Introduction

As a critical regulator of the cryosphere, snowpack governs surface–atmosphere energy exchange and regional hydrological cycles through its dual roles in albedo modulation and water storage1,2,3. Seasonal snow accumulation and ablation drive the temporal redistribution of water resources, serving as the primary source of river runoff and groundwater recharge in arid regions during spring-summer transitions4. This function is critical across the Tibetan Plateau (TP)—the “Water Tower of Asia”—where snowmelt sustains the headwaters of ten major Asian river systems (e.g., Yangtze, Indus, and Ganges), providing essential freshwater to over 1.5 billion downstream inhabitants5. This high-elevation cryosphere reservoir, however, is undergoing unprecedented transformations under global warming6,7. Climate-driven alterations to snowpack dynamics manifest through two concurrent changes on the TP: accelerated snowmelt from rising temperatures (0.34 °C per decade since the 1980s)8 and precipitation phase shifts favoring rainfall over snowfall9. These changes (Fig. 1) disrupt traditional snow hydrology through earlier melt onsets and reduced snow water equivalent (SWE), leading to snow drought. This phenomenon, characterized by a significant reduction in snowfall or premature snow ablation, contributes to regional water scarcity and has garnered increasing attention due to its widespread adverse effects on terrestrial ecosystems10.

Fig. 1: Overview of the Tibetan Plateau (TP).
figure 1

a Lakes, glaciers, and major river basins in the TP. Spatial distribution of the multi-year trend of b temperature and c snow water equivalent (SWE) for the period 1979–2022, respectively. d Intra-annual variations in mean SWE (left y-axis, orange) and 44-year cumulative changes in monthly SWE (right y-axis, green), with error bars representing the standard deviation of SWE for each month during 1979–2022. Filled bars indicate statistically significant (p < 0.05) results.

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Snow drought is generally classified into two types, dry snow droughts (DSD) and warm snow droughts (WSD). DSD is caused by a significant reduction in snowfall during the cold season, and WSD is caused by rising temperatures leading to premature snowmelt or a shift from snowfall to rainfall11. Recent studies have indicated that, under the context of global warming, the frequency and severity of snow droughts have shown an increasing trend in high-latitude regions of the Northern Hemisphere12,13. Snow drought not only directly affects the regional water supply but may also trigger a series of extreme hydroclimatic events by altering the surface energy balance and hydrological cycle processes. For example, the reduction of snowpack and the advancement of the snowmelt peak can significantly change the seasonal distribution of runoff and increase the risks of floods and droughts14,15. More critically, the reduction in surface albedo with snowpack melting increases solar radiation absorption and thus accelerates surface warming, creating a positive feedback loop known as the “melt-warming” effect. Observational evidence indicates that 35% of snow drought events between 1981 and 2020 triggered heatwaves across global snow-dominated regions16, highlighting the detectable contribution of snow droughts to compound disasters. Additionally, the hydroclimatic consequences resulting from snow droughts vary between the two types. WSD typically increases winter runoff peaks (potentially raising flood risks) while reducing spring and summer runoff, exacerbating seasonal water shortages. In contrast, DSD tends to lower annual runoff peaks, leading to insufficient water supply throughout the year17.

Extreme hydrological events rarely manifest in isolation, and they predominantly result from interactions among meteorological and hydrological drivers18, namely compound climate extremes19. Emerging evidence suggests that snow droughts can trigger compound climate extremes through coupled physical processes. Snow-substrate feedback mechanisms significantly alter surface energy budgets, while reduced snow albedo enhances shortwave radiation absorption, amplifying local warming20,21. This process forms a positive feedback mechanism with the exponential growth of the atmosphere’s water-holding capacity as the temperature rises (following the Clausius–Clapeyron (C–C) theory)22. The chaotic nature of the climate system causes actual precipitation changes to deviate from the ideal thermodynamic laws23,24. Precipitation anomalies under climate warming (especially water deficits or surpluses that exceed the expectations of the C–C theory) significantly increase the co-occurrence probability of drought (or pluvial) through land–atmosphere coupling25. The impact of such compound climate extremes on eco-hydrological systems exhibits a strongly amplified, non-linear effect, far exceeding the simple sum of individual extremes26,27. These events threaten the TP’s role as Asia’s freshwater reservoir, with cascading risks to downstream irrigation28, terrestrial ecosystems29, groundwater recharge15, and water resource management30.

Although existing studies have established the relationship between snowpack changes and individual extreme events, there is still a lack of systematic understanding of the cascading effects of snow drought events and compound extreme hydroclimatic processes. In particular, for key scientific issues such as the physical mechanisms of snow drought triggering compound climate extremes in high-altitude areas (e.g., the snow–atmosphere–soil coupling process), the spatiotemporal propagation patterns, and their eco-hydrological aftereffects, a complete theoretical framework has not been formed yet. This hinders an in-depth understanding of how the cascading effects of snow drought reshape the regional hydro-climatic pattern.

Based on multi-source datasets, we systematically investigate the mechanisms by which different types of snow droughts drive compound climate extremes across the TP from 1979 to 2022. The study focuses on the following three aspects: (a) analyzing the spatiotemporal distribution characteristics and evolution patterns of different types of snow droughts; (b) examining the response characteristics of compound climate extremes to different types of snow droughts; and (c) revealing the triggering mechanisms and physical processes by which snow droughts induce compound climate extremes. By establishing a nexus framework between snow droughts and compound climate extremes, we aim to advance predictive capacity for cascading climate risks in high-elevation regions while informing adaptive strategies for vulnerable snow-impacted watersheds.

Results

Spatiotemporal divergence of snow droughts

Based on the SWEI and SPI values, we classified snow droughts into two types: DSD and WSD. Figure 2a–d shows the spatial distributions of the annual mean frequency and severity of DSD and WSD from 1979 to 2022. The frequency of WSD (0.19 times per year on average) was higher than that of DSD (0.13 times per year on average) on the plateau, while the severity of DSD (1.17 on average) exceeded that of WSD (1.04 on average). Note that both DSD and WSD exhibited similar spatial patterns in severity, whereas the occurrence frequency showed distinct spatial heterogeneity. Specifically, the hotspots of DSD (WSD) were primarily concentrated in the northwestern (southeastern) part of the TP. The complex topography of the TP imposed pronounced controls on snow drought typology, as evidenced by our altitudinal zonation analysis (500 m intervals). As shown in Fig. 2e, low-elevation areas (below 3000 m) were the hotspots for DSD occurrence, while WSD prevailed at mid-high elevations, and peaked at 4000–4500 m. Furthermore, regional moisture conditions significantly influenced snow drought types (Fig. 2f). The higher frequencies of DSD were mainly exhibited in arid regions, particularly in hyper-arid and arid zones (0.15 times). Conversely, WSD was more common in dry sub-humid regions (0.21 times). As shown in Fig. S1, statistical analysis revealed a significant negative correlation between DSD frequency and AI (slope = –0.05, p < 0.05), while WSD frequency showed a significant positive correlation with AI (slope = +0.05, p < 0.05).

Fig. 2: Occurrence frequency and severity of dry snow drought (DSD) and warm snow drought (WSD) from 1979 to 2022.
figure 2

Spatial distribution of annual mean occurrence frequency (a, c) and severity (b, d) of DSD and WSD, elevation dependency of snow drought occurrence frequency (e), and snow drought frequency under different moisture conditions (f).

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Long-term trend analysis revealed divergent trajectories in snow droughts evolution across the TP from 1979 to 2022, characterized by intensifying drought severity alongside shifting spatial dominance. As shown in Fig. 3a and b, both DSD and WSD exhibited significant intensification, with the severity increasing at 0.05 per decade and 0.04 per decade (p < 0.05), respectively. However, their spatial footprints demonstrated inverse patterns, DSD-affected areas contracted by –0.15% per year, while WSD coverage expanded by +0.43% per year. Linear regression identified the year 1991 as the critical breakpoint in snow drought regime shifts across the TP. Before 1991, DSD exhibited a higher areal proportion than WSD, whereas WSD became the dominant type of snow drought events after 1991. For a more intuitive presentation, we counted the proportions of the areas where the two types of snow drought occurred before and after 1991 (Fig. 3c). Before 1991, the proportions of DSD and WSD were 15.46% and 13.47%, respectively. After 1991, the proportions of DSD and WSD were 12.04% and 20.86%, respectively. The area with WSD expanded rapidly, and the area without snow drought decreased from 71.07% to 67.1%. That is, the area with snow droughts on the TP was increased, and WSD was the dominant type.

Fig. 3: Changes in the severity and area of different types of snow droughts.
figure 3

a Interannual variations in the severity of snow droughts (i.e., absolute value of SWEI) from 1979 to 2022. b Interannual variations in the proportion of the area affected by snow droughts. A vertical dashed line marks 1991, the key breakpoint determined through linear regression. c Proportion of the area affected by different types of snow droughts during the entire study period, before 1991, and after 1991.

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Cascading dynamics between snow droughts and compound climate extremes

With the increasing frequency and severity of snow droughts and compound climate extremes on the TP (Fig. S2), it is crucial to investigate the potential link between these phenomena. To explore this relationship, we analyzed two compound climate extremes, CDHE and CPHE, occurring within 1–3 months after the end of the DSD and WSD. Through coincidence analysis, we found that DSD (WSD) was more closely associated with CDHE (CPHE) (Fig. 4a). Although DSD and CPHE, as well as WSD and CDHE, exhibited statistically significant coincidence rates, the coincidence areas for both pairs remained below 5%. Consequently, our subsequent analysis prioritized the impacts of DSD on CDHE and WSD on CPHE (Fig. 4b, d). Specifically, the significant coincidence rates for the paired DSD–CDHE and WSD–CPHE were highest within 1 month after snow disappearance, reaching 0.68 and 0.58 (p < 0.05), respectively. As the lag time increased, the significant coincidence rates gradually decreased (Fig. S3), and the proportion of affected areas also correspondingly declined, from 33.4% to 2.3% for DSD–CDHE, and from 30.5% to 6.7% for WSD-CPHE, across the snow-covered regions of the plateau. Additionally, the coincidence rates for DSD–CDHE mostly ranged between 0.5 and 0.8 (Fig. 4c), whereas those for WSD–CPHE were relatively lower, mainly ranging from 0.4 to 0.7 (Fig. 4e). Compared to the relationship of WSD–CPHE, the coincidence rates of DSD–CDHE were more pronounced.

Fig. 4: Relationships between snow droughts and compound climate extremes.
figure 4

a Maximum proportion of the area and magnitude of significant coincidence rates (p < 0.05) between different types of snow droughts and compound climate extremes within 1–3 months after snowpack disappearance. Spatial distribution of significant coincidence rates (colored regions) between DSD and CDHE at 1-month lag (b) and between WSD and CPHE at 1-month lag (d). Kernel density distributions of significant coincidence rates for DSD and CDHE (c) and WSD and CPHE (e) at lags of 1–3 months, where the noted ‘Proportion’ and ‘Mean’ indicate the area proportion and mean value of the significant coincidence rates for each lag.

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We further quantified how the severity of compound climate extremes responds to the severity of snow droughts (Fig. 5a, b). Here, the sensitivity was calculated as the change in the severity of compound climate extremes caused by a per-unit change in the severity of snow droughts. The results showed that, for both DSD to CDHE and WSD to CPHE, more than half of the areas (with significant coincidence) exhibited positive sensitivity. That is, as the severity of snow droughts increased, the severity of compound climate extremes also increased. Specifically, 18.5% (17.3%) of the area exhibited positive sensitivity of CDHE (CPHE) severity to DSD (WSD) severity. Overall, the severity of CDHE (CPHE) increased by 0.09 (0.10) for every unit increase in the severity of DSD (WSD). Furthermore, both sensitivities exhibited distinct latitudinal patterns (Fig. 5c, d), characterized by significant declining trends from south to north in the sensitivity of CDHE to DSD (slope = –0.01, p < 0.05) and CPHE to WSD (slope = –0.02, p < 0.05).

Fig. 5: Sensitivity of compound climate extremes to the snow droughts with 1-month lag.
figure 5

Spatial distribution of the sensitivity of CDHE to DSD (a) and CPHE to WSD (b), where insets show the proportion of areas with positive and negative sensitivities. Values 66.6% and 69.5% indicate the spatial coverage with non-significant coincidence rates for DSD & CDHE and WSD & CPHE associations, respectively (p ≥ 0.05). Latitudinal gradient of the sensitivity of CDHE to DSD (c) and CPHE to WSD (d).

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Mechanisms of compound climate extremes triggered by snow droughts

To elucidate the causal relationships between snow droughts and compound climate extremes, we analyzed surface and atmospheric conditions of the first snow-free month during the snow droughts, such as soil moisture, surface albedo, ET, and surface sensible heat flux. We compared soil moisture and surface albedo in grids where snow droughts occurred with those in all grids. As shown in Fig. S4, when snow droughts occurred, regardless of whether they were DSD or WSD, surface albedo generally decreased. The surface albedo in grids experiencing snow droughts was significantly lower than that in all grids. However, DSD (WSD) tended to reduce (increase) soil moisture, which may be attributed to the offsetting effect of near-normal or above–normal precipitation associated with WSD. As shown in Fig. 6, snow droughts significantly altered surface conditions and may have further influenced local and regional hydrological and climatic conditions through energy and water cycle processes.

Fig. 6: Schematic diagram illustrating the impacts of snow droughts on land surface and atmospheric conditions.
figure 6

This figure demonstrates how DSD (a) and WSD (b) induce contrasting compound climate extremes (CDHE and CPHE) by altering surface albedo and soil moisture, thereby modifying energy and water fluxes. Snow drought reduces surface albedo, increasing net solar radiation absorption and enhancing sensible heat flux. The distinct pathways of DSD and WSD further diverge through their impacts on evapotranspiration and precipitation formation. Symbols (−) and (+) denote decreased and increased changes in variables, respectively.

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Based on heat fluxes and water vapor fluxes, we constructed a PLS-SEM model to systematically reveal the physical mechanisms by which snow droughts trigger compound climate extremes. As shown in Fig. S5, in the heat flux pathway, the reduction in surface albedo caused by the occurrences of DSD and WSD significantly increased net shortwave radiation (path coefficients corresponded to –0.85 and –0.86, respectively). The enhanced net shortwave radiation further drove an increase in surface sensible heat flux, directly exacerbating near-surface temperature rise, thereby increasing the likelihood of CDHE. In the water vapor pathway, DSD (Fig. 6a) directly reduced soil moisture, limiting ET and further decreasing atmospheric specific humidity, ultimately suppressing precipitation and showing a significant negative correlation with the significant coincidence rate of DSD and CDHE (path coefficient was –0.23). In contrast, WSD (Fig. 6b), due to its near-normal or above-normal precipitation, maintained higher soil moisture, promoting ET and atmospheric water vapor content, thereby increasing the probability of precipitation and showing a significant positive correlation with the significant coincidence rate of WSD and CPHE (path coefficient was 0.18).

Discussion

This study revealed significant changes in snow droughts in the TP over the past 44 years: DSD and WSD both had increasing trends in severity, but their frequency exhibited distinct temporal variations and spatial heterogeneity. Around 1991, WSD superseded DSD as the dominant snow drought type, consistent with observed patterns26. Snowpack exhibits strong temperature dependence, as temperature governs both the proportion of precipitation falling as snow and the rate of snowmelt. The transition from DSD to WSD likely results from regional warming. In recent decades, the TP has experienced accelerated warming (approximately reaching twice the global average)8,24. Consequently, even under normal precipitation conditions, rising temperatures induce non-precipitation-related SWE losses through accelerated snowmelt and precipitation phase shifts (snowfall-to-rainfall transition), thereby establishing temperature-dominated droughts.

Furthermore, the frequency of snow droughts was significantly modulated by regional wet-dry conditions. In arid regions, DSD’s frequency was higher, closely related to scarce and highly variable precipitation. In contrast, WSD was more frequent in humid regions, where relatively abundant precipitation makes temperature the dominant factor controlling SWE changes. This nonlinear response stems from hydrothermal coupling effects: precipitation thresholds constrain snowpack formation in arid regions, while sufficient precipitation renders the snowpack system more vulnerable to thermal perturbations in humid zones. These are manifested through prolonged snowmelt duration and increased frequency of rain-on-snow events under warming conditions, collectively compromising snowpack storage capacity27. This finding further confirms the strong linkage between DSD (WSD) and precipitation (temperature) sensitivity.

We comprehensively revealed the close connections between DSD (WSD) and CDHE (CPHE). Our analysis demonstrated that the coincidence rates and spatial extents of these compound climate extremes peaked one month following snow drought onset. Our conclusions align with a recent observational study31, though their work linked snowpack mass and duration extremes to compound climate extremes. They found that compound climate extremes are more sensitive to changes in severity compared to changes in the probability of snowpack extremes, suggesting severity variations could serve as an effective new indicator for improving early predictions of such events. Our results also demonstrated that amplified snow drought severity intensifies compound climate extremes. Moreover, the sensitivity of compound climate extremes severity to snow droughts severity weakened with latitude. This latitudinal gradient in sensitivity may have arisen because high-latitude regions exhibited buffered snow–atmosphere coupling due to stable snowpack and permafrost thermal inertia, whereas low-latitude areas experienced amplified impacts via albedo feedback and Rossby wave dynamics32,33.

Mechanistically, DSD and WSD drove compound climate extremes through distinct water–energy coupling pathways. DSD reduced the spring snowmelt recharge, leading to a significant decrease in soil moisture34,35. Meanwhile, the decrease in surface albedo enhanced the surface absorption of solar radiation, promoting an increase in surface sensible heat flux36,37,38. The coupling effect between the soil and the atmosphere further exacerbated the accumulation of surface heat. Coupled with an insufficient water supply, CDHE was ultimately triggered. In contrast, WSD was usually accompanied by an increase in soil moisture, which might be related to the advance of the snowmelt period and the precipitation compensation effect. According to the C–C theory, the increase in temperature led to an enhanced water-holding capacity of the atmosphere, thus creating a high-humidity environment24,39. Such an environment, through the coupling of latent heat release and convective activities, further drove the occurrence of CPHE. Notably, both snow drought types shared a common pathway: surface energy imbalance (surface albedo reduction → net shortwave radiation increase → surface sensible heat flux rise). However, divergent moisture conditions (soil moisture/ET increases or decreases) dictated the dry–wet differentiation of extremes (Figs. S6 and S7).

Under future warming and wetting scenarios, the frequency of WSD are projected to further increase, while DSD may decrease26. These changes will significantly alter the evolution of compound climate extremes through multiple feedback mechanisms and profoundly impact watershed hydrological processes. Specifically, the increased occurrence of WSD events will lead to earlier snowmelt, advancing spring peak runoff. When superimposed with concurrent heavy precipitation events, the combined effects of soil moisture saturation and rapid runoff generation will substantially elevate flood risks4. Meanwhile, although the reduction in DSD may mitigate spring drought stress, the persistent decline in snowpack’s buffering capacity will lead to insufficient summer baseflow recharge. This deficit will synergistically interact with heat extremes and, through reduced soil infiltration rates, establish a positive feedback loop that intensifies hydrological drought40. In a climate-sensitive region such as the TP, the drastic variations in snowpack storage not only directly alter river baseflow composition but also reshape the severity of the Asian monsoon by modulating plateau thermal forcing, thereby amplifying the complexity and uncertainty of global climate change. These findings provide new scientific insights into the coupled mechanisms of the snowpack–climate–hydrology system under climate change and offer critical guidance for regional water resource management and disaster prevention and mitigation.

There are still some limitations in this study. Although the utilized state-of-the-art ERA5-Land and TPMFD datasets feature high spatiotemporal resolution and demonstrate high reliability on the TP, compared to other datasets, the interpolation with a uniform spatial resolution inevitably introduces errors. In addition, there currently exist two predominant approaches for classifying snow droughts. The first method directly categorizes snow droughts into dry and warm types11,16, while the second introduces an additional warm–dry snow drought classification under concurrent warm and dry conditions17,41. The former classification has become more widely adopted due to its straightforward physical interpretation and computational efficiency, which guided our methodological choice. To assess the robustness of our findings, we conducted parallel analyses using both classification schemes for snow drought types (Figs. S8–S10). The results confirm that our core conclusions remain consistent regardless of whether dry snow droughts are combined with warm–dry snow droughts or analyzed as separate categories, demonstrating methodological stability in our findings.

Moreover, when exploring the physical connections between snow droughts and compound climate extremes, we mainly focused on the local-scale hydro-thermal balance mechanisms (e.g., changes in soil moisture and surface albedo). However, large-scale atmospheric circulation (such as the westerly jet and the South Asian monsoon) and teleconnections (such as ENSO and the Arctic Oscillation)42,43,44, which modulate the climate of the plateau, may have a significant impact on the occurrence and severity changes of snow drought and subsequent extreme events. It is necessary to further combine numerical simulations and land–surface models to clarify these complex interactions. Meanwhile, it is recommended to incorporate snow drought dynamics into the key parameters of regional climate models. By integrating future scenario data to analyze the evolution trajectory of WSD–CPHE under the “warming and wetting” scenario, the predictive ability of compound disaster risks on the plateau can be enhanced.

Methods

Study area

The TP (Fig. 1), with an average elevation exceeding 4000 m, is renowned as the “Third Pole” and the “Asian Water Tower”45. Its unique cryosphere topography, interacting with the monsoon–westerlies coupled circulation system, sustains a vital freshwater reservoir dominated by glaciers, snow cover, and lakes5,46. As a sensitive amplifier of global climate change, the TP dynamically regulates the East Asian monsoon, South Asian monsoon, and global atmospheric circulation patterns through its pronounced dynamic and thermal effects47,48,49. However, over the past half-century, the TP has undergone accelerated warming at twice the global average rate50,51,52 that has triggered cascading environmental responses, such as glacier retreat, snowpack depletion, lake expansion, and vegetation greening53. The imbalance in the solid–liquid–ecological water systems not only jeopardizes regional hydrological stability but also intensifies multi-scale coupling effects of extreme events, establishing the TP as a hot spot in global climate risk cascades.

Datasets

We utilized the monthly precipitation and temperature data from the high-resolution near-surface meteorological forcing dataset (TPMFD)54 for the Third Pole region from 1979 to 2022, with a spatial resolution of 1/30°. The TPMFD dataset is generated by fusing high-resolution gridded meteorological data with in-situ observational data, using methods such as random forests and kriging. The high-resolution gridded meteorological data is produced through downscaling the ERA5 reanalysis data using a convolutional neural network (CNN)-based downscaling model. The TPMFD features extensive temporal coverage and high spatial resolution, offering greater accuracy than widely used reanalysis datasets55,56, which makes it a crucial resource for climate analysis and land surface process modeling in the Third Pole region.

Compared to other SWE products such as GlobSnow, Snow-CCI, and GLDAS, ERA5-Land demonstrates superior reliability both in the Northern Hemisphere and TP31,57. Therefore, we selected the ERA5-Land reanalysis product (1979–2022, 0.1°)58 for analyzing snow droughts and the associated surface and atmospheric conditions during their occurrence. The hydrometeorological variables used from the ERA5-Land included monthly SWE, surface albedo, 0–100 cm soil moisture, net solar radiation, surface sensible heat flux, evapotranspiration (ET), and specific humidity.

Additionally, the MODIS/Terra monthly snow cover dataset (MOD10C1)59 from 2001 to 2022 (0.05°), along with the monthly SWE data from ERA5-Land, were used to mask snow-covered regions and determine the first snow-free month (see next section). Aridity Index (AI) data were obtained from the Global Drought Index and Potential Evapotranspiration Database v3 (30 arc-seconds)60 were utilized to classify the TP into five categories (as shown in Fig. S11): hyper-arid (AI < 0.03), arid (0.03 ≤ AI < 0.2), semi-arid (0.2 ≤ AI < 0.5), dry sub-humid (0.5 ≤ AI < 0.65), and humid (AI ≥ 0.65). All datasets were aggregated to a spatial resolution of 0.1° × 0.1° before analyzing.

Identification of snow-covered regions on the TP

Based on the proposed methodologies in refs. 12,16, we first calculated the 3-month average snow cover climatologies by using MODIS/Terra data, and extracted the snow-covered areas by setting a snow cover threshold of 5%. Subsequently, we further refined the snow-covered areas by combining with ERA5-Land SWE data. Specifically, for a given month in all years, at least 75% of the area in the region was required to be covered by snow (i.e., the SWE value was non-zero). In addition, the first snow-free month was defined as the month when the snow cover climatology was below 5% for the first time based on the MODIS/Terra snow cover data, and the month preceding it was regarded as the last snow-covered month61.

Definition of snow droughts and compound climate extremes

For each grid cell, we used the Gringorten plotting position formula62 (Eq. (1)) to calculate the empirical probability of 3-month cumulative SWE (Eq. (2)) for the last snow-covered month. Accordingly, we standardized the results and converted the empirical probability into the standardized snow water equivalent index (SWEI; Eq. (3)).

$$p\left({A}_{m,i}\right)=\frac{i-0.44}{N+0.12}$$
(1)
$${A}_{m,i}={{{S}}{{W}}{{E}}}_{m-2}+{{{S}}{{W}}{{E}}}_{m-1}+{{{S}}{{W}}{{E}}}_{m}$$
(2)
$${{S}}{{W}}{{E}}{{I}}({A}_{m,i})={\Phi }^{-1}[p({A}_{m,i})]$$
(3)

where \(i\) is the rank of the non-zero variable (from smallest to largest); \(N\) is the sample size; \(m\) denotes the last snow-covered month; \({A}_{m,i}\) provides an integrated measure of both the SWE amount and its persistence over each set of 3 months; \({{{{\rm{SWE}}}}}_{m}\) is the SWE value of the last snow-covered month; and \({\Phi }^{-1}\) is the inverse standard normal distribution. The calculation methods for the standardized precipitation index (SPI) and standardized temperature index (STI) are similar to those for SWEI.

When the SWEI of the last snow-covered month was ≤–0.5, a snow drought was considered to have occurred11,63. To explore the different characteristics of snow drought in more depth, we further divided snow drought into two categories: DSD and WSD. Here, DSD is caused by insufficient precipitation, with SWEI ≤ –0.5 and SPI ≤ –0.5; and WSD occurs when precipitation is close to or higher than normal levels, i.e., SWEI ≤ –0.5 and SPI  > –0.516. The WSD is caused by warm temperatures, resulting in precipitation falling as rain rather than snow. We emphasize that the precipitation deficit is not the only factor for triggering DSD based on our classification. Less precipitation is generally accompanied by higher temperature, which may also contribute to the occurrence of DSD41,64. In this study, the severity of snow droughts was defined by the absolute SWEI value during drought episodes, with the occurrence frequency calculated as the proportion of drought years relative to the total observational duration at individual grid cells.

We defined two types of compound climate extremes, including compound dry–hot extremes (CDHE) that were characterized by concurrent dry and hot conditions, as well as compound pluvial-hot extremes (CPHE) that were characterized by concurrent pluvial and hot conditions, using the SPI and the STI to characterize hydrothermal conditions. Here, we employed the Blended Dry and Hot Events Index (BDHI) and the Blended Pluvial and Hot Events Index (BPHI)65 to depict the CDHE and CPHE, respectively. The full names of these abbreviations were provided in Table S1 for clarity. Before calculating the compound indices, the SPI and STI series were divided into negatively dependent sequences (i.e., (SPI ≤ 0, STI > 0) and (SPI > 0, STI ≤ 0)), corresponding to dry/hot and wet/cold conditions) and positively dependent sequences (i.e., (SPI ≤ 0, STI ≤ 0) and (SPI > 0, STI > 0)), corresponding to dry/cold and wet/hot conditions). Studies have shown that BDHI and BPHI are effective in monitoring the occurrence of compound climate extremes25,66. Supposing the SPI and STI as two random continuous variables X and Y, respectively, the specific formulas for BDHI and BPHI are as follows:

$$\begin{array}{rcl}P(X\le x\vee Y > y)=\Bigg\{\begin{array}{cc}1-\nu +{C}^{-}(u,\nu ) & {{i}}{{f}}\,[({{S}}{{P}}{{I}}\le 0,{{S}}{{T}}{{I}} > 0) \& ({{S}}{{P}}{{I}} > 0,{{S}}{{T}}{{I}}\le 0)]\\ 1-\nu +{C}^{+}(u,\nu ) & {{i}}{{f}}\,[({{S}}{{P}}{{I}}\le 0,STI\le 0) \& (SPI > 0,STI > 0)]\end{array}\end{array}$$
(4)
$${{BDHI}}={\Phi }^{-1}[F\{P(X\le x\vee Y > y)\}]={\Phi }^{-1}[F\{1-\nu +{C}^{\pm }(u,\nu )\}]$$
(5)
$$P(X > x\vee Y > y)=\left\{\begin{array}{cc}1-{C}^{-}(u,\nu ) & {{if}}\,\left[({{SPI}}\le 0,{{STI}} > 0) \& ({{SPI}} > 0,{{STI}}\le 0) \right] \\ 1-{C}^{+}(u,\nu ) & {{if}}\,\left[ ({{SPI}}\le 0,{{STI}}\le 0) \& ({{SPI}} > 0,{{STI}} > 0)\right]\end{array}\right.$$
(6)
$${{BPHI}}={\Phi }^{-1}[F\{P(X > x\vee Y > y)\}]={\Phi }^{-1}[F\{1-{C}^{\pm }(u,\nu )\}]$$
(7)

where \(u=P\left(X\le x\right),v=P(Y\le y)\), \(C\left(u,v\right)=P\left(X\le x\wedge Y\le y\right)\). \({C}^{\pm }\left(u,\nu \right)\) contains the joint cumulative distribution functions (CDFs) of copulas with respect to the positive dependence (denoted as \({C}^{+}\left(u,\nu \right)\)) and negative dependence (denoted as \({C}^{-}\left(u,\nu \right)\)) between SPI and STI combination subseries. The joint CDFs were selected from the pools of Gaussian, Clayton, Gumbel, Frank, Joe, Student t, BB1, BB6, BB7, and BB8 bivariate copulas. We used the maximum-likelihood estimation method to estimate the parameters of these bivariate copulas. The best bivariate copula of \({C}^{\pm }\left(u,\nu \right)\) was selected using the minimum Akaike information criterion. With the help of the empirical Gringorten plotting position formula and normal quantile transformation, \(P\left(X\le x\vee Y > y\right)\) and \(P\left(X > x\vee Y > y\right)\) were then transformed into BDHI and BPHI, respectively (Eqs. (5) and (7)). Here, a threshold of –0.8 was applied to identify the condition of compound climate extremes, i.e., BDHI ≤ –0.8 and BPHI ≤ –0.8.

Coincidence analysis of snow droughts and compound climate extremes

On the basis of an event coincidence analysis method67, we quantified the triggering coincidence rate between different types of snow droughts and compound climate extremes for each grid cell. The coincidence rate (\(r\)) is the ratio of the number of coincidences between snow droughts and compound climate extremes, which can be expressed as

$$r(\Delta T,\tau )=\frac{1}{{N}_{{{{\rm{S}}}}}}\displaystyle {\sum }_{i=1}^{{N}_{{{{\rm{S}}}}}}\Theta \left[\displaystyle {\sum }_{j=1}^{{N}_{{{{\rm{H}}}}}}{1}_{[0,\Delta T]}(({t}_{j}^{{{{\rm{H}}}}}-\tau )-{t}_{i}^{{{{\rm{S}}}}})\right]0\le r\le 1$$
(8)

where \(\Theta\) is the Heaviside function:

$$\Theta \; \left(x\right){{{\rm{:= }}}}\left\{\begin{array}{cc}1 & x > 0\\ 0 & x\le 0\end{array}\right.$$
(9)

and \({1}_{[0,\Delta T]}:\) (·) is the indicator function of the selected window [0, ΔT]:

$${1}_{\left[0,\Delta T\right]}\left(x\right) \; {{{\rm{:= }}}}\left\{\begin{array}{cc}1 & x\in \left[0,\Delta T\right]\\ 0 & x\notin \left[0,\Delta T\right]\end{array}\right.$$
(10)

Here, \({N}_{{{{\rm{S}}}}}\) and \({N}_{{{{\rm{H}}}}}\) represent the total number of snow droughts and compound climate extremes, respectively, while \({t}_{i}^{{{{\rm{S}}}}}\) and \({t}_{j}^{{{{\rm{H}}}}}\) denote the time series of snow droughts and compound climate extremes, respectively. The parameter \(\tau\) is a time lag, indicating \((({t}_{j}^{H}-\tau )-{t}_{i}^{S})\le \Delta T\).

Considering the possible lag effect of snow droughts, we selected a lag of 1–3 months after the end of snow droughts to analyze the coincidence rate of compound climate extremes. Meanwhile, to test the significance of the coincidence rate between snow droughts and compound climate extremes, we adopted the shuffle test method31. Specifically, we randomly changed the time series of snow droughts while keeping the time sequence of compound climate extremes unchanged. This process was repeated 1000 times to calculate the coincidence rate between snow droughts and compound climate extremes in each randomly generated time series. If the observed coincidence rate exceeded 95% in the 1000 randomly generated time series (i.e., p < 0.05), the result was considered statistically significant31. The coincidence analysis only considered statistically significant results.

Mechanistic analysis of snow droughts triggering compound climate extremes

To elucidate the mechanisms by which snow droughts trigger compound climate extremes, we employed structural equation modeling (SEM), as a statistical technique based on covariance matrix analysis of multivariate data, to further investigate how snow droughts influence the occurrence of compound climate extremes by altering hydrothermal fluxes. SEM is effective in revealing potential causal relationships among diverse variables68. We utilized partial least squares SEM (PLS-SEM) to examine the impact of snow droughts on hydrothermal fluxes and their pathway relationships with coincidence rates.