A multivariate probabilistic framework for tracking the regional tropical edges: analysis of inter-annual variations and long-term trends

In the present study, a multivariate probabilistic framework is used to identify the meridional positions of regional tropical edges (RTEs), which are based on two variables: sea level pressure and precipitation minus evaporation. This new defined metric effectively captures inter-annual variability and long-term trend of the commonly adopted zonal mean tropical edge based on meridional mass stream function and near-surface winds. Besides, pronounced RTE trends are primarily located over the oceanic regions, and the terrestrial areas exhibit substantial inter-annual variability. These results are consistent among three modern reanalysis datasets. Moreover, the impacts of climate modes on RTE are investigated. The El Niño-Southern Oscillation, the Atlantic multi-decadal oscillation, and the Southern Annular Mode are important both on the inter-annual variations and long-term trends of RTE. The Pacific Decadal Oscillation is more inclined to affect long-term contribution rather than inter-annual relationship, and the Pacific–North American teleconnection, the North Atlantic Oscillation, and the Arctic oscillation highlight the inter-annual relationship with RTE in the specific regions, such as North Pacific, North Atlantic, and North Africa, respectively.


Introduction
The tropical belt is the region manifested by the Hadley circulation, with updraft and intense rainfall near the equator, as well as downdraft and arid zones in the subtropics of each hemisphere.In the past few decades, numerous studies have utilized observations and reanalysis data to discover the tropical expansion trend since the late 1970s (Fu et al 2006, Hu and Fu 2007, Seidel et al 2007, Seidel and Randel 2007, Fu and Lin 2011, Davis and Rosenlof 2012, Allen et al 2014, Davis and Davis 2018, Grise et al 2018, 2019, Ma et al 2018, Staten et al 2018, Lau and Tao 2020), accompanied by the poleward shift of the subtropical jet, subtropical high, and sinking branch of the Hadley circulation (Fu et al 2006, Archer and Caldeira 2008, Fu and Lin 2011, Grise et al 2018), as well as the poleward movement of subtropical arid zone (Dai 2011, Cai et al 2012, Allen et al 2014, He and Soden 2016, Huang et al 2016, 2017, Tivig et al 2020).However, the poleward shift of the tropical edge is unlikely to occur at all longitudes and exhibits significant regional features (Nguyen et al 2017, Grise et al 2018, Staten et al 2020).Additionally, from the perspective of climatology, the contribution of regional Hadley circulation to zonal mean Hadley circulation also varies with region.For example, taking the tropical zonal wind field as an example, its intensity varies in different regions, and even the northerly winds appear in some regions (Hoskins et al 2020, Hoskins andYang 2021).
Previous studies have pointed out that tropical expansion can cause a series of regional climate changes in the subtropics, such as shifting the dry zones (Feng andFu 2013, Schmidt andGrise 2017), altering the regions of oceanic upwelling (Rykaczewski et al 2015), modifying typhoon tracks (Kossin et al 2014, Rykaczewski et al 2015, Sharmila and Walsh 2018, Studholme and Gulev 2018, Anjana and Kumar 2023), and exacerbating wildfires (Zhang et al 2020).The changes of regional tropical edges (RTEs) may serve an important role in linking tropical expansion and regional climate impacts.
At present, there are still limited studies on the changes of RTE, owing to the difficulty in defining the RTE.Previous studies have made some attempts, but more works are needed.Chen et al (2014) selected six regions globally and studied the long-term trends of tropical edges in these regions by using an outward longwave radiation (OLR) based metric.Their results showed that the expansion was more obvious in the Northern Hemisphere (NH) than the Southern Hemisphere (SH), with a greater contribution from the eastern Pacific.However, the OLR based metric strongly depends on the selection of thresholds (Davis and Rosenlof 2012, Nguyen et al 2017), and OLR is not only affected by atmospheric circulation dynamics but also related to atmospheric thermal properties (Waugh et al 2018), leading to the considerable uncertainty in the obtained results.In addition, Schwendike et al (2015) attempted to use the improved mass stream function by applying the meridional component of the divergent winds instead of the conventional meridional winds at each longitude and examined the RTE at 500 hPa.According to this metric, the interannual variability of RTE over the Asia-Pacific sector dominantly governs that of the tropical edge in the SH (Nguyen et al 2017), and the long-term trend over the eastern Pacific showed the largest contribution in the recent tropical expansion (Staten et al 2019).Due to the use of mass stream function, this metric captures the dynamic characteristics of RTE and its close relationship with near-surface climate.However, on one hand, it is not clear whether this metric can be a true representation of the RTE (Hu et al 2018).On the other hand, the obtained edges are not continuous over the entire latitude circle and cannot represent the changes in certain specific regions.For the metric by using sea level pressure (SLP), the long-term changes of RTE were found to be mainly located in oceans, and the SLP based metric could not reflect edge changes on land (Schmidt and Grise 2017).Moreover, the results of the above three metrics in each region are not completely consistent (Chen et al 2014, Schmidt and Grise 2017, Staten et al 2019), indicating that the single metric has strong metric dependency.
Therefore, it is necessary to select more suitable variables and develop new metrics for better representing the location of RTE and exploring the possible influencing factors for RTE variations.The rest of the text is organized as follows: section 2 provides data and methods.Section 3 provides the main results, including the inter-annual variations and long-term trends of the new defined multivariate metric at zonal mean and regional scale, as well as the potential influence of climate modes on RTE.Section 4 provides a concluding summary.In the main body of this paper, the primary conclusions are drawn from ERA5, which has the highest resolution.The other two reanalysis datasets are employed for validation purposes, and the relevant results are shown in supplemental materials.

Metrics measuring the zonal mean tropical edge
From the perspective of zonal mean, there are various metrics to measure the meridional position of the tropical edge.The most used metrics are defined according to the position of vertical descent in the mid-layers of the troposphere between the Hadley circulation and Ferrel circulation, specifically the zerocrossing of meridional mass stream function in the subtropics at 500 hPa (PSI metric;Hu and Fu 2007), which holds clear dynamical implication.
The other metrics can be categorized into upperatmospheric and lower-atmospheric metrics.Upperatmospheric metrics include the latitude where OLR equals 250 W m −2 (Hu and Fu 2007) and the position of the tropopause break (Davis and Rosenlof 2012, Xian and Homeyer 2019).Lower-atmospheric metrics include the position where precipitation minus evaporation firstly reaches minimal or zero (PE metric;Lu et al 2008, Davis andBirner 2017), the latitude of maximum SLP (SLP metric;Hu et al 2010, Choi et al 2014), the latitude where the near-surface winds become westerlies (UAS metric; Davis and Birner 2017, Grise and Davis 2020), and the latitude of eddydriven jet (EDJ metric;Schneider 2006, Kang andPolvani 2011).Compared to the upper-atmospheric metrics, lower-atmospheric metrics more effectively capture the inter-annual variations and long-term trends of the PSI metric (Solomon et al 2016, Davis and Birner 2017, Waugh et al 2018).This is mainly due to the fact that lower-atmospheric metrics are more readily constrained by the momentum transport associated with the Hadley circulation, while upper-atmospheric metrics reside in the free atmosphere and primarily follow thermal wind relationships (Grise et al 2019, Staten et al 2020).Moreover, Waugh et al (2018) mentioned the forced response of the EDJ metric may substantially differ from direct measures of tropical width, such as the PSI metric.Thus, the present study primarily focuses on the rest three lower-atmospheric metrics based on PE, SLP and UAS.
The computational modules for zonal mean tropical edge calculations are derived from the Tropicalwidth Diagnostics software package (Adam et al 2018).These modules encompass functionalities dedicated to the determination of latitudinal points where zero-crossings, thresholds, and maxima of variables are satisfied.The statistical methods including the regression and correlation analysis are used in sections 3.1 and 3.2 to investigate the inter-annual and long-term characteristics of the new defined metric, while the partial regression and partial correlation analysis are used in section 3.3 to explore the influence of climate modes on RTE.The significance level is estimated based on the standard two-tailed Student's t-test.

A multivariate probabilistic framework
In the present study, we applied a multivariate probabilistic framework, which was first proposed by Mamalakis and Foufoula-Georgiou (2018) to precisely identify the regional features of the intertropical convergence zone.The framework relies upon statistical copula models, employing mathematical methods to access multiple variables simultaneously.The movement of the tropical edge is accompanied by the adjustments of various variables (Allen et al 2012, 2014, Birner et al 2014, Brönnimann et al 2015, Lau and Kim 2015, Lau and Tao 2020), and this method provides a more comprehensive approach to consider the interactions between different variables at fine-scale.
Based on Mamalakis and Foufoula-Georgiou (2018) and Mamalakis et al (2021), the location of the RTE at a longitude l under the multivariate probabilistic framework can be obtained through the following steps: (1) Select N variables (in this study, N = 2), and calculate annual average of these variables.Take a latitudinal window ([l − w/2, l + w/2], where w is the window width, and w = 15 • in this study, consistent with the latitudinal window width used in Choi et al (2014) and Totz et al (2018).
The zonal average of N variables is calculated within this window.
(3) Calculate the cumulative distribution function (CDF) of the copula function, and the latitude where its CDF exceeds a certain threshold a (in this study, a = 70%) is considered as the position of the RTE for that specific longitude l and year t.Noted that this result corresponds to the latitude where maximum SLP and minimum PE occur simultaneously.
Utilizing this framework, we can develop a new metric that considers the physical relationship among two or more variables simultaneously.Additionally, by varying the parameters w, N, and a, the sensitivity of the results for different problem scenarios could be tested.Note that the obtained position of the RTE exhibits a limited sensitivity to parameter selection.Slight adjustments to the parameters of w and a only influence the specific numerical values of the CDF and do not alter the major conclusions of the present study.

Selection of variables for RTE
Consistent with the zonal mean tropical edge, the location of RTE should be close to the areas that correspond with the downdraft and the arid zone (Staten et al 2019).Therefore, among the three loweratmospheric variables, a metric defined under the multivariate probabilistic framework to reasonably delineate the RTE is enabled: (1) the variable should well capture the maximum downward movements, as the minimum of PE (Davis and Birner 2017); (2) the variable should have a connection with the aridity, as the maximum SLP (Lau and Kim 2015).
A more informed selection process to discern the disparities of the three lower-atmosphere metrics is shown in figure 1  larger meridional differences over Central and West Asia and Australia.For maximum SLP, some high outliers are observed over North America and Central Eurasia, accompanied by high pressure caused by sealand thermal difference.Compared with maximum SLP, minimum PE tend to locate more equatorward.Note that although the spatial distribution of the zero-crossing of UAS roughly matches with that of maximum SLP, large differences can be observed over the North American and Asian monsoon regions with the other two metrics.Moreover, the definition of UAS metric derived from the balance between the vertically averaged eddy momentum flux convergence and surface drag on the zonal mean surface zonal wind (Davis and Birner 2017), but their regional-scale implications remain ambiguously.Thus, SLP and PE are used to construct the multivariate RTE metric, which is denoted as the SLPPE metric.
where V d is the meridional component of the divergent winds calculated by Helmholtz decomposition, a is the radius of Earth, and g is the gravitational acceleration.Consistent with previous results, the RTE obtained by the Ψ φ at 500 hPa are not latitudinally continuous and located between the maximum SLP and the minimum PE at most longitudes (figure 1(a)), further supporting the reasonable selection of these two variables.
The climatological joint CDF between SLP and PE is shown in figure 1(b), generally encompassing the RTE obtained by Ψ φ metric.This characteristic is further reproduced in MERRA2 and JRA55 (figure S1 in the supplemental material).Additionally, the RTE depicted by SLPPE are located between the maximum SLP and minimum PE both at regional (figures 1(a) and (b)) and zonal mean scale (figure 1(c)), and the high CDF values between SLP and PE correspond well with the maximum SLP and minimum PE (figure S2 in the supplemental material), indicating that the SLPPE metric effectively considers both of them.The advantage of SLPPE metric is that it not only allows to cover the location of RTE represented by the divergent winds but also the location where the divergent winds are not well determined.

Inter-annual variability and long-term trend of zonal mean edge
The reliability of the positions of RTE based on SLPPE metric under the probabilistic framework is first inspected for its consistency with the conclusions drawn from previous studies concerning zonal mean metrics at both inter-annual and long-term timescale.Considering the variables used for the copulas function, it is reasonable to infer the RTE defined by SLPPE metric exhibit a strong correlation with SLP and PE.Figures 2(a) and (b) show the time series of zonal mean positions of RTE based on the SLPPE metric, compared with the time series of positions of tropical edge based on the four zonal mean metrics based on PSI, PE, SLP, and UAS in the NH and SH, respectively.The temporal evolution of the SLPPE metric shares a similar inter-annual variation feature with the four zonal mean metrics both in the NH and SH.In particular, the correlation coefficients between the SLPPE metric and the four metrics exceed 0.42 except for the PSI metric in the NH (figure 2(c)), reaching the 99% confidence level.The lowest correlation coefficient between the SLPPE metric and PSI metric in the NH stands at 0.34, which is still significant at the 95% confidence level.
Besides, the poleward shifts of the tropical edge are observed both in the NH and SH (figures 2(a), (b) and (d)).In the SH, the expansion trend is remarkably consistent in all metrics, at approximately 0.2 degrees per decade.However, in the NH, there is a larger discrepancy of trends among metrics.The longterm trends of PSI and SLP metric exhibit comparatively weaker tendencies, whereas more significant tropical expansion trends can be seen in PE and UAS (Baldassare et al 2023).The trends of the SLPPE metric are notably stronger in both hemispheres compared to the other four metrics.This phenomenon may arise from the methodology employed for its calculation, due to that the time series of the SLPPE metric involves regional calculations first and then the zonal mean calculations, potentially excessively emphasizing the trends within certain regions.The time series of the SLPPE metric based on JRA55 and MERRA2 closely resemble those from ERA5, with correlation coefficients exceeding 0.7 both in the NH and SH (figure S3 in the supplemental material).Therefore, the SLPPE metric well captures the interannual variability and long-term trend of zonal mean tropical edge based on the four zonal mean metrics, and the inter-annual variability and long-term trend of RTE are further investigated in the next subsection.In the SH, the most pronounced poleward trends of RTE can be seen over the South Pacific, South Atlantic, South Africa, and southeastern Australia, at approximately 0.4 • -0.7 • per decade (figure 3(c)), contributing to the drying trend of SH semi-arid regions (Cai et al 2012).Note that the trends are not significant over 170 • W-130 • W of South Pacific due to the vigorous inter-annual variations (figure 3(c)).The poleward trend is around 0.15 • per decade over the South Indian Ocean.Similar to the RTE trend over North America, the RTE in South America also tends to move equatorward with strong inter-annual variability (figures 3(b) and (c)).

Inter-annual variability and long-term trend at regional scale
The RTE expansion over the central and eastern Pacific in the NH and SH are consistent with the results obtained by using Ψ φ (Staten et al 2019) and OLR metric (Chen et al 2014), and the poleward shift of RTE over the Atlantic in both hemispheres are also seen in the RTE trends of SLP metric (Schmidt and Grise 2017).Generally, the pronounced RTE trends are primarily located over the oceanic regions (Schmidt and Grise 2017), while the terrestrial areas exhibit substantial inter-annual variability.Furthermore, the spatial distribution depicting the linear trend of joint CDF between SLP and PE from JRA55 and MERRA2 are further inspected in the supplemental figure S4.The CDF trends from JRA55 and MERRA2 are highly close to those from ERA5 (figures 3(a) and S4), demonstrating that there is no data dependency in the results.2007, Son et al 2009, Polvani et al 2011a, 2011b, Min and Son 2013, Lucas and Nguyen 2015).These modes are also recognized as drivers of RTE change, with their own distinct geographical contributions.Besides, the North Atlantic Oscillation (NAO) and the Pacific-North American teleconnection (PNA) have the potential contributions to RTE change, as that they have significant variability and are important to regional climate (Loon and Rogers 1978, Horel and Wallace 1981, Hoerling et al 1997, Wang et al 2021, 2023, Li et al 2023a).To better understand their complex interplay, the global RTE are divided into 12 regions with six regions in each hemisphere according to the regional feature of inter-annual variability and long-term trend in section 3.2, and their interaction with the seven climate modes and the significant results are summarized in figure 4   The partial correlation coefficients between the seven climate mode indices and RTE locations over the six regions in the NH and SH are summarized in supplemental tables S2 and S3, respectively.In the NH, the ENSO, PNA, NAO, AO, and AMO emerge as prominent contributors, exerting substantial influence over multiple geographical regions (table S2 in the supplemental material).AMO has a noteworthy impact on the inter-annual variation of RTE over North America and the North Atlantic, and ENSO and NAO exert a pivotal role in the RTE over the North Atlantic with correlation coefficients at −0.36 and 0.38, respectively.Besides, the RTE over the North Pacific experiences a discernible effect from PNA, and AO highly correlates with the RTE over North Africa at 0.57.

Interaction with the climate modes
In the SH, ENSO exhibits substantial impacts, particularly over regions such as South Africa, the Indian Ocean, and the South Pacific (table S3 in the supplemental material).AMO shows high correlation coefficient with the RTE over the South Indian Ocean, South America, and South Africa.Additionally, the influence of the SAM is paramount in shaping the climate patterns in the SH, with pronounced effects observed in the South Pacific and Australia.Note that although the correlation coefficient between AO and RTE over the South Pacific is significant, this relationship reflects rather than a case of direct 'cause and effect' .Certain regions, such as the South Atlantic and Central Asia, are not significantly influenced by the seven modes.
The contribution of seven climate modes to the trends of RTE locations over the six regions in the NH and SH are summarized in supplemental tables S4 and S5, respectively.In the NH, ENSO plays a crucial role in driving the expansion trends of the RTE, with substantial contributions from the six regions except the North Pacific (table S4 in the supplemental material).Although PDO does not exhibit significant inter-annual correlations with the RTE in most regions, it considerably contributes to the RTE expansion.It is particularly evident in the North Pacific and North Atlantic, where also experience the most significant expansion (figure 3(b)).Additionally, no less than 10% of RTE trends in the six regions are modulated by AMO, and AO's impact mainly appears over North Africa.
Similar to the SH, ENSO, PDO, and AMO exhibit pronounced contributions to RTE trends of most regions over the SH (table S5 in the supplemental material).SAM is another important climate mode, which largely influences the RTE trends over Australia, the South Pacific, and the South Atlantic.These findings indicate that the relationship between climate modes and RTE depends on the geographical location, and the SLPPE metric provides a more convenient way to investigate their regional connections.

Conclusion
In this study, the RTE are investigated with two appropriate variables: maximum SLP and minimum PE, which are combined to form a new metric under the multivariate probabilistic framework.This SLPPE metric effectively captures inter-annual variations and long-term trends in the zonal mean tropical edge.The zonal mean positions of RTE based on the SLPPE metric closely correlate with the positions of tropical edge based on PSI and UAS metric, with the correlation coefficient no less than 0.34.Besides, the poleward shifts of the tropical edge are observed both in the NH and SH, but their expansion rates are larger than those of PSI and UAS metric.
Furthermore, the SLPPE metric offers a flexible approach for capturing inter-annual variations and long-term trends of RTE locations at regional scale.In the NH, the significant poleward shifts of RTE over the North Pacific and North Atlantic are observed, and the strong inter-annual variability of RTE are located at most regions of Eurasia and North America.In the SH, pronounced poleward RTE trends were evident over the South Pacific, South Atlantic, and South Africa, and the inter-annual variations of RTE are vigorous over the part of the South Pacific and South America.Therefore, the pronounced RTE trends are primarily located over the oceanic regions, while the terrestrial areas exhibit more substantial inter-annual variability.
This study also investigates the relationship between climate modes and RTE from the interannual variations and long-term trends perspective.In the NH, ENSO and AMO exert widespread impacts across most regions on inter-annual variations and long-term trends of RTE, and PNA, NAO, and AO highlight the inter-annual relationship with RTE in the specific regions, as North Pacific, North Atlantic, and North Africa, respectively.In the SH, besides that ENSO and AMO wield notable influence over most regions on inter-annual variations and longterm trends of RTE, SAM also plays an important role in the RTE change around Australia and continuing eastward to the South Atlantic.Furthermore, the impacts of PDO on RTE are more inclined to long-term contribution rather than inter-annual relationship.
Note that although there is no benchmarking to directly measure the SLPPE metric's superiority and reliability, which are guaranteed in the present study by multiple reasons.Firstly, the selection of SLP and PE is grounded in a physical understanding to establish a robust link between the new metric and RTE.Secondly, the multivariate probabilistic framework enriches the relationship between variables, and avoids dependency of a single variable.Thirdly, the RTE obtained by SLPPE metric is continuous in zonal direction, and can captures the common features of climatology, inter-annual variations, and long-term trends of RTE based on the other single variable metrics in previous studies.By considering these aspects comprehensively, the SLPPE metric shows clear physical implication, and the conclusions by using the SLPPE metric are robust.
For the probabilistic framework in the present study, it innovatively offers the ability to use multiple variables to define the positions of RTE, and this method considers the physical relationship among multiple variables simultaneously, providing a valuable tool for climate change assessment studies involving model simulations.Besides, the annual positions of RTE are calculated based on the annual average of the two variables in the present study, and the same framework could be used for obtaining the monthly and seasonal positions of RTE crossing all longitudes to analyze their monthly and seasonal dynamics.All these will significantly contribute to the understanding of the Hadley circulation and its impacts on the associated regional climate.

References
Due to the diverse performance of reanalysis datasets in depicting the variations of the Hadley circulation (Allan and Soden 2007, Stachnik and Schumacher 2011, Nguyen et al 2013, Davis and Birner 2017, Davis and Davis 2018, Grise et al 2018, Lau and Tao 2020), and modern reanalyses can better capture the tropical expansion characteristics (Grise et al 2019).Therefore, monthly mean output for the period of 1979-2020 from three modern reanalyses are used in this study: fifth generation of European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalyses of the global climate (ERA5; Hersbach et al 2020), the National Aeronautics and Space Administration's Modern-Era Retrospective Analysis for Research and Applications 2 (MERRA2; Gelaro et al 2017), the Japanese Meteorological Agency's Japanese 55 year Reanalysis (JRA55; Kobayashi et al 2015).
(a), which presents the climatological spatial distribution of RTE based on PE, SLP, and UAS metric.The maximum SLP and minimum PE at each longitude have almost identical latitude differences of approximately 10 • , and show the apparent

Figure 1 .
Figure 1.(a) Climatological spatial distribution of RTE based on Ψ φ , PE, SLP, and UAS metric spanning the period from 1979 to 2020, represented by the zero-crossing areas of 500 hPa Ψ φ (shading from red to blue indicates from positive value to negative one), areas of bottom 10% PE (red dots), areas of top 10% SLP (blue dots), and the areas of zero-crossing point with ±2.5 • deviation of near-surface winds in the transition from easterlies to westerlies (green dots).(b) Spatial distribution of climatological joint CDF between SLP and PE spanning the period from 1979 to 2020 (shading).The black dotted line demarcates the position of the maximum joint CDF between SLP and PE.(c) Zonal mean of PE (red line), SLP (blue line), and SLPPE (black line).
For comparison, the metric based on horizontally divergent winds is used (Keyser et al 1989, Schwendike et al 2015, Staten et al 2019), and the overturning stream function at each longitude is calculated as: Figure 3(a) shows the climatology and linear trend of joint CDF between SLP and PE from 1979 to 2020, and figures 3(b) and (c) present the interannual variations and long-term trends of RTE in the NH and SH, respectively.In the NH, significant poleward shifts of RTE are observed over the North Pacific and North Atlantic at about 0.3 • -0.6 • per decade and with relatively low inter-annual variability (figure 3(b)), accompanied by the anomalous decreasing and increasing trends of CDF in the south and north of the climatological RTE, respectively (figure 3(a)).Due to the strong inter-annual variability of terrestrial RTE, the long-term trends in most regions of Eurasia and North America are not significant.Consistent with the recent weakening intensity of the monsoon and the expansion of the transition zone (Dabang and Huijun 2005, Zhang and Zhou 2015, Wang et al 2016, Seth et al 2019), the RTE in East Asia shows some poleward trends, although the trends in most regions do not reach the confidence level (figures 3(a) and (b)).

Figure 2 .
Figure 2. Time series of zonal mean positions of RTE based on SLPPE metric for the (a) NH and (b) SH, along with the time series of positions of tropical edge based on the four zonal mean metrics based on PSI, PE, SLP, and UAS.(c) Correlation coefficients of detrended time series of SLPPE metric with those of the other four metrics in the NH and SH.(d) Linear trends (degrees/decade, poleward positive) of the time series of the five metrics in the NH and SH.* * denotes significance at the 99% confidence level, while * denotes significance at the 95% confidence level.
The expansion of the zonal mean tropical edge is typically influenced by climate modes, such as the Pacific Decadal Oscillation (PDO; Grassi et al 2012, Allen et al 2014, Brönnimann et al 2015, Amaya et al 2017), the Atlantic Multi-decadal Oscillation (AMO; Brönnimann et al 2015, Cao et al 2020), the El Niño-Southern Oscillation (ENSO; Lu et al 2008, Feng and Li 2013, Nguyen et al 2013, Adam et al 2014, Rollings and Merlis 2021, Li et al 2023b), the Arctic oscillation (AO; Hu et al 2019, Moon and Ha 2019), the Southern Annular Mode (SAM; Previdi and Liepert . The 12 regions are shown as 12 boxes in figure 4, including North Africa (10 • W-50 • E), Central Asia (50 • E-85 • E), East Asia (85 • E-125 • E), North Pacific (125 • E-120 • W), North America (120 • W-80 • W), and North Atlantic (80 • W-10 • W) in the NH, as well as South Africa (15 • E-40 • E), South India Ocean (40 • E-110 • E), Australia (110 • E-180 • ), South Pacific (180 • -80 • W), South America (80 • W-40 • W), and South Atlantic (30 • W-10 • E) in the SH.Seven climate modes mentioned above are chosen, ensuring a comprehensive representation of major global climate variabilities and considering their importance to contribute RTE change.The sources of the seven modes are listed in supplemental table S1.The instantaneous states of the seven climate modes are represented by a monthly index, and then the annual average of each mode is calculated to obtain the yearly time series.Partial correlations are used to measure the linear relationships between each climate mode and

Figure 3 .
Figure 3. (a) Spatial distribution illustrating the linear trend of joint CDF between SLP and PE (shaded), with the black dashed line denoting the location of the climatological maximum joint CDF between SLP and PE.(b) Linear trends (black line, degrees/decade, poleward positive) and inter-annual variations (red line) of the RTE in the NH.(c) Same as (b) but in SH.Dots in (a) and blue crosses in (b) and (c) denote significance at the 95% confidence level, and green stars in (b) and (c) denote significance at the 90% confidence level.

Figure 4 .
Figure 4. Summary of the relationship between climate modes and RTE.The first, second, third, and forth line in each box denote the name of the region, major inter-annual influencing factors, major long-term contribution factors, and RTE expansion rates, respectively.
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