https://orcid.org/0000-0002-2810-5109
ABSTRACT
Aquatic vegetated habitats are disappearing rapidly, and restoration projects are crucial for maintaining their ecological functions. The success of these projects hinges on the ability to retain sediment, necessitating a comprehensive understanding of sediment transport in vegetated areas. Therefore, this study builds a model based on the flow turbulent kinetic energy near the riverbed to predict bedload transport. The turbulent kinetic energy in the model comprises turbulent terms generated by the riverbed and vegetation, which can be further divided into a shear turbulent term induced by the velocity difference and a stem turbulent term induced by the vegetation stem. The experimental data confirmed that turbulent kinetic energy can predict the bedload transport rate more effectively than bed shear stress. The average relative error of the predicted bedload transport based on bed shear stress was within 632%, and the average relative error of the predicted value based on turbulent kinetic energy was within 97%. When the dimensionless submerged vegetation density was less than a threshold, the bedload transport rate increased with increasing vegetation density. These findings yield crucial insights into the interaction mechanisms among vegetation, flow, and sediment and provide a promising direction for predicting sediment transport in the future.
HIGHLIGHTS
Turbulent kinetic energy (TKE) is adopted to predict the bedload transport rate.
TKE in the vegetated flows is caused by the riverbed TKEr and vegetation TKEv.
Vegetation density has a significant impact on bedload transport.
INTRODUCTION
Schematic diagram of bedload transport in natural riverbeds covered by the submerged and emergent vegetation.
Schematic diagram of bedload transport in natural riverbeds covered by the submerged and emergent vegetation.
Previous studies on the bedload transport rate focused primarily on rivers without vegetation, and scholars proposed state-of-the-art methods to predict the bedload transport rate, including a bedload transport rate model based on the physical process of sediment movement, a model combining hydraulics and statistical methods, and empirical or semi-empirical models established using a large amount of measured data (Braithwaite 2023). Lammers & Bledsoe (2018) established new transport rate formulae for bedload and total sediment based on the Bagnold (1966) power method. Sharma et al. (2019) proposed an empirical expression for the bedload transport rate based on experimental data on non-uniform sediments and seepage velocity. Deal et al. (2023) suggested that particle shape affected the sediment transport rate. By considering the influence of particle shape on flow resistance, it was predicted that sediment incipient motion and sediment transport efficiency are dependent on the average resistance coefficient and volume friction coefficient of sediment (Deal et al. 2023). Stipić et al. (2022) established a model that could fully simulate the interaction between water flow and sediment in natural rivers using the lattice Boltzmann method while considering the riverbed deformation and particle exchange processes.
Compared with conditions in the absence of vegetation, those in the presence of vegetation reduce the sediment incipient velocity, promote sediment transport, and increase the sediment transport rate (Kothyari et al. 2009; Tang et al. 2013; Liu et al. 2024). The previous formula for the bedload transport rate based on a bare bed is no longer suitable for direct application to the prediction of bedload transport in vegetated rivers. Bedload transport rates in vegetated channels are first investigated by shear stress. Jordanova & James (2003) divided the total resistance of rivers based on the effects of the riverbed and emergent vegetation and used the bed shear stress to calculate the bedload transport rate. Based on the vertical distribution of velocity and the Manning friction law, Romdhane et al. (2018) proposed a method for dividing the total shear stress of rivers and used it to improve the formula for the sediment transport rate. Duan & Al-Asadi (2022) experimentally studied the effects of vegetation on bedform resistance and the bedload transport rate and obtained an empirical formula for these parameters in vegetated channels. They found that bedform resistance increased, whereas the bedload transport rate decreased with an increase in vegetation density.
Previous bedload transport models have predominantly relied on bed shear stress as a key predictor. However, these models have shown limitations when applied to vegetated environments, where the presence of vegetation significantly complicates the flow dynamics (Yang et al. 2016; Wu et al. 2021; Zhao & Nepf 2024). Firstly, extensive experimental and field observations have demonstrated that vegetation within water channels induces notable changes in the flow structure, giving rise to complex turbulence patterns (Huai et al. 2019a). These turbulence patterns are not adequately accounted for by traditional models based on bed shear stress. Previous research has highlighted that sediment transport models founded on bed shear stress exhibit inaccuracies in vegetated regions. In particular, when the bed shear stress is held constant, an increase in vegetation density has been observed to result in an increase in the measured bedload transport rate (Yang & Nepf 2018). This phenomenon suggests that the influence of vegetation-induced turbulence, which intensifies with increasing vegetation density, plays a significant role in enhancing bedload transport. Such findings imply that the inaccuracies of traditional models may stem from their failure to incorporate the impact of vegetation-induced turbulence. Secondly, in the quest for more accurate predictors of sediment transport rates, a comparison of different parameters was conducted. It was found that turbulence-related parameters, such as turbulent kinetic energy, exhibit stronger correlations with the observed sediment transport rates, especially in vegetated regions (Shan et al. 2020). Sumer et al. (2003) increased the flow turbulence by adding a horizontal pipe and a series of grids under the condition of constant shear stress; their results showed that the sediment transport rate increased exponentially with an increase in turbulence intensity. Additionally, when considering the same level of near-bed turbulent kinetic energy, the measured bedload transport rates in both bare and emergent vegetated channels were found to agree within the margin of uncertainty. This observation indicates that turbulent kinetic energy could potentially serve as a more reliable and generally applicable predictor compared with bed shear stress, thereby providing a more accurate approach for understanding and predicting sediment transport processes across different conditions. Moreover, from a theoretical perspective, as noted by Sumer et al. (2003), turbulence plays a crucial role in redistributing momentum and energy within the flow. This redistribution directly affects the lift and drag forces acting on sediment particles, thereby influencing their transport. In conclusion, the focus on turbulence in this study is driven by both empirical evidence from experiments and a comprehensive theoretical understanding of fluid dynamics and sediment transport mechanisms. We are motivated by the need to develop a more accurate and comprehensive model for sediment transport in vegetated channels, as understanding the complex physical processes in such systems is essential for various applications and for advancing our knowledge of natural water systems. This study thus aims to bridge the existing gaps in the literature and provide new insights into sediment transport dynamics under vegetated conditions by emphasizing the role of turbulence.
Previous investigations have predominantly focused on predicting the bedload transport rates of bare riverbeds or riverbeds with emergent vegetation. A significant portion of these studies has placed emphasis on utilizing the shear stress of the riverbed as a key factor for such predictions. However, the existing body of research has not adequately accounted for the bedload transport rates in riverbeds covered by submerged vegetation, where the flow structure and turbulence exhibit a high degree of complexity. In response to this gap in the literature, the present study introduces an innovative sediment transport model. This model is centered around turbulent kinetic energy (TKE) as its core parameter. The overarching objective of this model is to accurately forecast the sediment transport in open vegetated channels. It is specifically designed to handle scenarios involving both submerged and emergent vegetation, thereby providing a more comprehensive and accurate understanding of sediment transport processes in vegetated river systems. The arrangement of sections in this study is as follows: Section 2 proposes the bedload transport rate model and TKE calculations, Section 3 presents the induction and sortation of experimental data, Section 4 describes model verification, and Section 5 presents the discussion and limitations of the bedload transport rate model.
THEORY
Models for the bedload transport rate
is the bedload transport rate (g/m/s), 
is the bed shear stress, 
is the critical incipient shear stress of sediment in the bare bed (Julien 2010), and k and m are empirical parameters that depend on the sediment characteristics. Although the physical mechanism of Equation (1) is clear, the dimensions are not harmonious and it can only be applied in the bare bed. The focus of this study was mainly on river channels containing aquatic vegetation; in such cases, the accuracy of the shear stress in Equation (1) calculated for the riverbed is often insufficient. In the process of fluid flow, there will be turbulent bursts or vortices. These turbulent phenomena can generate an uplifting force on the sediment particles located at the bottom. This uplifting force will cause the sediment particles to start moving, thus initiating the process of sediment transport. Smart & Habersack (2007) attributed the sediment lift-up to a local, vertical adverse pressure, or lift force, generated by the turbulence. Dittrich et al. (1996) and Batchelor (1951) suggested that the uplifting force in turbulent flow is mainly generated by pressure fluctuations caused by turbulence passing over the particles and the pressure fluctuations caused by the turbulence is proportional to the square of the velocity fluctuations. TKE is the sum of the velocity fluctuations in the streamwise, lateral, and vertical directions. Yang & Nepf (2018) claimed that the measured 
increased with increasing turbulence generated by vegetation at the same bed shear stress and demonstrated that the TKE generated by vegetation is the main driving factor that stimulates bedload transport, although the shear stress of the riverbed is also important (Lefebvre et al. 2010; Zong & Nepf 2010; Ortiz et al. 2013; Tinoco & Coco 2016; Yang & Nepf 2018, 2019). Hence, the dimensionless bedload transport rate was predicted by considering turbulence as follows:
is the dimensionless bedload transport rate; 
is the dimensionless turbulent kinetic energy. 
represents the sediment density; 
represents the flow density; g represents the acceleration of gravity; 
represents the median diameter. 
considered in this study is much smaller than the water depth and river width, but larger than the mean free path of water molecules (Falkovich et al. 2001); K and M are empirical coefficients; 
is the critical incipient turbulent kinetic energy of the sediment; and 
is the near-bed turbulent kinetic energy, which can be calculated as:
) and the other from the riverbed (
). Next, 
, 
, and 
are analyzed in Sections 2.2–2.4, respectively.Prediction of turbulent kinetic energy from vegetation TKEv
. In a fully developed flow, all the hydraulic parameters remain unchanged along the flow direction and the turbulence generation and energy dissipation terms reach equilibrium. The dissipation rate is assumed to be proportional to 
(Nepf 2012). The expression for 
can be written as follows:
is the shear turbulent term occurred near the top of the submerged vegetation. 
is the wake turbulent term generated by each vegetation and Pstem can only be generated when the stem Reynolds number is greater than 120 (Liu & Nepf 2016). 
is an empirical parameter. d is the vegetation diameter. The prerequisite for the establishment of Equation (4) is that the vegetation diameter is smaller than the distance between two adjacent vegetation stems (Nepf 2012).
is caused by differences in the velocity. As the presence of vegetation undoubtedly increases the resistance to water flow, the longitudinal velocity in the vegetation layer decreases, whereas the velocity in the non-vegetation layer increases, resulting in a large velocity difference between the two layers (Huai et al. 2019a; Zhang et al. 2021). It should be noted that 
is mainly concentrated near the top area of the submerged vegetation, namely 
. For flows with emergent vegetation, a shear turbulent term near the vegetation top does not exist. The boundary layer is generated near the bottom of the riverbed, where there is a distinct velocity difference, thus leading to the formation of near-bed shear turbulence (
) (Conde-Frias et al. 2023). Specifically, for submerged vegetation:
Therefore, the expression of 
must be discussed separately for both emergent vegetation and submerged vegetation.
Shear turbulent term Pshear within emergent vegetation
within emergent vegetation only occurs near the riverbed, that is, only 
occurs. The expression of 
along the water depth is expressed as follows:
is the Reynolds shear stress; z denotes the vertical direction, with 
= 0 at the riverbed; 
and 
denote the longitudinal and vertical fluctuating velocities, respectively, and the horizontal line over these parameters indicates the time average. The 
averaged along the boundary layer depth 
is calculated as follows:
can be deduced as follows:
is an empirical coefficient (Chen et al. 2013); 
is the vertically averaged velocity above the boundary layer, which is uniform in a flow with emergent vegetation; 
is the vertically averaged velocity in the boundary layer (Figure 2(a)). 
is the friction velocity of the riverbed. We assume that the velocity decays linearly from a peak at 
to zero at 
. Hence, 
and 
, where Q is the flow discharge, H is the flow depth, and B is the flow width. Ghisalberti (2009) proposed that 
; thus,
Profiles of the streamwise velocity in emergent vegetation flows (a) and submerged vegetation flows (b). The direction of the water flow from left to right is consistent with the positive direction of the x-axis. The green columns represent vegetation and the yellow spheres near the riverbed represent sediment particles. The black curved thin solid line represents the distribution of velocity along the water depth and the blue shaded zone represents the depth-averaged velocity. The area between the curved dashed lines represents the shear layer and the blue broken line represents the vortex structure within the shear layer.
Profiles of the streamwise velocity in emergent vegetation flows (a) and submerged vegetation flows (b). The direction of the water flow from left to right is consistent with the positive direction of the x-axis. The green columns represent vegetation and the yellow spheres near the riverbed represent sediment particles. The black curved thin solid line represents the distribution of velocity along the water depth and the blue shaded zone represents the depth-averaged velocity. The area between the curved dashed lines represents the shear layer and the blue broken line represents the vortex structure within the shear layer.
is mainly confined to the boundary layer and linearly distributed in the boundary layer. Therefore, from the top of the boundary layer 
to the riverbed 
, the depth-averaged shear turbulent term 
is as follows:
Shear turbulent term Pshear within submerged vegetation
consists of 
in the boundary layer and 
near the vegetation top. The shear vortices at the vegetation top reach their maximum size in the region where the flow is fully developed, and these shear vortices invade the vegetation layer in the vertical direction. The length scale of this invasion, 
(Figure 2(b)), decreases with increasing vegetation density (Ghisalberti 2009), which can be calculated as follows (Nepf et al. 2007; Konings et al. 2012):
is the drag coefficient of vegetation; 
refers to the frontal area of vegetation within a unit volume of the water body; and n is the number of vegetation per riverbed area.
in Equation (12), we assume that 
decreases linearly from 
at the vegetation top to zero at 
. Therefore, the depth-averaged 
can be expressed as follows:
is the velocity averaged over the free-surface layer and 
is the velocity averaged along the vegetation layer (Figure 2(b)), which can be obtained using the following two-layer model (Chen et al. 2013) as follows:
is the cross-sectional average velocity; 
is an empirical parameter; 
is the solid volume fraction; and 
is obtained from the flow continuity as follows:
can be calculated as follows:
Therefore, 
within submerged vegetation can be obtained by the sum of Equations (14) and (17). Notably, when the intrusion depth 
of the shear layer at the top of submerged vegetation is less than the vegetation height 
, 
does not touch the riverbed. In such cases, the influence of 
on bedload transport may be negligible.
Stem turbulent term Pstem
, which can be calculated as follows along the vegetation layer (Tanino & Nepf 2008; Yang et al. 2016):
is an empirical parameter with a value of 1.07 ± 0.09 (Tanino 2008) and 
is the velocity inside the vegetation layer. The depth-averaged 
is as follows:
, vegetation number per riverbed n, and vegetation diameter d can be assumed to be constant along the water depth direction for uniform vegetation. In addition, 
. Then, Equation (19) can be rewritten as follows:
Prediction of the turbulent kinetic energy from riverbed TKEr
) has a certain linear relationship with the shear stress 
of the riverbed, shown as (Stapleton & Huntley 1995; Yang & Nepf 2019):
is the empirical coefficient, which is generally 0.2 ± 0.01 (Soulsby 1981); 
is the shear stress generated by the riverbed and is expressed as follows:
is the drag coefficient of the riverbed and can be calculated as follows (Julien 2010):
Prediction of the critical incipient turbulent kinetic energy TKEc of sediment
enhancing sediment resuspension. This phenomenon suggests that the TKE is likely to be the dominant factor affecting sediment movement (Lefebvre et al. 2010; Zong & Nepf 2010; Ortiz et al. 2013; Tinoco & Coco 2016). The concept of the critical turbulence kinetic energy (
) of sediment has attracted international attention. In this study, we adopted the linear relationship between the 
and critical shear stress 
, shown as:
is the critical incipient velocity of the sediment in the vegetated rivers, shown as (Tang et al. 2013):
in unvegetated rivers can be obtained according to the Shields curve and can be expressed as follows:
is the Shields parameter. Zhang et al. (2020) combined Equations (27) and (25) and obtained 
m2/s2 for a sediment size of 
. When using Equations (25) and (26), 
m2/s2. This difference may be because adsorption occurs among the particle surfaces for small particle sizes, which results in an inaccurate prediction of 
, although it may also be due to uncertainty in the Shields parameter (Buffington & Montgomery 1997). However, it is encouraging that the 
values calculated using the different methods are not significantly different, which demonstrates that Equation (25) is reasonable and accurate to a certain extent and is more convenient and simpler.
DATA SUMMARY
Experimental setup in previous studies
Based on previous research results, we sorted and summarized a comprehensive experimental dataset on bedload transport rates in vegetated flows and divided it into two categories according to whether the vegetation was submerged. Detailed information on the different experimental data is provided below.
Bedload transport rate in emergent vegetation flows
Wu et al. (2021) used a 12 m × 0.6 m × 0.6 m slope adjustable flume. The simulated emergent vegetation region had a length of 5 m. The vegetation diameter (
) was 7.8 and 10 mm. The channel bottom was uniformly covered with 10 cm thick quartz sediment, with a median diameter of 
= 0.931 mm. A tapered collector at the exit gathered sediment.
Jordanova & James (2003) conducted experiments in a 15 m × 0.38 m glass sided flume. They used 5 mm cylindrical metal rods for vegetation. The sediment was non-cohesive sand with a median diameter of 
= 0.45 mm.
Yager & Schmeeckle (2013) ran 12 sets of experiments. They used cylindrical rigid vegetation with 1.3 mm diameter. The median diameter in the riverbed was 0.5 mm. Bedload transport was measured by varying vegetation density, flume slope, and flow rate, and a high-speed video camera was used.
Zhao & Nepf (2021) employed a 1 m × 10.4 m recirculating flume. The modeled emergent vegetation had different cylinder diameters (
= 0.64, 2.5 cm). A 9 cm layer of flattened sand with a median diameter of 
= 0.6 mm was added to the channel bed. The bedload transport rate was measured using a butterfly valve to divert the flow from the sediment recirculation pipe to the mesh bag.
Kothyari et al. (2009) conducted experiments in 12 m flumes (0.15 and 0.20 m wide). Stainless-steel cylinders with a diameter of 2–5 mm were fixed on the channel bed covering a total length of 9 m. Six sizes of quartz sand representing medium and coarse sand and gravel were adopted.
Yang & Nepf (2018) used a 10 m × 1 m recirculated flume. They used aluminum dowels with a diameter of 
= 6.3 mm to simulate rigid emergent vegetation. A 4 cm thick layer of sediment, with a median diameter of 
= 0.5 mm, was manually flattened on the channel bed and was collected in a mesh bag.
Duan & Al-Asadi (2022) conducted 18 experiments in a 12.2 m × 0.6 m flume. The vegetation stems were simulated using emergent PVC rods at 16 mm in diameter. Two groups of non-uniform sediment, with 10 cm depth, were used with median diameters of 
= 0.45 and 1.6 mm.
Bedload transport rate in submerged vegetation flows
Bouteiller & Venditti (2015) conducted experiments in a 15 m × 1 m tilting flume. The staggered submerged flexible vegetation was 6 m. The flexible vegetation consisted of six plastic blades bundled into the bottom. Sediments, with a median diameter of 
= 150 
, were distributed evenly at depths of approximately 2 and 3 cm. Bedload rates were measured by collecting and weighing sand in a mesh box at the flume's bottom and outlet.
Lv et al. (2016) used a 12 m × 0.42 m × 0.7 m circulated rectangular channel with smooth glass walls. A flexible PVC cylinder with a total height of 12 cm was used to simulate the vegetation. Sediment, with a median diameter of 
0.67 mm, was evenly spread around the vegetation with a thickness of 5 cm. Vegetation's exposed height to the flow was 6 cm and the actual height was 3.5–5.5 cm under flow).
Clarification of experimental parameters
The above-mentioned experimental settings and vegetation attributes are summarized in Supplementary Appendix Tables 1 and 2. Supplementary Appendix Table 1 lists the parameters of the bedload transport rate for emergent vegetation, and Supplementary Appendix Table 2 for submerged vegetation. The height of the vegetation listed in Supplementary Appendix Tables 1 and 2 is the height of the flexible vegetation after bending, which is represented by 
. The height of rigid vegetation is expressed by 
. To be clear, the parameters in Supplementary Appendix Tables 1 and 2 are expressed as follows: D represents the diameter of vegetation; 
represents the median diameter; n represents the number of vegetation per unit channel area; 
represents the solid volume fraction; H stands for the water depth; B stands for the channel width; 
indicates the bedload transport rate per channel width per second; U stands for the cross-sectional velocity; and 
is the drag coefficient of vegetation. The instantaneous bedload transport rate 
, defined as the mass of sand passing through the channel cross-section per second per unit width, was calculated as the mass of the collected sand divided by the time required to collect the sand and the width of the flume.
RESULTS
Bedload transport rate model in emergent vegetation flows
in emergent vegetation flow could reasonably predict the dimensionless 
by utilizing the experimental data presented in Supplementary Appendix Table 1. The relationship between 
and 
is shown in Figure 3(a). The thin black solid line in Figure 3(a) shows the functional form based on the experimental data, which is shown as:
for 
is satisfactory and that the 
proposed in this study can be used as a key factor in the prediction model for 
in emergent vegetation flows. Scholars have used bed shear stress to predict 
(Jordanova & James 2003). Equation (1) is non-dimensionalized and calibrated using experimental data in Supplementary Appendix Table 1. The following equation is obtained:
and 
is shown in Figure 3(b). The thin solid line represents Equation (29). A comparison between the 
∼ 
(Equation (28)) and 
∼ 
models (Equation (29)) revealed that the correlation coefficient R2 of Equation (28) was 0.84 and that of Equation (29) was 0.79. The prediction of 
using 
was not as accurate as that using 
. To clarify the accuracies of the two models, i.e., Equations (28) and (29), an error analysis of the predicted and measured data was performed. The average relative error 
is expressed as follows:
is small, it indicates that the predicted values of the model are relatively close to the experimental values, and the model has high accuracy. Conversely, it means that there are significant deviations in the model's predictions. The average relative error of the predicted 
based on 
was within 632%, and the average relative error of the predicted 
based on 
was within 97%, which is better than that of the former. The findings further showed that 
is more suitable for predicting the bedload transport rate than the shear stress 
. 
(a) Relationship between dimensionless turbulent kinetic energy (
) and bedload transport rate (
) and (b) relationship between dimensionless bed stress (
) and bedload transport rate (
) in emergent vegetation flows. Data in the figure are from previous studies (Kothyari et al. 2009; Jordanova & James (2003); Yang & Nepf 2018; Wu et al. 2021; Zhao & Nepf 2021; Duan & Al-Asadi 2022).
(a) Relationship between dimensionless turbulent kinetic energy () and bedload transport rate () and (b) relationship between dimensionless bed stress () and bedload transport rate () in emergent vegetation flows. Data in the figure are from previous studies (Kothyari et al. 2009; Jordanova & James (2003); Yang & Nepf 2018; Wu et al. 2021; Zhao & Nepf 2021; Duan & Al-Asadi 2022).
Bedload transport rate model in submerged vegetation flows
and 
was obtained according to data from the study of Bouteiller & Venditti (2015) and is shown in Figure 4; the functional relationship expression is as follows:
Relationship between dimensionless turbulent kinetic energy (
) and bedload transport rate (
) in submerged vegetation flows. Data in the figure are from Bouteiller & Venditti (2015).
Relationship between dimensionless turbulent kinetic energy () and bedload transport rate () in submerged vegetation flows. Data in the figure are from Bouteiller & Venditti (2015).
The correlation between 
and 
had a value of R2 = 0.92, and the model in Equation (2) further confirms that 
can be used as an important index to predict 
in submerged vegetation flows.
and 
is shown in Figure 5, where data are from Lv et al. (2016). The vegetation density in Figure 5 is 
. Shear vortices occur near the top of submerged vegetation and significantly contribute to turbulence within and above the vegetation (Ghisalberti & Nepf 2002). Shear vortices penetrate downward into submerged vegetation over a penetration scale 
set by the vegetation density, shown in Equation (13). According to Equation (13), when 
with 
< 0.23 and 
= 1, the shear vortices will affect the riverbed, thus creating a highly turbulent condition over the entire vegetation height. The correlation between 
and 
is strong for 
from Figure 5, with R2 values of 0.85, indicating that the 
is still a satisfying factor for predicting the 
. 
Relationship between dimensionless turbulent kinetic energy (
) and bedload transport rate (
) in submerged vegetation flows. Data in the figure are from Lv et al. (2016).
Relationship between dimensionless turbulent kinetic energy () and bedload transport rate () in submerged vegetation flows. Data in the figure are from Lv et al. (2016).
). The vegetation density is characterized by the total frontal area of vegetation per vegetation volume, a. The extent to which flow is altered by the vegetation depends on the non-dimensional vegetation density 
. Nepf (2012) demonstrated that there are two limits of flow behavior for the submerged vegetation. First, for sparse vegetation (ah ≪ 0.1), the flow resistance associated with the vegetation is small compared with the flow resistance associated with the bed, and the flow velocity profile, is not significantly altered by the vegetation and follows a turbulent boundary layer profile, similar to that observed over a bare bed. Although the flow velocity is not significantly altered, stem turbulence is produced in the wakes of the individual vegetation. In this sparse vegetation limit, turbulence increases with increasing vegetation density 
, and the elevated turbulence level inhibits fine particle deposition (Liu & Nepf 2016). Alternatively, in the dense vegetation limit, there is a significant decrease in flow velocity within the vegetation. In this case, the discontinuity in drag at the top of the vegetation generates a region of rapid velocity change (shear), and the shear generates shear turbulence (via Kelvin–Helmholtz instability). The initiation of shear turbulence occurs near 
= 0.1 proposed by Nepf (2012). The shear vortices contribute significantly to turbulence within and above the vegetation (Ghisalberti & Nepf 2002). The shear vortices penetrate downward into the vegetation over a penetration scale (
) that is set by the vegetation density, shown as Equation (13). The greater the vegetation density (
), the smaller the penetration scale (
). The shear turbulence penetrates to the bed, 
>
, creating a highly turbulent condition over the entire vegetation height. Vegetation for which 
<
(Figure 2(c)) shield the bed from strong turbulence and turbulent stress. Because turbulence near the bed plays a role in resuspension, these dense vegetation cases are expected to reduce resuspension and erosion. Overall, as the vegetation density increases, the turbulence increases, and at this time, the sediment transport rate increases. However, when the vegetation density exceeds a certain threshold, the turbulent kinetic energy will decrease, and the sediment transport rate will decrease accordingly. Therefore, in the flow with submerged vegetation, the bedload transport rate first increases and then decreases with the increase of vegetation density, as shown in Figure 6. The threshold value of vegetation density can be found to be 0.105 according to Figure 6. Consistent with this, Moore (2004) observed that resuspension within the seagrass Zostera marina was reduced relative to that in adjacent bare bed areas only when 
> 0.4 (Luhar et al. 2008). Lawson et al. (2012) measured bedload erosion in seagrasses with different vegetation densities, shown in Figure 5 of Lawson et al. (2012). From 
= 0.08 to 0.3, erosion increased with increasing vegetation density because stem turbulence augmented the near-bed turbulence and increased erosion. However, above 
= 0.4, erosion was eliminated. Research from Moore (2004) and Lawson et al. (2012) also showed that above a certain vegetation density threshold (
), the near-bed turbulence becomes too weak to generate resuspension. It is worth noting that the vegetation density threshold obtained by Lawson et al. (2012) was 
= 0.4, which differed from that for flexible vegetation obtained by Lv et al. (2016), which was 
= 0.1. This discrepancy was because Lawson et al. (2012) used vegetation with four leaves per vegetation stem, a leaf width of D = 3 mm, and a leaf length of 
= 8 cm, and they obtained 
=4nD
. However, the vegetation used in the study by Lv et al. (2016) was a flexible PVC cylindrical stem with only one leaf and then its 
=nD
. Therefore, the density threshold obtained by Lawson et al. (2012) was likely four times that obtained by Lv et al. (2016). 
Variation of bedload transport rate (
) with vegetation density (
) under different water depths (
). The vegetation density threshold is represented by the vertical dashed line, which is 
.
Variation of bedload transport rate () with vegetation density () under different water depths (). The vegetation density threshold is represented by the vertical dashed line, which is .
Research on the vegetation density threshold is highly valuable and can be expanded with practical recommendations. In flows with submerged vegetation, the bedload transport rate changes with vegetation density, reaching a threshold where the turbulent kinetic energy and the bedload transport rate decline. This protects the riverbed from scouring and allows nutrients like carbon to be stored in vegetation patches, enabling efficient capture of fine particles. The vegetation density threshold is crucial for carbon storage in riverbeds, as seen in seagrass meadows, which are global carbon storage hotspots. Their high carbon storage capacity comes from high primary production and particle filtering abilities. However, over 50% of the carbon in seagrass soils comes from outside the meadow, and the storage capacity depends on hydrodynamic conditions. The sensitivity to vegetation and flow interaction leads to significant differences in carbon burial rates, causing large variability in carbon stocks across different seagrass habitats. This variability creates uncertainty in assessing global seagrass carbon stocks. This vegetation density threshold can control the bedload transport rate and fine particle retention within the vegetation region, which is a precursor for creating carbon stock in seagrass soil (Zhang et al. 2020). Future work will test the hypothesis that vegetation density controls carbon sequestration potential in vegetation patches and clarify the relationships between seagrass carbon stock, meadow morphology, and hydrodynamic conditions.
DISCUSSION
Extension of the TKE*-based model
TKE*-based model applied in bare bed
is better at predicting the bedload transport rate 
in vegetated channels than 
. In open channel flows with bare beds, researchers often use 
to predict 
. DuBoys (1879) proposed that the prerequisite for bedload movement is that the applied shear stress 
from the riverbed must be greater than the critical incipient shear stress 
of the sediment. Meyer-Peter & Muller (1948) developed a complex gravel-based transport rate formula based on the median diameter 
. Einstein (1942) demonstrated that the dimensionless bedload transport rate 
and the Shields parameter are functionally related and proposed the expressions of 
and Shields parameter. Here, the functional relationships between 
and 
, 
and 
were analyzed according to the bedload transport rate data measured in bare beds, as shown in Figure 7. These data were obtained from laboratory experiments conducted by Shan et al. (2020), Wu et al. (2021), Yang & Nepf (2018), and Yager & Schmeeckle (2013). 
(a) Relationship between dimensionless turbulent kinetic energy (
) and bedload transport rate (
) and (b) relationship between dimensionless bed stress (
) and bedload transport rate (
) in bare beds. Data in the figure are from previous studies (Yager & Schmeeckle 2013; Yang & Nepf 2018; Shan et al. 2020; Wu et al. 2021).
(a) Relationship between dimensionless turbulent kinetic energy () and bedload transport rate () and (b) relationship between dimensionless bed stress () and bedload transport rate () in bare beds. Data in the figure are from previous studies (Yager & Schmeeckle 2013; Yang & Nepf 2018; Shan et al. 2020; Wu et al. 2021).
As shown in Figure 7, both 
and 
present better predictions of 
in bare beds than in vegetated beds, and the prediction accuracy by 
with R2 = 0.99 is slightly better than that by 
with R2 = 0.97. On the one hand, this confirms that bed shear stress can be applied to determine the bedload transport rate in bare bed rivers, which is consistent with previous conclusions (Yang & Nepf 2018). On the other hand, it can be concluded that aquatic vegetation greatly affects the dominant role of shear stress on the bedload transport rate; therefore, the prediction of the bedload transport rate using 
is more advantageous in vegetated flows.
TKE*-based model applied in a vegetated channel with a random pattern
-based model to predict the bedload transport rate in open channel flows with a random distribution of vegetation. Shan et al. (2020) explored the impact of vegetation patchiness on channel-averaged turbulence and bedload transport. Vegetation stems were clustered into 16 randomly distributed circular patches that were randomly distributed in the test section with an area of 2.4 m2. In an open channel with patchy vegetation, the 
generated by the riverbed consists of two parts – 
caused by the non-vegetated riverbed and 
generated by the riverbed inside the vegetation patch. Therefore, 
is expressed as follows:
refers to the flow velocity in the non-vegetated region, 
refers to the flow velocity inside the patch, and 
refers to the patch area fraction (Shan et al. 2020).
and 
is shown in Figure 8 with R2 = 0.84. The improved 
-based model that considers patchy vegetation had a good power function relationship with the sediment transport rate 
, which verified the extension of Equation (2). It should be emphasized that in flows with patchy vegetation, the patch density should be determined and introduced into the calculation of 
. 
Relationship between 
and 
in flows with patchy vegetation.
Model ecological implications and limitations
Currently, aquatic vegetated habitats such as wetlands and mangroves suffer from severe degradation. Their ecological functions are crucial for fisheries, water quality, and riverbed stability. The effectiveness of ecological restoration work depends on the ability to retain sediment. However, the understanding of sediment transport in vegetated areas is limited, which has led to the failure of many restoration projects. The new bedload transport model proposed in this study takes into account the role of turbulence generated by vegetation. It can accurately predict sediment transport, assist in formulating scientific and reasonable river restoration plans, ensure the success of restoration work, and safeguard the health and stability of river ecosystems. Therefore, it holds significant ecological restoration significance.
Because of the presence of vegetation, the turbulent term near the riverbed consists of the TKE caused by the riverbed and vegetation, which are influenced and restricted by each other. According to the 
-based model (Equation (2)) proposed in this study, the turbulent items are all linear superpositions, and the interactions between them are not fully considered. Additionally, it is generally accepted that the relationship between the turbulent term and bedload transport rate follows a power law. The parameters, K and M, in the model are obtained by using the regression analysis method. However, these two exponential parameters K and M are associated with the morphological characteristics of the riverbed and vegetation. These characteristics include the median particle size, sediment density, as well as the diameter and density of vegetation. So far, a unified quantitative analysis of these exponential parameters has not been carried out. Moreover, a substantial amount of measured data is still needed to calibrate these parameters. When predicting the bedload transport rate based on bed shear stress, this parameter uncertainty has been noted. For example, Jordanova & James (2003) conducted the bedload transport rate experiments involving emergent vegetation and found that the values of K and M were 0.017 and 1.05, respectively. In contrast, Meyer-Peter & Muller (1948) carried out experiments with unvegetated flow and obtained values of 8 and 1.5 for K and M, respectively. For future research, an area of focus could be the development of a piecewise function for varying levels of turbulent kinetic energy within the 
-based model, along with further refinement of the parameter values.
The bedload transport rate discussed in this study is based on a uniform bedload, and the movement law of non-uniform bedload is different from that of uniform bedload. The fine particles in non-uniform bedload are blocked by coarser particles, leading to that the fine particles in non-uniform bedload are more difficult to move than those in uniform bedload. In contrast, the coarser particles in non-uniform bedload tend to be exposed to the bed surface and thus move more easily than those in uniform bedload. Therefore, the calculation of non-uniform bedload transport rates is more complicated than that of uniform bedload rates. Future studies should focus on more complex conditions that are consistent with the natural environmental conditions, such as sediment uniformity.
CONCLUSION
This study presents a novel approach for predicting the bedload transport rate by utilizing the surplus value of TKE, specifically the difference between the TKE and the critical TKE for sediment initiation. In vegetated flows, TKE is primarily generated by the riverbed and vegetation. The turbulence induced by vegetation consists of a stem-related turbulent term and a shear turbulent term. In submerged vegetation flows, shear-related turbulence occurs not only at the riverbed boundary layer but also at the top of the vegetation. Both bed shear stress and TKE prove to be effective predictors of bedload transport rates in bare bed channels. However, in vegetated flows, TKE demonstrates a superior predictive ability compared with bed shear stress. This indicates that incorporating TKE in sediment transport models can lead to more accurate predictions in the presence of vegetation. The influence of vegetation density on the bedload transport rate is mediated through the TKE induced by vegetation. There exists a threshold vegetation density. When the density is below this threshold, an increase in vegetation density leads to a gradual rise in TKE and subsequently an increase in the bedload transport rate. Conversely, when the density exceeds the threshold, further increases in vegetation density result in a decrease in TKE and the bedload transport rate. Future studies should focus on complex natural conditions and explore the hypothesis that vegetation density controls carbon sequestration potential in vegetation patches. The model will guide future river restoration projects, helping engineers accurately predict sediment transport, formulate scientific restoration plans, and safeguard river ecosystem health and stability.
FUNDING
This study received funding by the National Natural Science Foundation of China (Grant Nos. 42377078, 52279075), the Joint Open Research Fund Program of the State key Laboratory of Hydroscience and Engineering, the Tsinghua—Ningxia Yinchuan Joint Institute of Internet of Waters on Digital Water Governance (Grant No. sklhse-2023-Iow06), and Young Talent Fund of Association for Science and Technology in Shaanxi, China (Grant No. 20230461).
ETHICAL APPROVAL
This article does not contain any studies with human participants or animals performed by any of the authors.
INFORMED CONSENT
Informed consent was obtained from all individual participants included in the study.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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