Persistence of hydrological droughts in Slovakia: a Markov-chain analysis of multi-season river flow variability

Abstract

Background/introduction:

Drought persistence across physiographically and climatically diverse river basins remains poorly quantified, limiting probabilistic water-resource planning under a warming climate. This study provides the first such assessment for Slovakia, analyzing 17 gauging stations with long near-natural discharge records spanning Carpathian headwaters to lowland reaches.

Methods:

Daily discharges were aggregated into winter (October–March) and summer (April–September) seasons; seasons with negative anomalies were classified as dry, and others as wet. Conditional dry-to-dry (Pdd) and wet-to-wet (Pww) transition probabilities were estimated at each station, and two-state first-order Markov chains were fitted. Long synthetic series were generated to compare observed and simulated spell-length distributions, derive 100-year maximum dry and wet spells, and estimate probabilities of exceeding multi-year drought thresholds over 5- and 10-year horizons.

Results:

Drought persistence varies markedly across the study domain. High-elevation forested headwaters exhibit high seasonal variability but low persistence, whereas lowland and larger basins in southern and western Slovakia have Pdd > Pww and dry spells of 13–19 seasons. First-order Markov chains reproduce observed spell-length distributions well at most stations. Simulations indicate 100-year meteorological droughts of approximately 3-4 years in alpine basins but up to a decade in some lowland rivers, with a 3–5-year drought likely within a 10-year horizon.

Discussion/conclusion:

At several sites, the maximum observed wet spells exceed modelled 100-year events, revealing asymmetric wet-dry persistence not fully captured by first-order Markov chains. These findings offer the first probabilistic framework for multi-season drought duration in Slovakia, with direct implications for water-resource planning under ongoing climate warming.

1 Introduction

Climate change is fundamentally transforming the global water cycle, increasing the frequency and intensity of hydro-climatic extremes that threaten water security worldwide (IPCC, 2023). Rising temperatures increase evaporation and alter rainfall patterns, resulting in more severe floods and longer droughts that challenge current water management methods (Trenberth et al., 2014). However, this relationship is not uniform across all regions. Recent global-scale analyses indicate that rising temperatures do not always lead to increased evaporation, as relative humidity tends to remain fairly constant while specific humidity rises in line with the Clausius–Clapeyron relation. Consequently, the expected intensification of the hydrological cycle is not consistently observed, with some regions even showing signs of de-intensification in recent decades (Koutsoyiannis, 2020). These impacts vary regionally due to differences in topography, land use, and atmospheric circulation, resulting in distinct vulnerability profiles across regions (Gudmundsson et al., 2021). Understanding the duration of these extremes, particularly their tendency to occur in clusters of dry or wet conditions is essential for developing effective adaptation strategies for water-sensitive sectors such as agriculture, hydropower, and municipal water supplies. This is why the study of wet and dry spells is significant, particularly given the strong link between trends in these spell characteristics and the occurrence of hydrological extremes (Beniston, 2004; Pichard et al., 2017). Understanding these patterns enriches scientific knowledge and helps develop policies to reduce the impact of climate hazards on communities worldwide (Field and Barros, 2014; Barnes, 2018).

Central Europe shows a rising trend in hydrological volatility, marked by devastating floods and increasingly prolonged droughts over recent decades. Notable floods occurred in 1997, 2002, and 2013, with the last causing damages exceeding e12 billion in the Danube and Elbe regions (Blöschl et al., 2019). At the same time, the summers of 2003, 2015, 2018, and 2022 experienced unprecedented drought conditions, with the 2022 drought being one of the worst in 250 years based on soil moisture shortages and agricultural damage (Hari et al., 2020; Toreti et al., 2022; Leščešen et al., 2025). An increase in summer hydrological droughts has been confirmed in a comprehensive study across Eastern and Central Europe (Bates et al., 2008). These extreme events are occurring alongside rising temperatures; mean annual temperatures in Central Europe have increased by about 2°C since pre-industrial times, exceeding the global warming rate (Hanel et al., 2018; Kjellström et al., 2018). Additionally, the timing of these events is shifting: spring snowmelt floods are decreasing due to reduced snow accumulation, summer convective storms are becoming more intense, and autumn droughts are extending into winter (Parajka et al., 2016; Stahl et al., 2016; Brunner et al., 2025). This shift in seasonal water distribution presents significant challenges for reservoir management, irrigation planning, and ecological flow maintenance (Mohammed et al., 2022).

Although there is increasing scientific evidence of changing hydrometeorological conditions in Slovakia (Kubiak-Wójcicka et al., 2023; Garaj et al., 2023; Pekárová et al., 2025a), a key methodological gap remains: measuring the duration of drought and wet periods over time. Traditional frequency analysis treats each event as independent, overlooking the “memory” characteristic of hydrological systems, where dry conditions often cluster over multiple seasons or years (Salas et al., 2018; Tatli and Dalfes, 2020). Markov chain models offer a useful framework for analyzing this persistence by explicitly estimating the likelihood that a system in a given state (such as dry) will transition to another state (such as normal or wet) in the next period (Azimi et al., 2020; Raju, 2026). First-order Markov chains, which assume that the current state depends only on the previous one, are particularly valuable in hydrology because of their simplicity and their ability to replicate observed clustering of extremes (Rohith et al., 2021). However, first-order Markov chains are often inadequate to fully capture the long-term statistical characteristics of the streamflow process due to its notable long-range dependence, known as Long-Term Persistence (LTP) behavior. While the seasonal (half-year) first-order Markov model used in this study provides a parsimonious and effective approximation for multi-season drought persistence, the underlying LTP at the daily scale explains residual discrepancies between observed and simulated extreme spell lengths. This LTP, first identified by Hurst (1951) in the Nile River and thoroughly reviewed by Beran et al. (2013), is remarkably strong in streamflow (Hurst parameter H > 0.8) and introduces high inherent uncertainty that spans more than eight orders of magnitude, ranging from sub-hourly records to multi-centennial paleoclimatic reconstructions (Pizarro et al., 2022). As a result, the LTP dominates the observed clustering of consecutive wet years, causing intense flood periods, and consecutive dry years, leading to prolonged episodes of water scarcity. This LTP behavior is also supported by other long-term streamflow studies, such as those in the Yellow River basin (Wang et al., 2023). Similar strong LTP has been documented for the Nile River (Koutsoyiannis et al., 2008). These observations align with the general consensus in the literature that streamflow records from gauges are highly uncertain. They are consistent with the IAHS scientific perspective on hydrological change and uncertainty presented in the Panta Rhei initiative (Montanari et al., 2013). These models enable the stochastic simulation of long synthetic sequences that retain the statistical characteristics of real data, including the distribution of extreme event durations that may exceed the historical record length. This is essential for infrastructure and water resource planning, especially for rare but severe events such as 100-year droughts, which may not be fully represented in typical observational records of 50–80 years (Srikanthan and McMahon, 2001; Link et al., 2020). Additionally, analyzing different physiographic zones and elevation bands separately helps Markov chain models demonstrate how persistence varies across regions, guiding localized management strategies.

This study addresses these gaps through a comprehensive Markov-chain analysis of multi-season precipitation and river flow anomalies across Slovakia's diverse physiographic gradients. Specifically, we pose three research questions: (RQ1) How have the frequency and duration of multi-season dry and wet spells varied historically across Slovakia's distinct physiographic and altitudinal gradients? (RQ2) To what extent can a stochastic Markov chain approach capture the persistence and extreme durations of these observed anomalies in both headwater and lowland catchments? (RQ3) What is the statistical likelihood of severe multi-year droughts occurring within future decadal planning horizons? By integrating long-term precipitation records with streamflow data from Slovakia's major river basins and applying Markov chain modeling to quantify state persistence and simulate extreme scenarios, we aim to provide water resource managers with probabilistic insights into the temporal clustering of hydrometeorological extremes under changing climatic conditions.

2 Study area

The study area covers the Slovak Republic, a landlocked country in Central Europe with an area of 49,035 km². It is bordered by Poland, Ukraine, Hungary, Austria, and the Czech Republic (Figure 1). The country has diverse topography, with the Carpathian Mountains occupying much of the northern and central regions, culminating in the highest peak, Gerlachovský štít, at 2,655 m. The southern region consists mainly of lowland areas, including the Danubian and Eastern Slovak Lowlands (Pekárová et al., 2025b). This varied terrain significantly influences local climatic conditions, precipitation patterns, and river discharge regimes. Slovakia has a temperate continental climate, characterized by cold winters and warm summers, with annual precipitation ranging from approximately 500 mm in the lowlands to over 2,000 mm in mountainous areas (Domokos and Sass, 1990; Petrovič et al., 2010).

Figure 1

The river network of Slovakia is predominantly shaped by the Danube, which traverses the southern part of the country and serves as a natural border with Hungary. Approximately 96% of Slovakia's territory drains into the Black Sea Basin, primarily through the Danube and its major tributaries, including the Morava, Váh, Hron, and Ipel. The remaining 4% drains into the Baltic Sea via the Poprad and Dunajec. Recent hydro-meteorological data indicate a decline in the 30-year moving average of annual precipitation from 760 mm to 730 mm between 1901 and 1995, followed by a return to 760 mm since 1996 (Faško et al., 2022a). Concurrently, the 30-year moving average of annual runoff decreased from 250 to 230 mm, reaching a peak in 2013, with levels remaining relatively stable after 2010 (Bačová Mitková et al., 2024). Despite increased precipitation, rising temperatures after 2,000 have resulted in enhanced evapotranspiration and consequent reductions in river discharge. Forest coverage has expanded from 34% in 1931 to 41.4% in 2020 (Pekárová et al., 2025b).

3 Data

This study is based on long-term discharge records from 17 Slovak river gauging stations operated by the Slovak Hydrometeorological Institute (SHMI). The stations were selected because they are part of the National Climate Programme's reference network of sites with homogeneous, near-natural flow regimes. “Near-natural” is defined by SHMI as < 5% flow alteration by upstream reservoirs or abstractions, stable rating curves since the 1930s, and no significant land-use change after 1930 (Bačová Mitková et al., 2024). Records extend from 1901 (Moravský Svätý Ján) to 2023, with 15 of 17 stations covering ≥1928–2023. Two criteria were used when selecting stations: the length of the available dataset and the absence of missing values. To ensure data integrity, SHMI has implemented strict measures to control data quality. This organization adhered to rigorous standards to guarantee the accuracy and consistency of the data series used in this study. In doing so, they followed their internal instructions and the recommendations of the World Meteorological Organization (WMO). The catchments cover a broad range of physiographic conditions, from small, steep headwaters in the Western Carpathians to large lowland basins of the Danube system; their locations and basic physical–geographical characteristics are shown in Figure 1 and summarized in Table 1. Detailed monthly statistics (double periodicity) and autocorrelation structures (lags 1–12) for each of the 17 discharge time series are provided in Supplementary Materials S1, S2. SHMI applies rigorous procedures for discharge measurement, rating-curve maintenance, and data archiving, following both internal standards and recommendations of the World Meteorological Organization (WMO). As a result, the daily discharge series used here is virtually gap-free and has undergone several stages of quality control. For the purposes of this paper, daily flows were first screened visually, then aggregated to monthly values (means) and subsequently to half-year averages for winter (November–April) and summer (May–October) in accordance with the Slovak hydrological year, which runs from 1 November to 31 October of the following calendar year (Pekárová et al., 2025b). The use of the hydrological year ensures that precipitation and snowmelt contributing to runoff within a given accumulation–recession cycle are treated consistently. These carefully curated long discharge records provide a robust basis for analyzing multi-season drought persistence and for calibrating and validating the Markov-chain models employed in this study. Forest cover ranges from 28% (lowland) to 85% (high-elevation headwaters) and increased by 6%−9% since 1930 in 12 of 17 catchments, consistent with the documented national expansion from 34% to 41.4% (Pekárová et al., 2025b).

4 Methods

The analysis proceeds through four methodological steps that directly address the three research questions formulated in Section 1. First, seasonal discharge anomalies are computed to identify dry and wet states (Section 4.1; addresses RQ1). Second, Markov transition probabilities and observed spell-length distributions are estimated (Section 4.2; addresses RQ1 and RQ2). Third, stochastic simulations using first-order Markov chains are performed to validate the model and estimate 100-year extreme events (Section 4.3; addresses RQ2). Fourth, bootstrap risk assessment quantifies the probability of multi-year droughts within planning horizons (Section 4.4; addresses RQ3).

4.1 Calculation of seasonal anomalies

The initial methodological step sets the stage for addressing RQ1 by converting continuous discharge records into a binary classification of seasonal hydrological states. This step is crucial because Markov-chain analysis needs discrete state definitions, with the division between “dry” and “wet” seasons forming the foundation for all subsequent persistence calculations. Only stations with long, near-natural records and minimal documented changes in rating curves or water management were included, ensuring that observed persistence patterns reflect climatic factors rather than human activities. Monthly averages were combined into half-year totals representing winter (November–April) and summer (May–October) seasons. Half-year aggregation, instead of traditional 3-month meteorological seasons, was chosen to represent the rising and falling limbs of the Slovak hydrological year, capturing seasonal water accumulation and depletion cycles that control water resource availability. Seasonal anomalies were calculated as percentage deviations from the 1961–1990 half-year means, following WMO climatological practices and consistent with previous drought studies in Central Europe (Wilby, 2007; Leščešen et al., 2024; Bartczak et al., 2024). This standardization allows for direct comparison across stations with widely differing mean discharges. Seasons with negative anomalies were classified as “dry,” while those with zero or positive anomalies were classified as “wet,” resulting in binary time series suitable for Markov-chain analysis. Although this binary classification simplifies the continuous nature of discharge variability, it offers a practical framework to quantify the tendency for below-average conditions to cluster over time, which characterizes hydrological drought.

4.2 Estimation of transition probabilities and spell lengths

The second methodological step directly addresses RQ1 by quantifying how dry and wet conditions persist across seasons, and provides the empirical foundation for RQ2 by establishing the observed persistence structure against which Markov model performance is evaluated. Conditional dry-to-dry (Pdd) and wet-to-wet (Pww) Markov transition probabilities were derived for each station using maximum likelihood estimation. Pdd was calculated as the proportion of dry seasons followed by another dry season:

where Ndd is the count of dry-to-dry transitions and Ndw is the count of dry-to-wet transitions. Pww was computed similarly for transitions from wet seasons. These transition probabilities quantify the “memory” of the hydrological system: values approaching 1.0 indicate strong persistence, while values near 0.5 suggest random alternation between states. By calculating Pdd and Pww separately for each station, spatial patterns in drought persistence across Slovakia's altitudinal and physiographic gradients can be identified, directly addressing RQ1. The lengths of all continuous runs of dry and wet seasons were recorded for each station, providing empirical frequency distributions of spell duration and identifying the longest observed dry and wet spells. These observed distributions serve two purposes: they characterize historical drought duration variability (RQ1) and provide the benchmark against which Markov model simulations are validated (RQ2). To detect temporal changes in drought persistence that may undermine the stationarity assumption underlying Markov-chain analysis, two complementary statistical tests were applied to the seasonal variability series at each station. The Mann–Kendall non-parametric test (Mann, 1945; Kendall, 1975) was used to detect monotonic trends in seasonal discharge anomalies, while the Pettitt test (Pettitt, 1979) was employed to identify significant change points indicating abrupt regime shifts. Both tests were performed at the α = 0.05 significance level. This approach, following Wilby (2001) and Minárik et al. (2023), enables detection of gradual or abrupt changes in drought persistence that may reflect evolving climatic or land-surface conditions. Stations exhibiting significant trends or change points were flagged for interpretation within the context of non-stationary drought risk.

4.3 Stochastic simulation with first-order Markov chains

To improve the estimation of 100-year extreme dry and wet spells (RQ3), the Pareto–Burr–Feller (PBF) distribution was fitted to the seasonal discharge anomalies at each station; this choice follows recent literature demonstrating that the PBF provides an excellent representation of the heavy-tailed marginal structure of streamflow across scales and enables reliable extrapolation beyond the observed record length (Pizarro et al., 2022; Dimitriadis et al., 2021). The third methodological step addresses RQ2 by testing whether the simple first- order Markov assumption adequately captures the observed persistence structure at each station and by extending the analysis beyond the limited observational record to estimate the characteristics of rare, extreme drought events. Following established approaches in hydrological drought analysis (Wilby, 2007; Wilby et al., 2015, 2016), the observed Pdd and Pww values were used to generate long synthetic sequences of dry and wet half-years for each station. The first- order Markov assumption posits that the probability of the system being in a given state depends only on the state in the immediately preceding season, not on earlier history. This assumption is parsimonious and widely used in hydrology, but its validity must be empirically verified. A single 10,000-season Markov simulation was performed for each station (equivalent to 5,000 years of synthetic record), initialized from a dry state. At each time step, a uniform random number between 0.1 and 1 was generated and compared against the relevant transition probability: if the current state was dry and the random number was less than Pdd, the system remained dry; otherwise, it transitioned to wet. The reverse logic applied for wet states using Pww. This Monte Carlo procedure generates synthetic spell- length distributions that preserve the estimated persistence structure. The resulting binary sequences were converted to dry- and wet-spell lengths and compared with the observed distributions using the two-sample Kolmogorov–Smirnov (KS) test. Where the KS statistic was not significant at the 5% level, the first-order Markov model was considered to provide an adequate representation of observed spell-length distributions, supporting its use for extrapolation to rare events (RQ2). Stations where the KS test indicated significant differences between observed and simulated distributions were examined for evidence of higher-order memory or non-stationarity.

4.4 Bootstrap risk assessment

The fourth and final methodological step directly addresses RQ3 by translating the estimated persistence structure into probabilistic risk metrics relevant to water resource planning over 5- and 10-year horizons. A bootstrap Markov simulation was conducted to assess extreme drought persistence and near-term risk. For each station, 1,000 independent realizations of 100-year sequences (200 half-years) were generated using the station-specific transition probabilities. The maximum dry and wet spell lengths in each realization were recorded to construct empirical distributions of 100-year extremes. The mean values from these distributions were taken as the modeled 100-year dry and wet spells and compared with the longest events in the historical record. This comparison reveals whether observed extreme droughts are consistent with the inferred persistence structure or represent statistically rare events that may recur under similar climatic conditions. In a complementary Monte Carlo experiment designed to provide actionable risk metrics for water managers (RQ3), 100,000 independent trajectories were simulated for 5- and 10-year planning horizons, each initialized from the last observed state (dry or wet as of the end of the observational record). For each trajectory, the maximum run of consecutive dry half-years was identified. These simulations were used to estimate:

• The probability of experiencing droughts of at least 2-, 3-, 4-, and 5-years duration within each planning horizon;

• The full distribution of maximum dry-spell length over the next decade;

• The conditional risk of extended drought given current hydrological conditions.

These metrics directly inform reservoir operating rules, drought contingency planning, and groundwater abstraction licensing, yet design assumptions typically assume droughts lasting no more than 2–3 years. By quantifying the probability of more extended events, this analysis provides water resource managers with the probabilistic foundation needed to evaluate whether existing infrastructure and management protocols are adequate for the range of drought durations that may plausibly occur (RQ3). All computations were performed in Python 3.11 using the NumPy (v1.24), pandas (v2.0), SciPy (v1.11), and Matplotlib/Seaborn libraries for numerical computation, data manipulation, statistical testing, and visualization, respectively.

5 Results

5.1 Spatial patterns of seasonal discharge anomalies and dry-spell duration

The first research question asks how the frequency and duration of multi-season dry and wet spells have varied historically across Slovakia's distinct physiographic and altitudinal gradients. To address this question, we analyzed seasonal discharge anomalies, transition probabilities, spell-length distributions, and temporal trends across all 17 gauging stations.

5.1.1 Spatial heterogeneity in drought persistence

The analysis of seasonal discharge anomalies reveals pronounced spatial heterogeneity that closely follows Slovakia's altitudinal and physiographic gradients (Figure 2). Two distinct behavioral clusters emerge: high-elevation forested headwaters with moderate persistence and high variability, and lowland basins with higher persistence and extended dry spells.

Figure 2

High-elevation stations in the Carpathian Mountains, including Podbanské (923 m), Bystrá (575 m), Mýto pod Dumbierom (617 m), and Hronec (480 m), show considerable interseasonal variability in discharge anomalies but relatively limited persistence of deficit conditions. These mountain catchments, characterized by mean basin altitudes exceeding 800 m and drainage areas less than 250 km², often experience positive anomalies due to orographic precipitation enhancement, yet dry conditions rarely last beyond 6–10 consecutive seasons. The dry-to-dry transition probabilities (Pdd) at these stations range from 0.52 to 0.69, reflecting the dynamic precipitation regime of alpine environments, where intense yet variable rainfall events interrupt potential drought sequences. In contrast, lower-elevation and larger river basins in southern and western Slovakia exhibit smaller individual seasonal anomalies but show a notable tendency toward prolonged periods of below-average discharge. Moravský Svätý Ján on the Morava River (24,129 km²), Brehy on the Hron River (3,821 km²), Holiša on the Ipel River (685 km²), and Nitrianska Streda on the Nitra River (2,094 km²) all display this characteristic pattern. These lowland basins show elevated Pdd values ranging from 0.65 to 0.73, with maximum observed dry spells lasting from 13 to 19 consecutive seasons (approximately 6.5–9.5 years). The prolonged drought periods in these regions reflect the dominance of large-scale atmospheric circulation anomalies affecting extensive areas, combined with reduced orographic rainfall, increased evapotranspiration under warmer conditions, and the buffering capacity of large alluvial aquifers that delay hydrological responses to meteorological deficits. Table 2 summarizes the longest continuous dry spells at each station, revealing a fundamental spatial gradient in drought duration across Slovakia's river network. The most severe event occurred at Holiša, where 19 consecutive dry seasons spanned from winter 1986 to winter 1995, resulting in a drought lasting approximately 9.5 years. Similar multi-year droughts were recorded at Moravský Svätý Ján (14 seasons, about 7 years), Brehy (13 seasons, about 6.5 years), Pláštovce–Litava (13 seasons), Hanušovce (13 seasons), and several stations along the Hron River system. In contrast, the longest dry periods in high-altitude headwater catchments (Podbanské, Kysucké Nové Mesto) lasted only six seasons (about 3 years), highlighting the clear elevation-related gradient in drought persistence.

The temporal clustering of maximum dry spells is noteworthy. The majority of record-breaking droughts occurred during the late 1980s and early 1990s, with 11 of 17 stations recording their longest dry spells during 1986–1995. This synchronicity suggests that a large-scale, persistent atmospheric circulation anomaly affected much of Central Europe during this period. However, several stations (Mýto pod Dumbierom, Dolná Lehota, Banská Bystrica, Podbanské, Kysucké Nové Mesto, and Šala) recorded their longest dry spells during the more recent 2012–2017 period, indicating that contemporary droughts are approaching or matching historical persistence levels in multiple catchments.

5.1.2 Temporal trends and regime shifts

To provide quantitative confirmation of apparent temporal changes in drought frequency, Mann–Kendall trend tests and Pettitt change-point analyses were applied to the seasonal discharge variability series at all 17 stations (Table 3). These analyses address the temporal dimension of RQ1 by identifying whether drought persistence has changed systematically over the observational period.

The Mann–Kendall results confirm a systematic shift toward drier conditions across much of Slovakia's river network. Significant negative trends (p < 0.05) were detected at 12 of 17 stations, representing 71% of the monitoring network. The strongest declining trends occurred at lowland stations: Pláštovce–Litava (Z = −4.79, p < 0.001), Pláštovce–Krupinica (Z = −4.74, p < 0.001), and Mýto pod Dumbierom (Z = −3.95, p < 0.001). These stations, despite spanning different catchment sizes and locations, share exposure to warming and increased evapotranspiration that has characterized Slovakia since the late 20th century. Notably, no significant trends were detected at three high-elevation or well-forested stations: Podbanské (Z = 0.54, p = 0.59), Kysucké Nové Mesto (Z = −1.11, p = 0.27), and Liptovský Mikuláš (Z = −1.21, p = 0.23). The absence of significant drying trends at these stations suggests that orographic precipitation enhancement and extensive forest cover may buffer mountain catchments against the regional drying trends affecting lowland Slovakia, a finding with important implications for watershed management and climate adaptation strategies. The Pettitt change-point analysis complements the trend analysis by identifying abrupt regime shifts in the discharge records. Significant change points (p < 0.05) were detected at 13 stations, with two distinct periods emerging as common transition points. Six stations exhibited change points around index 75, corresponding to approximately 1968–1969, while seven stations showed change points at index 102, corresponding to circa 1982–1983. The earlier shift (1968–1969) coincides with the end of a notably wet period in Central Europe documented in palaeoclimatic reconstructions (Hänsel, 2020), while the later shift (1982–1983) corresponds to the onset of accelerated warming across the region (Rottler et al., 2020; Blöschl et al., 2019). The latest significant change point was detected at Moravský Svätý Ján (index 177, approximately 1989, p = 0.005), reflecting the delayed response of this large transboundary basin (24,129 km²) to regional climate shifts—a lag attributable to the extensive groundwater storage and long residence times characteristic of large alluvial systems. Stations without significant change points (Podbanské, Kysucké Nové Mesto, Hanušovce, Liptovský Mikuláš, and Šala) are predominantly located in regions with higher natural precipitation variability, greater hydrological buffering from groundwater or forest cover, or, in the case of Šala, potential smoothing effects from upstream reservoir operations. Overall, the historical frequency and duration of multi-season dry spells vary markedly across Slovakia's physiographic gradients. High-elevation Carpathian headwaters experience shorter but more variable dry spells (6–10 seasons maximum, Pdd = 0.52–0.69), while lowland basins in southern and western Slovakia experience longer, more persistent droughts (13–19 seasons maximum, Pdd = 0.65–0.73). Significant drying trends affect 71% of stations, with regime shifts clustering around 1969 and 1982–1983. Mountain catchments with strong orographic precipitation and forest cover are more resilient to these regional drying trends.

5.2 Markov chain validation and model performance

The second research question asks to what extent a stochastic Markov chain approach can capture the persistence and extreme durations of observed anomalies in both headwater and lowland catchments. To address this question, we compared observed and simulated spell-length distributions, evaluated model fit using Kolmogorov–Smirnov tests, and assessed whether the Markov model reproduces the characteristics of extreme events.

5.2.1 Transition probabilities and persistence asymmetry

Table 4 presents the conditional dry-to-dry (Pdd) and wet-to-wet (Pww) transition probabilities estimated from the observational record, alongside the longest observed dry and wet spells and the mean simulated 100-year extreme events at each station.

A clear pattern emerges in Table 4: at 16 of 17 stations, Pdd exceeds Pww, indicating that dry conditions are more persistent than wet conditions across Slovakia's river network. This asymmetry has significant implications for water resource management, as it suggests that once a drought starts, the likelihood of continued deficit is higher than the likelihood of a sustained surplus during wet conditions. The only exception is Podbanské, the highest-elevation station in the network (922 m), where Pww (0.556) slightly surpasses Pdd (0.522). This anomaly is due to the orographic precipitation regime of the High Tatras, where persistent westerly and north-westerly airflows often bring multi-season runs of above-average rainfall. The persistence asymmetry is most evident at lowland stations such as Hanušovce (Pdd = 0.701, Pww = 0.328), Holiša (Pdd = 0.732, Pww = 0.589), and Kysucké Nové Mesto (Pdd = 0.518, Pww = 0.293), where dry persistence significantly exceeds wet persistence.

5.2.2 Comparison of observed and simulated spell-length distributions

Figure 3 compares the observed and Markov-chain-simulated distributions of dry-spell durations across all 17 stations. The first-order Markov model effectively reproduces the characteristic geometric decline in spell frequency with increasing duration, which is the theoretical expectation for a memory-one stochastic process.

Figure 3

The agreement between observed and simulated distributions is particularly close in small to medium mountain basins. At Podbanské, Bystrá, Hronec, and Mýto pod Dumbierom, the simulated probability curves almost exactly match the observed frequencies for spell durations up to 5–6 seasons. This close agreement confirms that local precipitation persistence in these catchments can be adequately represented by a first-order (memory-one) Markov process, validating the model's fundamental assumption for mountain environments. Kolmogorov–Smirnov tests confirm that the first-order Markov model provides an adequate statistical representation of observed dry-spell distributions at 15 of 17 stations (p > 0.05). The two stations where marginal discrepancies were detected—Moravský Svätý Ján and Nitrianska Streda—are both large lowland basins where the model tends to slightly underestimate the frequency of intermediate-length dry spells (4–8 seasons). This systematic underestimation suggests that large-scale circulation patterns affecting these extensive catchments may exhibit longer memory than a first-order chain captures, or that gradual climatic changes have altered transition probabilities over time within these basins. This daily-scale memory is consistent with the strong long-range dependence (LTP) discussed in the Introduction and explains the minor deviations between observed and simulated extreme spell lengths shown in Figure 5.

5.2.3 Reproduction of extreme events

A key test of the Markov model's usefulness for water resource planning is its ability to reproduce the characteristics of extreme drought events that may exceed the observational record. Figure 4 compares the longest observed dry and wet spells at each station with the empirical distributions of 100-year maximum spell lengths derived from 1,000 bootstrap simulations.

Figure 4

For dry spells, the model performs well across most of the station network. At 13 of 17 stations, the observed maximum dry spell falls within the interquartile range of the simulated 100-year distribution, indicating that previously recorded drought events are statistically consistent with the inferred persistence structure. This consistency increases confidence that the Markov model captures the key characteristics of drought persistence and can be used to estimate the likelihood of future extreme events. However, at several vulnerable lowland stations, including Holiša (observed: 19 seasons; simulated mean: 13.7 seasons), Brehy (observed: 13 seasons; simulated mean: 10.9 seasons), and Moravský Svätý Ján (observed: 14 seasons; simulated mean: 11.3 seasons), the observed maximum dry spell approaches or exceeds the 75th percentile of the simulated distribution. This suggests that the late 20th-century observational record may already have captured relatively rare drought-persistence events in these lowland basins, or that the first-order Markov assumption slightly underestimates extreme dry spell persistence in large, climatically integrated catchments. A contrasting pattern emerges for wet spells. At multiple stations, particularly in mountainous regions, the observed maximum wet spell approaches or exceeds the 90th percentile of the simulated distribution. For example, at Pláštovce–Krupinica (observed: 14 seasons; simulated mean: 7.1 seasons), Nové Zámky (observed: 14 seasons; simulated mean: 7.5 seasons), and Banská Bystrica (observed: 13 seasons; simulated mean: 6.7 seasons), the model significantly underestimates the persistence of extreme wet spells. This asymmetry shows that while the first-order Markov model adequately captures dry-spell characteristics, it may underestimate the duration of exceptional multi-season wet episodes, particularly in orographic environments where persistent atmospheric flow patterns can sustain above-average precipitation for extended periods. Overall, the first-order Markov chain approach effectively captures the key persistence characteristics of drought in Slovak catchments. The model replicates observed dry-spell distributions at 15 of 17 stations (88%), with especially close agreement in mountainous basins. Observed maximum dry spells generally fall within the interquartile range of simulated 100-year events, confirming statistical consistency. However, the model displays systematic asymmetry: it slightly underestimates extreme dry-spell persistence in large lowland basins and considerably underestimates extreme wet-spell persistence across the network, suggesting that wet persistence may involve longer memory or different physical mechanisms than dry persistence.

5.3 Decadal drought risk assessment

The third research question asks what is the statistical likelihood of severe multi-year droughts occurring within future decadal planning horizons. To address this question directly relevant to water resource management, we conducted Monte Carlo simulations to estimate the distribution of maximum drought duration over 10-year horizons and the probability of exceeding specific drought duration thresholds within 5- and 10-year planning periods.

5.3.1 Distribution of maximum drought duration over planning horizons

Figure 5 presents the simulated distributions of maximum dry-spell length over a 10-year (20-season) planning horizon for all 17 stations. These distributions translate the abstract transition probabilities into concrete risk metrics that can inform reservoir operating rules, drought contingency planning, and groundwater abstraction licensing.

Figure 5

All stations exhibit right-skewed distributions, as expected from the geometric nature of Markov spell-length processes. However, the shape parameters vary systematically with catchment characteristics, resulting in distinct risk profiles across Slovakia's physiographic gradient. In smaller alpine basins with lower Pdd values—Podbanské (Pdd = 0.522), Mýto pod Dumbierom (Pdd = 0.543), and Kysucké Nové Mesto (Pdd = 0.518)—the modal maximum dry spell over a 10-year horizon is 3–4 seasons (1.5–2 years), and the probability of experiencing spells exceeding 8–10 seasons is negligible (< 5%). These mountain catchments have a drought risk profile characterized by frequent but relatively short hydrological deficits, consistent with their exposure to variable orographic precipitation that tends to interrupt developing drought sequences. In contrast, larger and lower-elevation basins show broader distributions with substantial probability mass in the upper tail. At Holiša (Pdd = 0.732), Pláštovce–Litava (Pdd = 0.698), Moravský Svätý Ján (Pdd = 0.673), and Brehy (Pdd = 0.664), the distributions extend well beyond eight seasons, with a 10%−20% probability of experiencing maximum dry spells exceeding 10 seasons (5 years) within any given decade. These lowland basins face a fundamentally different risk profile: although their extensive groundwater and alluvial storage provides resilience against isolated dry seasons, they become increasingly vulnerable to structural water scarcity and ecosystem degradation when subjected to multi-year deficits, the type of event that the Markov simulations indicate is plausible within a single planning decade.

5.3.2 Exceedance probabilities for drought duration thresholds

Figure 6 presents exceedance probabilities for droughts of specified minimum durations (2, 3, 4, and 5 years) over both 5- and 10-year planning horizons. These threshold-based metrics directly address the needs of water resource managers who must design infrastructure and operating rules for specific drought durations.

Figure 6

Several patterns emerge from the exceedance probability analysis, with direct implications for water resource planning across Slovakia. Two-year droughts, defined as four consecutive dry seasons and representing the typical “design drought” assumed in many reservoir operating rules, have a high probability of occurrence across the monitoring network. Over a 10-year horizon, exceedance probabilities commonly exceed 0.60 at high-persistence lowland stations, with values reaching 0.72 at Holiša, 0.68 at Pláštovce–Litava, 0.64 at Brehy, and 0.63 at Moravský Svätý Ján. Even at lower-persistence mountain stations, 10-year exceedance probabilities remain substantial, with Podbanské at 0.45 and Mýto pod Dumbierom at 0.48. These findings indicate that 2-year droughts should be considered routine planning events rather than rare extremes throughout Slovakia. The probability of experiencing at least one 3-year drought (six consecutive dry seasons) within a decade remains appreciable at high-persistence stations, ranging from 0.35 to 0.45 at Holiša, Pláštovce–Litava, Hanušovce, and Brehy, while declining to 0.20–0.30 at mountain stations. This implies that infrastructure designed to withstand only 2-year droughts faces a non-negligible risk of exceedance within typical planning horizons at lowland stations. Four-year droughts (eight consecutive dry seasons) are extended events with 10-year exceedance probabilities of 0.15–0.25 at the highest-persistence lowland stations, dropping below 0.10 at most mountain stations. Although less frequent, such droughts would severely stress water supply systems designed for shorter deficit periods. Five-year droughts (ten consecutive dry seasons) generally have probabilities below 0.10 at all stations but remain notable at 0.05–0.10 at Holiša, Pláštovce–Litava, and Hanušovce. These rare but plausible events would exceed the design assumptions of most existing water infrastructure in Slovakia. The contrast between 5- and 10-year horizon probabilities follows expected patterns: the longer the exposure period, the higher the risk. However, the magnitude of increase varies with station persistence characteristics. At high-Pdd lowland stations, 10-year probabilities are approximately 1.5–2.0 times the corresponding 5-year probabilities, while at low-Pdd mountain stations the ratio approaches 2.0–2.5, reflecting the more stochastic nature of drought occurrence in variable precipitation regimes.

To further examine the suitability of the first-order Markov model for capturing persistence (RQ2), we computed the lag-1 autocorrelation coefficients of the original daily discharge series for all 17 stations. Values ranged from 0.76 (Kysucké Nové Mesto) to 0.98 (Moravský Svätý Ján), with a mean of 0.90. These high daily lag-1 autocorrelations are fully consistent with the global streamflow statistics reported by Dimitriadis et al. (2021), where the long-term persistence (Hurst) parameter averages 0.78 (range 0.67–0.86). The pronounced daily-scale memory revealed here explains the minor underestimation of extreme dry spells in large lowland basins and the occasional exceedance of simulated 100-year wet spells in orographic headwaters observed in Figure 5.

5.3.3 Implications for water resource planning

The probabilistic risk metrics derived from the Markov model have direct implications for water resource management across Slovakia's diverse hydrological regions. In lowland agricultural regions (Holiša, Pláštovce–Litava, Nové Zámky), these areas face the highest drought persistence and the longest expected maximum spells. Reservoirs and irrigation systems should be designed with 3–5-year drought contingency reserves, and groundwater abstraction licensing should account for multi-year recharge deficits. The probability of 0.35–0.45 for experiencing at least one 3-year drought within any given decade indicates that such events should be considered in routine operational planning rather than treated as exceptional emergencies. In large lowland rivers (Moravský Svätý Ján, Brehy, and Nitrianska Streda), while these systems benefit from extensive groundwater buffering that provides resilience against isolated dry years, their high Pdd values indicate a substantial risk of multi-year droughts that could deplete storage reserves. Operating rules for major reservoirs on the Morava, Hron, and Nitra rivers should incorporate a 10%−15% probability of 4-year droughts into planning horizons. In mountain headwaters (Podbanské, Mýto pod Dumbierom, and Hronec), these catchments face lower persistence risk but higher variability. Drought management strategies should focus on seasonal low-flow alerts and short-term contingency measures rather than multi-year strategic reserves. The low probability (< 5%) of droughts lasting 4 years or more suggests that existing infrastructure is generally adequate for the drought risk profile of mountain regions. The Markov-based risk assessment reveals substantial spatial variation in the probability of severe multi-year droughts across Slovakia. Over 10-year planning horizons, 2-year droughts have a probability greater than 60% at high-persistence lowland stations, 3-year droughts have a 35%−45% probability at these same stations, and even 5-year droughts retain a non-negligible probability (5%−10%) in the most vulnerable basins. These quantitative risk metrics provide the probabilistic foundation needed to evaluate whether existing water infrastructure and management protocols are adequate for the range of drought durations that may plausibly occur under current climatic conditions, and to design adaptation measures where current capacity is insufficient.

6 Discussion

This study presents the first comprehensive probabilistic assessment of multi-season drought persistence across Slovakia's diverse river network, revealing pronounced spatial gradients in drought duration with significant implications for water resource management under changing climatic conditions. By integrating long-term discharge records with first-order Markov chain modeling, we quantify how drought persistence varies systematically with elevation, basin size, and physiographic setting, addressing a critical methodological gap in Central European hydrology.

The results show a clear elevation-dependent gradient in drought persistence across Slovakia. High-elevation Carpathian headwaters exhibit moderate dry-to-dry transition probabilities (Pdd = 0.52–0.69) and maximum dry spells of 6–13 seasons, while lowland basins in southern and western Slovakia display substantially higher persistence (Pdd = 0.65–0.73) with dry spells extending to 13–19 consecutive seasons. This spatial pattern reflects fundamental differences in the hydroclimatic processes governing drought development across physiographic zones. Mountain catchments experience intense but variable precipitation from orographic uplift over complex terrain, with frequent interruptions to developing drought sequences by convective storms and frontal passages (Segadelli et al., 2020; Handwerker et al., 2025). In contrast, lowland basins integrate large-scale circulation anomalies over extensive areas, and their substantial alluvial aquifer storage buffers short-term precipitation deficits while allowing moderate seasonal deficits to accumulate into multi-year hydrological droughts (Tallaksen and van Lanen, 2004; Hellwig et al., 2020; Kalu et al., 2025). These findings closely align with drought persistence patterns documented elsewhere in Europe. In the United Kingdom, Wilby et al. (2015) reported Pdd values exceeding 0.65 and 100-year dry spells of 8–12 seasons in lowland English basins, values that closely match those found here for Moravský Svätý Ján, Brehy, and Holiša. Similarly, Vicente-Serrano et al. (2018) documented comparable dry-spell persistence in continental Spanish basins experiencing Mediterranean-Continental climatic influences. In contrast, alpine headwaters in the European Alps (Azimi et al., 2026) and monsoonal catchments in the Western Ghats of India (Raju, 2026) show lower persistence (Pdd ≈ 0.50–0.55), mirroring the moderate values observed in Slovak Carpathian headwaters such as Podbanské and Mýto pod Dumbierom. The consistency of these elevation–persistence relationships across diverse climatic regions confirms that the elevation-controlled memory effect is a general hydrological phenomenon rather than a regional peculiarity, strengthening confidence in the transferability of Markov chain approaches for drought persistence assessment globally (Marchand et al., 2025; Liu et al., 2021). This is consistent with the identification of a common stochastic structure for streamflow across eight orders of magnitude, based on the Pareto–Burr–Feller marginal distribution and a generalized Hurst–Kolmogorov (HK) dependence structure that confirms strong long-range dependence as a fundamental property of the process (Pizarro et al., 2022). The observed persistence asymmetry, with Pdd exceeding Pww at 16 of 17 stations, has important implications for understanding drought dynamics under climate change. This asymmetry indicates that once drought conditions are established, they are more likely to persist than wet conditions, creating a systematic bias toward drought intensification in warming climates where evapotranspiration increases (Spinoni et al., 2018). Only Podbanské, the highest-elevation station influenced by persistent orographic precipitation, exhibits the reverse pattern (Pww > Pdd), suggesting that high-altitude catchments may serve as relative refugia from regional drying trends.

The Mann–Kendall and Pettitt test results provide robust statistical confirmation of systematic changes in Slovakia's hydrological regime. Significant negative trends were detected at 71% of stations, with the strongest drying signals at lowland stations in southern Slovakia (Pláštovce–Litava: Z = −4.79; Pláštovce–Krupinica: Z = −4.74). These trends align with documented increases in evapotranspiration and declining runoff coefficients across Slovakia since the late 20th century (Minárik et al., 2023; Pekárová et al., 2025a), and are consistent with broader European patterns of increasing drought frequency and severity (Spinoni et al., 2018; Hanel et al., 2018). The clustering of Pettitt change points around two distinct periods, 1968–1969 and 1982–1983, provides insight into the temporal evolution of Slovakia's drought regime. The earlier transition coincides with the end of a notably wet period in Central Europe documented in palaeoclimatic reconstructions (Brázdil et al., 2018), while the later shift corresponds to the onset of accelerated warming across the region (Hanel et al., 2018). These regime shifts align with the IAHS Panta Rhei initiative, which emphasizes the need to explicitly account for change and uncertainty in hydrological systems when interpreting observed shifts in drought and flood regimes (Montanari et al., 2013). Similar synchronous transitions driven by large-scale atmospheric forcing have been documented in major European river systems (Blöschl et al., 2019).

Similar regime shifts have been identified in other Central European River systems, including the Danube and Elbe basins, where the early 1980s marked a transition toward increased drought frequency (Blöschl et al., 2019; Leščešen et al., 2025). The detection of these synchronous change points across multiple Slovak catchments suggests that large-scale atmospheric forcing, rather than local factors, drove these hydrological transitions. Notably, stations without significant trends or change points (Podbanské, Kysucké Nové Mesto, and Liptovský Mikuláš) share common characteristics: high elevation, extensive forest cover, and strong orographic precipitation enhancement. This pattern suggests that mountain catchments with intact forest ecosystems may exhibit greater resilience to regional drying trends, a finding with important implications for watershed management and nature-based climate adaptation strategies (Berghuijs et al., 2017).

The first-order Markov chain approach proved highly effective at reproducing observed drought persistence characteristics across Slovakia's diverse catchments. Kolmogorov–Smirnov tests confirmed adequate model fit at 15 of 17 stations (88%), with particularly close agreement in mountain basins where precipitation variability is dominated by short-memory stochastic processes. This high success rate validates the fundamental assumption that seasonal drought transitions can be adequately represented by a memory-one process for most practical applications. However, systematic discrepancies emerged that warrant discussion. In large lowland basins such as Moravský Svätý Ján and Nitrianska Streda, the model slightly underestimates intermediate-length dry spells (4–8 seasons), suggesting that large-scale circulation patterns affecting these extensive catchments may exhibit longer memory than a first-order chain captures. This finding is consistent with research demonstrating that persistent atmospheric blocking patterns and ocean-atmosphere teleconnections can induce multi-seasonal memory in European drought sequences (Folland et al., 2015; Kingston et al., 2015). Future model improvements could incorporate climate indices such as the North Atlantic Oscillation (NAO) or Atlantic Multidecadal Oscillation (AMO) as conditioning variables, following the mixture model approach of Kiely et al. (1998) and Wilby (2001). The presence of exceptionally strong long-range dependence (Hurst parameter H > 0.8) across scales from minutes to centuries indicates that first-order Markov chains should be extended with generalized Hurst–Kolmogorov processes to fully capture the long-term memory and inherent uncertainty of streamflow (Pizarro et al., 2022; Koutsoyiannis et al., 2008). The model also showed marked asymmetry in reproducing extreme wet and dry spells. While observed maximum dry spells generally lay within the interquartile range of simulated 100-year distributions, observed maximum wet spells often approached or exceeded the 90th percentile of simulations. This asymmetry suggests that wet persistence may involve different physical mechanisms from dry persistence, potentially related to persistent westerly flow regimes and orographic precipitation enhancement, which are not fully represented by the symmetric Markov framework.

The probabilistic risk metrics derived from Markov simulations have direct implications for water resource infrastructure design and drought contingency planning in Slovakia. The finding that 2-year droughts have a greater than 60% probability of occurring within any 10-year period at high-persistence lowland stations indicates that such events should be considered routine planning scenarios rather than exceptional emergencies. More concerning, 3-year droughts retain probabilities of 0.35–0.45 at the most vulnerable stations (Holiša, Pláštovce–Litava, Brehy), substantially exceeding the design assumptions of most existing reservoir operating rules, which typically assume maximum drought durations of 2–3 years. These findings reflect experiences from other regions that have faced multi-year drought sequences. The Australian Millennium Drought (1997–2009) demonstrated that prolonged drought, combined with continued abstractions, can trigger apparent regime shifts in riverine ecosystems and deplete groundwater reserves beyond recovery timescales (Fowler et al., 2022; Van Dijk et al., 2013). Similar concerns apply to Slovakia's lowland basins, where our simulations indicate non-negligible probabilities (0.05–0.10) of 5-year droughts within decadal planning horizons. Such events would severely stress water supply systems and could cause lasting ecological damage to flow-dependent ecosystems. The spatial differentiation in drought risk profiles suggests that regionally tailored management strategies are essential. Lowland agricultural regions (Holiša, Pláštovce–Litava, and Nové Zámky) require infrastructure designed for 3–5-year drought contingencies, while mountain headwaters need primarily seasonal low-flow alerts rather than multi-year strategic reserves. The Markov-derived exceedance probabilities can be directly incorporated into drought early warning systems operated by the Slovak Hydrometeorological Institute, providing threshold-based triggers for irrigation restrictions, reservoir drawdown protocols, and inter-basin transfer activation.

Research on persistent droughts in Central Europe has identified several atmospheric mechanisms contributing to multi-year dry spells. The North Atlantic Oscillation, Atlantic Multidecadal Oscillation, and remote Pacific forcing such as ENSO all influence seasonal rainfall and runoff deficits across the region (Fraedrich, 1990; Wilby, 1993, 2001; Folland et al., 2015; Leščešen et al., 2025). Positive NAO phases and warm North Atlantic Sea surface temperatures are particularly associated with reduced winter precipitation and increased drought persistence in Central and Eastern Europe (Seager et al., 2020; Fuentes-Franco et al., 2023). However, as Folland et al. (2015) and van der Wiel et al. (2023) emphasize, no single driver explains all multi-year droughts; rather, regional circulation anomalies, land–atmosphere feedbacks, and oceanic conditions interact in complex ways to produce extended dry periods. Long-term persistence in major river basins such as the Yellow River further confirms that strong LTP is a globally relevant driver of prolonged dry and wet clustering (Wang et al., 2023). Non-climatic influences also affect drought sequences in Slovak catchments, particularly along regulated rivers such as the Váh, Hron, and Morava. River regulation, hydropower operations, inter-basin transfers, and groundwater abstraction can either amplify or attenuate low-flow persistence depending on operational protocols and demand patterns (Wen et al., 2023; Zhao et al., 2023; Bačová Mitková et al., 2024). Our sensitivity analysis excluding three stations with documented upstream regulation (Moravský Svätý Ján, Šala, Nové Zámky) showed negligible effects on transition probabilities and 100-year spell lengths (< 0.02 and < 1 season change, respectively), indicating that regulation effects are minimal in the present dataset. Nevertheless, future studies should systematically incorporate reservoir operating rules and land-use change metadata, including the documented expansion of forest cover from 34 to 41.4% since 1930 (Pekárová et al., 2025b).

Extended low-flow sequences pose significant risks to riverine biota in Slovakia. While shorter droughts cause temporary macroinvertebrate population declines followed by recovery, ecosystem resilience to multi-year droughts remains poorly understood and is likely compromised by concurrent stressors including pollution, channel modification, and temperature extremes (Monk et al., 2008; Gething, 2024; Döll and Zhang, 2010). Protecting hydrological refugia such as deep pools, hyporheic zones, and spring-fed tributaries is essential for maintaining ecological integrity, particularly in regulated lowland rivers where natural refugia may be diminished. We acknowledge several limitations in our modeling framework. The assumption of stationary, first-order Markov chains with fixed parameters ignores potential multi-decadal changes in transition probabilities linked to sea surface temperature variations or shifts in atmospheric circulation. Additionally, we model drought duration but not intensity, although impacts depend critically on both dimensions. Extending the framework to incorporate severity distributions would provide a more complete picture of drought risk. The hydrological cycle exhibits fluctuations rather than monotonic intensification, with de-intensification prevailing across many variables in the 21st century, underscoring the need for stochastic Hurst–Kolmogorov frameworks that explicitly incorporate long-term persistence for robust drought-risk assessment (Koutsoyiannis, 2020). Finally, our analysis uses station-based observations; applying similar approaches to gridded datasets or climate model outputs would enable assessment of spatial drought coherence across Slovakia and neighboring countries, addressing the broader regional context of Central European drought dynamics.

7 Conclusion

This investigation utilized long-term, quality-controlled discharge records from 17 gauging stations across Slovakia to analyse the spatial and temporal variability of multi-season drought duration, providing the first comprehensive probabilistic assessment of drought persistence across the country's diverse physiographic gradients. By integrating half-yearly discharge anomalies with first-order Markov chain modeling, we have developed quantitative tools that offer water resource managers actionable insights for planning under changing climatic conditions. The analysis reveals pronounced spatial heterogeneity in drought characteristics that closely follows Slovakia's altitudinal and physiographic gradients. High-elevation Carpathian headwaters, including Podbanské, Mýto pod Dumbierom, and Hronec, exhibit moderate dry-to-dry transition probabilities ranging from 0.52 to 0.69, with maximum observed dry spells of 6–13 seasons. This pattern reflects the dynamic orographic precipitation regime characteristic of alpine environments, where intense but variable rainfall events frequently interrupt developing drought sequences. In marked contrast, lowland basins in southern and western Slovakia display substantially higher persistence, with Pdd values of 0.65–0.73 and dry spells extending to 13–19 consecutive seasons. The most severe recorded drought occurred at Holiša on the Ipel River, where 19 consecutive dry seasons spanned nearly a decade from winter 1986 to winter 1995. Mann–Kendall trend tests confirmed significant negative trends at 71% of stations, while Pettitt change-point analysis identified regime shifts clustering around 1968–1969 and 1982–1983, periods that correspond to documented transitions in Central European climate. Notably, high-elevation forested catchments showed no significant drying trends, suggesting these environments may possess greater resilience to regional climate change impacts.

The first-order Markov chain approach proved highly effective at capturing the essential persistence characteristics of drought across Slovakia's diverse catchments. Kolmogorov–Smirnov tests confirmed adequate model fit at 15 of 17 stations, with particularly close agreement in mountain basins where precipitation variability follows short-memory stochastic processes. Consistent with previous research conducted in the United Kingdom, Spain, and other temperate regions, these models demonstrate that drought persistence can be approximated by a geometric distribution, providing a straightforward yet robust framework for estimating the likelihood of rare events. Comparison of observed maximum dry spells with bootstrap-simulated 100-year distributions revealed that most catchments have experienced droughts consistent with their inferred persistence structure. However, at several vulnerable lowland stations, including Holiša, Brehy, and Moravský Svätý Ján, observed maximum dry spells approach the upper quartiles of simulated distributions, suggesting that the late 20th-century record may have already captured relatively rare persistence events that could recur under similar climatic conditions. These persistence characteristics translate directly into quantifiable risk metrics with immediate relevance for water resource planning. Monte Carlo simulations indicate that 2-year droughts, representing the typical design drought assumed in many reservoir operating rules, have probabilities exceeding 0.60 over 10-year horizons at high-persistence lowland stations. More concerning, 3-year droughts retain probabilities of 0.35–0.45 at the most vulnerable stations, substantially exceeding current infrastructure design assumptions that typically consider maximum drought durations of 2–3 years. Even 5-year droughts, which would severely stress existing water supply systems, maintain non-negligible probabilities of 0.05–0.10 at stations such as Holiša, Pláštovce–Litava, and Hanušovce. Mountain headwaters face a fundamentally different risk profile, with modal maximum dry spells of only 3–4 seasons over decadal horizons and negligible probability of events exceeding 8–10 seasons.

The practical implications of these findings are significant and vary by region. Reservoirs on lowland rivers like the Hron, Nitra, and Ipel should include drought contingency plans that allow for 3–5-season deficits, which have about a 0.35 probability of occurring within any 10-year planning period. In contrast, high-elevation headwater catchments primarily need seasonal low-flow alerts rather than multi-year strategic reserves. The exceedance probabilities derived from the Markov model can be directly integrated into drought early-warning systems managed by the Slovak Hydrometeorological Institute, offering threshold-based triggers for irrigation restrictions, reservoir drawdown protocols, and inter-basin transfer activation. Catchments with Pdd exceeding 0.65 should be prioritized for infrastructure investments in additional storage or demand management measures, due to their heightened risk of multi-year drought sequences.

While Slovak water management discussions often focus on major reservoirs along the Váh and Hron rivers, our findings reveal that smaller lowland basins with limited storage capacity can be equally or more vulnerable to persistent hydrological deficits. Long-lasting low-flow periods pose significant threats to aquatic ecosystems already under pressure from river regulation, water abstraction, pollution, and channel modifications. Human-induced climate change adds further uncertainty regarding ocean-atmosphere interactions and drought persistence in the coming decades, especially since recent warming has already decreased runoff despite stable or increasing precipitation totals across much of Slovakia.

This study opens several promising pathways for future research. Potential enhancements to the Markov framework include integrating drought onset seasonality or large-scale climate indices like the North Atlantic Oscillation as conditioning factors, extending analyses to simulate multi-year drought intensities along with durations, and incorporating low-frequency oceanic forcing mechanisms into model parameters. Applying similar approaches to gridded datasets would enable assessment of spatial drought coherence across Slovakia and neighboring Central European countries. Additionally, comparing regional climate model outputs with observed spell-length distributions would boost confidence in future projections. Ultimately, key policy questions remain about the best strategies for balancing ecological, societal, and economic interests during multi-season and multi-year droughts, which our analysis indicates are physically plausible within typical infrastructure planning horizons.

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