Stochasticity in the estuarine sediment erosion processes
Introduction
Sediment transport is a key element in the study of the evolution of rivers and coasts. The fate of sediment is determined by sediment transport, erosion and deposition in the seabed, which has a direct impact on the evolution of estuarine geomorphology (Brand et al., 2010). Particularly, in some extreme events (e.g., storms), excessive deposition in the harbor pools and waterways may obstruct ship navigation and strong erosion may hasten the threats to the coastal shoreline (Mercier and Delhez, 2007). Besides, pollutants and nutrients that cling to sediments can directly affect the ecology and environment in estuaries and coasts (Geyer and Ralston, 2018).
One of the most important properties influencing sediment transport is its erodibility. Two main erosion modes have been reported and widely accepted: Type I erosion (limited erosion mode) and Type II erosion (infinite erosion mode) (Amos et al., 1997). These two modes are distinguished from each other by whether the erosion rate decreases gradually with time (Type I) or not (Type II) (Sanford, 2008). Amos et al. (1992) and Amos et al. (1997) further divided Type I erosion into two sub-types (Type Ia erosion and Type Ib erosion). Type Ia erosion occurs in the surface layer of the seabed (on a millimeter scale, where the sediment is in poor physical consolidation) and is most common when the bed shear stress is less than 1 Pa. Type Ib erosion, on the other hand, occurs deeper in the seabed, where sediment is well consolidated, and mostly under a much higher bed shear stress (greater than 1 Pa) (Amos et al., 1992).
A linear formula is frequently used to calculate the erosion rate (or erosion flux) (Partheniades, 1965; Kandiah, 1974; Owen, 1977):where E is the erosion rate of sediment per unit area, τ is the applied bed shear stress, τc is the critical shear stress of sediment, and M is the erosion coefficient, which is variable with depth in the seabed and is related to the sediment properties.
In contrast to the previously mentioned deterministic nature, sediment erosion exhibits a high degree of stochasticity (Qian and Wan, 1983; Van Prooijen and Winterwerp, 2008). This stochasticity can be caused by the randomness in the bed shear stress and/or the critical shear stress for erosion (Hofland and Battjes, 2006). The instantaneous bed shear stress fluctuates around the time-averaged bed shear stress, which will exert force to the seabed and thus exhibits temporal variability. As a result, even though the average bed shear stress is below the critical value, the instantaneous turbulent bed shear stress may still exceed it, resulting in incipient motion (Van Prooijen and Winterwerp, 2008).
On the other hand, the critical erosion shear stress (τc) also exhibits some randomness (Qian and Wan, 1983). In the past, it was usually considered to be a constant for sediment with uniform grain size. However, Shields (1936) observed that it is not a constant, but rather varies over a range for a variety of factors. The incipient motion of sediment, according to Van Prooijen and Winterwerp (2010), is noted to be a stochastic process that follows the probability law in statistics.
Hofland and Battjes (2006) provided a detailed summary of the probability distribution function (PDF) of bottom shear stress. Several shear stress distributions have been proposed for the natural turbulent bottom boundary layer (Miyagi et al., 2000; Hofland and Battjes, 2006): (1) The simplest form is a single value distribution. Only the mean bed shear stress is considered in this mode, while the effect from stochasticity is ignored (Qian and Wan, 1983). (2) Gaussian distribution. The bed shear stress is assumed to have a Gaussian distribution in this mode (Van Rijin, 1993). (3) Log-Gaussian distribution. This distribution was once used by Bridge and Bennet (1992). Though it is useful for describing the skewed distribution and corresponds well with the actual experimental results, it is an empirical distribution and has not been well studied or accepted. (4) Velocity-derived distribution. This distribution is more consistent with observations because it is based on the assumption that bed shear stress is proportional to the square of the instantaneous velocity (Hofland and Battjes, 2006).
The study of stochasticity in sediment has a long history. In 1965, Partheniades (1965) observed sediment stochasticity in erosion experiments and first considered the temporal distribution of shear stress. Several decades later, Van Prooijen and Winterwerp (2010) investigated the stochasticity in sediment incipient motion from the probabilistic view. In their study, they took into account the stochasticity in bottom shear stress as well as in critical shear stress for erosion. Though the role of the turbulence in the erosion process has been thoroughly investigated (Schmeeckle, 2014), further research into the factors influencing the stochasticity in sediment erosion (e.g., the effects of the current stochastic parameters on the bottom shear stress, the combined effects of stochasticity caused by both the bottom shear stress and the critical shear stress for erosion) is needed.
In this study, the velocity and bed shear stress of turbulence were studied using the in-situ data collected with an observation tripod in the Pearl River Estuary (PRE), China. Typical sediment samples composed mostly of mud (silt and clay) were selected for the Gust erosion microcosm system (GEMS) experiments (Gust and Muller, 1997). Based on the obtained data, a numerical model of sediment erosion was developed. The effects of stochastic parameters on sediment erosion were studied using numerical experiments.
The remainder of this paper is organized as follows. Section 2 introduces the study methods. Section 3 presents the results of the in-situ observation and numerical models. Section 4 discusses the effects of stochastic turbulence in the erosion processes, and Section 5 provides a summary.
Section snippets
Data collection
In 2015, in-situ observations in the western part of the PRE were conducted using a tripod equipped with an acoustic Doppler velocimeter (ADV), a pulse-coherent acoustic Doppler profiler (PC-ADP), an optical backscatter sensor (OBS), a conductivity-temperature-depth sensor (CTD), and an acoustic wave and current profiler (AWAC) (Huang et al., 2020). The observation sites were located in a sub-channel (Denglong Waterway) and a sub-shoal (Entrance Shoal) of the PRE (Fig. 1).
Fig. 2 depicts the
In-situ turbulence parameters
Statistical analysis was performed on the observations. We noted that the turbulent velocity probability distribution agreed well with the Gaussian distribution of Eq. (2), which is consistent with Hofland and Battjes (2006). Several functions (such as linear relationship, quadratic relationship, power-law relationship, exponential law relationship, logarithm relationship, etc.) were used to fit the relationship between the mean velocity (μ) and standard deviation of velocity (σ). The best
Relationship between the PDF of turbulent velocity and bed shear stress
For any finite change du in the velocity, the change in integrated probability is dP = p(u)du. According to Eq. (7), a one-to-one link exists between the bed shear stress and the velocity. Thus, their integrated probabilities are equal (dP = dQ), where dQ = r(τ)dτ is the integrated probability of bed shear stress, where r(τ) is the PDF of the bed shear stress (Hofland and Battjes, 2006).
Therefore, the PDF of instantaneous bed shear stress can be obtained as
Conclusions
In this study, three methods (in-situ data analysis, GEMS experiments, and numerical experiments) were used to study the stochastic effects on Type Ia erosion in the surficial erodible sediment in the estuarine seabed. The following conclusions were obtained:
- (1)
The relationship between the stochastic parameters (mean velocity μ and standard deviation of velocity σ) was found to vary within a limited range instead of being linear. In-situ data collected via an observation tripod in the PRE show a
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was funded by the National Natural Science Foundation of China [grant numbers 51761135021, 41890851] and the Guangdong Provincial Water Conservancy Science and Technology Innovation Project (grant number 201719). We would like to thank our graduate students from Sun Yat-sen University, including Zhongyuan Lin, Lianghong Chen, Guang Zhang, Rui Zhang, Yuren Chen, Jiawei Qiao, for their help in fieldwork and sediment sample analysis in the laboratory.
References (39)
- et al.
In situ erosion measurements on fine-grained sediments from the Bay of Fundy
Mar. Geol.
(1992) - et al.
The stability of fine-grained sediments from the Fraser River Delta. Estuarine
Coast. Shelf Sci.
(1997) - et al.
Erodibility of cohesive sediment: the importance of sediment properties
Earth Sci. Rev.
(2011) - et al.
In-situ study of the spatiotemporal variability of sediment erodibility in a microtidal estuary
Estuar. Coast Shelf Sci.
(2020) - et al.
Size sorting of fine-grained sediments during erosion: results from the western Gulf of Lions
Continent. Shelf Res.
(2008) - et al.
Diagnosis of the sediment transport in the Belgian coastal zone. Estuarine
Coast. Shelf Sci.
(2007) - et al.
Statistical analysis on wall shear stress of turbulent boundary layer in a channel flow using micro-shear stress imager
Int. J. Heat Fluid Flow
(2000) - et al.
Estimation of bed shear stress using the turbulent kinetic energy approach—a comparison of annular flume and field data
Continent. Shelf Res.
(2006) Modeling a dynamically varying mixed sediment bed with erosion, deposition, bioturbation, consolidation, and armoring
Comput. Geosci.
(2008)- et al.
A unified erosion formulation for fine sediments
Mar. Geol.
(2001)