1 Introduction

In recent years, there has been a series of heavy rainfall disasters that are highly likely to be affected by climate change (e.g [1,2,3]). In August 2016, three typhoons made landfall in the Hokkaido region of Japan within one week for the first time in recorded history. Immediately thereafter, another intense typhoon approached very close to Hokkaido, causing severe floods and sediment disasters (e.g., [4, 5]). One notable sediment-related disaster associated with this event was that residential areas and infrastructure along narrow and steep streams were severely damaged by extreme bank erosion and meandering development in the Tokachi River Basin in the eastern part of Hokkaido (Fig. 1). To prevent such damage, we need to, for example, protect riverbanks with revetments and enlarge the river width. However, the total length of the river makes it impractical to protect all riverbanks. Instead, it would be beneficial to estimate the location and amount of riverbank erosion in advance to take the necessary countermeasures for such weak points or reaches of the river. Considering that climate change will increase flood peak discharge (e.g [6]), it is important to estimate the amount of riverbank erosion under future climate conditions.

At least two components are needed to assess the amount of riverbank erosion and the related risks under climate change: a mechanical predictor of meander amplitude, which is closely related to the amount of riverbank erosion, and an input dataset regarding future discharge under future climate scenarios. Using numerical models of the morphological change of the riverbed with the bank erosion model, Nagata et al. [7], Eke et al. [8], and Schuurman et al. [9] showed that the formation of bars induces meandering development in sand- and gravel-bed rivers and that the meander amplitude increases with the magnitude of discharge. Asahi et al. [10] developed a free-meandering model that considered the riverbank erosion model proposed by Parker et al. [11] and the land accretion model, reasonably reproducing the complex natural processes of meandering development with cut-offs.

Fig. 1
figure 1

Aerial view of the Pekerebetsu River after the August 2016 flood (image courtesy of the Construction Department, Hokkaido Government)

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A dataset of spatiotemporal rainfall and water discharge was developed using climate models in the context of climate change research. In Japan, the Database for Policy Decision making for Future Climate Change (d4PDF) [12] was developed for this purpose and has been used to estimate river discharge under climate change conditions. Masuya et al. [13] showed significant changes in the probability distribution of maximum annual precipitation due to climate change in the Tokachi River basin in the Hokkaido region of Japan using rainfall data from d4PDF. Tanaka et al. [14] assessed the flood risk in numerous main river basins in Japan using massive ensemble rainfall data from d4PDF with the runoff model.

As described above, various studies have been conducted to estimate either meander amplitude or river discharge under climate change conditions. However, it remains unclear how future changes in river discharge affect river channel geometry and its dynamics. As river channel geometry changes, the discharge capacity and frequency of inundation also change [15, 16]. Redolfi et al. [17] estimated the future change in alternate bar geometry under climate change by coupling the future discharge dataset and a mathematical model of alternate bar dynamics. They showed that there is no significant change in bar shape as discharge changes in rivers with wide channels, whereas bar height and size change in narrow channels. Chassiot et al. [18] studied the effects of freeze-thaw cycles and river ice on riverbank erosion in cold regions. However, it is also important to understand future changes in meander amplitude and river width under climate change to prevent floods and sediment-related disasters associated with river meandering.

In this study, we first used the numerical model of free-meandering rivers proposed by Asahi et al. [10] to understand the dynamics of river meandering under various hydraulic and channel geometry conditions (e.g., discharge, channel slope, and initial channel geometry). The numerical model was validated by comparing certain channel characteristics (i.e., width-depth ratio, bankfull shear stress, and sinuosity) under bankfull conditions between the model results and the observed data of natural meandering rivers. We then combined the simulated results of meander dynamics with a dataset of river discharge under past and future climate conditions obtained from the massive rainfall dataset, d4PDF, to assess future changes in meander amplitude, which may indicate sediment-related risk along rivers under climate change. We applied the above framework to the case of the Pekerebetu River in Japan, which was highly damaged by meandering during the 2016 flood.

2 Study site

Our study site is the Pekerebetsu River, which joins the Sahoro River, a first-order tributary of the Tokachi River (Fig. 2). The Pekerebetsu River is a steep river with a basin area of 48.2 km2, a bed slope of approximately 0.01 to 0.025, an average grain size of approximately 60 mm to 120 mm, and a river width of approximately 15–35 m. In August 2016, heavy rainfall totaling 388 mm over three days resulted in a peak flow discharge of 250 m3/s in the Pekerebetsu River [19]. Consequently, a relatively straight channel developed into a meandering channel. Both banks continuously collapsed near the urban area, causing damage to bridges, house outflows, and inundation of the urban area of Shimizu Town. The meander amplitude, which averaged approximately 45 m (minimum: 35 m, maximum: 75 m) before the flood, increased to an average of 135 m (minimum: 90 m, maximum: 200 m) after the flood.

Fig. 2
figure 2

Location of the Pekerebetsu River basin

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3 Overview of free meandering model

Because the details of the free-meandering model used in this study have been described by Asahi et al. [10], we provide only an overview of the model in this section. The model proposed by Asahi et al. [10] includes models of the flow field, bed deformation, and changes in the channel planform owing to bank erosion and land accretion. The hydrodynamic model is a two-dimensional unsteady shallow water flow model in a general coordinate system that considers the movement of the left and right bank boundaries owing to bank erosion and land accretion [10, 20]. We only considered bedload as a mode of sediment transport because Pekerebetsu River is a typical gravel-bed river. The bed material was assumed to be uniform sediment, and the bedload transport rate was estimated using the Meyer-Peter and Müller formula [21]. The bed elevation was then updated using the two-dimensional Exner equation. Bank erosion and land accretion were calculated using the models of Parker et al. [11] and Asahi et al. [10], respectively. The land accretion model assumes that the calculated area in which the bed surface is not inundated for a certain period is a newly constructed land area. The model then removes this area from the computational domain and reconstructs the calculation grid by fitting a new channel shape. This process promotes the development of meanders because the development of the inner bar enhances the corresponding outer bank erosion (bar-push mechanism [8]).

We simulated 29 cases with different flow discharges and initial river widths, with a bed slope \(i=0.01\) and grain size of 75 mm, assuming representative values for the downstream reach of the Pekerebetsu River and a bed slope of \(i=0.001\) assuming a relatively mild river (Table 1). A bed slope of \(i=0.001\) was simulated to verify the applicability of the model. In the cases with \(i=0.01\), we gave ten wavelengths of meanders with a meandering angle of 5° for a total channel length of 2000 m (i.e., an initial wavelength of 200 m). In the cases with \(i=0.001\), we gave 10 wavelengths of meanders with a meandering angle of 5° for a total channel length of 20,000 m (i.e., an initial wavelength of 2000 m). Because the Pekerebetsu River was a flat riverbed and straight channel prior to 2016, a flat riverbed and straight channel were used as the initial channel shapes in the simulations. However, the initial meandering wavelength and angle were used as triggers for meandering development. In each case, we continue the morphodynamic calculation until the meandering amplitude approximately reached the dynamic equilibrium state (the state in which the meanders migrate with time, but the value of the meander amplitude does not change significantly), which was 80 h real time for the cases with i = 0.01 and 3000 h real time for the cases with \(i=0.001\). We used periodic boundary conditions for the upstream and downstream ends so that the water and sediment were recirculated in the computational domain. It should be noted that we also calculated the bank erosion that supplies sediment from the bank area (i.e., outside the computational domain) into the channel (i.e., computational domain). This means that the sediment in the system increased as the meander developed, and the average bed elevation increased during the calculation. The bank height is always given by the bankfull condition during the simulations (i.e., the bank height equals the calculated flow depth).

Table 1 Computational conditions
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4 Results of free-meandering simulations

4.1 Dynamic equilibrium state

Figure 3a, b, and c show the temporal changes of the channel planforms of Runs 1, 3, and 26, respectively. All the runs showed significant meander development from the initial channel and eventually reached an equilibrium amplitude of channel planform; however, the process of the meander development depended on the computational conditions. In Run 1, the channel first widened after the calculation started, and, accordingly. Point bars formed in the inner bank of the initially given meander channel. Subsequently, meanders develop as the bar height increases and land accretion occurs. In contrast, in Run 26, the channel width first decreased at the beginning of the calculation, and meanders began to develop after that. Run 3 showed a more complex development of the meandering channel; that is, the channel widened similar to that of Run 1. However, the meander then developed after the channel width temporarily decreased. Furthermore, in Run 3, the position of meander development was not dependent on the initial meander shape, and the meanders migrated significantly downstream over time.

Fig. 3
figure 3

Contour map of calculated water depth, (a) Run 1 (bed slope i = 0.01, grain size d = 75 mm, flow discharge Q = 40 m3/s, initial channel width = 5 m), (b) Run 3 (bed slope i = 0.01, grain size d = 75 mm, flow discharge Q = 40 m3/s, initial channel width = 15 m), (c) Run 26 (bed slope i = 0.001, grain size d = 1 mm, flow discharge Q = 1,000 m3/s, initial channel width = 200 m). A is meander amplitude, L1/L2 is sinuosity

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Figure 4a shows the temporal change in the channel width-water depth ratio for the 25 cases with i = 0.01. At the beginning of the calculation, this ratio showed a large variation among the simulated runs; however, the average value of this ratio asymptotically reached a constant value of approximately 20 after 40 h. Figure 4b shows that the average values of the width-depth ratios for the four cases with \(i=0.001\) reaching a constant value after 1000 h, and the average value also converged to approximately 50. Figure 4a and b also show the temporal change in the channel sinuosity for the runs of i = 0.01 and i = 0.001, respectively. The sinuosity first increased and asymptotically reached equilibrium. This trend was similar to that of the width-to-depth ratio. A remarkable thing about this result is that the milder cases (Fig. 4b) demonstrated relatively small differences in sinuosity behavior among the runs. In contrast, the steep slope case showed a large scatter around the averaged value.

Fig. 4
figure 4

Time variation of calculated channel width-water depth ratio and sinuosity, (a) Runs 1–25 with bed slope i = 0.01, (b) Runs 26–29 with bed slope i = 0.001. The squares indicate average values and error bars indicate maximum and minimum values

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4.2 Comparison with field observation data

To confirm that the width-depth ratios calculated under the conditions shown in Table 1 are generally consistent with those in natural rivers, we compared them with the observed results at bankfull conditions in natural rivers published in the database of Li et al. [22] (hereinafter, “observed values”). The database contains discharge, channel width, depth, bed slope, and grain size under bankfull conditions for 231 rivers worldwide. The data are distributed in various sizes, ranging from 0.34 to 216,340 m3/s in flow discharge, 2.3 to 3,400 m in river width, 0.22 to 48.12 m in water depth, 0.00000875 to 0.052 in bed slope, and 0.01 to 167.5 mm in grain size. In the calculation results, we used the values obtained after reaching a dynamic equilibrium state (hereafter, " calculation values “). Specifically, we used time-averaged values of 40 to 80 h in runs with \(i=0.01\), and 2,500 to 3,000 h in runs with \(i=0.001\).

Figure 5 shows the relationship between the flow discharge and the width-depth ratio for the observed and calculated values. We also plotted the initial width-depth ratio for each run. The observed width-depth ratio exhibited a power-law relationship with the flow discharge, and the calculated values were plotted in the range where the observed values were distributed. In addition, we found that the calculated values shifted toward the approximate line of the observed values as they approached the dynamic equilibrium state from the initial state.

Fig. 5
figure 5

Relationship between the flow discharge and the width-depth ratio for observation and calculation values. The solid line represents the approximate line of observed values

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According to Li et al. [22] and Czapiga et al. [23], the following empirical relationship between dimensionless shear stress \({\tau }_{*}\), dimensionless grain size \({D}_{*}\) and bed slope \(i\) is satisfied under hydraulic conditions at bankfull discharge:

$${\tau }_{*}=\frac{\beta {i}^{m}}{{D}_{*}}$$
(1)
$${D}_{*}=\frac{{\left(Rg\right)}^{1/3}}{{\nu }^{2/3}}D$$
(2)

where \(\beta =1220\), \(m=0.365\), \(R\) is the specific gravity of sediment in water, \(g\) is gravity acceleration, \(D\) is grain size, and \(\nu\) is kinematic viscosity. Figure 6 suggests that the calculated values were consistent with the observed values. In this figure, milder slope cases (i.e., 𝑖=0.001) showed no significant change from the initial to the dynamic equilibrium states because the initial state we provided for these runs was close to the approximate line of observed values. In runs with \(i=0.01\), however, the calculation values satisfied the observed feature among \({\tau }_{*}\), \({D}_{*}\) and \(i\) regardless of the initial state. It suggests that the calculation results generally agreed with the empirical relationship regarding bankfull river geometry.

Fig. 6
figure 6

Relationship between the dimensionless shear stress to slope ratio and the dimensionless grain size for observation and calculation values. The solid line represents the approximate line of observed values

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4.3 Relationship between the sinuosity and flow discharge

A larger width-depth ratio led to a smaller sinuosity in the dynamic equilibrium state (Fig. 7a). In runs with i = 0.01, the sinuosity in the dynamic equilibrium state ranged from 1.2 to 1.6, with an average of 1.4. However, in runs with i = 0.001, the sinuosity was lower than that in the steep-slope case. The decrease in sinuosity with decreasing bed slope may be due to the difference in the width-depth ratio under the dynamic equilibrium conditions, as shown in Fig. 4. Schumm [24] showed that the sinuosity of natural rivers tends to decrease from approximately 1.6–1.2 as the width-depth ratio increases from 20 to 50. This may suggest that sinuosity is affected by changes in the formation and migration of free bars as the width-depth ratio increases because the width-depth ratio is a controlling factor in the dynamics of free bars.

Many natural rivers have sinuosity values exceeding 2. According to results of Schumm [24], the width-depth ratios of these rivers are often less than 10. In our simulations, the width-depth ratio was approximately 15–20 in runs with \(i=0.01\), and approximately 40–60 in runs with \(i=0.001\). Although there are natural rivers with width-depth ratios below 10 even under the flow discharge used in this study, the width-depth ratio was never below 10 in the dynamic equilibrium state in our calculation results (Fig. 5). Changes in bank erosion rates due to cohesive materials, changes in land accretion rates due to differences in climate and vegetation, and changes in sediment supply, which were not considered in our calculations, may affect the width-to-depth ratio of a natural river. In addition, the grain size of the bed material strongly influenced the width-depth ratio, as shown in Fig. 7b. A more in-depth analysis of the impact of these factors on the width-to-depth ratio and sinuosity is an exciting topic for future research.

The meandering amplitude-width ratio approximately ranged from 3 to 6, with no significant differences between i = 0.01 and i = 0.001 (Fig. 7c). In contrast, the meander amplitude increased with the flow discharge in runs with \(i=0.01\). However, the amplitude did not appear to depend on the flow discharge in runs with \(i=0.001\) (Fig. 7d). Note that the average meander amplitude in runs with \(i=0.01\) was approximately 80 m, whereas the runs with \(i=0.001\) showed a much larger amplitude of approximately 650 m. As explained above, the sinuosity of the milder channel cases was smaller than that of the steep channel cases. However, the absolute value of the amplitude was larger in the case with a milder bed slope. This is because the steep-slope case caused a much shorter wavelength of the meandering channel, resulting in a larger sinuosity but a smaller amplitude than that of the milder case.

The discharge dependence on the meander amplitude is unclear when focusing on either milder or steeper cases. However, from a macroscopic perspective, as shown in Fig. 7d, all the results with different slopes are summarized, and there is a strong positive relationship between the flow discharge and meandering amplitude. However, from a microscopic perspective, the change in amplitude with increasing flow discharge was more acute for steep-slope rivers than for mild-slope rivers. As shown in Fig. 4, the timescales of meander development also differed significantly between the steep and gentle-slope rivers. In steep rivers, meanders develop within short periods of flooding. This suggests that if the flow discharge increases by several 10% due to climate change, the risk of bank erosion can increase, especially in steep rivers.

Fig. 7
figure 7

(a) Relationship between the calculated sinuosity and channel width-water depth ratio, (b) Relationship between the calculated width-depth ratio and grain size-depth ratio, (c) Relationship between width-depth ratio and meandering amplitude-width ratio, (d) Relationship between the calculated meandering amplitude and flow discharge. The circles represent Runs 1–25 with bed slope i = 0.01, the triangles represent Runs 26–29 with bed slope i = 0.001. The dotted line represents the power-law approximation of calculated values

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These results suggest that the proposed numerical model of river meandering can capture the meander characteristics of natural rivers. In addition, the results indicate the crucial role of water discharge in meander amplitude and sinuosity. This may provide important insights into the risk assessment of river meandering since amplitude or sinuosity is the main indicator of the hazard due to meander development during flood events. In the following sections, we roughly assess the risk caused by river meandering by coupling physics-based meander dynamics and past and future discharge characteristics predicted by a climate model.

5 Dataset of flood hydrographs in past and future climate scenarios

To evaluate the meander amplitude and its future changes under climate change, we need flood hydrographs that have taken place until the present and will likely occur in future conditions in addition to the free-meandering model. Observed hydrological data on some decadal scales are generally available for large rivers, but the amount of data is limited, and sometimes, the data are not available for small streams, such as the Pekerebetsu River. In addition, future changes in hydrological characteristics due to the effects of climate change need to be evaluated. Extremely massive ensemble calculation data for rainfall have recently been developed using several types of climate models (e.g., UKCP09 [25] and d4PDF [26]) to overcome such difficulties. These datasets include spatiotemporal rainfall data for thousands of years and several expected future climatic conditions. Thousands of flood hydrographs for past and future climate conditions can be estimated using rainfall data with the help of runoff models.

In this study, we used discharge datasets developed by Hoshino et al. [27] and Hoshino and Yamada [6]. Hoshino et al. [27] downscaled 3000 cases (60 years x 50 ensembles) in past climate conditions, 3240 cases (60 years x 6 sea surface temperature (SST) x 9 ensembles) in 2 K increase scenario in temperature from before the industrial revolution (+ 2-K), and 5400 cases (60 years x 6 SST x 15 ensembles) in 4 K increase scenario in temperature from before the industrial revolution (+ 4-K), from rainfall data at 20 km horizontal resolution available in d4PDF for the period when the annual maximum basin-averaged rainfall occurs at the Obihiro in the Tokachi River basin. Then, runoff calculations were performed to evaluate river water discharge under past and future climatic conditions using year-maximum rainfall event data [6]. A lumped-type storage function runoff model proposed by Kimura [28], which is widely used in Japan, was used to estimate the flow discharge. The Hokkaido Development Bureau, an organization under the Ministry of Land, Infrastructure, Transport, and Tourism in Japan (MLIT), applied this model to the Tokachi River Basin. Their analysis divided the Tokachi River Basin into several sub-basins, and the model parameters were adjusted for each basin. The reproducibility of their analysis was confirmed at the Memurobuto Flow Observatory (Fig. 2), where large floods have been observed several times in the past. Hoshino et al. [27] and Hoshino and Yamada [6] described the details of the rainfall and discharge datasets.

Because of the distance between Obihiro and the Pekerebetsu River Basin, the predicted periods of maximum annual precipitation do not always coincide. Therefore, we extracted only those periods in which the periods of maximum annual precipitation predicted at the Obihiro site (the period for which downscaling data were available) coincided with the period of maximum annual precipitation predicted in the Pekerebetsu River Basin. Therefore, the number of calculated cases was slightly lower than that downscaled by Hoshino et al. [27].

Figure 8 shows the relative and cumulative relative frequencies of the peak flow discharge in the Pekerebetsu River obtained from the runoff calculations for all cases. The maximum flow discharge obtained from runoff calculations was 262 m3/s under past climate conditions, 430 m3/s under the + 2-K future climate scenario, and 416 m3/s under the + 4-K future climate scenario. Although the maximum flow discharge in the + 4-K future simulation is smaller than that in the + 2-K future simulation, Fig. 8 shows that the frequency of large-flow discharge tends to increase as the temperature rises due to climate change. The 99% tile values were 147 m3/s in the previous simulation, 193 m3/s in the + 2-K future simulation, and 274 m3/s in the + 4-K future simulation.

The flow discharge that gives a bed shear stress larger than the critical value for bedload motion in the case of uniform grain size was estimated to be approximately 50 m3/s using a simple uniform flow calculation, based on the conditions of the Pekerebetsu River before August 2016: 35 m width, 0.015 bed slope, 0.030 Manning roughness coefficient, and 90 mm grain size. In this study, we excluded cases with a peak flow discharge of 50 m3/s or less, which did not cause sediment transport, since we focused on meander amplitude development.

Fig. 8
figure 8

Relative and cumulative relative frequencies of peak discharge in the Pekerebetsu River in the past, + 2-K, and + 4-K climate scenarios

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The number of cases in which the peak flow discharge exceeded 50 m3/s was 266 out of 2434 cases in past climate conditions, 497 out of 3046 cases in the + 2-K future climate scenario, and 985 out of 4075 cases in the + 4-K future climate scenario; the median flow discharge was 77 m3/s in past climate conditions, 82 m3/s in the + 2-K future simulation, and 97 m3/s in the + 4-K future simulation. The 99% tile values of the flow discharge were 231 m3/s, 303 m3/s, and 353 m3/s for the past, + 2-K future, and + 4-K future climate scenarios, respectively. This indicates that the flow discharge that induces sediment transport tends to increase by approximately 30% in the + 2-K future climate scenario and by approximately 50% in the + 4-K future climate scenario compared to past climate conditions. The annual exceedance probability of a 250 m3/s flood event, the peak flow discharge at the time of the August 2016 flood was approximately 1/180 in the past climate simulation, 1/30 in the + 2-K future simulation, and 1/20 in the + 4-K future simulation. Therefore, such rare events may occur more frequently in the future.

The duration of a flow discharge of 50 m3/s or more also increases as the discharge increases owing to climate change.

6 Assessment of meandering amplitude under climate change

The most straightforward way for estimating meander amplitude and its change due to climate change is to apply the physics-based free-meandering model such as Asahi et al. [10] to the Pekerebetsu River geometry using the all the discharge data for several climate scenarios (e.g., the past, + 2-K, and + 4-K). However, it is difficult to conduct numerical calculations of the morphological changes in rivers for large ensemble data. Thus, we propose a simple method to understand the sensitivity of meander amplitude development to changes in hydrograph characteristics under climate change using a simple relationship between meander amplitude and flow discharge derived from a physics-based morphological model. Our numerical calculation provides the following equation between the meander amplitude and flow discharge for a bed slope of 0.01, as shown in Fig. 9:

$$y=18.647{Q}^{0.3842} \left(3\right)$$
(3)

where \(y\) is the meander amplitude (m), and \(Q\) is the flow discharge (m3/s). The maximum meander amplitude in each run can be encompassed by adding approximately 100 m to Eq. (3). Similarly, the minimum meander amplitude can be obtained by subtracting approximately 50 m from Eq. (3). For example, substituting the peak discharge of 250 m3/s for the August 2016 flood into Eq. (3) yielded an average meander amplitude of 150 m, with a maximum of 250 m and a minimum of 100 m. The meandering amplitude of the Pekerebetsu River averaged approximately 135 m, with a minimum of 90 m and a maximum of 200 m, due to this flood. These estimated values approximately represent the meandering amplitude of the Pekerebetsu River after the 2016 flood. Eq. (3) was developed from simulations with a discharge of 40 m3/s to 70 m3/s, bed slope of 0.01°, and grain size of 75 mm, so that the general applicability of this system should be verified in future studies.

Fig. 9
figure 9

Relationship between the calculated meandering amplitude and flow discharge in case of i = 0.01 (Runs 1–25). The dotted line represents the power-law approximation of calculated values. The Solid line is the maximum-minimum envelope

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Equation (3) is obtained from the numerical results under different flow discharges and initial river widths, assuming steady flow discharge (shown in Sect. 4). However, meander development in a natural river occurs under unsteady flow conditions. Generally, meander development is related to the magnitude and duration of flow discharge (e.g., [29, 30]). Thus, we incorporated the effect of flood duration into the estimation of the meander amplitude based on the numerical results of meander development. Figure 4a shows the time variation in sinuosity for rivers of different widths and flow discharges with a bed slope of 0.01. The average value increased until 30 h and then reached a steady state. This means that a hydrograph shorter than this threshold timescale will not allow the equilibrium meander amplitude predicted by Eq. (3). To consider this effect, the correction factor indicated by the red line in Fig. 4a was applied to the value calculated using Eq. (3). For example, substituting the peak flow of 250 m3/s for the August 2016 flood into Eq. (3) yielded a meander amplitude of 150 m. However, in the August 2016 flood, the duration of sediment transport above 50 m3/s was 20 h. Therefore, a correction factor of 0.66 (20 h / 30 h to reach an equilibrium state) was multiplied, and the estimated meander amplitude was 100 m. The meandering amplitude of the Pekerebetsu River averaged approximately 135 m, with a minimum of 90 m and a maximum of 200 m, owing to this flood. The predicted values, both with and without correction factors, were within the meander amplitude of the Pekerebetsu River after the 2016 flood and approximately replicated the meander amplitude of the Pekerebetsu River. Strictly speaking, the meander amplitude without a correction factor (i.e., 150 m) slightly overestimates the average meander amplitude of the Pekerebetsu River after the 2016 flood; however, for practical purposes, this will yield a safer amplitude. The small effect of the correction factor that represents the flood duration on the meander amplitude is due to the short morphodynamic timescale for steep, gravel-bed rivers such as the Pekerebetsu River (i.e., the dynamic equilibrium state can be reached within a relatively short timescale). The correction factor may be more important for low-slope rivers where meander amplitudes take longer to reach dynamic equilibrium.

Because Eq. (3) roughly reproduces the meandering amplitude of the Pekerebetsu River after the August 2016 flood, we evaluated the meandering amplitude that is likely to occur in future climate scenarios by substituting the peak flow discharge into Eq. (3). We estimated the meander amplitude with and without the duration correction factor and discussed the effects of increasing the peak flow discharge and duration.

Using Eq. (3), we calculated the meandering amplitudes for 266 cases of past climate conditions, 497 cases of the + 2-K future climate scenario, and 985 cases of the + 4-K future climate scenario. Figure 10 shows that the occurrence frequency of large meander amplitudes increased with climate change. The 99% tile values without the correction factor were 151, 167, and 178 m for the past, + 2-K, and + 4-K future climate scenarios, respectively. The 99% tile values with a correction factor were 128, 143, and 155 m for the past, + 2-K, and + 4-K future climate scenarios, respectively. Figure 10 shows that the results with and without correction factors are significantly different. Because the 2016 flood had a sufficiently large and long water discharge that could almost reach the dynamic equilibrium state, the meander amplitudes were roughly reproduced both with and without correction. However, smaller floods with relatively small discharges and shorter flood durations cannot induce intense sediment transport for a long time; therefore, it is likely that a dynamic equilibrium state is reached in a few cases. The reproducibility of meander amplitudes in floods with short sediment transport periods should be verified in future studies.

It is apparent that the correction factor that represents the hydrograph duration gives a smaller meander amplitude; however, this does not mean that the hydrograph duration has a negligible effect on the future increase in meander amplitude. To demonstrate this, we calculated the 99% tile value ratio between past and + 4-K future climate scenarios with and without the correction factor. This ratio is an indicator of a future increase in meander amplitude (thus, a risk related to bank erosion around the river), and we find that considering the correction factor increases this ratio from 1.18 to 1.21. This suggests that the increase in flood duration associated with climate change could also be an important factor in increasing meander amplitude.

Fig. 10
figure 10

Relative and cumulative relative frequencies of meandering amplitude in the past, + 2-K, and + 4-K climate scenarios, (a) without the correction factor, (b) with the correction factor

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In addition, the annual exceedance probability of the meandering amplitude of 150 m during the August 2016 flood, calculated from the cumulative relative frequencies without correction, was approximately 1/80, 1/20, and 1/15 for the past, + 2-K, and + 4-K future climate scenarios, respectively, indicating that the probability of occurrence increased with the progress of climate change.

We estimated the meander amplitude of the Pekerebetsu River under future climatic conditions based on an empirical equation obtained from numerous numerical results. Thus, Eq. (3) does not consider the characteristics of individual rivers or the development of local meanders. In addition, because the Pekerebetsu River is steep and its meander development reaches a steady state in a short time, we evaluated the effect of flood duration in a simplified manner by linear interpolation. However, it is necessary to study the relationship between the duration and meander amplitude in more detail to estimate the meander amplitude widely in other rivers.

Although the meandering amplitude assessment proposed in this study considered the effects of the peak flow discharge and flood duration, it did not consider the effects of various hydrograph patterns. Huang et al. [31] showed that the equilibrium states of free-migrating alternate bars obtained under unsteady flow conditions (e.g., the interval between repeated hydrographs or the rate at which the flow discharge increases and decreases) are different from the equilibrium states obtained under steady flow conditions. Because the formation and migration of free bars affect channel migration, meander amplitudes can vary depending on the hydrograph pattern. Periodic boundary conditions were used in the numerical calculations. In this calculation, the sediment supply from the banks was considered; however, we did not include changes in the sediment supply from mountainous upstream areas. Many researchers have suggested that increased rainfall due to climate change affects sediment supply rates and flow discharges (e.g [32,33,34]). At the Pekerebetsu River, the target site of this study, the sediment supply rate from the upstream mountains under the + 4-K scenarios might be 3–5 times greater than that under past conditions [35]. An increased sediment supply encourages bar development, which, in turn, facilitates channel meandering (e.g [5]). In some cases, this may trigger a transition from meandering to braided channels (e.g [36]). The numerical model in this study assumes a flat floodplain composed of non-cohesive alluvial materials without any restriction on channel widening in the cross-sectional direction. However, many naturally steep rivers have riverbanks partially composed of bedrock [37] or cohesive materials [38] or flanked valley walls (narrow valleys with constrained river width) [39]. In such cases, the bank erosion model for alluvial banks proposed by Parker [11] is not applicable, and the meander amplitudes would yield different results. In addition, this will be the same for the case in which bank erosion is strongly constrained by river training structures such as spur dikes and revetments (e.g [40]). Although this study evaluated changes in alluvial meanders owing to climate change under simple conditions, it has great potential for future development.

7 Conclusion

In this study, we attempted to quantify the risks due to river meandering during floods and the impact of climate change on such risks using a physics-based morphodynamic model of free meandering and a dataset of hydrographs under past and future climate conditions. We first tested the numerical model of free-meandering proposed by Asahi et al. [10] to determine whether it could capture the characteristics of natural rivers by conducting various meandering simulations under different flow discharges, initial bed slopes, and initial river widths under steady flow conditions. The simulation results were reasonably consistent with the observed characteristics of natural rivers, specifically the channel width-water depth ratio, bankfull shear stress, and sinuosity, as reported in the literature. The results suggest that sinuosity decreases with increasing river width-depth ratio and that there is a strong positive correlation between meandering amplitude and flow discharge, which is more evident in steep-slope rivers.

An empirical equation for estimating meandering amplitude was derived to evaluate the risks associated with meander development based on the numerical results. The discharge dataset prepared by Hoshino and Yamada [6] was incorporated into this empirical relationship between meander amplitude and flow discharge to understand the probabilistic characteristics of meander amplitude under past and future climate conditions. The results show that the meander amplitude increased significantly with climate change, even for the same probability of occurrence. We also found that the annual exceedance probability of the 150 m width of the Pekerebetsu River during the August 2016 runoff was approximately 1/80 for past climate conditions, but 1/20 and 1/15 for the + 2-K and + 4-K future climate scenarios, respectively, suggesting a significant future increase in risks related to river meandering in steep gravel-bed rivers.

The results of this study were applied only to the conditions of bed slope \(i=0.01\), so that we mainly assessed the risks related to bank erosion and meandering development in steep gravel-bed rivers. In addition, some natural rivers have cut-offs owing to the development of meanders. Analyzing the relationship between the flow discharge and meandering amplitude under a wide range of conditions with more computational examples is an exciting future challenge.