The study area is the Slovak–Hungarian reach of the Danube River, which is a free-flowing section of the river (Figure 1). Along this reach, the main channel width, at mean flow, varies between 200 and 500 m, with an average water depth of approximately 5 m. The mean river slope is 10 cm/km; however, within the study area, the Danube transitions from an upper-reach character to a middle-reach character (Török and Baranya, 2017), with the average slope decreasing from 20 cm/km to 7 cm/km. The riverbed material is predominantly gravel, with an average grain diameter of 0.009 m. Significant river regulation interventions have been carried out in this area since the late 19th century. The banks were stabilized using riprap, groyne fields were constructed, and flood protection levees were built in the floodplain (Nyiri and Török, 2024).

In the mid-20th century, extensive industrial gravel extraction took place, followed by the construction of two hydropower plants (HPP) near the upstream end of the study area in the early 1990s (Holubová et al., 2004). These interventions have led to the deepening of the riverbed, reaching up to 5 m in some places (Rákóczi, 2000). According to previous studies, over the last decades, the average riverbed degradation has been around 2 cm/year (Nyiri and Török, 2024). However, due to the complex bed topography and the interacting effects of the various interventions, bed degradation is not uniform, on some shorter sections, local aggradation has also been observed instead (Rákóczi, 2000; Török and Baranya, 2017). The deepening of the riverbed has resulted in several adverse effects, including the sedimentation of side branches, a decrease in groundwater levels, a lowering of water levels in tributaries, and the emergence of navigational issues (Habersack et al., 2019), making it a representative European large-river section for evaluating sediment-based restoration measures.

The modelling system couples a quasi-steady 1D flow module with a bedload-based 1D morphodynamic module for the Danube section between rkm 1710–1810. The novelty lies in parameterizing both flow conveyance and sediment transport using effective widths derived from a calibrated 2D hydrodynamic model. This preserves computational efficiency for multi-decadal simulations while incorporating key spatial information on hydraulically active channel zones.

For model calibration and validation, field measurements of water levels, bed elevations, and bedload transport rates were used. A detailed longitudinal water level survey was conducted in 2018 during a low-flow period, and these measurements could directly be compared to the water levels computed by the model. Additionally, for validating the flow model under mean flow conditions, we used water surface profiles simulated by the previously established 2D hydrodynamic model.

To validate the morphodynamic model, results from repeated bathymetric surveys were used. Bed surveys conducted in 2005, 2010, and 2018 along the study reach provided observed bed level changes for model validation. The bathymetric measurements were performed using a single-beam echo sounder with a longitudinal spacing of approximately 100 m, which is substantially finer than the model cross-section spacing of 500 m. This resolution allowed for a direct and robust comparison between simulated and observed bed level changes. As with the modeling approach, the cross-sectionally averaged bed levels were computed over the effective flow conveying parts of the channel, based on the previously described method.

To further validate the sediment transport model, we used the results of a bedload measurement campaign carried out in recent years at a selected cross section of the study reach. The multi-year dataset was used to establish a flow discharge-bedload transport relationship, i.e., a bedload rating curve, across a relatively wide range of discharges. These relationships were developed using a combination of direct and indirect measurement methods (Baranya, 2024).

As described previously, the model was developed based on the 2018 riverbed geometry measurements and the results of the 2D hydrodynamic model under mean-flow conditions. Based on the simulated specific discharge field, zones with values below 1 m²/s were excluded, as preliminary analysis indicated that these areas do not contribute significantly to flow conveyance (Figure 3).

Based on the 2D map, cross-sectional width data were extracted at 500-m intervals. On average, the effective cross-sectional width relevant for discharge transport (B _eff) was reduced to about 70% of the original width (Figure 4). The model was run under low-flow (Q = 950 m³/s) and mean-flow (Q = 2,200 m³/s) conditions, assuming steady-state conditions. The calibration parameter of the flow model was the channel roughness factor C _f , which was uniformly set to 0.0033 along the entire reach.

The simulated water surface profiles (Figure 5) clearly show that the studied reach can be divided into an upper section with steeper slope and a lower section with lower slope, with the break point around rkm 1785, similarly to the results previously presented by, e.g., Nyiri and Török (2024). Over the entire study reach, the model satisfactorily reproduced the measured water levels (low-flow case) and the 2D-modelled water levels (mean-flow case). For the mean-flow condition, the difference between the two water surface profiles is negligible. In the low-flow condition, the overall agreement is good, although under- and overestimation can be observed over short sections. The largest deviation is found in the upper reach, where the model underestimates the measured levels by up to 20 cm. The mean absolute error (MAE) is 0.18 m for the low-flow case and 0.07 m for the mean-flow case, respectively.

The morphodynamic model was run with the already calibrated hydrodynamic model. Unsteady simulations were run for the period 2005–2018, with the initial riverbed geometry defined based on the 2005 riverbed geometry survey and using the 2D model-based delineation introduced above.

As the model operates with a time-stepping approach, daily discharge time series and corresponding downstream water levels were defined as boundary conditions for the periods 2005–2018. The use of a daily time step ensures that the full range of hydrological conditions occurring during this period, from low-flow to high-flow events, is explicitly represented in the simulations. As for the sediment transport boundary condition at the upstream boundary zero transport was described, because the model domain starts immediately downstream of the artificial diversion channel of the Gabčíkovo HPP, where coarse sediment is effectively intercepted by the impoundment. In this configuration, the bedload continuity from the upper Danube is altered (Habersack et al., 2019), and the reach receives negligible upstream bedload supply under typical operations. Note that a test for close-to-zero bedload transport boundary values was also performed showing low sensitivity on the resulted morphodynamics.

For bed level change computations, the model incorporated the effective channel widths for sediment transport (B _eff,sed). In each cross-section, only those parts were considered where the bed shear stress computed by the 2D model exceeded the critical Shields parameter at the mean flow case. For the entire reach, the effective channel width for sediment transport averaged about 40% of the total channel width (Figure 6).

The sediment transport model was calibrated using the critical Shields stress value and the exponent of the MPM formula, as described above. The best agreement, both in terms of bed level changes and bedload transport rates, was achieved with the τ _c ​^* = 0.03 for the critical shear stress, and 2.1 for the exponent, respectively. These parameters should not be regarded as universally fixed, but rather as site-specific and subject to calibration, as also suggested by Habersack and Laronne (2002). The model adequately reproduces both the longitudinal and temporal behavior of the bed changes. Local deviations are within the order of 10 cm, but overall, the model captures the patterns of erosion and deposition well. The longitudinal bed level changes simulated by the model were compared to two sets of measured bed change data (Figure 7), collected in 2010 and 2018, respectively, corresponding to time spans of 5 and 13 years (for the latter, only the upper 40 km of the study reach was available). The bed change plots clearly indicate that dynamic bed changes take place in the steeper section of the Danube reach (up to around rkm 1785), while downstream of this point, the riverbed can rather be considered stable. Bed incision is evident between rkm 1810 and 1795, whereas downstream from this section, sediment deposition occurs due to erosion upstream. The maximum depth of incision after 5 years is approximately 0.5 m, increasing to 1 m after 13 years. The maximum deposition already reaches 1 m after 5 years, and later the magnitude remains similar, but the longitudinal extent increases. Moreover, similar to the measurements, the model only indicates dynamic morphological changes in the upper ∼25 km of the reach, while the downstream sections are identified as morphologically stable. For the 5-year period, the MAE of simulated bed level changes was 0.14 m, with a slight negative bias (−0.04 m). For the 13-year period, MAE was 0.24 m, respectively, and the bias was small (+0.03 m). These values are within the accuracy range expected for large-river bathymetric surveys and confirm that the model reproduces both the magnitude and the spatial pattern of observed morphological change. Importantly, the model correctly captured the pattern of alternating incision and deposition, which is critical for the intended sediment-management applications.

As for the bedload transport values, the bedload rating curved established by Baranya (2024) for the monitoring section at rkm 1790.6 was used as model validation data. The model reproduces the observed non-linear increase in transport rate with discharge, spanning over two orders of magnitude (Figure 8). Although some scatter is present in the measurements, reflecting natural variability and measurement uncertainty, the agreement between measured and simulated values is satisfactory, particularly in the mid-to high-flow range (>2000 m³/s), where morphodynamic processes take place.

While validation at a single cross-section is a limitation, it reflects the reality of gravel-bed monitoring in the Danube and similar large rivers, where bedload measurements are difficult and spatially sparse (Habersack et al., 2019). In the context of the present large-scale, long-term application, the combination of longitudinal bathymetric validation and bedload rating curve validation is unique and provides a robust basis for confidence in model behaviour.

To assess the role of the 2D-derived effective width approach in the morphodynamic model, four long-term simulations (2005–2018) were compared (Figure 9):

v1: Validated version–spatially varying effective flow conveyance widths (B _eff ) derived from 2D hydrodynamics, as well as the spatially varying effective sediment transport width (B _eff,sed ).

The results show that the 2D-derived effective-width parameterization is critical for reproducing observed morphodynamic patterns and magnitudes. In the most active reach (rkm 1795–1810), the validated model predicts persistent, substantial erosion, consistent with observations. When the total channel width is used (v2), the morphodynamic signal is severely distorted: erosion amplitudes are damped and, critically, local deposition appears where only erosion is expected. This inversion arises mainly because the total width counts groyne-field dead zones in the conveyance, reducing unit discharge and shear and thus altering transport gradients. Groyne fields commonly behave as recirculating areas with limited downstream conveyance and intermittent exchange with the main flow, as shown by Uijttewaal (2001) and later laboratory studies on groyne-layout effects and exchange processes (Uijttewaal, 2005; Weitbrecht et al., 2008; Yossef and de Vriend, 2010).

The constant-width case (v3) performs even worse, essentially suppressing the morphodynamic response along the entire study reach. In fact, by homogenizing the specific discharge, it removes local transport capacity gradients, leading to Δz values close to zero and a complete loss of realistic bed change patterns.

In contrast, omitting the effective sediment transport width approach (v4) has only a minor effect. The spatial patterns remain nearly identical to the validated run, with only small amplitude differences in high-erosion zones. This suggests that the τ-based correction fine-tunes the results, whereas the main added value comes from using spatially variable effective flow conveyance widths that reflect the actual hydraulics. Notably, Camenen et al. (2011), also in a Danube case study, demonstrated that computing bed-load from a mean cross-sectional shear stress is generally unsatisfactory, and that introducing cross-sectional variability of τ, with an analytical approach in their case, substantially improves 1D bed-load predictions. The weaker sensitivity to the τ-correction in this study likely stems from the fact that the 2D-derived effective widths already consider much of the lateral conveyance and dead-zone effects, such as groyne fields, so the additional correction acts as a second-order refinement rather than a first-order control.

Overall, the analysis demonstrates that 2D-based effective width parameterization is crucial for realistic morphodynamic modelling. It preserves the spatial variability in flow conveyance that controls sediment transport gradients, ensures correct prediction of erosion/deposition locations, and prevents the large-scale qualitative and quantitative errors that arise when cross-sectional variability is neglected.

Using the field-validated model, 30 years of morphodynamics was simulated, as the observed 13-year period followed by a 17-year forecast. For the sake of simplicity, the inflow boundary conditions for the future simulations, the 13-year inflow hydrograph was repeated and, consistent with the validation setup, the upstream bedload boundary was set to zero. While climate change is expected to influence future flow regimes, including the occurrence of hydrological extremes, the large uncertainties associated with regional discharge projections currently limit their direct application in long-term morphodynamic simulations. Consequently, climate change impacts were not explicitly addressed in this study. To assess long-term management effects, three variants were evaluated: (R) do-nothing reference; (W) channel widening, effective flow conveyance width increased by a factor of 1.5 along the study reach as a proxy for groyne-field removal; and (F) targeted sediment feeding of 10,000 m³/yr at rkm 1804, the local incision hotspot. Measures (W, F) were implemented after year 13, and their effects were evaluated over the subsequent 17 years.

Focusing only on the dynamic section of the river, the longitudinal profiles for the reference model variant (R) show that the most dynamic reach 1790–1810 remains dominated by erosion (Figure 10 top). To assess the temporal behavior of the bed changes, time series of Δz at rkm 1802 was investigated. Similarly to the model validation runs, this section exhibits a persistent incisional trend that decelerates over time, approaching a quasi-equilibrium after ∼20–25 years. By year 30, incision is about 0.8 m, suggesting a mean bed incision of 2.7 cm/yr.

As for the increased river width model variant (W), the analyzed measure indeed reduces specific flow discharge and bed shear stress, thereby lowering transport capacity that drives bed incision processes. This is evident in the longitudinal profile that indicates smaller amplitudes of the erosional peaks around rkm 1795–1810 (Figure 10 middle). At rkm 1802, widening roughly halves the long-term incision to about 0.4 m by year 30 (equal to 1.3 cm/yr). Longitudinal extension of sediment deposition patterns downstream of rkm 1795 also decreases compared to the reference version, suggesting more sustainable navigational channel.

Artificial feeding, or nourishment, of sediments (model variant F) produces a locally significantly reduced erosion, centered on rkm 1804, resulting in almost zero bed level change there (Figure 10 bottom). Also, sediment feeding reduces downstream incision, but the effect is weaker than the channel widening at the same site. By year 30, rkm 1802 shows ∼0.55–0.60 m of erosion, i.e., ∼2.0 cm/yr, showing an improvement of ∼0.2–0.25 m relative to the reference variant. The measure was implemented after year 13 at rkm 1804, and the first clear impact at rkm 1802 appears only around year ∼20, implying a propagation time of roughly 7 years between the sites. Time series of bed changes suggests partial stabilization after ∼20–25 years, with modest recovery toward the end, indicating that the annual 10,000 m³ input seems to be insufficient to fully offset the local transport capacity under the recurring hydrograph.

The model results confirm that the upper section of the Hungarian Danube continues to experience a persistent sediment deficit and associated hydromorphological stress, manifested by pronounced bed incision between rkm 1810–1795 and relative downstream stabilization. This spatial pattern is consistent with decades of prior assessments and reflects the combined effects of upstream hydropower impoundment, historical sediment extraction, and channelization through groyne fields, which constrain lateral channel mobility and reduce sediment supply, leading to “hungry water” conditions and channel deepening (Kondolf, 1997). Such downstream attenuation of incision is characteristic of river reaches below hydropower reservoirs, where rapid post-impoundment degradation progressively adjusts toward a new dynamic equilibrium in slope and transport capacity (Kondolf, 1997; Brandt, 2000). Comparable long-term observations from the Danube east of Vienna similarly report persistent but decreasing degradation rates, accompanied by management interventions such as sediment feeding, bed coarsening, and channel widening aimed at stabilizing bed levels (Klasz et al., 2016; Habersack et al., 2019).

These morphological changes extend beyond geomorphology, as progressive channel incision lowers water levels and groundwater tables and reduces the frequency and duration of side-channel as well as floodplain inundation, thereby influencing key ecological processes, including (i) hydrological and lateral connectivity between the main channel, side channels, and floodplains; (ii) groundwater–surface water interactions; (iii) habitat availability and quality for fish, particularly spawning and nursery habitats; (iv) substrate composition and stability for benthic communities; and (v) organic matter and nutrient exchange within the river corridor (Shields et al., 1994; Loheide and Booth, 2011; Habersack et al., 2016). In this sense, incision is not solely a geomorphic phenomenon but a driver of freshwater ecosystem stress. The validated model’s ability to reproduce these patterns provides a quantitative basis for diagnosing the degree and extent of hydromorphological pressure in large, regulated rivers.

The clear difference between the active, erosion-prone upper reach and the more stable downstream section is particularly relevant for restoration planning. It indicates where interventions such as sediment feeding or morphological reactivation will have the greatest restorative leverage and where they may have limited effect. Identifying and prioritizing such “pressure hotspots” is fundamental for the cost-effective implementation of large-river restoration measures under European policies such as the Water Framework Directive and the EU Biodiversity Strategy.

The comparison of model variants (introduced in Section 3.3) highlights the central role of spatially variable effective width in realistically reproducing the reach-scale morphodynamic signal. The use of total geometric width systematically underestimates incision and produces deposition where none is observed. The constant-width variant suppresses nearly all morphodynamic activity. Only the 2D-informed effective-width approach captures the observed magnitude and spatial distribution of erosion and deposition.

These differences illustrate an important conceptual point: the morphodynamically active river width is not the geometric width, especially in highly regulated rivers. Groynes, low-flow sections, and recirculation zones are hydraulically and morphologically inactive for much of the flow regime. Treating them as active artificially dilutes shear stress and sediment transport capacity across the full cross-section, leading to quantitatively and sometimes qualitatively incorrect results.

In restoration science, distinguishing between structural characteristics (the constructed corridor) and functional characteristics (the hydraulically active corridor) is crucial (Ward, 1989; Jungwirth et al., 2002). Effective width therefore represents a bridge between process-based hydraulics and practical large-scale modelling: it preserves the lateral variability of hydraulic conditions without requiring computationally intensive multidimensional simulations for long time horizons.

The proposed hybrid 1D–2D approach thus offers an important methodological advance for rivers where (i) decadal predictions are needed, (ii) hydromorphological pressures are strong, and (iii) scenario testing must be efficient yet realistic. This is particularly relevant for European rivers with long histories of channel training, where restoration strategies often aim to reactivate or reconnect portions of the lateral corridor (Schmutz and Sendzimir, 2018).

The modelling results highlight two complementary process-based pathways for mitigating long-term bed degradation in large, trained rivers: reducing sediment transport capacity through channel widening and increasing sediment supply through sediment feeding. Channel widening increases the hydraulically active corridor, thereby lowering unit stream power and sediment transport capacity (Surian and Rinaldi, 2003). This mechanism is consistent with experiences from European rivers, where providing additional space has promoted gravel-bar formation, enhanced hydraulic and substrate heterogeneity, and improved lateral connectivity—key attributes associated with ecological resilience and the recovery of side-channel habitats (Surian and Rinaldi, 2003; Wohl et al., 2015). These findings align with the recently published Danube Sediment Management Guidance, which explicitly recommends moderate channel widening to alleviate bed degradation (Habersack et al., 2019), as well as post-project monitoring from rivers such as the Thur (Switzerland) and the alpine Ahr, where widening resulted in gravel-bar development and increased bed elevations (Campana et al., 2014; Martín et al., 2018).

Sediment feeding directly addresses sediment discontinuity, a pervasive pressure in regulated rivers where upstream trapping and channel confinement limit the downstream supply of coarse material. International case studies demonstrate that appropriately designed replenishment measures can locally raise bed levels, stabilize incision hotspots, and improve substrate conditions, particularly where spawning or benthic habitats have been degraded (Ock et al., 2013; Mörtl and De Cesare, 2021; Chardon et al., 2021). Long-term experiences from the Rhine river highlight the importance of sufficient volumes and appropriate grain-size selection (Chardon et al., 2021; Czapiga et al., 2022), while recent sediment management activities on the Danube similarly emphasize targeted bedload additions as a key measure to counter persistent bed incision (Klasz et al., 2016; Habersack et al., 2019).

Together, these insights highlight the value of combining measures that reduce capacity with those that increase supply as was shown by, e.g., Chardon et al. (2018). Widening creates a geomorphic template capable of retaining added sediment, while feeding accelerates morphological and ecological recovery where sediment deficits are most acute. In European rivers shaped by decades of channel training and confinement, such integrated, process-based strategies are increasingly central to restoring lateral dynamics, reactivating floodplain interactions, and meeting hydromorphological objectives under the Water Framework Directive. The hybrid 1D–2D modelling framework provides a practical tool for exploring these interactions over decadal scales and for informing the strategic design of restoration interventions.

Beyond the specific Danube case, the study demonstrates how hybrid modelling can support freshwater restoration across Europe. The approach: i) enables screening of multiple restoration measures over multi-decadal horizons, ii) identifies geomorphic pressure points, iii) provides quantitative expectations of long-term bed evolution, iv) can be applied to other regulated rivers where computational constraints limit 2D/3D modelling.

This aligns with continental-scale initiatives such as the Danube River Basin Management Plan and the EU Nature Restoration Law, both of which emphasize restoring sediment continuity and hydromorphological processes.

Although effective for long-term assessment, the model simplifies some processes important for ecological interpretation, such as unsteady hydrodynamics, mixed-size sediment sorting, bank erosion, and fine-sediment transport. Future developments could incorporate: i) fractional transport and active-layer dynamics, ii) event-scale unsteady hydraulics for disturbance-driven habitat processes, iii) evolving width or bank-erosion modules for systems with mobile banks, iv) dynamic updating of effective widths as morphology changes, v) linking morphodynamic outputs with habitat or ecological models. Such extensions would enhance the value of the modelling framework for integrated river restoration and freshwater ecosystem assessments.