging from 1.25 to 6.36%, and declines slightly in a small number of cities. For PRD, PM2.5 in all cities decreases significantly, ranging from 7.78 to 32.44%. Except Shenzhen, O3 in other cities shows an upward trend, ranging from 2.60 to 33.99%. This result shows that PM2.5 decreased while O3 increased in most cities due to the impact of COVID-19. The rise in O3 is not only related to the decrease in PM2.5, but is also influenced by the decrease in the concentration of its precursors NOx, which reduced the titration of O3.

Fig. 3
figure 3

Variations of PM2.5 and O3 in each cities of BTH and PRD during the COVID-19 period compared with the non COVID-19 period

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Figure 4 shows the comparison of variances above of PM2.5 and O3 in BTH and PRD. In general, there is no significant difference in concentrations variations of PM2.5 between BTH and the PRD. PM2.5 decreases by 14.09% and 16.30% in BTH and PRD respectively. Comparatively, the increment of O3 in PRD is much higher than that in BTH. The average O3 concentrations increase by 1.72% in BTH, but 19.39% in PRD. This result suggests that there is obvious regional difference in the responses of PM2.5 and O3 to the reduction of primary emission. The increase of O3 in the PRD is much larger than that in BTH, which may be influenced by meteorological differences between two regions.

Fig. 4
figure 4

Averaged variations of PM2.5 and O3 in BTH and PRD during the COVID-19 period compared with the non-COVID-19 period

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Long-term cross-correlation between PM2.5 and O3 during the COVID-19 period

In order to explore the long-term cross-correlation between PM2.5 and O3 in BTH and PRD, taking Beijing and Guangzhou as examples, the DCCA result during the COVID-19 period is shown in Fig. 5. As shown in the figure, there is a linear relationship between the covariance fluctuation function \(F(s)\) and time scales \(s\) with logarithm. The \(\alpha\) values are 0.84 and 0.89 in Beijing and Guangzhou respectively, indicating that the cross-correlation between PM2.5 and O3 during the COVID-19 period shows strong long-term persistence characteristics.

Fig. 5
figure 5

DCCA plot of PM2.5 and O3 in Beijing and Guangzhou during the COVID-19 period

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In order to verify whether DCCA method truly reflects the long-term persistence characteristics of the cross-correlation between non-stationary series, this paper applies the same way to analyze the random shuffled series. According to Markov process theory, if the time series \(x_{i}\) and \(y_{i}\) of the two pollutants are completely random, the correlation function will decay exponentially with time, and the correlation coefficient is 0 at a specific time. In this paper, the original PM2.5 and O3 are randomly shuffled and then analyzed by DCCA, which show an obvious power-law relationship of the two cities at the whole time scale, and its \(\alpha\) values are 0.53 and 0.51 respectively, which indicate that there is no correlation between the randomly shuffled series. The comparative result shows that DCCA method can effectively reveal the long-term persistence of the cross-correlation between original PM2.5 and O3 series at a long time scale.

Based on the above analyses, DCCA calculation of PM2.5 and O3 concentration series during COVID-19 and non-COVID-19 period in BTH and PRD is further carried out. It can be seen from Fig. 5 that the \(\alpha\) values of BTH and PRD fluctuate between 0.78 and 0.99, which means the correlation between PM2.5 and O3 series in each city shows strong long-term persistence. Besides, the \(\alpha\) values during the COVID-19 period is lower than that during the non-COVID-19 period in most cities. Table 1 shows the statistical analysis of \(\alpha\) values during the COVID-19 and non-COVID-19 period in the two regions.

Table 1 The main statistical values of DCCA exponents \(\alpha\) during COVID-19 period and non-COVID-19 period in BTH and PRD
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The Shapiro–Wilk test is performed to test the normality of \(\alpha\) values series. The results show that the \(\alpha\) values obey the normal distribution. The T-test is used to compare the difference of \(\alpha\) values between the two periods, which show that the P values of BTH and PRD are 0.02 and 0.02 respectively, which are less than the set significance level (0.05). Therefore, the null hypothesis that there is no significant difference in \(\alpha\) values between the COVID-19 and non-COVID-19 period is rejected, that is, the \(\alpha\) values are significantly different during the two periods. As shown in Fig. 6, the \(\alpha\) values of most cities during the COVID-19 period decrease significantly compared with that in the non-COVID-19 period. The average \(\alpha\) values are 0.91 and 0.85 respectively in BTH (decreasing by 4.40%), while are 0.87 and 0.83 respectively in PRD (decreasing by 2.35%). The results indicate that the long-term persistence of PM2.5-O3 correlation has weakened during the COVID-19 period in both regions.

Fig. 6
figure 6

Comparison of DCCA exponents \(\alpha\) of PM2.5 and O3 between COVID-19 period and non-COVID-19 period in each cities of BTH and PRD

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Long-term cross-correlation between high PM2.5 and O3 concentrations

The spatiotemporal distribution of PM2.5 or O3 high pollution events, in which the daily average concentration of PM2.5 is above 75 \(\mu g/m^{3}\), and the daily maximum 8-h average ozone (MDA8 O3) concentration is above 160 \(\mu g/m^{3}\), in BTH and PRD during the non-COVID-19 period and COVID-19 period are compared at first. As is shown in Fig. 7, the frequency of high pollution events in BTH is higher than that in PRD. Specifically, for PM2.5, the frequency of high pollution events is higher during the non-COVID-19 period. For O3, the frequency of high pollution events is higher during the COVID-19 period.

Fig. 7
figure 7

Days of high pollution in BTH and PRD during the non-COVID-19 period and COVID-19 period

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Further, taking 1 h (original concentration series), 4 h, 8 h, 12 h, 16 h, 20 h, 24 h and 28 h as interval length, the cross-correlation between high PM2.5 and O3 is analyzed by VM-DCCA method. Its VM-DCCA exponents \(\alpha_{VM}\) varying with the intervals are shown in Fig. 8.

Fig. 8
figure 8

Variations of average VM-DCCA exponents \(\alpha_{VM}\) with interval length in BTH and PRD during the non-COVID-19 period and COVID-19 period

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It can be seen from Fig. 8 that \(\alpha_{{{\text{VM}}}}\) values during the non-COVID-19 period are always higher than that during the COVID-19 period at different interval length both in BTH and PRD. It indicates that the long-term cross-correlation between high PM2.5 and O3 during the COVID-19 period is weaker than that in the non-COVID-19 period. Secondly, the \(\alpha_{{{\text{VM}}}}\) values in BTH are always higher than that in PRD, which illustrates that the long-term cross-correlation between high PM2.5 and O3 in BTH is stronger than that in PRD. Moreover, with the increase of interval length, \(\alpha_{{{\text{VM}}}}\) values in BTH fluctuate between 0.91 and 1.0 during the non-COVID-19 period, between 0.85 and 0.90 during the COVID-19 period, while those in PRD continuously decreases from 0.85 to 0.65 during the non-COVID-19 period, from 0.83 to 0.64 during the COVID-19 period. It reveals that with the increase of interval length, the long-term cross-correlation of the two high pollutants weakened in PRD, while remain stable in BTH.

Discussion

The SOC theory is used to explain the source of long-term persistence in the evolution of complex systems from a perspective of macro integrity. In the previous studies, Shi and Liu (2009) explained the scale invariant structures in the long-term evolution of PM10, SO2 and NO2 in Shanghai based on SOC theory. Further, Shi et al. (2015) established a numerical sand pile model of pollutant evolution based on SOC theory to quantitatively explain the formation mechanism of stable long-term persistence of PM2.5 during a typical haze period in Chengdu. Besides, Chelani (2016) pointed out that the emergence of long-term persistence in the temporal evolution of air pollution can be absolutely explained by SOC theory.

The evolution of the cross-correlation between high PM2.5 and O3 is highly similar to the known SOC system. Firstly, the atmosphere is an open dissipative system in which primary pollutants emitted from human production and life provide material and energy. For example, primary pollutants NOx and VOCs generate PM2.5 and O3 through various atmospheric photochemical reaction pathways, a process analogous to the continuous input of sand grains into a sand pile. At the same time, some dissipative processes of these secondary pollutants through various pathways happen in atmospheric system. For example, high O3 enhances atmospheric oxidation and promotes photochemical reactions, which rapidly depletes O3. PM2.5 pollutants can be transported on microscopic scales by diffusion or convection through rainfall deposition or wind diffusion, etc. This process is analogous to sand grains slipping in a sand pile. Secondly, pollutant components in the atmosphere can form short-range interactions through a variety of physical and chemical mechanisms. O3 can influence the formation of NO3−, SO42− and secondary aerosols by affecting the concentration of various oxidants. Correspondingly, PM2.5 can affect O3 formation directly or indirectly by influencing atmospheric dynamics, photolysis rates, cloud optical thickness and non-homogeneous reaction processes. This process is analogous to the mutual extrusion stress between sand grains. Thirdly, under stationary weather conditions, the interaction between PM2.5 and O3 will drive the entire atmospheric system to evolve spontaneously to a critical state and maintain it for a long time. Here, the short-term correlation between PM2.5 and O3 may cause a chain of forces involving the whole system, so as to evolve the long-term persistence. This process is analogous to the spontaneous evolution of a sand pile system to a SOC state due to the action of localised extrusion stresses between sand grains in a sand pile. Finally, in a critical state, the temporal correlation function of the atmospheric system to dissipation of external material or energy is power-law. The critical state of air pollution will continue to be locked as long as the external meteorological conditions of the stationary weather do not change, which is determined by the long-term persistent characteristics of the interaction between the pollution components (PM2.5 and O3) in the air pollution complex. This process is analogous to the power law distribution between the magnitude and frequency of sand pile collapse.

In Fig. 7, \(\alpha_{{{\text{VM}}}}\) shows lower values during the COVID-19 period and varies with the interval length in BTH and PRD, which can be explained by the SOC behavior. Driven by the input of primary and secondary pollutants, the cross-correlation of high PM2.5 and O3 experiences a stable internal dynamic process. In this case, external interference may destroy the short-term correlation between pollutants, thus affecting the long-term persistence of the cross-correlation between high PM2.5 and O3. At this time, critical state may be damaged and instability may occur. Regional meteorological factors and COVID-19 outbreak can be regarded as the external interference of atmospheric system. In general, higher temperature is accompanied by stronger solar radiation, resulting in stronger oxidation capacity and more active atmospheric photochemical reaction. In the PRD region, despite higher O3 concentrations, the rapid depletion of O3 itself and its precursors makes it difficult to maintain a lasting influence in the atmosphere. At the same time, higher RH and Pre may accelerate the coagulation and sedimentation of particulate matters, making it difficult for PM2.5 to maintain a lasting impact in the atmosphere. The rapid consumption of high PM2.5 and O3 results in low \(\alpha_{{{\text{VM}}}}\) values in PRD and decreases continuously with the increase of time scales. Besides, the increase of atmospheric oxidation capacity (AOC) caused by the rise of O3 levels during COVID-19 may lead to low \(\alpha_{{{\text{VM}}}}\) values. That is, the enhanced AOC accelerates the depletion of PM2.5 and O3 themselves and their precursors in atmospheric oxidation reactions, making it difficult to have lasting impacts in the atmosphere. Previous references have drawn some conclusions that can support this view. For example, Wang et al. (2020) found that the average O3 concentrations increased by 20.1 \(\mu g/m^{3}\) during the COVID-19 period in China. Huang et al. (2021) showed that the significant emission reduction of NOx during the COVID-19 promoted the increase of O3 and the formation of HNO3 free radicals at night, resulting in the increase of AOC. Qin et al. (2022) reported that the enhancement of AOC greatly affected PM2.5 and O3 and their relationship. Therefore, the long-term persistence of the cross-correlation between high PM2.5 and O3 during the COVID-19 period has weakened.

The greater the long-term persistence, the longer the cross-correlation between two pollutants. In this way, PM2.5 (O3) at a certain time will inevitably affect the formation of high O3 (PM2.5) at a certain time scale. This irresistible effect will increase the difficulty of coordinated control of PM2.5-O3 DHP in the future. The low \(\alpha_{{{\text{VM}}}}\) values in PRD are conducive to achieve the coordinated control of PM2.5-O3 DHP in some extent. Wu et al. (2021a) and Zhang et al. (2022) also found that PM2.5-O3 coordinated control is easier in Guangzhou, due to different meteorological factors, which is consistent with our results.

Therefore, the long-term persistence in PM2.5-O3 cross-correlation is closely related to the feasibility of coordinated control strategies. The relevant conclusions of this paper are helpful to the risk assessment and coordinated control of PM2.5-O3 DHP in different city clusters, which is of great significance in compound pollution prevention.

Conclusions

Based on the COVID-19 outbreak, this paper establishes a novel VM-DCCA model to study the long-term persistence of the cross-correlation between the high PM2.5 and O3 BTH and PRD.

The results show that the PM2.5 levels are observed to have a decrease in most cities while the O3 levels increased during the COVID-19 period, and the increase range of O3 in the PRD (19.39%) is significantly greater than that in BTH (1.72%). Secondly, the DCCA results indicate that the cross-correlation between PM2.5 and O3 shows strong long-term persistence. The average DCCA exponents \(\alpha\) of BTH is higher than PRD, and the non-COVID-19 period is higher than COVID-19 period. Finally, the VM-DCCA exponents \(\alpha_{VM}\) between high PM2.5 and O3 during the COVID-19 period are generally higher than non-COVID-19 period, and BTH are higher than PRD. The average VM-DCCA exponents \(\alpha_{VM}\) in PRD experience a significant downward trend with the increase of interval length, while remaining stable in BTH.

The above results showed that the spatiotemporal evolution of cross-correlation between high PM2.5 and O3 can be regarded as a typical SOC case. Based on the dynamic mechanism of SOC, the different response characteristics of PM2.5 and O3 pollutants in different regions during the COVID-19 period can be scientifically explained. The construction of VM-DCCA method related to high PM2.5 and O3 at different interval length is helpful to achieve coordinated control of pollutants.