Introduction

The mean altitude of the Qinghai–Tibet Plateau (QTP) exceeds 4000 m. Wang et al.1 and Xu et al.2 discovered that the low-density air conditions over the QTP are more favorable to initiating convective clouds over the plateau and lead to unusually strong convective activity and active cloud precipitation. Statistics show that the proportion of convective clouds over the QTP to the total cloud cover ranges from 4.0% to 21.0%, which is about five-fold that of the world3,4. The latent heat emitted by clouds and precipitation is the chief source of thermal forcing on the QTP5,6,7 and leads to the formation of atmospheric “heat islands” and “wet pools” over the plateau8. These meteorological conditions facilitate the high–low layer anticirculation pattern over the QTP, which is like the self-excited feedback mechanism of typhoons9. This pattern can excite the convection development in the region and help active cloud precipitation, which highlights the plateau’s role as a “window” for global energy exchange and water vapor transport8,10. However, the warming climate and the interaction between the westerly and the Indian monsoon are making the QTP change into a warming and wetting climate11,12,13,14,15. This warming–wetting climate produces a more unstable atmosphere and causes increased convective activities and precipitation over the QTP16.

In the climate system, the QTP actively interacts with climate variabilities worldwide17,18,19,20,21, among which the Atlantic climate variability can remotely affect the QTP’s climate and environment and drive the warming–wetting climate of the plateau16,22,23,24. The examination from the World Climate Service confirms that in the northern hemisphere, the QTP is generally warmer and wetter than normal during the strongly positive phase of the North Atlantic Multiyear Oscillation (AMO), whereas the QTP is probably cooler and drier than normal during the strongly negative phase of the AMO25. The Atlantic climate variability is closely related to the Atlantic sea surface temperature (SST)26,27. The Atlantic SST can affect the Asian climate by controlling teleconnection wave trains28,29,30,31. Teleconnection or remote response, as well as atmospheric low-frequency oscillations, have common characteristics. They demonstrate wave train-like features on a sphere, and their ray paths follow the large circle theory presented by Hoskins et al.32. Xu et al.33 extended the Hoskins theory to mathematical models with underbase flow and terrain-forcing conditions with westerly profiles. They discovered that external forcings such as SST and terrain influence the propagation of low-frequency waves and can form abnormal responses to local wave flow patterns, which result in weather and climate anomalies34,35,36.

Past studies have mainly investigated the effects of the QTP’s variability on surrounding regions and tropical monsoon systems over interannual or shorter timescales37,38. However, the response of the interdecadal convective activity over the QTP to the global climate change, as well as its impact mechanism for the regional–global circulation systems, remain poorly understood, particularly under the warming–wetting climatic background. Hence, our paper studies this core issue and investigates the following four topics. Is the convective activity over the QTP related to the North Atlantic SST and the interdecadal variability of AMO? Against the backdrop of intensified climate warming, can the continuously rising North Atlantic SST function as the “thermal forcing” and drive the interdecadal variability of the convective activity over the QTP? If so, how does the “oscillation source,” formed by the SST change in the high-impact areas of the North Atlantic, steer the interdecadal change of meridional wave train phase, which further prompts the response of the Hadley Cell (HC) in South Asia to the interdecadal change of the convective activity over the QTP? Does a mutual feedback mechanism exist between the convective activity and HC in the South Asian monsoon zone?

Results

Characteristics of the interdecadal change of the convective activity over the QTP

Statistics reveal that the distribution characteristics of low cloud cover anomalies on the QTP during the summers of 1961–1990 (hereinafter referred to as the former 30 years) and 1991–2018 (hereinafter referred to as the latter 30 years) were opposite, specifically in the northeast and central western parts of the QTP (Fig. 1a, b). In addition, the low cloud cover grew remarkably in the latter 30 years. This development suggests a clear interdecadal reversal in convective activity on the QTP, and the latter 30 years were more active. Thus, (30.5°–35.5° N, 81.5°–90° E) and (30.5°–39° N, 90°–103° E) were chosen as sensitive area A for convective changes in the QTP. The anomalies field of 200 hPa vorticity and atmospheric circulation structure (Fig. 1c, d) present that the vorticity and atmospheric circulation structure in the former 30 years and the latter 30 years also demonstrated an antiphase distribution characteristic. In the former 30 years, most of the QTP was regulated by cyclones, whereas in the latter 30 years, most parts of the QTP, which displays a structural characteristic of strong divergence in the upper troposphere, were ruled by anticyclones. The distribution of changes in vorticity and atmospheric circulation structure at 200 hPa (Fig. 1e, f) also reveals that the vorticity and atmospheric circulation structure exhibited opposite trends in the former and latter 30 years, and strong divergence was noted in the upper atmosphere of the QTP in the latter 30 years. Comparing the structure anomalies and the variation trend of 200 hPa vorticity and atmospheric circulation finds that in the latter 30 years, both displayed a significant “+ − + −” Rossby wave train structure spreading eastward from the North Atlantic to the QTP, whereas in the former 30 years, both demonstrated the opposite “− + − +” Rossby wave train structure. Thus, is the distribution difference of low cloud cover over the QTP during the former and latter 30 years linked to the interdecadal reversal of atmospheric meridional wave structures?

Fig. 1: Interdecadal variation characteristics of convective activity and atmospheric circulation structure over the QTP.
figure 1

Anomalies distribution characteristics of summertime low cloud cover (%) over the QTP during (a) 1961–1990 and (b) 1991–2018 (compared to the average value of 1961–2018), the area within the black box represents the sensitive zone A (30.5°–35.5° N, 81.5°–90° E) and (30.5°–39° N, 90°–103° E) for convective changes on the QTP; summertime vorticity (10−6 s−1) at 200 hPa and atmospheric circulation anomalies structure during (c) 1961–1990 and (d) 1991–2020 (compared to the average value of 1961–2020); 200 hPa vorticity (10−6 s−1) and atmospheric circulation variation trend during (e) 1961–1990 and (f) 1991–2020, the white-dotted area exceeds 0.05 significance level; (g) transport characteristics correlated between summertime low cloud cover in Zone A of the QTP and 200 hPa wind field vortices during 1961–2020 (The flow field is the integration of low cloud cover and u–v correlated vectors in Zone A, and the coloring is the integrated vorticity of low cloud cover and u–v correlated vectors in Zone A). The area within the deep pink box represents the high–impact area B of North Atlantic SST changes (75°–43° W, 25°–52.5° N) in (c–g).

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The examination of the correlation between low cloud cover in Zone A and the 200 hPa atmospheric circulation field reveals that most of the QTP was controlled by the anticyclone system (Fig. 1g), which facilitated convection39. Thus, the interannual change of summer convection over the QTP was linked to the interdecadal phase transition of the Rossby wave train structure spreading eastward from the North Atlantic to the QTP. In the latter 30 years, the anomalous high-pressure circulation at 200 hPa in the upper atmosphere of the QTP provided more beneficial conditions for the generation of active convection.

Interdecadal response of convective activity in the QTP to the oscillation source of SST westerly waves in the high–impact area of the North Atlantic

Numerous studies have discovered that at the basin scale, the North Atlantic SST displays substantial decadal-scale variation characteristics and alternating warm and cold phases40,41,42. Comparing the global SST trends from 1961 to 1990 and from 1991 to 2020 (Fig. 2a, b) reveals that the SST in the North Atlantic region (75°–43° W, 25°–52.5° N) stayed in a considerable warming stage (warm phase) in the latter 30 years, which was opposite the trend in the former 30 years. Statistics reveal that the SST in this area exhibited synchronous unusual changes with the low cloud cover and vorticity in Zone A of the QTP. Moreover, from 1991 to 2020, the SST displayed a remarkable upward trend, which was consistent with the low cloud cover and vorticity in Zone A (Fig. 2c). Thus, this region was selected as Zone B with a high impact on the SST changes in the North Atlantic.

Fig. 2: Interdecadal response of convective activity over the QTP to the oscillation source of SST westerly wave in the high–impact area of the North Atlantic.
figure 2

Distribution characteristics of summertime SST (°C) variation trend (a) from 1961 to 1990 and (b) from 1991 to 2020, the white-dotted area exceeds 0.05 significance level; (c) annual variation of low cloud cover (%) and 200 hPa vorticity (10−6 s−1) over the QTP and the SST in the high–impact area B of the North Atlantic during 1961–2020; the transport structure correlated between the SST in the high–impact area B of the North Atlantic and 200 hPa circulation vortices in the summers of (d) 1961–2020, (e) 1961–1990, and (f) 1991–2020 (the flow field is the integration of SST in Zone B and u–v correlated vectors, and the coloring is the integrated vorticity of SST in Zone B and u–v correlated vectors). The area within the deep pink box represents the high–impact area B of North Atlantic SST changes (75°–43° W, 25°–52.5° N) in (a, b, d–f).

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Additional calculation proves substantial westerly wave rays and their latitudinal teleconnected wave train structure in the transport structure correlated between the SST and 200 hPa circulation vortices (Fig. 2d). Figures 2e and 2f reveal that during the cold (1961–1990) and warm (1991–2020) phases of the SST in Zone B, the structures linked to circulation vortices displayed Rossby wave train structures with alternating high- and low-pressure circulation centers of “+ − + − +” and “− + − + −,” respectively. They spread eastward along the North Atlantic to the QTP, and an unusual high-pressure circulation center was formed in the upper atmosphere (200 hPa) of the plateau. The structure of the teleconnection wave train exhibited interdecadal “antiphase” changes in the former and latter 30 years and had wave train structure features similar to the anomalies and variation trend field of the 200 hPa vorticity and atmospheric circulation structure in the former and latter 30 years, respectively. The 200 hPa over the QTP was regulated by a cyclonic circulation (convergence) system in the former 30 years and then converted into an anticyclonic circulation (divergence) system that was favorable to convective activity on the QTP in the latter 30 years. This situation indicates the weak-to-strong transition of convective activity on the QTP, which precisely explains the cold-to-warm phase change of the SST in the high-impact area of the North Atlantic. Hence, the local response of convective activity on the QTP to the interdecadal phase transition of the SST in the high-impact area of the North Atlantic is disclosed. The Rossby source, also called the “oscillation source,” formed by the SST changes in the upper troposphere of the high-impact area of the North Atlantic, was a vital “thermal driving source” for the interdecadal variation of convective activity on the QTP. The latitudinal teleconnection wave train structure triggered by the warming/cooling of the sea surface in the high-impact area of the North Atlantic showed the opposite phase. Moreover, atmospheric circulation teleconnection wave trains associated with interdecadal changes in convective activity on the QTP spread latitudinally from west to east. They reveal that unusually warm SSTs in the North Atlantic stimulated Rossby wave trains to spread downstream from the North Atlantic to the QTP. This discovery further discloses the thermal driving influence of the “oscillation source” SST anomalies on the propagation of westerly wave rays and wave trains. Thus, under external forcing, the nonlinear interactions between vortical currents and wave currents of the atmospheric motion and the low-frequency wave activity may cause sustained anomalies in local circulation structures.

Difference in the interdecadal variations of the oscillation source of the westerly wave in AMO

AMO manifests the interdecadal variability of the North Atlantic SST43. Thus, does the convective activity on the QTP have the same response mechanism to the interdecadal variation of AMO? Fig. 3a and 3b reveal that the AMO and the summer SST anomalies in the high-impact area exhibited synchronous changes, and a 30-year antiphase change started in 1990. Particularly, the AMO was negative phase from 1961 to 1990 and became positive phase from 1991 to 2020. According to the interdecadal variation features of AMO, the transport characteristics of AMO and 200 hPa circulation vortices before and after 1990 were calculated (Fig. 3c, d). During the negative and positive phases of AMO, the structures linked to the AMO and circulation vortices displayed Rossby wave train structures of “− + − + −” and “+ − + − + −,” respectively, and spread eastward from the North Atlantic to the QTP. However, unlike the meridional teleconnection wave train structure triggered by the SST phase inversion in the high-impact area of the North Atlantic, the meridional teleconnection wave train structure excited by the AMO phase inversion was controlled by the anticyclonic system over the QTP and did not exhibit the anomalous reversal consistent with the AMO phase. This finding suggests that the phase reversal of the AMO cannot steer the interdecadal variation of convective activity on the QTP, which may be linked to the definition range of AMO (75°–7.5° W, 0°–60° N)43, which is greater than the range of the North Atlantic SST High-Impact Zone B.

Fig. 3: Difference in the interdecadal variations of the oscillation source of the westerly wave in AMO.
figure 3

a The summertime SST anomalies (℃, compared to the average value of 1961–2020) in the key are of the North Atlantic and (b) the annual variation of AMO from 1961 to 2020; the transport characteristics correlated between AMO and 200 hPa wind field vortices in the summers of (c) 1961–1990 and (d) 1991–2020 (the flow field is the integration of AMO and u–v correlated vectors, and the coloring is the integrated vorticity of AMO and u–v correlated vectors). The area within the deep pink box represents the high–impact area B of North Atlantic SST changes (75°–43° W, 25°–52.5° N) in (c, d).

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Feedback mechanism between the convection activity over the QTP and the HC in the South Asian monsoon region

In summer, the effect of the large-scale thermal forcing over the QTP generally causes a counterclockwise rotating “small circulation” within the 90° E longitude circle, which rises over the QTP and sinks in the Indian Ocean44. The airflow rises from the QTP, turns southward in the upper air, and sinks at low latitudes. The sinking airflow returns to the QTP in the lower air and forms a counterclockwise meridional circulation loop, famous as the South Asian Monsoon Circulation Loop44,45,46,47, which can be considered the overlap and interaction between the regional HC and the planetary-scale Walker circulation48. The “small circulation” is the special form of HC in the special geographical conditions of the QTP, i.e., the HC in the South Asian monsoon region. In addition to terrain effects, the latent heat released by cloud precipitation processes on the QTP was the main source of thermal forcing on the plateau5,6,7. Thus, do the interdecadal-scale changes in convective activity on the QTP influence the HC in the South Asian monsoon region?

The development of convection on the QTP is generally accompanied by strong upward motion1,2. Calculating the structure of the 90° E atmospheric vertical velocity anomalies in the former and latter 30 years finds that the vertical motion of the QTP exhibited remarkable interdecadal changes and displayed an antiphase distribution in the former and latter 30 years (Fig. 4a, b). In the latter 30 years, a significant positive anomalies was observed over the QTP and the southern slope, which indicates upward movement (Fig. 4b). The meridional regression analysis of the vertical transport structure of atmospheric water vapor and low cloud cover in Zone A during summer reveals a similar counterclockwise rotating meridional correlated vertical circulation circle between 0° and 30° N on the 90° E and a significant positive correlation zone between low cloud cover and water vapor in Zone A, pointing vertically from the QTP to the zenith (Fig. 4c). According to the interdecadal reversal characteristics of the 90° E atmospheric vertical velocity anomalies structure calculated earlier for the former and latter 30 years (Fig. 4a, b), under conditions of abundant water vapor, the remarkably enhanced upward motion of the QTP in the latter 30 years created a beneficial environment for the increase of convective activity on a large scale, which indicates that the convective activity on the QTP is highly related to the HC in the South Asian monsoon region.

Fig. 4: Feedback mechanism between the convection activity over the QTP and the HC in the South Asian monsoon region.
figure 4

The anomalies structure of the vertical speed (-10−2 Pa/s) at 90°E (compared to the average value of 1961–2020) during (a) 1961–1990 and (b) 1991–2020; (c) meridional regression analysis on the low cloud cover in Zone A of the QTP and the vertical transport structure of atmospheric water vapor at 90° E in the summers of 1961–2018. The white spotted area exceeds the significance level of 0.1 (the flow field is the integration of the correlated vectors between the low cloud cover in Zone A and v-ω×-10−2, and the coloring is the correlation distribution of low cloud cover and specific humidity in Zone A); (d) deviation of the vertical transport structure of the atmospheric water vapor at 90°E between 1991–2020 and 1961–1990 (the flow field is integrated from v-w×-10−2 vectors, and the coloring is the deviation of specific humidity (g/kg) between the former and the latter 30 years); (e) deviation of the variation trend (the flow field is integrated from v-w×-10−2 vectors, and the coloring is the deviation of specific humidity (g/kg) between the former and the latter 30 years). The black dashed arrow represents the meridional circulation and uplift transport in (c–e).

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The convective activity on the QTP was in an active stage from 1991 to 2020 and demonstrated considerable interdecadal enhancement characteristics. From the view of vertical movement on the QTP, water vapor can be transported to the upper troposphere through strong mesoscale convective activities and the atmospheric heat transfer driven by the “chimney effect”10,49. The latent heat emitted by the condensation of convective activity can form an atmospheric “heat island” and a “wet pool” over the plateau10, causing the air in the lower atmosphere to be sucked upward and then move from the upper atmosphere to the surrounding areas, and develop a high–low anticirculation pattern similar to the self-excited feedback mechanism of a typhoon9. The strengthening of convective activity on the QTP can generate more latent heat through condensation and enhance the “heat pump” effect on the plateau5,6,7,9,50,51. Thus, are the interdecadal-scale changes in convective activity on the QTP, through the “heat pump” effect of the QTP, the driving force for the northward movement of the HC in the South Asian monsoon region?

To answer this question, the deviation fields of the vertical transport structure anomalies and variation trend of atmospheric water vapor from 1991 to 2020 and from 1961 to 1990 were calculated to examine the interdecadal evolution of the HC in the South Asian monsoon region further. Figure 4d presents a substantial counterclockwise rotating meridional vertical circulation circle between 5° and 30° N on the deviation fields of the atmospheric water vapor vertical transport structure anomalies and significantly positive deviations in the water vapor at 850–300 hPa in the QTP and its southern region were noted. Equally, similar vertical structural features were observed in the deviation fields of the variation trend of the atmospheric water vapor vertical transport structure in two stages, that is, a significant counterclockwise rotating meridional vertical circulation circle was between 5° and 30° N, and the water vapor in the central region of the circulation circle at 850–300 hPa showed a significant positive deviation (Fig. 4e). This outcome indicates that the HC in the South Asian monsoon region has experienced a substantial northward shift, and more water vapor is transported from the southern boundary of the QTP to the QTP through the HC in the South Asian monsoon region, thus increasing the content of water vapor over the plateau. The HC not only brings abundant water vapor to the QTP; its vertical ascending branch also forms a large-scale beneficial environment for convective activity on the plateau, which stimulates convective activity in the region. Conversely, the convective activity that can generate more latent heat further promotes the northward shift of the HC in the South Asian monsoon region. This outcome validates the mutual feedback between the convective activity on the QTP and the HC in the South Asian monsoon region.

Discussion

The interactions between the atmosphere and ocean20,52, as well as atmospheric dynamical processes53,54, specifically Rossby wave dynamics, support the connectivity among different regional climate systems. This paper aims to investigate the response of interdecadal changes in convective activity on the QTP to global climate change and its influence mechanism on regional–global circulation systems and to uncover the linkage mechanism caused by the SST interdecadal variations in the high-impact area of the North Atlantic.

Our research results reveal that convective activity on the QTP significantly grew during the summers from 1991 to 2020, and the plateau was in a period of active convective activity, which can be traced back to the North Atlantic. The Rossby source, also called the “oscillation source,” formed by the SST changes in the upper troposphere of the high-impact area, can serve as an essential “thermal driving source” for the interannual variation of convection on the QTP. The interdecadal phase reversal of atmospheric meridional wave trains in the mid–high latitudes is linked to the interdecadal phase reversal of the SST in key areas of the Atlantic Ocean. The meridional teleconnection wave train structure excited by the warming (1991–2020)/cooling (1961–1990) of the sea surface in the key areas of the mid–high latitudes of the North Atlantic is the opposite. The Rossby wave train that spreads from the North Atlantic to the QTP during the warming phase is “− + − + −” and has a substantial anticyclonic structure (strong divergence) over the QTP (200 hPa), which is like the wave train structure features of the 200 hPa vorticity and atmospheric circulation structure anomalies and variation trend field in the latter 30 years. The remarkable anticyclonic structure stimulates active convective activity over the QTP. However, the convective activity on the QTP does not display the same response mechanism to the interdecadal variation of AMO, which is possibly due to the variance in the range between the AMO and SST high-impact area in the North Atlantic. The calculation results reveal that the convective activity on the QTP is highly related to the HC in the South Asian monsoon region. On the QTP, the interdecadal variation of vertical movement is substantial. From 1991 to 2020, under the conditions of abundant water vapor, the significantly improved upward motion on the QTP created a conducive environment for the increase of convective activity on a large scale. The strengthening of the convective activity on the QTP can generate more latent heat through condensation, which improves the “heat pump” effect of the QTP5,6,7,9,50,51, through which the HC in the South Asian monsoon region is driven northward. As a consequence of this northward movement through the HC, more water vapor is transported to the QTP from the southern boundary of the plateau, the content of water vapor is increased, and the convective activity in the region is stimulated. The active convective activity generates more latent heat and promotes the northward propagation of the HC in the South Asian monsoon region, which validates the mutual feedback between the convective activity over the QTP and the HC in the South Asian monsoon region.

According to the interdecadal response characteristics discussed above, this paper offers a comprehensive physical image that reveals the northward propagation of the HC in the South Asian monsoon region steered by the active convective activity over the QTP, which in turn is driven by the SST warming in the high-impact area of the North Atlantic (Fig. 5). The strengthened convective activity on the QTP is a key response to global climate change. Moreover, the changes in the convective activity on the QTP largely drive the modifications in the regional–global circulation systems. This interaction mechanism stresses the “hub” effect of the QTP in the physical quantity transport and exchange of atmospheric energy and water circulation between the low- and mid–high latitudes during the convergence of westerly and monsoon winds. This interaction mechanism is also a core issue in the study of regional and global “water, energy, and climate.”

Fig. 5
figure 5

Comprehensive physical image of the northward propagation of HC in the South Asian monsoon region driven by active convection over the Qinghai–Tibet Plateau triggered by SST warming of the North Atlantic.

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Data and methods

We use the low cloud cover data of 1961–2018 from more than 2000 stations provided by the monthly meteorological dataset of basic meteorological elements of China National Surface Weather Station (V3.0) from National Meteorological Information Centre, China Meteorological Administration. However, due to the high elevation of the QTP and its complex terrain, the ground observation sites are sparsely distributed in the western QTP. Therefore we need to adopt ERA5 monthly averaged 0.25° × 0.25° low cloud cover data on single levels from 1961 to 2020 (https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-single-lev els-monthly-means?tab=form) provided by European Centre for Medium Range Weather Forecasts (ECMWF). By optimizing the variational correction method, objective observation data from ground stations are effectively integrated with ERA5 reanalysis data to compensate for the shortcomings of conventional observations in the central and western QTP. A low cloud analysis field with both high resolution and objectivity is obtained to analyze the interdecadal variation characteristics of low cloud cover over the QTP from 1961 to 2020. This variational correction technique has been repeatedly proven to be useful for obtaining high-resolution and high–quality fusion data on the QTP, solving the problem of sparse ground conventional station observations55,56. The detailed method is as follows:

The fundamental problem of variational methods is to find the extremum of a function, which depends on multiple independent variables:

$${\rm{J}}\left[{\rm{U}}\left(x,y\right)\right]={\iint }_{\!\!G}F(x,y,U,\frac{\partial U}{\partial x},\frac{\partial U}{\partial y}){dx}{dy}$$
(1)

Here, U(x, y) must satisfy the Euler equation:

$${\rm{Fu}}-\left(\frac{\partial }{\partial {\rm{x}}}{\rm{F}}{u}_{x}+\frac{\partial }{\partial {\rm{y}}}{\rm{F}}{u}_{y}\right)=0$$
(2)

Let the ERA5 low cloud field be R* (x, y), and the corresponding low cloud field from the limited ground observation be R (I, J), in Eq. (2). Then, at the station coordinates (\(I,J\)), the difference between these two fields is defined as the error field:

$${C}_{r\,}^{ \sim }\left(I,J\right)={\rm{R}}\left(I,J\right)-{R}^{* }(I,J)$$
(3)

Because of the limited ground observation stations, it is necessary to construct a more generalized correction factor field function \({C}_{r}\left(I,J\right)\) for the entire field. To obtain \({C}_{r}\left(I,J\right)\), it must satisfy the following conditions:

$$\widetilde{J}={\iint }_{\!\!D}{({C}_{r}-{C}_{r}^{ \sim })}^{2}{dx}{dy}\,\to \min$$
(4)

Since the grid points of the low cloud cover field are limited, that is to say \(\sum _{i}\sum _{j}{({C}_{r}-{C}_{r}^{ \sim })}^{2}\) reaches the minimum value. Therefore, this problem is transformed into calculating the extremum of the function. For the above variational problem, it can be assumed to construct the function J:

$${\rm{J}}=\iint \left\{{({C}_{r}-{C}_{r}^{ \sim })}^{2}+\lambda \left[{\left(\frac{\partial {C}_{r}}{\partial x}\right)}^{2}+{\left(\frac{\partial {C}_{r}}{\partial y}\right)}^{2}\right]\right\}{\rm{dx}}{dy}$$
(5)

In Eq. (5), \(\lambda\) is the constraint coefficient. Calculating the first-order variation of the above equation yields:

$${\rm{\delta }}{\rm{J}}={\rm{\delta }}\sum \sum \left\{{({C}_{r}-{C}_{r}^{ \sim })}^{2}+\lambda \left[{\left(\frac{\partial {C}_{r}}{\partial x}\right)}^{2}+{\left(\frac{\partial {C}_{r}}{\partial y}\right)}^{2}\right]\right\}=0$$
(6)

The necessary condition for the function to attain the extremum is \({\rm{\delta }}{\rm{J}}=0\), which corresponds to the Euler equation:

$$\left({C}_{r}-{C}_{r\,}^{ \sim }\right)-\widetilde{\lambda }\left[\frac{{\partial }^{2}{C}_{r}}{\partial {x}^{2}}+\frac{{\partial }^{2}{C}_{r}}{\partial {y}^{2}}\right]=0$$
(7)

In Eq. (7), \(\widetilde{\lambda }\) is the shape constraint function, that is, the Lagrange multiplier, which is introduced to determine the conditional extremum. The numerical solution of the above equation can be obtained through iteration, and the new variational correction factor field \({C}_{r}\left(x,y\right)\) can be obtained. The original field plus this correction factor field form the corrected low cloud cover field:

$${\rm{R}}\left(x,y\right)={R}^{* }\left(x,y\right)+{C}_{r}\left(x,y\right)$$
(8)

In addition, we use the AMO (Atlantic Multidecadal Oscillation) Index (https://psl.noaa.gov/data/timeseries/AMO/) and global 1° × 1° monthly-mean SST data during 1961–2020 (https://www.psl.noaa.gov/data/gridded/data.cobe.html) provided by Japanese Meteorological Center (JMA). Here, the AMO Index is the unsmoothed and not detrended data of 1961–2020, which are calculated from the monthly-updated Kaplan SST dataset (5° × 5°). ERA5 monthly averaged data (0.25° × 0.25°) from 1961 to 2020 (https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-pressure-levels-monthly-means?tab=overview), provided by ECMWF, is also included in our study, including humidity, meridional wind, meridional wind, and vertical velocity and so on. With comprehensive diagnostic statistical methods such as linear tendency estimation and correlation analysis57, this study explores the interdecadal-scale response mechanism of convective activity on the QTP to meridional wave train phase adjustments triggered by SST anomalies and AMO phase reversal under the current warming–wetting. It also reveals the impact mechanism of interdecadal changes in convective activity over the QTP on the regional–global circulation systems.