Abstract

Migration of rivers can tremendously impact both human lives and natural landscapes by supporting biodiversity, or by causing hazards such as flooding and landslides. This necessitates the development of reliable models that can predict such events and potentially help in flood risk management and land use planning over the long term. However, existing models often focus on isolated morphological indicators and lack quantitative validation using long-term remote sensing data. Furthermore, these models either focused on idealized or synthetic river geometries and never been applied or validated for rivers in Indian subcontinent. In this work, a numerical model is proposed to address the spatial and temporal evolution of migrating rivers in an alluvial landscape. The model integrates morphodynamic and hydrodynamic components to predict several important geomorphological features of a river such as centerline migration, change in river length, formation of neck cutoff zone, and sinuosity evolution across decadal timescales. The model is validated using the Landsat image data for two Indian rivers with different morphological behaviors: the Kosi River in northern India and the Manu River in the northeast. To the authors’ knowledge, this is the first comprehensive study that quantitively validates multidecadal river migration model against the satellite image data in the Indian subcontinent, addressing several gaps (both methodological and regional applicability) found in existing literature. 

1 Introduction

Lateral migration of rivers plays a critical role in impacting the geomorphology of alluvial floodplains in many ways (Williams et al. 2020). It is not only responsible for supporting diverse habitats and enriching agricultural lands with essential nutrients (Kafle et al. 2020), but also for frequent flooding and landslides (Panhalkar and Jarag 2017), thus resulting in both useful and adversarial outcomes (Li et al. 2007; Panhalkar and Jarag 2017). This necessitates the need for comprehensive studies on lateral migration patterns of rivers to understand their dynamic behavior and interactions with ecosystems (Naim and Hredoy 2021). The findings from such studies help us to accurately predict and better manage flood risks (Donovan and Belmont 2019), make informed decisions in executing infrastructure projects (Agnihotri et al. 2020), and assess the impacts of human on the sustainability of river ecosystems (Nayak 1996; Parker et al. 2011).

One of the earliest reported investigations into river meandering and migration dates back to the late 19th century and early 20th century, when the Mississippi River Commission conducted two series of hydrographic surveys on the lower Mississippi River (Hudson and Kesel 2000). In that pioneering work, they studied the morphologic characteristics and dynamics of the alluvial river channel between 1877 and 1924 (Hickin 1983). A few years later, researchers performed another experimental study (Eakin 1935) to assess the parameters responsible for river shifting along the Mississippi River.

In the next few decades (1950–1980s), a shift towards studies investigating the empirical relations between river migration and morphology can be observed due to the high cost and time associated with the experimental studies. A notable contribution was made by (Leopold and Wolman 1960), where the researchers examined how the length of meanders influences the width of the channel and the curvature radius of the Mississippi River, and established an empirical relationship for the same. During the 1970s, a set of important studies (Hickin 1974; Hickin and Nanson 1975) examined the evolution of meanders in the Beatton River of Canada and established an empirical relationship between the channel curvature and the river channel migration. The late 1970s and the next few decades experienced a paradigm shift towards the development of numerical models with the advent of improved computing resources for studying river meander stability (Parker 1976), meander pattern, provision of neck cut-off and spatial propagation (Howard and Knutson 1984; Hooke 1995), and formation of oxbow lakes (Sun et al. 1996). In the last two decades, remote sensing and geographic information system (GIS) technology have gained immense popularity in studying riverbank erosion and channel shifting due to their accuracy and ability to provide detailed information (Thakur et al. 2017; Langat et al. 2019; Kumar et al. 2024). A few of the noteworthy rivers where satellite remote sensing was integrated with the GIS to study river migration/meandering are as follows: Yellow River delta in China (Yang et al. 1999), Brahmaputra River in India (Archana et al. 2012), and Karnali River Megafan in Nepal (Rakhal et al. 2021). In the last two decades, researchers have been particularly intrigued by the fact that external factors such as human interventions and climate change in addition to sediment transport, bank erosion, flow dynamics, etc., significantly affect the migrating behavior of a river (Gregory 2006; Gibling 2018; Yin et al. 2023; Basha et al. 2024). In parallel, recent studies have explored the use of machine learning techniques to predict river migration, offering a data-driven alternative or complement to traditional methods (Amini et al. 2024).

Given the range of approaches developed over the years, this study focuses specifically on numerical modeling for understanding river migration. Compared to field-based investigations, numerical models offer a cost-effective way to explore long-term channel dynamics while maintaining predictive accuracy. To develop such a model, it is essential to account for key hydromorphological factors that influence lateral migration. Prior studies (e.g., Zhu et al. 2018; Parker et al. 2011) have shown that migrating rivers tend to maintain almost a constant width over long periods, justifying their treatment as a fixed parameter in simplified numerical models. Neck cutoff dynamics, critical to river evolution, are driven by bend convergence and flow diversion, with resulting oxbow lakes influencing future migration patterns (Camporeale et al. 2005; Schwenk et al. 2015). Soil cohesion, vegetation, and erodibility also impact lateral movement (Perucca et al. 2007; Wickert et al. 2013), and these effects must be reflected in a model for realistic outcomes. To capture flow-induced migration, approaches such as simplified bank erosion driven models, curvature-driven flow models (e.g., Ikeda et al. 1981; Parker 1976; Zolezzi and Seminara 2001) have been developed.

Despite the progress, several key gaps exist in literature, which warrants further investigation. The existing numerical modeling studies in the literature have primarily focused on idealized or synthetic rivers and overlooked the unique hydromorphological behavior of rivers in the Indian subcontinent. Furthermore, while some of the existing models can capture general planform evolution and statistical signatures of meandering, none of them quantitatively validate specific migration features: such as river centerline shift, neck cutoffs, changes in river length, or sinuosity—using long-term remote sensing data. There is also a notable lack of studies across multiple rivers with different morphological features. Lastly, to the best of authors’ knowledge, no prior study has demonstrated the ability to retrospectively predict river migration across multiple decades and validate such predictions using Landsat images. These limitations highlight the need for regionally focused, field-data informed, quantitatively validated modeling frameworks that can support long-term flood risk assessment, river management, and geomorphic forecasting.

In this work, all the above-mentioned limitations of the existing literature have been addressed. To do so, the proposed framework integrates morphodynamic and hydrodynamic models that utilize initial river centerline from Landsat data, and several morphological data (e.g., grain size, slope, river discharge, etc.,) as primary model inputs. The model is then run to simulate lateral migration over time, and its predictions are compared against Landsat-derived centerlines for multiple time intervals. To demonstrate the general applicability of the model, two Indian rivers (i.e., Kosi River and Manu River) with distinct geomorphological features have been considered. The first one, Kosi River, flows through the northern part of the Indian subcontinent and historically served as an important source of life and diverse ecosystems in the corresponding region (Oldham 1894; Dhar and Nandargi 2002). It has often been dubbed the “Sorrow of Bihar” due to Kosi’s recurrent floods and significant changes in the course. Kosi has shown extremely dynamic behavior and has tremendous importance on flood risk management, land use planning, and the overall environmental wellbeing of the region (Sinha et al. 2019), and thus modeling and understanding the migrating behavior of Kosi is critical. The Manu River flows through the state of Tripura in North-East India and serves as a crucial waterway for the region’s agriculture, ecology, and human settlements. It is famous for its dynamic and winding nature (Debnath et al. 2019; Deb and Ferreira 2015), and exhibits significant lateral migration, seasonal flooding threatening the adjacent towns (Debnath et al. 2022). Similar to Kosi, modeling and understanding the meandering behavior and long-term migration of Manu is essential for sustainable land use planning, flood risk mitigation, and regional water resource management in northeast India—a region that is both ecologically sensitive and socioeconomically vulnerable.

The novelty of this work is mainly as follows:

1. Proposed a numerical model, which integrates morphodynamic and hydrodynamic components to predict spatial and temporal evolutions (i.e., lateral migration and meandering) of an alluvial river;
2. Unlike some previous works on other rivers, where researchers generally studied very few aspects of a migrating river in a particular work, we tried to cover several different important aspects (e.g., river centerline shift, neck cutoffs, changes in river length, or sinuosity) for characterizing the migrating behavior of a river;
3. Demonstrated the ability of the numerical model to retrospectively predict river migration across multiple decades and validate such predictions using Landsat images; and lastly
4. To authors’ knowledge, there is no such comprehensive study available in the literature for Indian rivers, whereas rivers have been an integral part of this subcontinent for ages.

The rest of the paper is structured as follows: Section 2 describes the numerical modeling approach; Section 3 presents the study areas, simulation results, and validation; and Section 4 discusses the findings and concludes with directions for future work.

2 Methods

This section outlines the numerical implementation of the proposed model for simulating lateral migration and planform evolution of meandering rivers. The model couples a physics-based hydrodynamic component, which computes near-bank excess velocity by solving linearized depth-averaged flow equations using Fourier decomposition, with a morphodynamic component that updates river centerline displacement using finite difference methods. This framework captures interactions between channel hydraulics and river morphology, including neck cutoffs. The section is organized into three parts:

1. definitions of geometric notations used to represent the floodplain and river centerline;
2. key physical factors influencing river migration; and
3. the discretization and computational steps involved in implementing the model.

2.1 Notations used to describe the floodplain

In Figure 1, we have presented a schematic diagram of the floodplain and a cross-section of the river. The spatial coordinates of the floodplain are described by Cartesian coordinates (x^∗, y^∗, z^∗), where x^∗ and y^∗ denotes the horizontal axes and z^∗ denotes the vertical axis. The spatial coordinates of the river in the floodplain are described by using orthogonal curvilinear coordinates (s^∗, n^∗, z^∗), where s^∗ represents the direction of water flow and n^∗ represents the direction of lateral shifting from the channel centerline. The curvature of the channel axis (C*) is defined by C*=1/R*=- dθ/; where local radius of curvature of the river and the local angle of deviation is represented by R^∗ and θ, respectively. Now, if we consider a cross-section of the river at AA’, the cross-section would be similar to the schematic diagram shown in Figure 1(b). Here, the bankfull channel width is represented by , the average depth of the channel is represented by , and the slope of the floodplain is assumed to be constant for developing the model. Some other important parameters for defining the cross-section of the river would be the local bed elevation w.r.t. a datum (denoted by η^∗), free water surface elevation w.r.t. a datum (denoted by H^∗), and the local river water depth (denoted by D^∗ = H^∗ − η^∗). 

Figure 1 Schematic diagram of the a) floodplain layout, and b) the lateral river cross-section.

2.2 Factors affecting riverbed evolution

Researchers have shown that the bank strength and width-to-depth ratio play a vital role in influencing the migrating pattern of a river (Zhu et al. 2018). Several factors like the probability of bank failure, composition of slumped blocks, and vegetation cover control the bank strength, thus affecting the width-to-depth ratio of a river. Therefore, in the proposed model, we have considered the width-to-depth ratio of the river channel to be one of the most important parameters. In the proposed model, a constant width of the river channel has been assumed, and it is well-known in the community that a migrating channel attains roughly constant width over an extended period, by integrating the consequences of multiple flood events and sediment transport processes in this timeframe. Although there can be some short-term variations, as the river’s sinuosity evolves, the channel maintains a stable constant width by reaching dynamic equilibrium (Parker et al. 2011).

Another important thing is to have a comprehensive understanding of the bank pull mechanisms and how they affect the channel width, as this in turn will impact the river dynamics, floodplain behavior, and the channel morphology (Wickert et al. 2013; Frascati and Lanzoni 2013). The combining effects of bank pull and push create a distinctive ridge-and-swale topographical feature on point bars, which typically characterizes the migration of a river and complex sedimentation patterns linked to it (van de Lageweg et al. 2014; Eke et al. 2014; Schuurman et al. 2016).

Cutoffs play an important role in determining the shape of meandering rivers, influencing various factors like the abandonment of certain channel sections and causing alterations in flow patterns, sediment transport, and floodplain dynamics. Understanding cutoff occurrences is crucial for accurately predicting the enduring characteristics of rivers’ meandering patterns (Camporeale et al. 2005; Frascati and Lanzoni 2010). Abandoned loops, identified by various researchers (Gagliano and Howard 1984; Gay et al. 1998; Hooke 2004; Constantine et al. 2010; Grenfell et al. 2012), play a role in sediment storage, influencing sediment distribution and dynamics within the river system. Oxbow lakes, also formed as a result of cutoffs, offer an historical record of the river’s past path and provide insights into migration patterns. Oxbow lakes also influence the future evolution of the river system, impacting hydraulic and sedimentary conditions, channel dynamics, flow patterns, and sediment transport (Schwenk et al. 2015). They act as sediment sinks, affecting the sediment balance and distribution within the channel-floodplain system (Camporeale et al. 2005). As it is evident that the cutoff is a major player in the river meandering process, in this modeling framework, we consider this as one of the important parameters in our proposed model. The present modeling framework primarily focuses on gradual neck cutoffs, where bend loop convergence and intersection are dominant, excluding chute cutoffs.

Soil properties (e.g., cohesion and friction angle) influence the resistance to erosion and in turn, impact the stability of alluvial riverbanks. In addition to that, the presence and characteristics of riparian vegetation, like root systems and density, contribute significantly to bank resistance against erosion, providing mechanical reinforcement and reducing the impact of the hydraulic forces (Perucca et al. 2007; Wickert et al. 2013; Motta et al. 2012). In our proposed model, we account for the erodibility, as it is evident from the preceding discussion that it plays a significant role in shaping the course of the river.

2.3 Mathematical modeling

With the knowledge of the most important factors affecting the migration of a river, we move on to discuss the proposed model. First and foremost, we define the displacement of the river across a floodplain, occurring perpendicular to the channel axis by ξ_n^∗(s^*, t^*). This can vary locally along the channel, and the local migration rate denoted by ζ^∗(s^*, t^*), represents how fast the displacement is changing with time t ^∗ (see Equation 1).

(1)

Where:

ζ*	=	local migration rate,
s*	=	direction of water flow,
t*	=	non-dimensional time, and
	=	displacement of the river centerline perpendicular to the channel axis.

In the case of a constant width channel (which is an assumption in the current model), the local migration rate ζ^∗ can also be expressed by introducing a dimensionless quantity, ζ, where ζ is the dimensionless local migration rate, obtained by dividing the actual rate (ζ^∗) by cross-sectionally averaged velocity (U₀^∗) (Ikeda et al. 1981). The dimensionless local migration rate ζ can also be expressed as a product of the long-term erosion coefficient (E) and the nondimensional excess velocity near the bank U_b, U_b = U_b^∗/U₀^∗. Both U_b and ζ vary with the longitudinal dimensionless coordinate (s), which is obtained by normalizing the actual coordinate (s^∗) by the characteristic width scale (B₀^∗).

We use a numerical approach (finite difference method) to solve the river migration problem posed in Equation 1. To do so, first, we discretized the river centerline as a polyline with N equally distributed points (P_i (x_i, y_i)) (see Figure 2), where (x_i, y_i) = (x_i^∗, y_i^∗)/B₀^∗. In each iteration, at every nondimensional time increment (t^k+1 = t^k + ∆t^k) (where, t = t^∗U₀^∗/B₀^∗), the migration of each node along the normal direction to s can be expressed by Equation 2 (Crosato 1990):

	(2a)
	(2b)

Where the updated positions for discretized points on the river centerline at each time step are (x_i^k+1, y_i^k+1) = (x_i (t^k+1), y_i (t^k+1)); and ζ_i^k = E_i^k U_bi^k. θ_i^k can be calculated using both backward and forward averaging (Lanzoni and Seminara 2006):

(3)

Where:

	=	coordinate of node i at time k + 1,
	=	coordinate of node i at time k,
	=	dimensionless local migration rate of node i at time k, and
	=	local angle of deviation of node i at time k.

Figure 2 A visual representation of how a point P, situated on the central axis of a channel, moves when the channel’s shape changes from its initial state defined by the configuration s(t) to a new state s (t + ∆t), where ∆t is a small time increment.

The dimensionless local curvature (C_i^k) can be obtained by discretizing the dimensionless distance ∆s_i^k between consecutive points along the river centerline:

(4)

Where:

v0	=	= curvature ratio (i.e., the relationship between curvature radius and river width),
	=	minimum value of R ∗ along the channel, and
	=	dimensionless distance between consecutive points.

To account for any potential numerical anomalies in the computation of local curvature, we applied the Savitzky-Golay smoothing filter (Frascati and Lanzoni 2009; Motta et al. 2012). To ensure computational efficiency and solution accuracy, while maintaining the stability of the solution, the time step size  is determined according to the following relationship:

(5)

Where:

α	=	threshold parameter between stable and unstable computations, approximately around 10-2, which is typically chosen empirically,
	=	erosion coefficient of node i at time k, and
	=	nondimensional excess near-bank velocity of node i at time k.

The selection of α involves a trade-off between the computational cost and the accuracy of the solution (Crosato 1990; Lanzoni and Seminara 2006). The dimensionless distance ∆s_i^k is generally assumed between 2/3 to 4/3, and the mesh of the river centerline is periodically reconstructed to ensure quasi-uniform spacing between nodes, adding or removing nodes as needed.

As mentioned earlier, cutoffs play an important role in determining the shape of meandering rivers, and thus we focus next on identifying potential neck cutoffs. To do so, we employ a similar technique proposed by Howard and Knutson (1984) and Sun et al. (1996) to examine the dimensionless distance between specific points. It utilizes a computationally efficient algorithm to detect potential neck cutoff locations based on the Cartesian distance between nodes. (Camporeale et al. 2005).

If the separation distance between nodes P_i and P_i+r (see Figure 3) goes below a predetermined threshold, the corresponding nodes are removed, thus simulating the formation of oxbow lakes. Subsequently, all the points P_i+j (j = 1 to r−1), are excluded from the computational grid, forming a new oxbow lake. Furthermore, some nodes upstream of P_i and downstream of P_i+r (e.g., P_i−q, P_i+r+q with q = 1, 2, 3) are also removed (see Figure 3 for an illustration of a natural neck cutoff event). Removal of these points prevents the formation of a highly curved segment of the river which is not expected to happen (Hooke 1995; Camporeale et al. 2008).

Figure 3 Illustration of a natural neck cutoff event that took place in the Kosi River, India, flowing from left to right: (a) Landsat image (resolution of 30 m) from March 6th, 1992; (b) Landsat image (resolution of 30 m) from March 10th, 2002; (c) and (d) schematic diagrams of the neck cutoff event.

In Figure 3, L^* is the river length along the central axis, l_x* is the straight line/Euclidean distance length of the river, r denotes the number of points considered when assessing the initial neck cutoff, and q represents the number of points removed to inhibit the formation of high-bend river stretches following a cutoff experience.

Another influential factor in modeling the riverbed evolution is the bank strength, which is governed mainly by the soil erosion coefficient. We assume separate erosion coefficients (i.e., relative susceptibility to erosion) for different geomorphic units (e.g., pristine floodplain, point bar complex, abandoned oxbow lake), and considered them as constant over time. In this model, we utilize the winding number algorithm (Hormann and Agathos 2001) to determine a point lies in which geomorphic unit, and then based on that, the appropriate erosion coefficient is assigned to the point. This method enables representing spatial variations in soil strength due to meandering dynamics, which allows us to capture the influence of heterogeneous geomorphic units on erosion and migration patterns.

One of the other crucial factors in determining the newly shifted position of the migrated river and the local migration rate is the excess near-bank velocity (U_b^∗), which requires studying the morphodynamics of river systems. To do so, the use of quasi-two-dimensional methodology is popular in the literature (Zolezzi and Seminara 2001). In this method, complex three-dimensional equations governing mass and momentum conservation are averaged over the flow depth, effectively reducing them to two dimensions. By doing this, the quasi-two-dimensional approach enables us to investigate river morphological changes, planform dynamics, etc., while reducing the computational complexity. Finding analytical solutions to complex equations describing river morphodynamics is always challenging, so to make things feasible and simpler, we considered a linear system of equations (facilitated by the assumption: perturbations of uniform flow and bed topography induced by spatial variations of channel curvature are small enough) and utilized the Fourier series to solve them (Zolezzi and Seminara 2001; Frascati and Lanzoni 2013). The Fourier series expansion creates a set of four ordinary differential equations of fourth order for each Fourier mode and helps represent the dynamic relations between different parameters. We utilize this concept, and calculate the excess near-bank velocity based on the Fourier expansion, with its functional form derived from the linearized system of equations:

(6)

Where:

	=	excess near-bank velocity,
	=	average velocity across the cross-section of a channel,
F	=	function,
	=	half-width-to-depth ratio, depicting the geometric characteristics of the watercourse,
Cf	=	friction coefficient,
τ∗	=	Shields number, quantifying the sediment transport capacity of the flow relative to the sediment grain size,
Rp	=	Reynolds particle number,
C	=	curvature of the channel axis,
λmj	=	characteristic exponents,
s	=	longitudinal dimensionless coordinate, and
ξ	=	displacement of the river across a floodplain.

Once the excess near-bank velocity is obtained, all the necessary inputs needed to calculate the updated river coordinates (and subsequently, channel migration, neck cutoff, etc.) using Equation 2 are available. The hydrodynamic component of the model follows boundary conditions based on the formulation by Zolezzi and Seminara (2001). The transverse velocity perturbation at the channel banks is set to zero, and the model assumes steady discharge and a uniform floodplain slope throughout the simulation. Equation 6 has various important river morphological parameters, and the following discussion will describe each of those parameters.

The Shields number is calculated by the following expression: , where ∆ is the submerged specific gravity, and expressed by ∆ = (ρ_s − ρ)/ρ: ρ_s and ρ stand for the density of the sediment and water, respectively; g is the acceleration due to gravity; and  is the characteristic grain size, i.e., average sediment grain size (referred by  as well), and expressed by  = d_s ×  (d_s denotes the particle size, a non-dimensional quantity representing the ratio of the sediment grain size to the channel depth).

The Reynolds particle number R_p = √(g∆)d_s^∗3/2/ν reflects the importance of particle inertia compared to viscous forces and depends on factors such as the sediment density and kinematic viscosity of the water (ν). In alluvial rivers (e.g., Kosi River, Manu River), when suspended load transport is more significant, the total load is predicted by using the approach proposed in (Engelund and Hansen 1967). The Reynolds particle number helps in deciding whether the riverbed is flat or covered with dunes, following the approach of (Van Rijn 1984). This information is important for computing the friction coefficient using the approach of (Engelund and Hansen 1967).

λ_mj (m = 0, ∞; j = 1, 4) denotes the four characteristic exponents, which are determined by solving the fourth-order ordinary differential equations for each Fourier mode. These exponents play an important role in deciding the morphodynamic behavior of migrating channels, affecting channel migration/meandering patterns, bedform development, and sediment transport processes.

As the river planform undergoes changes over time, total channel length and the bed slope change over time as well. Thus, it is necessary to establish appropriate relationships to ensure consistent steady flow and sediment transport conditions. This is done by numerically updating the pertinent physical factors (such as half width-to-depth ratio, Shields number, and particle size) from one time step to the next. While doing this, we assume a constant discharge and a constant floodplain slope over time, and the update is done based on the following equations:

(7a)

(7b)

(7c)

Where:

βk	=	half-width-to-depth ratio at time k,
	=	friction coefficient at time k,
	=	Shields number at time k,
	=	non-dimensional particle size at time k,
σT	=	river sinuosity, defined as the ratio of the curvilinear river length along the central axis of the channel (L∗) to the straight line/Euclidean distance length of the river (lx∗) (see Figure 3).

The sinuosity factor facilitates the update of total channel length and thus captures the morphological changes of the river.

To conclude this section, Figure 4 summarizes all the steps that need to be followed to predict river migration by a successful implementation of the proposed method.

Figure 4 Concise flowchart describing the major steps of the proposed model.

3 Results and Discussion

To evaluate the performance of the proposed numerical model, two alluvial rivers in India—the Kosi River (Bihar) and the Manu River (Tripura)—were selected as case studies. Both rivers exhibit active meandering and frequent morphological changes, making them suitable for model validation. In this section, we first describe the study areas of both the rivers, and how the input data was collected; followed by a discussion on the detailed validation of the proposed model in predicting lateral migration, neck cutoff zones, change in river length, and sinuosity of the river by comparing with the Landsat image data.

3.1 Study area and collection of input data

A 100 km segment of the Kosi River was selected, situated in the eastern part of Bihar in India, spanning from (25°40’N, 86°40’E) to (25°27’N, 87°20’E), and a 12 km stretch of Manu River situated in the Indian state of Tripura, extending from (24°16’N, 92°1’E) to (24°13’N, 92°3’E) (see Figure 5) as the study area.

Figure 5 (a) Study area shown in the map of India: K represents the Kosi River, and M represents the Manu River; (b) Kosi River planform configuration of the study area; and (c) Manu River planform configuration of the study area.

The Landsat image for Figure 5(b) was retrieved from the United States Geological Survey (USGS) website. The date of acquisition is 7th November 1972, spacecraft ID is Landsat 1, sensor ID: MSS, WRS Path: 150, WRS Row: 42, and Resolution: 80 m. The latitude and longitude span from 25^o40′N, 86^o40′E to 25^o17′N, 87^o20′E, and the flow of the river is from left to right.

The Landsat image for Figure 5(c) was also retrieved from the United States Geological Survey (USGS) website. The date of acquisition is 7th December 1992, spacecraft ID is Landsat 5, sensor ID: TM, WRS Path: 136, WRS Row: 43, and Resolution: 30 m. The latitude and longitude span from 24^o12′N, 92^o01′E to 24^o17′ N, 92^o03′E, and the flow of the river is from south to north.

The width and depth data of the Kosi River was collected from the Flood Management Improvement Support Centre (FMISC), Patna, Bihar, which surveyed seven sections along the river in this region. The data collected from FMISC, Patna indicates that the river width varies from 374 m to 744 m, and the depth varies from 0.5 m to 6 m in the study area. It is important to note that the width and depth mostly remain constant (approximately 500 m and 2.5 m, respectively) throughout the study area, and the extreme values are observed in very few places. Each year, the monsoon season (mostly from June to September) is the peak time for flooding, and the bank-full discharge ranges from 30 m³/s to 3500 m³/s. Predominantly, the selected study area has alluvial soil, which exhibits a texture ranging from coarse to finer and grayish color, with a characteristic sediment grain size = 0.1436 mm (Figure 6a shows the grain size distribution in the study area). The surrounding land is mostly used as cropland. The primary sources of water in the river are rain and/or melting of snow. For the selected study area, the terrain is mostly flat, and thus, we assumed the bed slope to be constant for the study area, and a value of 0.000315 (i.e., 1 m/3.17 km) was chosen based on the data obtained from FMISC, Patna at Kursela discharge endpoint. As Kosi is an alluvial river, we chose the erosion coefficient of 10−8, a value commonly assigned for the alluvial rivers in the literature (Frascati and Lanzoni 2009).

Figure 6 Bed sediment grain size variations in the studied stretch of the (a) Kosi River, and (b) Manu River.

A significant amount of information was obtained using GIS techniques. To do so, ArcMap10 was utilized for extracting geographical details, such as the spatial arrangement of the river axis (represented by the longitudinal curvilinear coordinates). First, the Landsat images of the study area (see Figure 5) (for the years 1972, 1992, 2002, 2013, and 2023) are obtained from the United States Geological Survey (USGS) server via Earth Explorer (https://earthexplorer.usgs.gov/). Next, for the computational domain of interest, the initial geometrical path of the river axis (obtained from Landsat images) was normalized by the half-width (), yielding normalized longitudinal and lateral coordinates of the river axis. For initial parameters for a representative simulation of Kosi River, considering the total bank-full width (2) as 500 m, the river depth () as 2.5 m, the slope of the channel (S) as 0.000315, friction coefficient (C_f ) as 0.01, and the characteristic grain size () as 0.1436 mm, it can calculate the following inputs for the numerical model: half width-to-depth ratio () is 100 and the Reynolds particle number () is 6.923. Assuming Manning’s coefficient (n) as 0.04 (commonly used value for alluvial river in the literature) and calculating hydraulic radius (R) as 2.47 m based on channel dimensions the cross-sectionally averaged velocity  = 0.81 m/s can be obtained. Once the  is determined, the Shields number () can be calculated as 3.328.

For the Manu River, the relevant information was obtained from the existing literature (Hossain et al.  2011; Deb and Ferreira 2015; Debnath et al. 2022). The data indicates that although the river width varies from 60 m to 140 m, and the depth varies from 0.3 m to 1.2 m, the width and depth mostly remain constant (approximately 100 m and 0.5 m, respectively) throughout the study area, and the extreme values are observed in very few places. The southwest monsoon significantly influences the regional climate of Tripura, contributing nearly 65% of the total annual rainfall during the months of May to September. Among these, June typically records the highest rainfall, closely followed by July. The Manu River exhibits pronounced seasonal variability in discharge. During peak flood conditions, discharge can reach up to 1,000 m³/s, while in the dry season, it can fall to as low as 3.0 m³/s (Deb and Ferreira 2015). The dominant soil type in the Manu River basin is sandy clay (Hossain et al. 2011), and the representative sediment grain size is d_s^∗= 0.119 mm (Debnath et al. 2022) (Figure 6b presents the grain size distribution of the study area). The surrounding area is primarily used for croplands and plantations. The main sources of water in the river are rainfall and limited snowmelt. The terrain is relatively moderate, and for the purpose of this study, the bed slope was assumed to be constant, with a value of 0.0005, as suggested by Debnath et al. (2023). Consistent with alluvial river modeling practices, an erosion coefficient of 10⁻⁸ was utilized.

A similar GIS-based approach was also followed for the Manu River. Using ArcMap 10 and Landsat images (see Figure 5), the river's spatial configuration was extracted and the river axis for the computational domain was normalized, as described for the Kosi River. The necessary input parameters for the numerical model, such as channel dimensions, slope, sediment size (see Figure 6b), Manning’s coefficient, and derived quantities like Shields number and Reynolds particle number, were computed using the same methodology. Table 1 summarizes all the relevant input information for Kosi and Manu Rivers.

Table 1 Collected information for the Kosi River and Manu River study area.

Input Information	Kosi River	Manu River
Length of the study area	100 km	12 km
Average river width	500 m	100 m
Average river depth	2.5 m	0.5 m
Average bankfull discharge	1012 m3/s	18 m3/s
Characteristic grain size	0.145 mm	0.119 mm
Bed slope	0.000315	0.0005
Erosion coefficient	10-8	10-8
Friction coefficient	0.01	0.02
Peak time for flooding	June to September	May to September
Land use	Mostly cropland	Mostly cropland and plantations
Sources of water	Snow and rain	Snow and rain

3.2 Prediction of lateral migration

Using the proposed numerical model, continuous yearly simulations of lateral migration (i.e., shift in river centerline) over the period from 1972 to 2023 were performed. First, the shift in river centerline was predicted for the years 1992, 2002, 2013, and 2023 based on the input data coming from 1972, 1992, 2002, and 2013, respectively. Additionally, the shift in river centerline for 2023 was predicted based on the input data from 1972. These two cases will help in evaluating the model for two interesting scenarios: (i) how accurate the model prediction is when the input data is coming from the preceding decade or so, and (ii) how accurate the model prediction is when the input data dates back half a century.

To quantify the accuracy of our model predictions w.r.t. the actual observation (obtained from Landsat images), two different metrics were used:

Mean Absolute Error (MAE): This is a widely popular statistical metric to assess accuracy of prediction by averaging difference between predicted and observed values. The mean absolute error (MAE) (ϵ) can be calculated using the following formula:

(8)

Where:

N	=	number of observation points,
Oi	=	observed value,
Pi	=	predicted value, and
||	=	absolute value of any quantity.

Hausdorff Distance (HD): This error metric quantifies the maximum spatial deviation between the predicted and observed river centerlines. Hausdorff distance is calculated using the following equation:

(9a)

(9b)

Hausdorff distance:

(9c)

Where:

O	=	observed centerline points,
P	=	predicted centerline points,
	=	Euclidean distance between points oϵO and pϵP.

These error metrics aid in getting comprehensive (both average and worst-case deviations) ideas about the accuracy of the model predictions.

Kosi River

In Figure 7, the numerically predicted vs the Landsat image data is plotted for Kosi River centerline to compare the numerically predicted river centerline vs the observed river centerline obtained from Landsat images. Figures 7a-d correspond to predictions done based on input data from the preceding decade, while Figure 7e represents the predictions done based on input data from 51 years ago. The qualitative analysis from these figures clearly shows a high degree of alignment between the predicted vs actual river centerlines. These figures also highlight the Hausdorff distance for each case (the numbers inside a small box, added in each subfigure), which varies from 2.85–3.99 km. While these values indicate the worst-case deviation, they still represent only about 3–4% of the study area length. Such errors are acceptable in river migration modeling, particularly over decadal timescales, where local anomalies—such as abrupt meander shifts, cutoffs, or data noise—can lead to isolated deviations.

Figure 7 Comparison between numerically predicted planimetric evolution of the Kosi River centerline with the data obtained from Landsat images over time: a) 1992, b) 2002, c) 2013, d) 2023, and e) 2023 based on the input from 1972.

To complement the Hausdorff Distance error metric, Figure 8 illustrates the Mean Absolute Error (MAE) across scenarios by plotting error distribution histograms. The plots clearly show that in all cases, not only is the mean absolute error low, but for most of the study area, the absolute error is also considerably negligible (evident from the histogram plots). It is observed that the mean absolute error is always less than 1 km (lies between 0.49–0.86 km) and for 67–87% of the points along the river, the error is less than 1 km. These statistics clearly demonstrate the goodness-of-fit for our proposed model, as 1 km is considerably small when discussing the length scale of a river like the Kosi.

Together, both the error metrics discussed above validate that the model performs reliably in both average and extreme cases of river migration over decadal and half-century timescales for the Kosi River.

Figure 8 Histogram plots showing the error distribution and mean absolute error (MAE) values for the comparison between the predicted and observed shift in Kosi River centerline for different time periods a) 1972–1992, b) 1992–2002, c) 2002–2013, d) 2013–2023, and e) 1972–2023.

Manu River

For the Manu River, the analysis is conducted in a similar way to the Kosi River, and the results are also assessed using the same error metrics to be consistent. Figure 9 presents the predicted centerline positions for the Manu River alongside the corresponding Landsat image-based data. The comparisons indicate a strong agreement for all the different scenarios. Like the Kosi River, the Hausdorff distances are also displayed, which vary between 0.26–0.5 km, i.e., the error is only ~ 2-4% of the study area length, which again falls in the acceptable range.

Figure 9 Comparison of numerically predicted planimetric evolution of the Manu River centerline with the actual data obtained from Landsat images over time: a) 1992, b) 2002, c) 2013, d) 2023, and e) 2023 based on the input from 1972.

Figure 10 displays the MAE for 5 different scenarios for the Manu River. The results demonstrate that the MAE typically remains around 0.1 km, whereas for over 80% of the evaluated points along the river path, the absolute deviation from the observed data is below 0.2 km. This metric demonstrates the excellent overall fit along the entire study area length of Manu River for both over decadal and half-century timescales.

Figure 10 Histogram plots showing the error distribution and mean absolute error (MAE) values for the comparison between the predicted and observed shift in the Manu River centerline for different time periods a) 1972–1992, b) 1992–2002, c) 2002–2013, d) 2013-2023, and e) 1972–2023.

3.3 Prediction of neck cutoff zone

Another important geomorphological feature that impacts the channel evolution and floodplain dynamics is neck cutoff. Thus, this section evaluates the accuracy of the numerical model in predicting neck cutoff zones for both the Kosi River and Manu River.

Kosi River

From the Landsat images (Figure 11) prominent neck cutoff formations can be seen in the Kosi River during 1992 and 2002, highlighted by red circles. The model aims to predict the neck cutoff zone formation for those years. In Figure 12, we compare observed river centerlines from Landsat images with model predictions. Landsat images in Figure 12a show that in 1972, the river centerline followed the red line, but in 1992, the river centerline followed the blue line, forming a neck cutoff zone. The numerical prediction from the model also shows a very similar trend (the green line), and it closely follows the actual river centerline, successfully predicting the formation of a neck cutoff zone. Figure 12b shows similar results for the year 2002.

Figure 11 (a) Landsat image for the Kosi River study area from the years 1972 (left), and 1992 (right); and (b) Landsat image for the Kosi River study area from the years 1992 (left) and 2002 (right). Red circles highlight the neck cutoff zones in the study area.

Figure 12 Comparison of the numerically predicted neck cutoff vs the observed neck cutoff for Kosi River study area from Landsat images: (a) for 1992; and (b) for 2002.

To quantitatively assess the accuracy of the predictions, the error distribution and mean absolute error (MAE) were plotted, as shown in Figure 13. The results indicate: (i) MAE values of 0.28 km (1992) and 0.42 km (2002), and (ii) 87–93% of river points had a prediction error of less than 0.6 km. Considering the complexities associated with river channel dynamics, this small variation observed between the simulated and Landsat data can be well accepted, and establishes the model’s ability to accurately predict the neck cutoff formation event for the Kosi River.

Figure 13 Histogram plots showing the error distribution and mean absolute error (MAE) values for the comparison between the predicted and observed neck cutoffs for the Kosi River in different periods: a) 1972–1992, and b) 1992–2002.

Manu River

For the Manu River, a similar analysis was also performed. For Manu, the neck cutoff is prominently observed in the 1992 Landsat image (Figure 14), again marked with red circles. Figure 15 compares the centerlines from 1972 (red) and 1992 (blue), showing the formation of a neck cutoff zone. The model prediction (green line) captures this trend effectively. The quantitative assessment shown in Figure 16 indicates: (i) a mean absolute error of less than 0.13 km, and (ii) for 100% of river points, the error remains below 0.25 km. Once again, the error is well within the acceptable limit and proves the efficiency of model in predicting the neck cutoff zone for Manu River.

Figure 14 Landsat image for the Manu River study area from the years 1972 (left), and 1992 (right). Red circles highlight the neck cutoff zones in the study area.

Figure 15 Comparison between the numerically predicted neck cutoff vs the observed neck cutoff for the Manu River study area from the Landsat image for 1992.

Figure 16 Histogram plots showing the error distribution and mean absolute error (MAE) values for the comparison between the predicted and observed neck cutoffs for Manu River during 1972–1992.

3.4 Prediction of river length

River length serves as a fundamental metric for understanding river morphology and tracking environmental changes. So, the numerical model was used to predict the lengths of the Kosi River and Manu River for the years 1992, 2002, 2013, and 2023, based on corresponding input data from 1972, 1992, 2002, and 2013, respectively. These predictions were then compared with the observed data obtained from Landsat images to evaluate model accuracy.

Kosi River

Table 2 summarizes the comparison of predicted river lengths with those derived from Landsat images for the Kosi River. For the Kosi River, the model performs very well for most years, with less than 3% error in 1992, 2002, and 2023. However, the errors rise to 10.5% in 2013, likely due to the presence of several abandoned river channels during that year, which are not considered in the current version of the model. This discrepancy may also come from the assumption of a planar catchment in the model, which does not account for existing old scroll bars or abandoned rivers. Future improvements could involve integrating such features into the model for better accuracy.

It can also be noted that both the predicted and Landsat data show that the length of the river is gradually decreasing, representing alterations of the river’s course over time. We also attempted to predict the river length in 2023 using only the 1972 input. In this case, the model yields a 17% error, which, although higher, still offers a reasonably acceptable estimate, especially in cases where more recent data might be unavailable.

Table 2 Comparison of numerically predicted and actual river length data for the Kosi River study area obtained from Landsat images (km).

Year	Kosi River
	Landsat Data	Numerical Prediction	% Error
1992	96.7	97.9	1.3
2002	90.8	93.3	2.7
2013	80.8	89.3	10.5
2023	78.5	79.8	1.7

Manu River

For the Manu River, the predicted river lengths closely match the Landsat-derived measurements for all years, as shown in Table 3. The maximum error across all years is 2.3%, demonstrating a high degree of accuracy and consistency in the model’s predictions for this river.

Table 3 Comparison of numerically predicted and actual river length data for the Manu River study area obtained from Landsat images (km).

Year	Manu River
	Landsat Data	Numerical Prediction	% Error
1992	11.5	11.6	1.2
2002	12.3	12.1	1.6
2013	12.3	12.6	2.3
2023	12.4	12.4	0.1

3.5 Prediction of river sinuosity

The sinuosity (defined as the ratio between the actual river centerline length and the straight-line distance between two points along the river) of a river is another important factor for understanding geomorphology and can act as an indicator of meandering behavior. Thus, in the following section, we aim to compare the numerically predicted river sinuosity with the actual data derived from the Landsat images.

Kosi River

To evaluate the sinuosity, the centerline of the Kosi River was divided into equal 10 km segments. For each segment, the sinuosity factor was calculated by taking the ratio of the river’s actual centerline length to the straight-line distance across the meander belt. This was done using ArcMap 10, and the analysis was conducted for the years 1992, 2002, 2013, and 2023.

In Figure 17, the numerically predicted sinuosity values are compared with those obtained from Landsat images. The key findings can be summarized as follows:

• If the sinuosity values are less than or equal to approximately 2.2; the numerical model prediction shows a pretty good agreement with the actual data derived from the Landsat images (except for segments 3 and 4 in some cases). This indicates that the proposed numerical model is mostly able to capture the overall twisty/meandering nature of the river over time.
• The relatively higher mismatch in segments 3 and 4 can be attributed to the presence of abandoned rivers in those regions, which were not considered in the current model.

Figure 17 Comparison between the numerically predicted sinuosity of the Kosi River with the actual data obtained from Landsat images over time: a) 1992, b) 2002, c) 2013, and d) 2023.

Manu River

For the Manu River, although the total river length in the study area is approximately 12 km, it was divided into three equal segments to evaluate sinuosity. The numerical model prediction shows a pretty good agreement with the actual data derived from the Landsat image. As shown in Figure 18, the predicted sinuosity values closely match those obtained from Landsat images. In segments 1 and 3, the sinuosity values remain close to 1 across all years, indicating relatively straight reaches. Segment 2, however, displays higher sinuosity ranging from 2.0 to 2.6, reflecting a more meandering pattern. Notably, the model successfully captures this elevated sinuosity, demonstrating its ability to reproduce tighter bends and localized curvature with high accuracy.

Figure 18 Comparison of the numerically predicted sinuosity of the Manu River with the actual data obtained from Landsat images over time: a) 1992, b) 2002, c) 2013, and d) 2023.

4 Conclusion

In this work, a numerical model was proposed for predicting the spatial and temporal evolution (i.e., lateral migration and associated planform evolution) of alluvial rivers and it was applied to two geomorphologically dynamic rivers in India: the Kosi River in Bihar, and the Manu River in Tripura. These rivers were selected due to their complex morphodynamics, historical significance, as well as their socio-environmental relevance. The model incorporates critical parameters, such as width-to-depth ratio, bed slope, sediment characteristics, etc., to reliably predict the lateral migration and associated geomorphological features of a river. To assess the model’s performance, the study focused on four key geomorphological indicators: river centerline migration, formation of neck cutoff zones, change in river length, and river sinuosity. Predictions were made for the years 1992, 2002, 2013, and 2023, using earlier input data, and were validated against Landsat image data obtained from USGS archives.

The model demonstrated good accuracy in simulating the centerline evolution (i.e., lateral migration) of both rivers over both decadal and half-century timescales. In addition to the visual/qualitative comparison, two different error metrics were utilized (i.e., Hausdorff Distance and Mean Absolute Error) to quantify the performance of the model. For the Kosi River, the Hausdorff distances between predicted and observed centerlines remain within 2.85–3.99 km (roughly 3–4% of the river length), while the mean absolute error ranges from 0.49–0.86 km. Notably, for 67–87% of the river segments, the errors remain below 1 km, which is quite acceptable for large-scale river modeling. The performance is even more precise in the case of the Manu River, where the Hausdorff distance varies between 0.26–0.5 km, and the mean absolute error remains around 0.1 km. For over 80% of the Manu River’s length, the centerline prediction error is under 0.2 km, indicating a very high degree of spatial accuracy.

The model also proved effective in predicting neck cutoff zones, with mean absolute errors consistently below 0.5 km for the Kosi River, and below 0.13 km for the Manu River. Additionally, over 87% of the points in Kosi, and 100% in Manu showed neck cutoff prediction errors below 0.6 km and 0.25 km, respectively. River length predictions also demonstrate high fidelity, with errors mostly under 3% in all cases—except for one scenario in the Kosi River (2013) where older scroll bars and abandoned channels introduced a higher error of 10.5%. Nonetheless, even in half century timescale predictions (e.g., predicting 2023 using 1972 data), the model provides reasonable estimates with an error of ~17%, showcasing its utility in data-sparse scenarios.

Lastly, the model successfully reproduces spatial variations in river sinuosity over time (i.e., twisty/meandering nature of the river over time), aligning closely with Landsat-derived data. For the Kosi River, model predictions aligned well with observed sinuosity when values were below 2.2, though deviations were observed in regions influenced by abandoned channels, an area that presents an opportunity for future model refinement. For the Manu River, sinuosity predictions showed excellent agreement, even in high-curvature zones, with little deviation from observed values. These results confirm the model’s capability to simulate complex meander evolution.

Despite these promising results, the current model has certain limitations. It assumes a uniform channel bed slope, fixed channel width and depth, and does not account for geomorphic complexities such as old scroll bars, abandoned channels, soil uplift, and subsidence in the river catchment area. Additionally, it does not account for abrupt environmental changes, including anthropogenic interventions (e.g., river engineering, land-use change) or extreme hydrological events (e.g., floods, droughts). These simplifications contributed to some local inaccuracies, particularly in the highly dynamic Kosi River. While the model was tested on two rivers with distinct scales and morphologies, the Kosi River study area (~100 km long, 500 m wide) and the Manu River study area (~12 km long, 100 m wide), future studies should further explore its applicability across a broader range of river systems to assess generalizability. Future research could also aim to incorporate spatially variable topography, evolving channel dimensions, additional geomorphic processes, and abrupt environmental changes.

Nevertheless, the proposed framework is a quantitatively validated approach to simulate lateral river migration over long timescales and demonstrates strong predictive capability across rivers with differing geomorphological characteristics. This highlights its practical potential in flood risk assessment, land-use planning, and sustainable river management, particularly in data-scarce but hydrologically sensitive regions.

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List of Variables

2B0*	Bankfull channel width
Cf	Friction coefficient
Cik	Dimensionless local curvature
ds*	Characteristic grain size
ds	Non-dimensional particle size
D0*	Average depth of the channel
D*	Local river water depth
E	Long-term erosion coefficient
g	Acceleration due to gravity
H*	Free water surface elevation w.r.t.  a datum
HD	Hausdorff distance
L*	River length along the central axis
lx*	Straight line/Euclidean distance length of the river
n	Manning’s coefficient
N	Equally distributed points along river centerline Pi (xi, yi)
Oi	Observed centerline points
Pi	Predicted centerline points
q	Number of points removed to the formation of high-bend river cutoff
r	Number of points considered when assessing the initial neck cutoff
R	Hydraulic radius
R*	Local radius of curvature
R0*	Minimum value of R* along the channel
Rp	Reynolds particle number
s	Longitudinal dimensionless coordinate
s*, n*, z*	Orthogonal curvilinear coordinates Where s* represents the direction of water flow, and n* represents the direction of lateral shifting from the channel centerline.
Δsik	Dimensionless distance between consecutive points
t	Non-dimensional time
Δtik	Time step size
Ub	Nondimensional excess near-bank velocity
Ub*	Excess near-bank velocity
U0*	Cross-sectionally averaged velocity
(xi, yi)	Non dimensional polyline coordinate
x*, y*, z*	Cartesian coordinates where x∗ and y∗ denotes the horizontal axes, and z∗ denotes the vertical axis
α	Threshold parameter between stable and unstable computations
β	Half-width-to-depth ratio
Δ	Submerged specific gravity
ϵ	Mean Absolute Error
ζ*	Local migration rate
ζ	Dimensionless local migration rate
η*	Local bed elevation w.r.t. a datum
ϑ	Local angle of deviation
λmj	Characteristic exponents
ν	Kinematic viscosity of the water
ν0	Curvature ratio
ξn*	Displacement of the river centerline perpendicular to the channel axis
ρs	Density of the sediment
ρ	Density of the water
σT	River sinuosity
τ*	Shields number