Morphological Characteristics of Constrained Meandering Rivers in the Loess Plateau

Abstract

In the Loess Plateau of China, loess is widely distributed and forms a unique geomorphic feature of the world. Meanwhile, the Yellow River water and sediment regulation system is under construction. Nonetheless, the morphological characteristics of constrained meandering rivers in the Loess Plateau are still unknown due to the difficulty of extracting the sediment-laden water body. An improved method is proposed based on Landsat 8 imagery, which automatically extracts the multi-band spectral relationship of high-sediment-concentration rivers in valleys. This study analyzes the morphological characteristics of constrained meandering rivers in the middle reaches of the Yellow River Basin, including their sinuosity, periodicity, curvature, and skewness based on the deflection points bend segmentation and continuous wavelet transform methods. These characteristics are then compared with those of other constrained meandering rivers and alluvial meandering rivers. The results show that the sinuosity of the constrained river bends is generally low (with an average of 1.55) due to limitations imposed by the riverbanks, which prevent full development. The average dimensionless curvature radius of the constrained rivers is 18.72, lower than that of alluvial rivers. The skewing angle of the constrained river bends typically inclines upstream, with a proportion reaching 59.44%. In constrained river bends, as the sinuosity increases, the proportion of bends skewing upstream initially increases and then gradually decreases. This indicates that constrained river bends can develop similarly to alluvial bends at lower sinuosity but are limited by the mountains on both sides at medium sinuosity. The analysis of river characteristics in regions with different geological structures reveals the effect of geological structures on the formation of constrained rivers in the Loess Plateau. These findings can provide a reference for selecting reservoir dam sites and are important for the dredging engineering layout in the middle reaches of the Loess Plateau. They also offer quantitative explanations for the meandering characteristics of these rivers.

1. Introduction

Meandering rivers are common in nature and exhibit certain regular patterns. Based on bed surface characteristics and surface environment, meandering rivers can be broadly classified into four types: alluvial rivers [1], bedrock–alluvial mixed meandering rivers [2], tidal rivers [3], and glacial rivers [4,5]. Previous studies have focused extensively on the characteristics of meandering morphology, hydrodynamics, and meandering migration in alluvial channels [6]. Kong et al. [7] developed a systematic and universal river morphology extraction framework to investigate changes in the planform of the Lower Yellow River between 1987 and 2017. Baniya et al. [8] analyzed the dynamics of channel planform and shifting of Koshi River using Landsat TM5 images and Sentinal images. He C et al. [9] presented a global dataset of the shape of drainage systems and provided information on the sinuosity of river channels. However, research on constrained bedrock–alluvial mixed meandering rivers in gorges remains insufficient. In terms of the elucidate the morphological characteristics of meandering bends, Dominguez et al. [10] proposed a meander statistics toolbox to calculate the river centerline’s sinuosity, meander length, chord length, meander amplitude, and meander orientation, while the data of the river centerline was drawn from upstream to downstream. Numerous efforts have been made to characterize large planforms’ dynamics using satellite images on GIS planforms [11], as well as on MATLAB (version 2016) and Python (version 3.6) platforms [12,13]. Recently, He Y et al. [14] based on the GEE (Google Earth Engine) platform (https://earthengine.google.com/, accessed on 22 September 2024), used multi-source remote-sensing images (Landsat 5/7/8 and Sentinel-2) for water segmentation of braided rivers and calculated the morphological characteristics parameters. Amini et al. [15] used machine learning algorithms to study the migration rate of simulated meandering rivers, such as multi-layer perceptron, eXtreme Gradient Boost, gradient boosting regressor, and decision tree to predict the migration rate. However, we need to explore a new method to extract the water body considering the high sediment concentration of the constrained rivers in the Loess Plateau.

The constrained meandering river mainly refers to the meandering river whose horizontal migration is constrained by valleys on both sides of the river and whose plane shape cannot be fully developed. The main difference between constrained meandering rivers and freely developed alluvial rivers is reflected in their plane shapes and migration characteristics [16]. The width of the bedrock valley is influenced by both sediment and water transport dynamics, as well as bedrock lithology as the river absence of significant tectonic activity [17]. Research on constrained bedrock–alluvial mixed meandering rivers to date has mainly focused on laboratory experiments and field investigations of the mechanisms of lateral and vertical erosion [18,19,20,21,22,23], as well as theoretical or numerical studies based on these physical processes [2,24,25,26]. Xia et al. [27] revealed through the morphological characteristics and evolution processes in the Lower Yellow River that these sharp bends in the Lower Yellow River had highly sinuosity and deformed rapidly, with curvature coefficient equal to 1.60–4.18. Nicoll and Hickin [16] studied the morphology and migration characteristics of constrained meandering rivers in 23 places on the Canadian prairie. Through comparing the relationships between the dimensionless wavelength, the curvature of the bends, and the migration rates of these rivers, they suggested that constrained meandering rivers should be regarded as a subset of meandering rivers.

The object of this study is the constrained meandering rivers in the Shanxi-Shaanxi gorge section of the Loess Plateau. The Cenozoic tectonic evolution varies across different regions. Based on differences in sedimentary processes and characteristics of Cenozoic tectonic movements, the Loess Plateau is divided into three Cenozoic tectonic movement zones: the Ordos stable block, the Longxi area, and the Fenwei Cenozoic rift [28], as shown in Figure 1. In the late Pliocene, the Ordos block was uplifted as a whole, resulting in strong river erosion, deep incision into the bedrock, and the development of mountain gullies on both sides of the river. These gullies are primarily characterized by a combination of loess landforms with varying degrees of erosion, such as tablelands, ridges, and hills. Consequently, we can conclude that the constrained meandering rivers in the Shanxi-Shaanxi gorge section of the Loess Plateau belong to the second type of Lewin’s classification [29]. This classification indicates that the width of the valley is less than the amplitude of the meandering river, the distribution of alluvial sediment is discontinuous, and the outer bank of the meandering river is close to the constrained gorge.

This paper analyzes the plane morphological characteristics of the constrained meandering river in the Shanxi-Shaanxi gorge section of the Loess Plateau. Since the limitation of channel migration is often related to the low erosivity of the outer bank, current research on constrained meandering rivers mainly focuses on the study of erosion mechanisms through flume experiments. These experiments utilize various types of flumes, including wide and straight flumes [30,31], constant curve flumes [21], and high-amplitude Kinoshita meandering flumes [18]. However, whether these flumes are widely applicable to constrained meandering rivers remains to be further studied. The planar morphological characteristics are important for selecting the dam and reservoir location, quantifying the reservoir desilting area, and planning sand mining in rivers. This paper attempts to analyze changes in sinuosity, curvature, bend-skewing angle, and wavelength of bends in the constrained meandering river of the Loess Plateau; it also discusses the differences between constrained meandering rivers and alluvial rivers. The quantification of the variability of meandering rivers morphometrics is necessary for scientific purposes, such as analyzing the interrelationship between meanders and riparian vegetation, the influence of sinuosity on surface and groundwater interrelationship, distinguishing natural from modeled channels, among others, and for practical purposes, such as the design of civil infrastructure projection, impact of reservoir on meandering planforms, and river restoration, etc. [32]. The internal regularity of the constrained meandering river’s morphology in the Shanxi-Shaanxi gorge section is explored, which has significant value for the landscape development and protection of natural river bends and can provide guidance for the construction of water conservancy projects, especially for the Yellow River water and sediment regulation system and the national water network.

2. Materials and Methods

2.1. Research Area

The Loess Plateau is a distinct regional geomorphic unit, extending from the western foot of the Taihang Mountains in the east to the Wushaoling Mountains and the eastern slope of the Riyue Mountains in the west, from the northern slope of the Qinling Mountains in the south to the Great Wall in the north. The Loess Plateau covers an area of about 64.00 × l0⁴ km², which is the largest loess area in the world, accounting for 6.6% of Chinese total area [33]. The Loess Plateau is located in the northern part of China, with a range of 33° N to 41° N and 100° E to 114° E (Figure 1). Due to the East Asian monsoon climate, the Loess Plateau is a typical semi-arid region affected by high-intensity monsoon activities. The annual average rainfall is about 420 mm, with 60–70% of the total rainfall occurring from June to September [34]. The geomorphic processes on the Loess Plateau are intense, leading to severe soil and water loss primarily due to erosion and denudation. These processes are both a cause and effect of geomorphic evolution, making the Loess Plateau the main source of sediment for the Yellow River.

2.2. Materials

Remote-sensing images can be used to obtain geographic information over a wide spatial range and large time scales. Guo et al. [35] used Landsat 8 images and Google Earth to collect data on meandering alluvial rivers and to analyze how the bend-skewing angle changes with increasing sinuosity. The data used in this paper are Landsat 8 images provided by the Earth Explorer USGS website (https://earthexplorer.usgs.gov, accessed on 14 July 2014). Considering the climate characteristics of the Loess Plateau, where rainfall is concentrated in summer, the imaging time was selected accordingly.

2.3. Methods of Water Extraction

The common methods used for water extraction are NDWI [36] and MNDWI [37]. The extraction effects of the above two methods are better in meandering alluvial rivers on the plains [38]. He Y et al. [14] used every MNDWI image and combination method of Local Otsu algorithm to extract the braided river water body. However, it is worth noting that it is difficult to achieve good extraction accuracy in canyon rivers using the above methods due to the disturbance of terrain objects, such as shadows, bare rocks, vegetation, and cultivated land. Therefore, the multi-band-spectral relationship method based on thresholds is better to be used in mountain areas [39]. Moreover, Li et al. [40] proposed an improved method based on Landsat 8 imagery, which automatically extracts the multi-band spectral relationship of high-sediment-concentration rivers in valleys, and the principle is as follows:

$$2\left(B3+B4\right)-\left(B5+B2\right)-3\left(B7+B1\right)GTN$$

(1)

where N represents the empirical threshold, and B1 through B7 represent bands 1 through 7 in Landsat 8, respectively, GT means greater than. The algorithm in Equation (1) is based on the statistical characteristics of the spectral curve distribution of the surface features on the Loess Plateau. After the river was extracted, we removed noise and filled in missing parts of the river. The central line of the river, equidistant from both sides, was also extracted (Figure 2).

2.4. Characteristic Parameters of the River Morphology

For the morphological parameters of rivers, Howard and Hemberger [41] suggested that the morphological parameters of alluvial meandering rivers could be divided into the following four factors: meander wavelengths, meander irregularity, sinuosity, and large-scale wandering of the meander axis. Zolezzi and Güneralp [42] suggested that further research should focus on studying the time evolution in continuous wavelet transformation of meandering rivers. The use of wavelet transforms of curvature series has been employed to describe meandering behavior to detect changes in meandering characteristics, analyzing the change at different supermeander scales, and to characterize the wavelengths of meandering [32,43,44]. Gutierrez and Abad [32] used wavelet transform to quantify meandering morphometrics such as planform curvature signal of meandering rivers. Vermeulen et al. [44] proposed a quantitative method based on continuous wavelet transforms to study the planform shape of meandering features. In the study of the relationship between sinuosity and the bend-skewing angle of 19 meandering alluvial rivers, Guo et al. [35] pointed out that under natural conditions, the bend tended to evolve from bends inclined downstream with low sinuosity to bends inclined upstream with high sinuosity.

In the analysis of the morphological characteristics of 23 constrained meandering rivers in Canada by Nicoll and Hickin [16], the sinuosity of the meandering rivers ranged from 1.1 to 1.8, which was lower than the average value of meandering alluvial rivers (2.0–3.0). The dimensionless curvature radius (the ratio of curvature radius to river width) ranged from 1.1 to 13, with an average value of 4.1, significantly higher than the average value of free meandering alluvial rivers (2.0–3.0). The average value of dimensionless curve wavelength (the ratio of curve wavelength to river width) was 17, significantly larger than that of meandering alluvial rivers (8.0–14.0). Therefore, considering the research focus of this paper on constrained meandering rivers in the Shanxi-Shaanxi gorge section of the Loess Plateau, their sinuosity, periodicity, skewness, and curvature radius were analyzed. Additionally, this paper discussed the differences and similarities among constrained meandering rivers in the Loess Plateau, those analyzed by Nicoll and Hickin [16], and the alluvial meandering rivers analyzed by Guo et al. [35].

2.5. Bend Segmentation

Table 1. The hydrology and plane characteristics of the study area river.

No	Name	Province	Lat/Lon	L*/m	B*/m	L	No. Bend	A/km2	S	Lc	Lsl	${\mathit{\mu }}_{\mathit{T}}$	R
1	Huangpu	Shaanxi	39.782 N, 110.772 E	88,903	150	593	64	3199	0.00242	18.52	14.72	1.26	31.28
2	Xiaonan	Shaanxi	39.301 N, 110.918 E	12,149	35	347	34	85	0.00963	21.04	17.77	1.18	55.60
3	Kuye	Shaanxi	38.789 N, 110.515 E	70,657	150	471	41	8645	0.00231	23.55	16.30	1.45	32.14
4	Tuwei	Shaanxi	38.431 N, 110.373 E	51,841	150	346	39	3253	0.00301	18.19	9.53	1.91	14.63
5	Babao	Shanxi	38.150 N, 110.699 E	23,102	30	770	73	131	0.01169	21.39	14.94	1.43	19.76
6	Tuban	Shanxi	38.013 N, 110.638 E	17,829	40	446	71	116	0.01212	12.73	8.44	1.51	14.48
7	Qingliangsi	Shanxi	37.980 N, 110.832 E	29,388	50	588	88	283	0.01099	13.51	11.10	1.22	19.87
8	Quchan	Shanxi	37.101 N, 110.808 E	28,590	60	476	68	1023	0.00829	14.22	10.84	1.31	15.10
9	Wuding	Shaanxi	37.512 N, 110.261 E	119,780	150	799	79	30,217	0.00183	20.48	9.23	2.22	14.85
10	Yan	Shaanxi	36.615 N, 109.457 E	159,167	70	2274	173	7671	0.00301	26.44	15.81	1.67	21.56
11	Wei	Shaanxi	34.576 N, 105.738 E	168,525	130	1296	107	30,661	0.00305	24.46	14.70	1.66	17.51
12	Malian	Shaanxi	35.982 N, 107.903 E	119,262	100	1193	88	20,400	0.00125	27.42	18.38	1.49	18.80
13	Beiluo 1	Shaanxi	36.580 N, 108.738 E	371,960	210	1771	152	25,165	0.00200	23.46	12.64	1.86	14.52
14	Huan	Shaanxi	36.773 N, 107.191 E	145,965	90	1622	116	10,400	0.00159	28.21	20.91	1.35	22.75
15	Beiluo 2	Shaanxi	36.737 N, 108.353 E	56,075	100	561	65	7366	0.00155	17.52	12.24	1.43	14.75
16	Weifen	Shanxi	38.405 N, 111.013 E	26,304	50	526	49	1476	0.00661	21.47	14.72	1.46	18.77
17	Luoyukou	Shanxi	38.224 N, 110.920 E	22,891	45	509	59	225	0.01459	17.54	13.33	1.32	21.99
18	Jialu	Shaanxi	38.128 N, 110.229 E	36,152	60	603	53	1121	0.00849	23.17	14.59	1.59	15.83
19	Jing	Shaanxi	35.020 N, 108.140 E	108,000	170	635	71	42,425	0.00226	18.15	8.22	2.21	8.55

Note(s): Lat/Lon means the starts point of the river’s latitude and longitude; L* and B* mean the total length and the width of the research rivers, respectively. L, L_c, L_sl represent the dimensionless river length, the dimensionless curve wavelength, and the dimensionless straight line wavelength, respectively. No. Bend shows the number of the research section’s bends. A is the drainage area of the research river. S is the surface slope, ${\mu }_T$ is the total sinuosity of the research river, R is the dimensionless curvature radius scaled by B*/2.

summarizes the hydrological information and morphological characteristics of the research rivers. By dividing the center line of the rivers at 60 m intervals, we calculate the corresponding curvature, sinuosity, and the relevant morphological characteristics of the rivers. Ruben-Dominguez et al. [10] proposed meander statistics toolbox to calculate the river centerline’s sinuosity, meander length, chord length, meander amplitude, and meander orientation, while the data of the river centerline were drawn from upstream to downstream. The calculation formula for curvature radius and curvature is as follows [44,45]:

$$R\left(s\right)={\left({\displaystyle \frac{d{x}_0}{ds}}{\displaystyle \frac{d^2{y}_0}{d{s}^2}}-{\displaystyle \frac{d{y}_0}{ds}}{\displaystyle \frac{d^2{x}_0}{d{s}^2}}\right)}^{-1},\mathrm{c}\left(s\right)={\displaystyle \frac{1}{R\left(\mathrm{s}\right)}}$$

(2)

In the above equations, R(s), c(s), and s represent the curvature radius, curvature, and the distance along the center line of the river, respectively, while dx₀ and dy₀ represent the small displacement along the X-axis and Y-axis of the Cartesian coordinate system, respectively. Due to the different bending directions, there are positive and negative signs of curvature from upstream to downstream. These turning points of curvature are called deflection points, and the straight lines connecting these deflection points divide the meandering river into connected bends (Figure 3).

There is much noise in the distribution of curvature along the channel. Guo et al. [35] smoothed the curvature distribution four times using an 11-point weighted average, while Perucca et al. [46] calculated the curvature using three adjacent points and filtered it with a low-pass filter. We combine both of these methods: the multi-point weighted average method is used to smooth the curvature distribution first, followed by further adjustment using a low-pass filter. The processes are shown in the figure below.

3. Results and Discussions

We compared the morphological characteristics of 19 constrained rivers on both sides of the Shanxi-Shaanxi gorge on the Loess Plateau to 19 original alluvial meandering rivers analyzed by Guo et al. [35], 3 alluvial meandering rivers analyzed by Zolezzi and Güneralp [42], and 23 constrained meandering rivers in western Canada analyzed by Nicoll and Hickin [16]. The comparison and analysis were made in four aspects: sinuosity, periodicity, curvature, and skewness.

3.1. Plane Geometric Features

From Table 1, we can see that the average width of rivers on the Loess Plateau ranges from 30 m to 210 m. The sinuosity distribution of the constrained meandering rivers ranges from 1.18 to 2.22, whose significance is less than that of alluvial meandering rivers ranges from 1.90 to 3.70 [35]. The distribution of the dimensionless curvature radius of the constrained meandering rivers on both sides of Shanxi-Shaanxi Gorge on the Loess Plateau ranges from 3.752 to 40.032, which is larger than that of alluvial meandering rivers ranges from 4.0 to 10.0 [35]. The corresponding dimensionless curvature ranges from 0.02 to 0.27, with an average value of 0.14. Therefore, it can be seen that there is a certain corresponding relationship between dimensionless curvature and sinuosity.

3.2. Sinuosity Characteristics in Different Type of Curved River

We consider the analyses of alluvial meandering rivers by Guo et al. [35] and Zolezzi and Güneralp [42] as representative of alluvial meandering rivers, then we analyze the different sinuosity types of alluvial meandering rivers and constrained meandering rivers. Howard and Hemberger [41] proposed the concept of total sinuosity (${\mu }_T$), full-meander sinuosity (${\mu }_W$), residual sinuosity (${\mu }_R$), an half-meander sinuosity (${\mu }_H$), as shown in Figure 4 below, and we can find that ${\mu }_T={\mu }_W{\mu }_H{\mu }_R$.

$${\mu }_T={\displaystyle \sum _{i=1}^n{l}_i}/L,{\mu }_W={\displaystyle \sum _{k=1}^{p}S{L}_k}/{\displaystyle \sum _{j=1}^{m}D{L}_j},{\mu }_H={\displaystyle \sum _{i=1}^n{l}_i}/{\displaystyle \sum _{k=1}^{p}S{L}_k},{\mu }_R={\displaystyle \sum _{j=1}^{m}D{L}_j}/L$$

(3)

In Figure 5a, we arrange the values of μ_T in order from the smallest to the largest. It can be seen from Figure 5 that the average μ_T of the constrained meandering rivers in the Loess Plateau is 1.55. The average value of μ_T in the constrained meandering rivers of western Canada is the lowest, at 1.42, while the average value of μ_T in alluvial meandering rivers is 2.53. All the μ_T values of the alluvial meandering rivers are higher than 1.5, but about half of the rivers in the Loess Plateau and western Canada have μ_T values lower than 1.5, which is considered to indicate low sinuosity.

In Figure 5b, the average values of μ_W in the plain alluvial meandering rivers and the constrained meandering rivers in the Loess Plateau are similar, equal to 1.08 and 1.05, respectively. The average values of μ_R in the above rivers are 1.30 and 1.09, respectively, and the average values of μ_H are 1.81 and 1.35, respectively. From these results, we can see that for different sinuosity parameters, the average values for alluvial meandering rivers are larger than those for constrained meandering rivers. There are many constrained rivers that belong to the transitional type between straight and meandering channels, with μ_T lower than 1.5. Many natural rivers exhibit pronounced wandering of the valley axis due to topographic and structural controls. The overall meander sinuosity is equal to one for a regular, symmetrical meandering path, and greater than one for regular but asymmetrical meander loops, or for meander paths with irregular or random components [41]. This result shows that mountains on both sides constrain the development of meandering rivers, and the meander paths are irregular.

In Figure 6, both alluvial bends and constrained bends account for a large proportion of low sinuosity (within 1.5). The percentages of alluvial bends and constrained bends are 42.52% and 70.68%, respectively, which means that compared with alluvial meandering bends, constrained meandering bends are more common in low sinuosity ranges. In the high sinuosity range (over 3.6), the percentages of alluvial bends and constrained bends are 5.31% and 1.24%, respectively. Constrained meandering bends are mainly concentrated in the low sinuosity range.

3.3. Periodicity Characteristics in Different Type of Curved River

In this part, we focus on the river curve wavelength and the bend curve wavelength. The former is the ratio of the curve length of the river to the number of periodic bends (equal to half the number of bends), while the latter is twice the length of the curve bend. Both of these wavelengths are normalized by the river width. Generally, rivers with larger widths tend to have longer wavelengths, while rivers with smaller widths usually have shorter wavelengths. The wavelength of meandering rivers is not only related to the flow rate but also to the grain size of bed sediment [47]. In most meander evolution models, although river width, flow rate, and valley slope are assumed to remain constant, the straight and curved wavelengths of the meander vary at different stages of bend development.

The average dimensionless curve wavelengths of the alluvial rivers, the constrained rivers in western Canada, and the constrained rivers in the Loess Plateau are 27.54, 16.80, and 20.60, respectively. We can see that the variation range of the dimensionless curve wavelength in the alluvial rivers is the largest, while that of the constrained rivers in the Loess Plateau is the smallest. If we analyze the dimensionless curve wavelength for both the alluvial river bends and the constrained river bends, we can see from Figure 7b that their average dimensionless curve wavelengths are 27.68 and 15.89, respectively. Regarding the average value of the dimensionless curve wavelength, the value for the alluvial river bends is 1.74 times that for the constrained river bends, and the distribution of the dimensionless curve wavelength for the constrained river bends is more concentrated near the lower values. In Figure 7b, the maximum values of the probability density curve for the dimensionless curve wavelength of the alluvial river bends and the constrained river bends are 22.67 and 18.67, respectively. In natural rivers, the relationship between curve wavelength and river width follows a certain pattern: the curve wavelength of a meandering river is approximately 12 times its width, while the straight wavelength is about 6 times the width. Our results revealed that there is no linear proportional relationship between the wavelength of meandering rivers and river width, due to the influence of many uncertain and random factors. The curve wavelength of constrained and alluvial river bends shows that the characteristics of the riverbed, bedload particle size, and slope significantly impact the river wavelength. When the riverbed is bedrock and bedload particles are larger, the river tends to erode and develop under the influence of the slope, resulting in a longer wavelength. This also explains why, in constrained rivers, the development of river width is restricted, and meander cutoffs do not occur, even at larger dimensionless curved wavelengths.

3.4. Curvature Characteristics in Different Type of Curved River

In this part, we consider the comparison of the dimensionless curvature radius of the alluvial rivers and the constrained rivers, which is scaled by the half of the river width.

The average value of the dimensionless curvature radius of each river’s bend within 200 is considered as the dimensionless curvature radius of the river. The limit value of 200 is chosen mainly because some of the river bends’ dimensionless curvature is too small (<0.005), which may cause significant errors. In Figure 8a, the dimensionless curvature radius of the alluvial rivers is larger than that of the constrained rivers, and particularly, the dimensionless curvature radius of the rivers in the Loess Plateau is also larger than that of the rivers in western Canada. The average values of the dimensionless curvature radius for the alluvial rivers, the constrained rivers in western Canada, and the constrained rivers in the Loess Plateau are 26.82, 8.08, and 18.72 (excluding the Xiaonan River), respectively.

Figure 8b shows that the distribution of the dimensionless curvature radius of the constrained river bends is concentrated in the low-value regions. The values of the dimensionless curvature radius of the constrained river bends within 14 account for 47.93%, and those within 46 account for 90.28%. However, in the alluvial river bends, the values of the dimensionless curvature radius within 20 account for 49.54%, and those within 44 account for 90.78%. The two types of bends are similar in the dimensionless curvature radius over 90% of the range, but quite different in 50% of the range. Li et al. [48] proposed that, unlike alluvial rivers, the width of bedrock bends is largely controlled by sediment flux rather than water discharge. Additionally, sediment flux and rock strength govern the incision of the river into the bedrock.

Gutierrez and Abad [32] and Gutierrez et al. [43] first introduced wavelet analysis into the study of meandering rivers. Gutierrez and Abad [32] used the Morlet wavelet function to perform the wavelet analysis of the natural meander’s dimensionless curvature-local abscissa. Zolezzi and Güneralp [42] used the continuous wavelet transform (CWT) method to analyze the dominant meander wavelength (DMW) of rivers and found that the difference between the results of the global wavelet spectrum (GWS) using azimuth sequence and curvature sequence is minimal. Then, we used the curvature sequence to analyze the DMW of the rivers in our research, and the program we used is the wavelet analysis software provided by Torrence and Compo [49] (http://paos.colorado.edu/research/wavelets/, accessed on 22 September 2024). We take the Jing River as an example, which is shown in Figure 9.

The study of the wavelet global time–frequency spectrum aims to reveal the frequency values at each point across the entire range of the river, thereby qualitatively understanding the spatial dynamics of periodicity along the meandering river. The wavelet basis used in this study is the Morlet wavelet. Through the distribution of GWS, the dimensionless curve control wavelength can be obtained (normalized by river width), and then the DMW corresponds to the peak value of energy in GWS. The dimensionless DMW equals 19.625, which is slightly larger than the curve wavelength of the Jing River (=18.15). Abad et al. [50] found that the lower dominant curve wavelength is about 6 times the river width, regardless of river type, while the higher wavelength ranges from 15 to 25 times the river width. The lower range corresponds to confined or constrained rivers with a single-phase, uniform-width channel. In contrast, composite rivers, exhibiting compound bends, develop intermittent middle wavelengths and show a wider frequency spectrum. The dominant meander wavelength and the curve wavelength of constrained rivers in the Loess Plateau are both larger than approximately six times the river width, which may indicate that, due to the presence of loess cover, the bends develop until they cut into the bedrock.

All research rivers’ DMWs are analyzed, and the corresponding Cartesian wave numbers ($=\pi {\mu }_H/DMW$) are calculated. The Cartesian wave number is the Cartesian distance between the upstream and downstream endpoints of the research meander [42]. The dominant meander wavelength is the wavelength associated with the highest oscillation energy, and the bend growth rate increases nonlinearly with decreasing value of Cartesian wave number, and is typically an O(0.1) number [42]. Blondeaux and Seminara [51] pointed out that the disturbance wave number value required for a river to transition from a straight to a bend typically falls within the range of 0.1 to 0.2. Luchi et al. [52] extended the range to 0.3, primarily considering the variations in curvature and river width along the channel.

The distribution range of Cartesian wave numbers of rivers on the Loess Plateau is primarily between 0.1 and 0.3, with the exception of the Jing River, whose Cartesian wave number is 0.35. This result suggests that although rivers on the Loess Plateau are constrained by the presence of loess tableland, loess mounds, and loess beams, the surface layer of loess is susceptible to erosion by flowing water, allowing for the development of river bends to some extent. However, it is evident that the Cartesian wave number of many rivers exceeds 0.2, indicating that the development of river bends remains somewhat constrained.

In Figure 10, the Cartesian wave numbers of the alluvial rivers are all below 0.3, with an average value of 0.15, while the average value of the Cartesian wave numbers of the constrained rivers is 0.2027. From the distribution of the Cartesian wave numbers, it is evident that those of the constrained rivers in the Loess Plateau range from 0.1 to 0.3. Therefore, we can conclude that the constrained meandering rivers in the Loess Plateau are quasi-stable under typical floodplain conditions. This quasi-stability is reflected in the oscillation of widths along the channel and in the observation that the widest part of the river bend moves downstream from the upstream [52]. The magnitude of Cartesian wave number is O(0.1) in both alluvial and constrained rivers, which indicates that the meander bend stability tends to select much longer meander wavelengths compared to the channel width.

3.5. Skewness Characteristics in Different Type of Curved River

The streamwise symmetrical sine-generated curve has been proposed as an ideal description of river meanders. However, it may not be suitable for asymmetric river bends. In such cases, the Kinoshita curve is used to describe the river bends by adding third-order terms, allowing for planforms with skewing and fattening [53].

$$\vartheta \left(\mathrm{s}\right)={\vartheta }_0\left(\sin ({\displaystyle \frac{2\pi }{\lambda }}s)+{{\vartheta }_0}^2\left(-J_F\sin (3{\displaystyle \frac{2\pi }{\lambda }}s)+J_S\cos (3{\displaystyle \frac{2\pi }{\lambda }}s)\right)\right)$$

(4)

Here, $J_S$ and $J_F$ are the skewness and flatness coefficients, respectively, ${\vartheta }_0$ is the maximum angular amplitude, $\lambda$ is the bends curve wavelength, s is the streamwise coordinate. The sketch of the skewness angle is illustrated in Figure 3d. Bend skewness is linked to hydrodynamics, bed morphodynamic regime, bank characteristics, riparian vegetation, and geological environment [50]. The average skewness angle of the alluvial rivers ranges from 0 to 20⁰, indicating that most of the alluvial rivers skew towards the upstream and account for 91% of rivers. In comparison with Figure 11a, the average skewness angle of the constrained rivers ranges from 0 to 40⁰, with 73.7% skewing towards the upstream, which is notably lower than that of the alluvial rivers.

We analyze the bends skewness of the alluvial and the constrained rivers as shown in Figure 12a, in which the proportion of bends skewing downstream is 40.56% for alluvial rivers and 42.07% for constrained rivers, respectively. Therefore, in terms of river bends, there are few differences between alluvial and constrained bends. However, the proportion of bends skewing upstream in alluvial bends is still larger than that in constrained bends, with a difference of 1.51% between the two.

The above analysis mainly focuses on the skewness of rivers and bends, but more attention should be paid to the characteristics of skewness in bends with different sinuosity. We still use the skewness angle to represent the bend skewness. We classify the sinuosity of river bends into intervals of 0.1 and then categorize them (Figure 12b). Due to the lack of bends with sinuosity greater than 3.6 in alluvial rivers, those with sinuosity higher than 3.6 are merged into the high sinuosity class for analysis. The results are compared with those of the constrained river bends in the Loess Plateau.

In Figure 12b, there is a sudden increase in the proportion of upstream skewness in the constrained river bends with sinuosity in the range of 2.8 to 3.1. The main reason for this is the insufficient number of sample bends, which can easily lead to significant deviations. If we ignore this range, we can conclude that with the increase in sinuosity, the proportion of upstream skewness gradually increases. Furthermore, in the sinuosity range of 1.8 to 2.7, the proportion of upstream skewness decreases. At a sinuosity of 2.7, the proportion of upstream skewness decreases to 50%, and at a sinuosity of 3.4, it decreases to 40%. However, what is interesting is that when the sinuosity is higher than 3.6 (high sinuosity), the proportion of upstream skewness rises to 65%. This condition is similar to that in the alluvial bends with high sinuosity, which is equal to 65.9%. Nonetheless, in Figure 12b, we can see that when the sinuosity of the constrained bends is larger than 2.0, there are not enough samples. For a more comprehensive comparison, we need to collect more effective bend samples for analysis. With the increase in sinuosity in the alluvial bends, the proportion of upstream skewness gradually increases. When the sinuosity is 1.1, the proportion is only 49.4%, but when the sinuosity is greater than 3.6, the proportion reaches 65.9%. Guo et al. [35] suggested that in alluvial meandering rivers, downstream skewing dominates when the bends are relatively straight, but upstream skewing increasingly dominates as bend sinuosity increases.

For the analysis of the skewness angle of the constrained and alluvial rivers at different sinuosity, when the sinuosity is in the relatively low range (1–1.8), the proportion of upstream skewness gradually increases. However, in the medium range (1.8–3.5), they exhibit different trends: the proportion of upstream skewness of the alluvial bends increases, while that of the constrained bends decreases. Interestingly, when the sinuosity is in the high range (greater than or equal to 3.6), the proportion of upstream skewness for both types of river bends reaches 65%, indicating their consistency with high sinuosity. Moreover, the proportion of upstream skewness of the oxbow lakes in the alluvial rivers further increases. Guo et al. [35] suggested that if the research river reach is sufficiently long, upstream- and downstream-skewed bends may occur with roughly equal frequency. Abad et al. [50] pointed out that natural rivers exhibit both upstream-skewed and downstream-skewed bend patterns along the same reach, independent of the morphodynamic regime. Nonlinear processes, such as width variation and bedform dynamics, affect the development of three-dimensional flow structures, which in turn influence erosional and depositional patterns, as well as lateral migration. The results reveal that large grain-sized sediment may drive constrained bends to skew downstream at low and medium sinuosity. In constrained bends, bedrock strength affects the lateral migration rate, and erosion rates increase at the upstream part of bends within low sinuosity. As sinuosity increases, the three-dimensional flow structures, such as secondary circulation and flow separation, alter the erosional and depositional patterns, changing the orientation of constrained bends.

We divide the skew angle, bend sinuosity, and dimensionless curvature radius into intervals of 30°, 0.5, and 10, respectively, and depict the distribution between bend sinuosity and skewness angle in Figure 12c, and the distribution between dimensionless bend curvature radius and skewness angle in Figure 12d. In Figure 12c, for low bend sinuosity (1.0–1.5), the percentage of upstream skew for constrained bends and alluvial bends is 56.6% and 51.4%, respectively, with skewness angles in the range of (60°, 90°), comprising 20.1% and 21.7%, respectively. With increasing sinuosity, the percentage of skewness angles in the range of (30°, 60°) also increases for both constrained and alluvial bends. However, at high bend sinuosity (>3.0), the dominant skewness angle range for constrained bends shifts to (60°, 90°), while for alluvial bends, it shifts to (0°, 30°). In Figure 12d, for low dimensionless curvature radius (R < 20), the percentage of upstream skew for constrained bends and alluvial bends is 61.1% and 63.8%, respectively, with most skewness angles falling within the range of (0°, 60°). However, as R increases, the percentage of upstream skew decreases for both constrained and alluvial bends, particularly in the range of (80, 90) for R. For high R (>90), the percentage of upstream skew for constrained bends is 20%, whereas for alluvial bends, it is 75%. The skewness of bedrock bends is not always constant and can vary from vertical to less than 25°. The skewness of constrained bends (or bank angle) influences the ratio of bank to bed erosion rates, and thus affects the predicted channel width [48]. Future research is needed to better understand the variation in skewness in constrained bedrock rivers and to measure all variables that control bedrock channel width [48].

3.6. Controlling Factors of the Morphological Characteristics and Practical Implications

There is a long history of the formation of the Yellow River, but the Yellow River does not follow the shortest path into the sea, so do its tributaries, which indicates that their developing processes are mainly controlled by geological tectonic movement. According to the division of Yuan et al. [28], the Loess Plateau can be divided into Ordos stable platform, Longxi area, and Fenwei Cenozoic Rift, as shown in Figure 1.

Ordos platform is the earliest continental core of the Chinese mainland and a stable small uplift area of Cenozoic. In the late Pliocene, the Ordos platform was uplifted as a whole, and the river erosion was strong. The river undercut bedrock river about 80–100 m, and the Neogene red soil accumulation was also cut and eroded, resulting in the pre-Quaternary paleo-topography. The present loess landform basically inherits the ancient landform before the Quaternary, but the later erosion makes the loess landform more complex and broken. After entering the Quaternary, erosion is still ongoing [28].

For Fenwei Cenozoic Rift Valley, it is a very special tectonic unit in the southeast of Loess Plateau [28]. Since Eocene, Fenwei Rift has been under the action of the extensional stress field. The extensional stress perpendicular to the longitudinal axis of the rift has become the basic driving force for the development of Fenwei Rift and the Neotectonic movement. The subsidence of the basin is accompanied by the rise of mountains on both sides of the rift and platforms on both sides of the river. The Loess landforms in this area are mainly loess tableland landforms. Quaternary Aeolian Loess is deposited on the platform composed of fluvial and lacustrine deposits in the rift valley, with a double-layer structure. The lower part is fluvial and lacustrine deposit, the upper part is Quaternary Aeolian Loess, and the top surface is flat. Because the basin is narrow, there are few large tributaries and the downcutting of the tableland is not strong [54].

Starting from the distribution characteristics of the river morphology, this paper also explains the differences between the river morphology in Shaanxi and that of in Shanxi. The rivers in Shaanxi are mainly located in the Ordos platform, and the rivers in Shanxi are mainly located in the Fenwei Cenozoic Rift Valley. Through the summary of previous scholars’ research results, we can find that there are some differences between the geological structure in Ordos platform and that in the Fenwei Cenozoic rift area, which mainly reflects in the different degrees of undercutting. There are rivers with deep bedrock in Ordos platform, while there are few large tributaries and insufficient river undercutting in the Fenwei Cenozoic Rift area.

In terms of the effect of the climate on the sediment discharge, we choose the Jing River Basin as example to reveal the change in sediment yield. Zhang et al. [55] suggested that water diversion works, soil and water-conservation measures, and variations in rainfall are the major contribution to the changes in sediment discharge. In the Jing River Basin, the average annual rainfall for the period from 1956–1969 was 534.6 mm, and from 1970–1979, 1980–1989, 1990–1999, and 2000–2015, it was 490.2 mm, 491.4 mm, 474.2 mm, and 522.2 mm. While the average annual rainfall from 1990–1999 decreased by 3.5% compared to 1980–1989, the sediment discharge increased by 28%. Therefore, Zhang et al. [55] pointed that in the Jing River Basin, the effect on sediment yield cannot be attributed to the average annual rainfall, and they concluded that despite the increase in annual rainfall, heavy rainfall has experienced a significant decrease, which greatly affects the sediment yield.

Climate change may lead to more frequent extreme weather events in the future, suggesting that heavy rainfall is likely to increase in the Yellow River Basin and result in higher sediment discharge. Therefore, Zhang et al. [55] suggested that the construction of the Dongzhuang Reservoir on the Jing River and the Guxian Reservoir in the middle reaches of the Yellow River should remain an essential consideration in the river’s runoff and sediment management system. Our study shows that the sinuosity of constrained rivers on the Loess Plateau is generally low, but there are still highly sinuous bends. The development of these bends is limited by the mountains on both sides, indicating that deposition is more pronounced than erosion when these rivers are within the influence range of a reservoir. The planform characteristics of the rivers studied provide some support for selecting reservoir sites and arranging dredging projects after reservoir construction. To enhance sediment flushing after reservoir construction, it is advisable to avoid areas with highly sinuous bends and tributary confluences, where complex hydrodynamic conditions may reduce the effectiveness of sediment discharge.

4. Conclusions

In this paper, the morphology of constrained meandering rivers on both sides of the Shanxi-Shaanxi gorge in the middle reaches of the Yellow River Basin in the Loess Plateau is analyzed. The sinuosity, periodicity, curvature, and skewness of the constrained meandering rivers are compared with those of the alluvial meandering rivers and the constrained meandering rivers in other areas.

The total sinuosity of the constrained meandering rivers in the Loess Plateau ranged from 1.18 to 2.22, indicating a transitional type between straight and meandering channels, influenced by surrounding mountains. The average dimensionless curve wavelength of the constrained bends in the Loess Plateau equal 27.68 is smaller than that of the alluvial bends, indicating that the bends cannot fully develop due to the constraints of the sidewalls. Regarding the small dimensionless curve wavelength, similarities exist between the constrained and alluvial bends, suggesting that these bends have developed within the valley and have not contacted both sides. The average dimensionless curvature radius in the constrained river bends is 18.72, and they are mainly distributed at low dimensionless curvature radius in constrained bends. The Cartesian wave number ranged from 0.1 to 0.35, which indicates that the meander bend stability tends to select much longer meander wavelengths compared to the channel width in constrained bends. The constrained meandering rivers on the Loess Plateau exhibit predominantly upstream skewness. In the constrained river bends, as sinuosity increases, the proportion of upstream skewness initially rises, then gradually decreases to its lowest point when the sinuosity is 3.4, and subsequently increases rapidly to 65% in the high sinuosity range (≥3.6). The analysis results indicate that constrained river bends can develop similarly to alluvial meandering rivers when sinuosity is low and as the sinuosity increases the three-dimensional flow structure, such as secondary circulation and flow separation, changes the erosional and depositional patterns which change the orientation of constrained bends. The geological structures of the Loess Plateau are not consistent, which lead to some differences in the morphology of rivers in different regions of the Loess Plateau and show some different from the morphology of the alluvial and constrained meandering rivers in other regions of the world as the river erosion the loess and cut into bedrock.

The research findings provide some guidance and support for reservoir construction and dredging in constrained rivers on the Loess Plateau. As the water and sediment regulation system in the Yellow River Basin continues to improve, along with the construction of key reservoirs, future research should focus on exploring the sedimentation processes in constrained river bends and selecting appropriate dredging sites based on the findings of this study. This would help extend the service life and preserve the effective storage capacity of the reservoirs in the Yellow River Basin, thereby promoting the high-quality development of the region.