Entrainment of coarse grains in turbulent flows: An extreme value theory approach


  • Manousos Valyrakis,

    1. Baker Environmental Hydraulics Laboratory, Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA
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  • Panayiotis Diplas,

    1. Baker Environmental Hydraulics Laboratory, Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA
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  • Clint L. Dancey

    1. Baker Environmental Hydraulics Laboratory, Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA
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[1] The occurrence of sufficiently energetic flow events characterized by impulses of varying magnitude is treated as a point process. It is hypothesized that the rare but extreme magnitude impulses are responsible for the removal of coarse grains from the bed matrix. This conjecture is investigated utilizing distributions from extreme value theory and a series of incipient motion experiments. The application of extreme value distributions is demonstrated for both the entire sets of impulses and the maxima above a sufficiently high impulse quantile. In particular, the Frechet distribution is associated with a power law relationship between the frequency of occurrence and magnitude of impulses. It provides a good fit to the flow impulses, having comparable performance to other distributions. Next, a more accurate modeling of the tail of the distribution of impulses is pursued, consistent with the observation that the majority of impulses above a critical value are directly linked to grain entrainments. The peaks over threshold method is implemented to extract conditional impulses in excess of a sufficiently high impulse level. The generalized Pareto distribution is fitted to the excess impulses, and parameters are estimated for various impulse thresholds and methods of estimation for all the experimental runs. Finally, the episodic character of individual grain mobilization is viewed as a survival process, interlinked to the extremal character of occurrence of impulses. The interarrival time of particle entrainment events is successfully modeled by the Weibull and exponential distributions, which belong to the family of extreme value distributions.

1. Introduction

[2] One of the fundamental objectives in Earth surface dynamics and engineering is to obtain a better understanding of the underlying dynamics of the interaction of turbulent flows and the bed surface that contains them, leading to the transport of coarse particles in fluvial, coastal, and aeolian environments. The precise identification of the critical flow conditions for the inception of sediment transport has many applications, ranging from the protection of hydraulic structures against scour to the assessment and regulation of flow conditions downstream of reservoirs to ecologically friendly stream restoration designs [e.g., Whiting, 2002; Lytle and Poff, 2004].

[3] The standard and widely employed method for the identification of incipient motion flow conditions is Shields' critical shear stress criterion. This is partly a reason why considerable effort has been spent to explain deviations from the Shields [1936] empirical diagram as well as to devise alternative plots for a variety of flow and sediment cases [Miller et al., 1977; Mantz, 1977; Yalin and Karahan, 1979; Bettess, 1984; Lavelle and Mofjeld, 1987; Wilcock and McArdell, 1993; Buffington and Montgomery, 1997; Shvidchenko and Pender, 2000; Paphitis, 2001; Paphitis et al., 2002]. Since then, many researchers have adopted a deterministic perspective [White, 1940; Coleman, 1967; Wilberg and Smith, 1989; Ling, 1995; Dey, 1999; Dey and Debnath, 2000]. However, the comprehensive review of Buffington and Montgomery [1997] shows a scatter of field and laboratory threshold of motion results in excess of an order of magnitude. This, in addition to the subjectivity inherent in precisely defining a threshold for mobilization of sediment grains [e.g., Kramer, 1935; Papanicolaou et al., 2002], shows that a deterministic treatment of the turbulent flow processes leading to particle entrainment based on time- and usually space-averaged criteria does not suffice to accurately describe the phenomenon.

[4] In recognition of the variability of the hydrodynamic forces as well as of the local bed microtopography and grain heterogeneities, many researchers have supported a stochastic approach for the description of incipient movement [Einstein and El-Samni, 1949; Paintal, 1971; Cheng and Chiew, 1998; Papanicolaou et al., 2002; Wu and Yang, 2004; Hofland and Battjes, 2006]. A statistical description of the critical flow conditions by means of probability distributions is necessary because of the wide temporal and spatial variability of the parameters that control it, such as relative grain exposure or protrusion [Paintal, 1971; Fenton and Abbott, 1977; Hofland et al., 2005], friction angle [Kirchner et al., 1990], local grain geometry [Naden, 1987], and bed surface packing conditions [Dancey et al., 2002, Papanicolaou et al., 2002]. Even for the simplified case of an individual particle resting on a fixed arrangement of similar grains, where the above parameters can be accurately specified, initiation of motion retains its probabilistic nature because of the variability of the near-bed turbulent stresses.

[5] The variability of the local grain configuration and bed surface geometry affects the features of the mechanisms that generate hydrodynamic forces on sediment particles, such as the turbulent flow structures [Schmeeckle et al., 2007] and bed pore pressure fluctuations [Hofland and Battjes, 2006; Vollmer and Kleinhans, 2007; Smart and Habersack, 2007]. These factors are essentially manifested in terms of fluctuating drag and lift forces, whose effect could generally be modeled with multivariate distributions. However, analysis of experimental results has shown that hydrodynamic lift has only higher-order effects and that high magnitude and sufficiently sustained drag forces are responsible for the case of particle entrainment by pure rolling [Heathershaw and Thorne, 1985; Valyrakis, 2011]. Then, neglecting such effects allows employing univariate distributions for probabilistic modeling of the phenomenon.

[6] The relevance of high-magnitude positive turbulence stress fluctuations in the vicinity of the boundary to the inception of particle entrainment was emphasized early in the literature [Einstein and El-Samni, 1949; Sutherland, 1967; Cheng and Clyde, 1972]. Recent detailed experiments and analyses have provided strong evidence for the significance of peak hydrodynamic forces for grain entrainment, particularly at low-mobility flow conditions [Hofland et al., 2005; Schmeeckle et al., 2007; Vollmer and Kleinhans, 2007; Gimenez-Curto and Corniero, 2009]. However, Diplas et al. [2008] demonstrated via carefully performed experiments that a rather small portion of the peak values results in particle dislodgement and, instead, proposed impulse, the product of force above a critical level and duration, as a more suitable criterion responsible for particle entrainment. Valyrakis et al. [2010] expanded and generalized the validity of the impulse criterion to a wide range of grain entrainment conditions by saltation and rolling. Analysis of experimental data indicates that turbulent force impulse values follow, to a good approximation, the lognormal probability density function [Celik et al., 2010]. However, their empirically derived critical impulse level, defined as the level above which the vast majority of impulses result in grain displacement, corresponds to the upper 3rd to 7th percentile of the entire distribution of impulses, for which the performance of the lognormal distribution is observed to decrease.

[7] Evidently of particular interest is the occurrence of relatively rare and extreme impulse events that are observed to dislodge a particle. In the following sections, impulse theory is reviewed (section 1), and the mobile particle flume experiments are described (section 3). The fundamental theorems and distributions of extreme value theory (EVT) are employed to develop an appropriate probabilistic framework for the stochastic analysis of impulses (section 2). Here the impulse events extracted from a series of experiments are shown to closely follow the Frechet distribution from the family of EVT distributions. This distribution is shown to be directly linked to a power law relation for the frequency and magnitude of occurrence of the impulses. The applicability of the peaks over threshold method is demonstrated for extracting the conditional exceedances of impulse data. For a sufficiently high impulse level the generalized Pareto distribution (GPD) is fitted to the distribution of the excess impulses. Finally, the response of an individual particle under different flow conditions is analyzed stochastically by means of reliability theory. The Weibull and exponential distributions are utilized to model the time between consecutive particle entrainments.

1.1. Impulse Theory

[8] Impulse is one of the fundamental physical quantities used to describe transfer of momentum. Bagnold [1973] was among the first researchers who employed the concept of “mean tangential thrust” to define the mean flow conditions required to sustain suspension of solids in the water column. Incipient displacement of a particle by rolling has been traditionally treated using a moments or torques balance [White, 1940; Coleman, 1967; Komar and Li, 1988; James, 1990; Ling, 1995], which also describes transfer of flow momentum to the particle. However, these static approaches refer to time-averaged quantities, thus being unable to incorporate the fluctuating character of turbulence. Recently, Diplas et al. [2008] introduced impulse Ii as the relevant criterion for the initiation of sediment motion. According to this concept, impulse is defined as the product of the hydrodynamic force F(t), with the duration Ti for which the critical resisting force Fcr is exceeded (Figure 1):

equation image

The proposed criterion accounts for both the duration and the magnitude of flow events, introducing a dynamical perspective for the incipient entrainment of coarse particles. Valyrakis et al. [2010] provided a theoretical framework for the incipient saltation and rolling of individual particles by impulses of varying magnitude and duration, validated by bench top mobile particle experiments in air. They derived isoimpulse curves corresponding to different particle responses ranging from incomplete movement (twitches) to energetic particle entrainment. In accordance with this theory, only impulses above a critical level Icr, which is a function of particle properties and local pocket geometry, are capable of the complete removal of a particle out of its resting position.

Figure 1.

Sketch of the temporal history of hydrodynamic force F(t) acting on a grain (the mean, Fm, and critical, Fcr, force levels are shown with the dotted and dashed horizontal lines). Impulse events (shaded areas) with a magnitude above a critical level Icr result in complete particle entrainment (vertical dotted line).

[9] An example illustrating the importance of duration in addition to the magnitude of a flow event is depicted in Figure 1 for the case of entrainment of a fully exposed grain by rolling (Figure 2). Two separate flow events with varying magnitude and duration and consequently different potential for momentum exchange impinge upon the particle under consideration. The instantaneous hydrodynamic force, parameterized with the square of the streamwise velocity, of the first flow event i peaks higher than the second flow event i + 1. However, the former is significantly more short-lived than the latter (Ti+1 > Ti). According to equation (1), the integral of the hydrodynamic force over the duration of the flow event (highlighted regions in Figure 1) is greater for the later impulse (Ii+1 > Ii), leading to a more pronounced response of the particle [Valyrakis et al., 2010]. If this impulse exceeds the critical impulse level (Ii+1 > Icr), then the particle will be fully entrained (denoted by the vertical dotted line in Figure 1). Thus, before the probabilistic framework of impulse and grain entrainment is attempted, the definition of ucr, used in extracting impulse events, as well as the theoretical impulse level for complete entrainment Icr has to be provided.

Figure 2.

Definition sketch for the local arrangement and forces exerted on a fully exposed mobile particle. The inset shows the top view of the particle configuration, illustrating the cut plane (dashed-dotted line).

1.2. Detection of Impulse Events and Determination of Icr

[10] As opposed to the traditional incipient motion identification techniques, the impulse concept provides an event-based approach, accounting for the dynamical characteristics of flow turbulence. In order to examine the statistical properties and distributions of impulse events and their effects on entrainment of coarse grains, the method employed for their identification must first be described. In the following, the applicability of the scheme proposed to extract impulse events and implementation of the theoretically derived critical impulse level for incipient rolling are critically reviewed.

[11] Typically, a simplified tetrahedral arrangement of well-packed spherical particles (Figure 2) is considered. Of interest is the response of the exposed particle, which is a function of the hydrodynamic and resisting forces, assumed to act through its center of gravity. Loss of initial stability may occur as a result of an impulse event imparting sufficient momentum to the particle. Over the duration of this event the sum of drag (FD), lift (FL), and buoyancy (Bf) force components along the direction of particle displacement, exceed the corresponding component of particle's weight W:

equation image

where equation image is the pivoting angle, formed between the horizontal and the lever arm (Larm in Figure 2) and equation image is the bed slope [Valyrakis et al., 2010]. Equation (2) describes the static equilibrium of forces or, equivalently, torques about the axis of rotation located at the origin of the polar coordinate system (D′ in Figure 2). Usually the effect of lift force for entrainment of completely exposed particles has been neglected without significant error [Schmeeckle and Nelson, 2003]. Inclusion of the hydrodynamic mass coefficient, equation image, where Cm is the added mass coefficient (equal to 0.5 for water [Auton, 1988]), equation image is the density of fluid, and equation image is the particle's density, increases the effect of the submerged particle's weight, and equation (2) becomes

equation image

where equation image, is the submerged particle's weight (assuming uniform flow), V is the particle's volume, and g is the gravitational acceleration. Equations (2) and (3) may be solved for the critical drag force, considering the equal sign, to define the minimum level above which impulse events capable of dislodging a particle occur (Fcr in Figure 1). For steady flows it is customary to parameterize the instantaneous drag force with the square of the streamwise local velocity component upstream of the exposed particle [e.g., Hofland and Battjes, 2006]. Then it is convenient to define the critical flow conditions directly in terms of the square of the local flow velocity:

equation image

where A is the particle's projected area perpendicular to the flow direction and CD is the drag coefficient, assumed here to be equal to 0.9. It can be shown that equation (4) is similar to the stability criterion suggested by Valyrakis et al. [2010] (if fh = 1, by neglecting Cm at this stage) and identical to the critical level proposed by Celik et al. [2010] (after the appropriate algebraic and trigonometric manipulations are performed).

[12] All of the detected flow impulses have the potential to initiate particle displacement. However, Valyrakis et al. [2010] predict complete removal of the exposed particle from its local configuration by rolling only when impulses exceed a theoretical critical impulse level, defined as the product of duration Troll of the impulse event with the characteristic drag force FD assumed constant over the duration of the flow event:

equation image


equation image

being a coefficient incorporating the effects of initial geometrical arrangement and the relative density of fluid and solid grain. Equation (5) is derived from the equation of motion of a rolling particle and has been validated for a range of particle arrangements via a series of laboratory experiments. This theoretical level corresponds to impulse events extracted using equation (2), neglecting hydrodynamic lift.

[13] Application of equation (5) allows for the a priori determination of physically based impulse levels (Icr), as opposed to their empirical estimation [Celik et al., 2010], which requires experimental identification of the impulses leading to entrainment. The two methods return equivalent results if the impulse values obtained by means of the former method are multiplied with an appropriate impulse coefficient CI. For the typically encountered cases of water flows transporting grains of specific density ranging from 2.1 to 2.6, fh is close to 1.4, and CI is determined to have a value of approximately 0.5 (by matching the theoretical and experimentally defined critical impulse levels).

[14] Even though for an individual particle and local bed surface arrangement the critical conditions for entrainment can be deterministically defined, the randomness of turbulent flow forcing renders a statistical description of the critical flow conditions more meaningful. Development of a complete and reliable probabilistic theory for inception of grain entrainment requires consideration of the impulse theory together with appropriate statistical distributions that account for the intermittent character of the modeled phenomenon. In section 2 a stochastic framework for the accurate identification of the probability of entrainment of coarse grains for low-mobility flow conditions is considered.

2. Stochastic Modeling of Impulses

[15] Many researchers have recently implemented and stressed the need for a stochastic approach to the initiation of sediment entrainment due to the action of near-bed turbulence [e.g., Dancey et al., 2002; Papanicolaou et al., 2002]. Central to such models is the assumption that the episodic removal of an individual particle from the bed surface is strongly linked to the occurrence of turbulent stresses exceeding a critical level (e.g., Figure 3a). Here, contrary to the past stochastic approaches, turbulence is treated as a discrete point process, where separate flow structures of varying magnitude and duration are modeled as impulse events Ii occurring at random instances in time (Figure 3b). Similarly, the sequence of conditional impulse exceedances equation image above a certain threshold Ithr describes the point process of peak impulses. Here Ithr (not to be confused with Icr, which depends on the grain and local microtopography parameters) refers to the impulse level above which the tail of the distribution of impulses is defined (e.g., about 90% quantile of the distribution of flow impulse events; see section 4.3.1).

Figure 3.

Representation of hydrodynamic forcing history on a coarse particle: (a) as a continuous time series of the drag force FDu2 and (b) as a discrete point process of corresponding impulse events and impulse exceedances equation image (denoted by the thick vertical lines above Ithr) randomly occurring in time. The threshold impulse level Ithr (<Icr) and impulse events associated with particle displacement (stars) are also shown.

[16] The probability of particle entrainment P may be approximated by the probability of occurrence of impulses in excess of the theoretically defined (e.g., equation (5)) critical level, PE = P(Ii > Icr). This concept is shown in Figure 4, where the very infrequent occurrence of such events, especially near threshold conditions, is evident. Since of interest are largely the extreme values of the distribution, its tail (region Ii > Ithr) may be modeled separately. Then the probability of particle entrainment may be found from the conditional probability that the critical impulse level is surpassed, equation image. Thus, it is important to find statistical distributions that accurately model the magnitude and extremal character of impulses and their conditional exceedances for low-mobility flow conditions. For this purpose, distributions from EVT are considered to provide an appropriate statistical tool.

Figure 4.

The probability of particle entrainment PE is approximated by the probability of occurrence of extreme impulse events above critical, defined from the probability distribution of impulses P(Ii > Icr) or from the conditional probability of impulse exceedances equation image, using equation (19).

2.1. Extreme Value Modeling

[17] Mobile particle flume experiments discussed by Diplas et al. [2008] revealed the significance of high-magnitude impulses for grain entrainment. In their work it was first observed that only a few of the most extreme impulses, those that exceed an empirically defined critical level, result in particle entrainment. These peak impulses represented a small portion, about 4.4%, of the entire sample and belong to the upper tail of the distribution of impulses. Celik et al. [2010] proposed that impulses follow the lognormal distribution. A good fit is visually observed for the core of the distribution, in the 1–2.5 range of normalized impulses (equation image, with Imean being the sample's mean [Celik et al., 2010, Figure 9]). However, particularly for equation image, the tail of the lognormal distribution falls quite faster than the distribution of the sample. The relatively high values of reported parameters such as the skewness and flatness support the observation that the impulse distribution has a heavy tail. It is also noted that the vast majority of impulses leading to particle displacement have values above an empirically defined critical level. Careful examination shows that for most experiments this level corresponds to equation image (e.g., using Icr = 0.0063 and Imean values from Table 1). This implies that the lognormal may not be the most suitable distribution in the range of interest, which may also affect the accuracy of the probability of particle entrainment estimations. EVT provides a flexible stochastic framework with the potential to model impulse events more accurately because it has the ability to capture the extremal character of turbulence-particle interactions for near-threshold flow conditions.

Table 1. Summary of Flow Characteristics for Mobile Particle Flume Experiments
 umean (m s−1)equation imageRepImean (m2 s−2)fI (events s−1)fE (entrainment s−1)

2.2. Generalized Extreme Value Distribution (GEV)

[18] EVT provides representative distributions that model the stochastic character of extreme values from the sequence of impulses Ii assumed to be independent and identically distributed (IID) [Gumbel, 1958]. The generalized extreme value (GEV) distribution unites the three types of extreme value distributions into a single family, allowing for a continuous range of possible shapes with a cumulative distribution function [e.g., Kotz and Nadarajah, 2000]:

equation image

where equation image is the shape parameter (determines the type of the extreme value distributions), equation image is the scale parameter, and equation image is the location parameter. For equation image, it corresponds to the Gumbel (type 1) distribution, for equation image, it corresponds to the Frechet (type 2) unbounded distribution, and for equation image it is the Weibull (type 3) distribution with an upper bound. Application examples are given in sections 4.1 to 4.3 to illustrate the utility of these distributions.

2.3. Generalized Pareto Distribution (GPD)

[19] The generalized Pareto distribution is an additional family of EVT distributions. It is used to model the distribution of exceedances above a threshold and has been widely applied to a broad range of fields ranging from finance to environmental engineering and engineering reliability [Gumbel, 1954; Ashkar et al., 1991]. For the case of flow impulses, modeling the distribution of extreme values (maxima) separately is of particular interest, considering that common models may be biased in the right tail due to the relatively lower density of data.

[20] According to the limit probability theory, the GPD is the appropriate distribution for exceedances (IiIthr), as it always fits asymptotically the tails of conditional distributions in excess of a sufficiently large threshold (Ii > Ithr) [Pickands, 1975]. The GPD is a right-skewed distribution parameterized by shape equation image and scale equation image parameters, with probability density function

equation image

Equation (7) provides an accurate representation of the tail of the distribution provided that the exceedances are statistically independent and the selected threshold is sufficiently high. Similar to the GEV, GPD is classified to the Frechet (type 2) and the Weibull (type 3) distributions on the basis of the shape or tail index for equation image and equation image, respectively. For equation image, GPD becomes the two-parameter exponential distribution:

equation image

While the shape parameter for the GEV and GPD has identical meaning and value, the scale parameters are interlinked with the threshold according to equation image [Coles, 2001]. The relation between the cumulative distribution functions of the two distributions is FGPD = 1 + ln(FGEV). GPD provides an adequate model, assuming that the threshold as well as the number of exceedances is sufficiently high, so that the asymptotic approximation of the distribution is not biased and is accurately estimated. The peaks over threshold (POT) method proposed by Davison and Smith [1990] is utilized to extract excess impulses above an appropriate threshold and fit the GPD model to the tail of impulses distribution.

[21] Extreme impulses extracted using the block maxima method (where the time series is split into blocks from which the maximum value is obtained [e.g., Coles, 2001]) could also be modeled by attempting to fit them to the GEV distribution. However, such modeling is not directly applicable for the phenomenon under investigation because extreme impulses do not occur at regular, easy to identify intervals. To the contrary, GPD utilizes only the peak impulses in excess of a high, but below critical, impulse threshold (Ithr < Icr). This renders GPD ideal for modeling the tail of distribution of impulses for low-mobility conditions since for such flow conditions, Icr is relatively large, allowing for a sufficiently high choice of Ithr without biasing the distribution. The utility of GPD is demonstrated through the application of a threshold-excess method after the description of the experimental method and setup employed to obtain a series of sample impulse distributions.

3. Description of Setup and Experimental Process

[22] Results from a series of incipient motion experiments (A. O. Celik, personal communciation, 2009, see also Diplas et al. [2010]) were used to provide synchronous time series of the local flow velocity and particle position over a range of flow conditions. For completeness a summary of the experimental setup and conditions is provided below. Incipient motion experiments were performed to obtain coupled data for the entrainment of a fully exposed Teflon® (specific gravity of 2.3) spherical particle in water equation image. The test section is located about 14.0 m downstream from the inlet of the 20.5 m long and 0.6 m wide flume to guarantee fully developed turbulent flow conditions. The sphere (12.7 mm diameter) rests on top of two layers of fully packed glass beads of the same size, forming a tetrahedral arrangement (Figure 5). The bed slope equation image remains fixed at 0.25% for all of the conducted experiments. The series of conducted experiments refer to uniform and near-threshold to low-mobility conditions. For those flow conditions the use of data acquisition techniques that do not interfere with the flow renders possible the identification of the impulse events as well as entrainment instances, with greater accuracy [Diplas et al., 2010].

Figure 5.

Demonstration of the experimental setup: (a) schematic of the laboratory flume showing the location of the test section with the mobile particle and (b) sketch (side view) of the experimental setup, illustrating the possible locations of the mobile particle and its local configuration with a retaining pin.

[23] The motion of the mobile sphere is recorded via a particle tracking system composed of a photomultiplier tube (PMT) and a low-power (25–30 mW) He-Ne laser source. As seen in Figure 5, the He-Ne laser beam is aligned to partially target the test particle. Calibration of the setup showed that the angular dislodgement of the targeted particle is a linear function of the signal intensity of the PMT, which changes proportionally to the light received. A continuous series of entrainments is made possible because of a restraining pin located 1.5 mm downstream of the mobile sphere (Figure 5), which limits the maximum dislodgement of the grain to the displaced position. The grain will not be able to sustain its new location for long and will eventually fall back to its initial position after the flow impulses are reduced below a certain level, without a need to interrupt the experiment to manually place the sphere back to its resting configuration.

[24] The time history of the streamwise velocity component one diameter upstream of the particle and along its centerline (u(t)) is obtained by means of laser Doppler velocimetry (4 W Argon ion LDV) at an average sampling frequency of about 350 Hz (Figure 5). These measurements are obtained simultaneously with the displacement signal, employing a multichannel signal processor. Utilizing equations (1) and (4), impulse events of instantaneous hydrodynamic forces exceeding a critical value can be extracted from the time series of FD = f(u2).

[25] A series of experiments (E1–E6) were carried out, during which coupled measurements of flow intensity and particle response were recorded for different low-mobility flow conditions. For each of the experimental runs the flow conditions were stabilized to achieve a constant rate of particle entrainment fE over long durations (about 2 h). Impulses are extracted from the about 15 min long time series of the local flow to allow for their statistical representation. The main flow and grain response characteristics such as particle Reynolds number, Rep, are shown in Table 1 for each experimental run. All of these experiments refer to near-incipient motion conditions of about the same mean local velocity umean, dimensionless bed shear stress equation image, and turbulent intensity (equal to 0.27). Contrary to the aforementioned flow parameters, which remain relatively invariant, the rate of occurrence of impulses, fI, and fE, change more than an order of magnitude (Table 1). Thus, estimation of the mean rate of particle mobilization is less sensitive if based on fI compared to using any of the above traditional flow parameters. In section 4 the relationship between the flow impulses and grain response is further explored under a probabilistic context.

4. Analysis and Results

4.1. Frequency-Magnitude Relationship

[26] Preliminary analysis of the impulse data obtained experimentally for near-threshold flow conditions showed that extreme events of both high magnitude and relatively low frequency of occurrence are linked to the instances of particle dislodgement. Thus, it is of interest to establish a relationship between the frequency of occurrence of extreme impulse events and their magnitude. To this purpose, the mean number of impulses per second fj(Ii) above a certain level (Ii) is found for practically the whole range of magnitude of impulses for all experimental runs (E1–E6, corresponding to j = 1–6). If the obtained pairs (Ii, fi) are plotted on a log-log scale, they are observed to closely follow an almost-straight line (Figure 6). This behavior is strongly indicative of power law dependence between the two variables:

equation image

where aj (>0) is the power law exponent and bj is the base coefficient for a certain flow condition defined by index j.

Figure 6.

Variation of magnitude of impulses normalized with the critical impulse level (Ii/Icr) with the mean frequency of their occurrence fj (for run E3) and fitted power law relationship (the lognormal fit is also shown for comparison).

[27] Equation (9) is fitted to the sample pairs, for each experiment, to acquire the values (aj, bj) that parameterize it (Table 2). For the range of flow conditions examined here the frequency of occurrence and magnitude of impulse events obey a power law relationship to a very good approximation, as confirmed by the high values of the coefficient of determination (R2 ∼ 0.92–0.98, Table 2). Here the scaling region, defining the range of applicability of power law, spans virtually the whole distribution of impulses. The value of the exponent remains almost constant (a = 1.82 ± 0.10) for the different experiments. The value of the coefficient bj obtains higher values with increasing flow strength (j from 1 to 6), implying that the peak impulses become more frequent. Comparison between the various flow conditions is facilitated by normalizing the impulse values with the critical impulse level, Icr = 0.0063. Use of the normalized impulses changes the value of base coefficient to bn,j = bjIcr−aj, while the exponent remains the same (Table 2). The effect of decreasing flow conditions (from E6 to E1) on the relation between frequency and magnitude of normalized impulses is clearly demonstrated in Figure 7. Such a representation is of practical importance and predictive value since it directly provides the expected frequency of occurrence of impulses at the critical (Ii/Icr) or multiples of it levels, which are of interest for particle entrainment.

Figure 7.

Illustration of the dependence of the frequency-magnitude power law relationship (equation (9)) of the normalized impulse values on the different flow conditions (E1–E6).

Table 2. Summary of Power Law and Frechet Parameters Characterizing the Magnitude-Frequency Relation and Distribution of Impulses
Runα = γbbnequation imageR2

[28] Power law models are attractive since they have the ability to describe a wide range of scale-invariant phenomena in Earth sciences [Schroeder, 1991; Turcotte, 1997], from the occurrence of rare natural hazards such as earthquakes [Bak and Tang, 1989; Rundle et al., 1996] to the self-similarity of channel networks and corresponding energy and mass distribution [Rodriguez-Iturbe et al., 1992; Rodriguez-Iturbe and Rinaldo, 1997]. Their superiority compared to more sophisticated models has been also demonstrated for the prediction of bed load transport rates [Barry et al., 2004]. Here the power law characterizes the momentum and energy contributed by flow structures toward particle entrainment.

4.2. Impulse Distribution

[29] In addition to their simplicity and effectiveness in expressing the frequency of high-magnitude impulses resulting in particle dislodgement, it may be shown that the proposed power law relation is statistically associated with the Frechet distribution [Leadbetter et al., 1983]. Assuming that the IID impulse events arrive according to a Poisson process with a mean frequency fj, then the probability that no impulses of magnitude greater than Ii occur in the unit time (s) is Fj(Ii) = exp[−fj(Ii)] or, using equation (9),

equation image

which is the cumulative density function of the Frechet distribution as a special case of the GEV (equation (6)) with shape parameter equation image and scale parameter equation image (or equation image if Ii is normalized with Icr; Table 2).

[30] Equation (10) together with the lognormal distribution are plotted against the data sample for one of the experimental runs for comparison purposes (Figure 8). Generally, both distributions are seen to have a good overall fit to the sample impulses. However, the Frechet distribution has a relatively heavier tail behavior compared to other distributions such as the lognormal [Mitzenmacher, 2003]. This may easily be observed in the log-log representation of normalized magnitude and frequency of impulses (Figure 6), where the region of extreme impulses (normalized with Icr) is emphasized. It is shown that the logarithmic distribution generally underestimates the frequency of occurrence of extreme impulses (Figure 6) or their probability (Figure 8) as opposed to the power law and corresponding Frechet distribution, respectively.

Figure 8.

Cumulative distribution function of impulse data F(Ii) (for run E3) and corresponding fitted models (equation (10)).

4.3. Distribution of Conditional Excess Impulses

[31] In sections 4.1 and 4.2 the extremal character of distribution of impulses has been demonstrated. Since out of the whole distribution it is rather the peak impulses that are linked to grain mobilization, separate modeling of the tail of the distribution is appropriate. Here, because of the heaviness of the tail, the POT method is employed to extract the conditional impulses equation image in excess of a threshold level Ithr and to fit the GPD. Since POT is threshold dependent, guidelines justifying the choice of Ithr are provided. The GPD parameters are evaluated for the range of examined flow conditions, using different methods of estimation, and the model's performance is accessed.

4.3.1. Application of POT

[32] For design applications, it is of interest to define the near-critical flow conditions for certain gravel properties and bed surface arrangements. The proximity to the critical flow conditions may be measured by the probability of grain entrainment PE, which can be estimated through the conditional probability of impulse exceedances equation image.

[33] Evidently, the goodness of fit as well as appropriateness of the GPD model depends on the choice of threshold value. A very low threshold may utilize more data, but it may violate the assumption of the asymptotic nature of the model, biasing the distribution. On the contrary, a very high threshold will increase statistical noise because of the high variability of the extreme values, affecting the accuracy of estimated parameters. In the case when the sample size of conditional excess impulses is large (e.g., several hundreds of data points) for high-impulse quantiles, the sensitivity of the method to the threshold selection is not high. However, for flow conditions very close to critical, the sample of equation image becomes relatively small, and the selection of an optimal threshold with which both statistical certainty and accurate parameter estimation are achieved requires further investigation.

[34] In practice, graphical diagnostic tools such as the mean excess over threshold plot are commonly employed in estimating a suitable Ithr [e.g., Davison and Smith, 1990]. This graph depicts the pairs of the threshold impulse and corresponding mean excess over threshold function, {Ithr, en(Ithr)}, for a range of threshold values. The mean excess over threshold function en(Ithr) is defined as the ratio of the sum of impulses in excess of the threshold Ithr over the number of those exceedances nthr:

equation image

with Ii > Ithr. Equation (11) provides an empirical estimation of the mean excess function E(IiIthr|Ii > Ithr) of impulses. The mean excess over threshold function is plotted for a range of thresholds for run E1 (Figure 9a). The distribution of excess impulses follows a GPD above a threshold impulse, when the mean residual excess plot shows a line with approximately constant gradient [Davison and Smith, 1990; Embrechts et al., 1997; Beirlant et al., 2004]. As an example, for experiment E1, this region corresponds to the range of 87.5%–97.5% quantiles of the impulse distribution. On the basis of the above observation and considering utilizing a relatively high number of data points (Figure 9b) for improved accuracy, a relatively low Ithr, such as the 90% quantile of the distribution, may be chosen (Ithr = 0.005, vertical dashed line in Figures 9a and 9b). This impulse level provides an acceptable threshold for all the uniform flow conditions examined here. Alternatively, a practical, physically sound threshold may be predefined considering a value for the ratio Ithr/Icr (<1). Here for the selected threshold this ratio varied from about 0.6 to 0.8. For a lower threshold, e.g., corresponding to the 85% quantile, it may range from 0.53 to 0.7.

Figure 9.

Graphical method for selection of optimal Ithr for run E1: (a) mean excess over threshold and (b) number of exceedances for a range of threshold impulses. Dashed lines correspond to the limits of 85% and 97.5% quantiles, defining the range of possible thresholds. The dotted vertical line corresponds to the chosen threshold Ithr = 0.005 (m2 s−2), corresponding to the 90% quantile.

4.3.2. Estimation of GPD Parameters and Model Performance

[35] For the previously defined Ithr the basic properties of the GPD model are satisfied. Thus, the shape equation image and scale equation image model parameters are estimated for various choices of the threshold (corresponding to the 87.5% and 90% quantiles) and different methods of estimation.

[36] The estimation of GPD parameters may be performed using a variety of methods such as the maximum likelihood method (MLE) and the method of moments (MOM). The maximum likelihood method, as discussed by Embrechts et al. [1997], is employed to obtain parameter estimations and their standard errors (with 95% confidence interval), for the distribution of conditional excess impulses normalized with the mean of the distribution. Normalizing the impulse exceedances in this manner essentially removes the effect of increasing magnitude of excess impulses for increasing flow conditions, allowing evaluation of whether changes occur in the shape of the tail of the distribution. The method of moments [Hosking and Wallis, 1987] employs the mean equation image and standard deviation equation image of impulse exceedances equation image to obtain the empirical estimates of equation image and equation image, respectively:

equation image
equation image

[37] The summary of the estimated parameters is shown in Table 3. In particular, the variation of the parameters computed for different thresholds and method of estimation is shown to enable their comparison. The relative precision of the estimates, indicated by the standard errors, decreases as the sample size of normalized equation image is reduced (E1 to E6). Both the tail and scale parameters remain positive and relatively constant with a mean of equation image and equation image for experiments E1E5 (Table 3). For the experiment closest to the critical flow conditions (E6) the shape estimate becomes negative, indicating a possible change in the form of the distribution (from unbounded to a distribution with an upper bound). However, the uncertainty for such an observation is relatively high, considering that the standard error is greater than the estimated value and the confidence intervals span above zero. This probably occurs because of the relatively small sample of impulse exceedances (only 29 for Ithr = 90% quantile). The within–sampling error invariance of the model parameters with threshold selection further justifies the appropriateness of the GPD for modeling impulses above the chosen threshold.

Table 3. Summary of the GPD Shape equation image and Scale equation image Parameters, Estimated Using Maximum Likelihood Method (MLE) and Method of Moments (MOM) for All Experimental Runsa
RunMLE (87.5%)MLE (90%)MOM (90%)
equation imageequation imageequation imageequation imageequation imageequation image
  • a

    Standard errors are shown in parentheses.

E10.09 (0.07)1.20 (0.12)0.05 (0.08)1.30 (0.14)0.041.31
E20.18 (0.08)0.94 (0.10)0.22 (0.10)0.91 (0.11)0.230.90
E30.23 (0.13)1.02 (0.15)0.10 (0.12)1.29 (0.20)0.071.33
E40.18 (0.14)1.12 (0.19)0.15 (0.15)1.22 (0.23)0.111.28
E50.13 (0.15)0.86 (0.17)0.15 (0.18)0.86 (0.20)0.100.89
E6−0.03 (0.23)1.11 (0.31)−0.18 (0.22)1.35 (0.39)−0.081.24

[38] The probability density function (equation (7)) and cumulative density function predicted by the GPD model are plotted against the empirical observations in Figures 10a and 10b for run E1. Despite the relative uncertainty in the estimation of the tail parameter, the GPD model provides an excellent fit to the tail of impulse events distribution, as assessed visually. The fitted distributions for all of the experiments are shown collectively in Figure 11, represented in a log-log scale, to emphasize the region of greater normalized equation image. In agreement with the previous observations and within statistical uncertainty the shape of the distributions remains invariant for the different examined flows. This implies lack of any significant trend for the shape parameter for uniform flow conditions of about the same turbulence intensity levels but different Reynolds numbers.

Figure 10.

Distribution of the tail of impulses for run E1 and threshold corresponding to the 90% quantile (Ithr = 0.005 (m2 s−2)): (a) probability of conditional excess impulses and best fit of the generalized Pareto distribution (GPD) model and (b) empirical cumulative distribution function (CDF) of conditional excess impulses and the corresponding model fit (equation (7)).

Figure 11.

Probability density functions predicted by the GPD model for all flow conditions.

4.4. Reliability of Grain Dislodgement

[39] In addition to the rate of occurrence of impulses in a flow, the interarrival times between the instances of entrainment may be further statistically studied under an EVT context. Specifically, considering the episodic nature of the phenomenon, the time to grain entrainment for low-mobility flow conditions is analyzed here utilizing reliability (or survival) theory. Under this framework, the complete entrainment of individual coarse particles may be viewed as a stochastic process, with a certain reliability or probability of survival of entrainment events, for a specific time interval and flow conditions. Similar concepts have been employed by Ancey et al. [2008], who considered the flux of coarse grains as a birth-death process, as well as Tucker and Bradley [2010], who investigated the probability of grain entrapment while moving along a transport path.

4.4.1. Empirical Estimation

[40] Consider a surface particle resting in its local configuration. Survival of the particle past time ti is defined as the probability that the particle remains in its position after time ti, without being entrained: S(ti) = P(Te > ti), with Te > 0 the random variable representing the time to entrainment. According to the multiplication rule for joint events, this probability may also be expressed as S(ti) = P(Te > ti|Teti)P(Teti), where the probability of entrainment occurring at least after ti equals the probability of surviving past the time ti−1: P(Teti) = P(Te > ti−1) = S(ti−1). By recursive application of these formulas, it is possible to express the survival from entrainment past time ti in terms of all the conditional probabilities for entrainment in times before ti, leading to the product-limit formula:

equation image

[41] For a particular flow condition, equation (13) implies that the survival function is a decreasing function of the time for grain entrainment.

[42] Equivalent to equation (13) is the Kaplan-Meier estimator [Kaplan and Meier, 1958], which is widely used for the nonparametric empirical estimation of the survival function. Utilizing the time series of particle dislodgement obtained from the He-Ne laser for different flow conditions, the interarrival times for each entrainment event can be defined as the time intervals between the instances of deposition and entrainment of the mobile grain. The order statistics for the interarrival times (ti, i = 1,…, m) for a total of m complete entrainment events may be obtained for each experiment. This is equivalent to having a population of m different particles, which get entrained at time ti, when a particular low-mobility flow condition is imposed. For such flows it is safe to assume statistical independence between different grain mobilization events. Defining ni as the number of particles that have “survived” entrainment just before time ti and ki as the number of dislodgements (or “deaths”) occurring at time ti, the Kaplan-Meier estimation of the survival function is expressed as

equation image

where t belongs to the duration interval over which the particle is expected to dislodge (ti < t < ti+1). Then the survival function S(t) indicates the percentage of grains that have “survived” entrainment by time t. As expected, for a fixed level of S, the higher the flow conditions, the lower the estimated time to displacement is (e.g., compare S(t) = 0.5 decreasing from E4 to E1; Figure 12).

Figure 12.

Plot of the empirical survival function S(t) estimated employing the nonparametric Kaplan-Meier estimator (equation (14)) and best fit curves predicted using the Weibull model (equation (15)).

[43] Estimates of the survivability of a grain to entrainment or of its reliability to not get entrained until past a certain time instance t are provided by equation (14) and plotted for various flow conditions in Figure 12. Usually, the right-hand tail of the survival function becomes unreliable when the number of grains remaining at risk for entrainment becomes small (e.g., less than 10 may be used as a rule of thumb). Consequently, it is not statistically meaningful to evaluate the survivability for experiments with a very low number of entrainments (E5 and E6). Similarly, it may be observed that longer sampling times may be required for accurate assessment of S(t) at very low mobility conditions and in accordance to the findings of Singh et al. [2009] and Bunte and Abt [2005], who observed the dependence of the bed load transport on the sampling interval in an experimental flume and real rivers, respectively. A useful parameter signifying the reliability of grain entrainment is the mean time between entrainments, which is the inverse of the mean frequency of complete grain entrainments (Table 4).

Table 4. Summary of Empirical Model Estimators and Weibull and Exponential Model Parameters and Mean Time to Displacement (EW and Ee), Characterizing the Survival Function for Various Flow Conditions (E1–E4)
 MTBEaequation imageequation imageEWR2equation imageEeR2
  • a

    Mean time between entrainments.


4.4.2. Parametric Modeling

[44] A number of parametric models for survival times have been proposed in the literature. Typically, distributions such as the exponential have been used to characterize the interarrival of rainfall events in hydrology [e.g., Adams and Papa, 2000]. Herein the application of Weibull and exponential distributions is demonstrated in modeling the extremal character of interevent times of particle entrainment.

[45] First, the two-parameter Weibull survivor function is defined as

equation image

the positive parameters equation image and equation image denote the scale and shape parameters of the Weibull model, respectively. The mean time to full grain dislodgement as predicted by the Weibull model may be analytically estimated as

equation image

where equation image( ) is the Euler Gamma function, defined as equation image. In addition to the survival function, the hazard rate function signifying the instantaneous particle entrainment rate may be of practical utility (e.g., section 5.3). Here the hazard rate h(t) may be defined as the conditional probability that a particle is entrained at time t. It may be calculated as the number of entrainments per unit time until time t divided by the number of particles that have survived entrainment:

equation image

The shape parameter defines the behavior of the hazard function, which is predicted to be monotonically increasing for equation image, decreasing for equation image, and constant otherwise. Best fit values for the Weibull model parameters are provided in Table 4. The coefficients of determination (R2 ∼ 0.99) indicate an excellent model fit. The mean estimated time to entrainment compares well with the empirical estimation. It is observed that the shape parameter for the different flow conditions is very close to 1, which within statistical uncertainty implies a constant hazard rate. This, in turn, confirms that entrainment of individual grains in low flow rates is a Poissonian-like process without memory, contrary to the case of higher flows when grains are entrained collectively [Ancey et al., 2008].

[46] Even though the Weibull model exhibits flexibility, which may be useful for higher-flow conditions as well, for near-threshold conditions it is more realistic and physically sound to assume that the survival times are sampled from a population with a constant hazard rate. To this purpose the one-parameter exponential distribution, a special case of the Weibull model with equation image, may be used. The survival function and density function of the exponential distribution are

equation image
equation image

with equation image being the time-independent hazard rate of the exponential model. The simplicity of the model is further showcased considering that the expected time to entrainment Ee is the reciprocal of the hazard rate. The performance of the exponential distribution is very good as assessed by the high values of the coefficient of determination (R2 ∼ 1). Practically, the curves predicted by the two models almost collapse on the same curve, providing a good fit to the empirical estimation, as may be assessed visually (Figure 12). Overall, the mean time to entrainment and hazard rate of the exponential model provide easy to calculate and efficient tools for characterizing the frequency of grain response to uniform and relatively low mobility flow conditions.

5. Discussion

5.1. Probability of Entrainment for an Individual Particle

[47] In the previous sections EVT models were fitted to the distribution of impulses and their tail (sections 4.1, 4.2, and 4.3 respectively). It was observed that any change in the Reynolds number is reflected mainly in the alteration of the scale parameter (in the case the fit is performed for Ii or equation image) or, equivalently, of Imean (when impulses or their exceedances are normalized with Imean). Herein the ability of the suggested models to predict the probability of entrainment for individual particles is evaluated assuming that impulses above the defined critical level are responsible for their mobilization.

[48] For the case of impulses that closely follow the Frechet distribution this probability is straightforwardly obtained from the probability of exceeding the theoretically derived critical level, P = P(Ii > Icr). Considering that the GPD model provides the conditional impulse exceedances above impulse threshold, the probability of grain entrainment is given by equation image, with

equation image

where the probability of exceedance at the threshold value (P(Ii > Ithr)) is estimated as the ratio of the number of impulses exceeding the threshold impulse over the impulses of the entire data set (for instance, P(Ii > Ithr) = 0.10 for Ithr corresponding to the 90% quantile).

[49] The probability of particle entrainment PE may be approximated employing various methods. For instance, Ancey et al. [2008] used for PE the ratio equation image, where equation image is the mean duration of entrainment and equation image is the average waiting time for entrainment to occur. This definition works well for the case when grains may freely dislodge downstream. For the case when the downstream motion of the particle is restrained (Figure 5), the same rationale may be implemented by approximating equation image as the average time the particle remains displaced. Then PE may be plotted against each model distribution P, as shown in Figure 13a.

Figure 13.

Comparison of the exceedance probability of impulses, P = P(Ii > Icr), predicted using different model distributions with the probability of entrainment of a single particle PE using various methods of estimation: (a) equation image, (b) PE = fI/fE, and (c) PE = f(Ii)/fC.

[50] In addition to the previous method, PE may be estimated by the relative frequency of impulses resulting in grain mobilization or, equivalently, the ratio of mean frequency of impulse events fI to the average rate of particle displacement fE [Celik et al., 2010]. The results obtained via this method are plotted in Figure 13b. This definition of PE accounts for all impulse events or, equivalently, the flow events for which u2 > ucr2, along with their observed impact on mobilizing the particle (from twitches to complete dislodgements). However, there exists some uncertainty associated with estimating the above frequencies (fE, fI). First, it is assumed that all particle mobilization events (even small displacements) are accounted for, while many of them may not be discernable because of being hidden by the small-scale, high-frequency fluctuations inherent in the displacement signal. Second, the estimation of fI or mean number of impulses per unit time is quite sensitive to the selection of ucr [Celik et al., 2010], implying that the estimation of PE will vary depending on whether very small (and possibly ineffective) impulses are accounted for or not.

[51] On the basis of the previous observations the exceedance probability P(Ii > Icr) may be more appropriately estimated as the ratio f(Icr)/fC, with fC being the rate of occurrence of complete particle entrainments and f(Icr) being the mean frequency of impulses above critical (Icr). Using the theoretical prediction of ucr (derived from equation (2)), the resulting PE is plotted against the various model predictions (P) in Figure 13c. Essentially, the latter estimation method of PE considers the effectiveness of the extreme impulse events (above a critical level Icr), avoiding some of the uncertainties associated with the previous method.

[52] Careful observation of Figures 13a, 13b, and 13c reveals the same trend for each model distribution, independent of the various methods of estimation of PE presented above. The overall performance of the Frechet and GPD distributions is very satisfactory and at least comparable to the lognormal distribution. Particularly, it is seen that the GPD has a high predictive ability when the model assumptions are satisfied (e.g., E1 and E4). For the case of relatively small data samples of impulse exceedances (E5 and E6) the uncertainty of statistical estimation increases, implying that the closer to threshold the flow conditions are, the longer the required flow sampling time needs to be.

[53] On the contrary, the Frechet distribution has a consistent and superior performance throughout the range of examined flow conditions. The predictions from this distribution fall very close to the line of perfect agreement (P = PE), returning a smaller error compared to the other models (Figure 13c). As opposed to the GPD, it can model the whole distribution of impulses.

5.2. Extension of the Utility of the Power Law Relation to Variable Grain and Flow Characteristics

[54] The connection of the proposed extreme value distributions to a power law relation for the magnitude and frequency of impulses renders it a tool of great utility in characterizing the impact of a particular flow on individual particles. Specifically, two particle arrangements may have varying critical impulse levels (I1 and I2, with ratio m = I2/I1,) because of a number of factors such as different geometrical characteristics of the local bed configuration and/or the grains composing it (see equation (5)). By applying equation (9) a relation between the power law coefficients (b1 and b2) describing the two critical flow conditions and the ratio of the critical levels may be derived. For approximately constant exponent a (which is demonstrated to be valid here for a relatively small range of flow conditions), the ratio of rate of occurrence of two different impulses is

equation image

Different threshold conditions are defined in terms of the same probability of entrainment (PE,1/PE,2 = 1, which is reduced to f1(I1)/f2(I2) = 1 if the efficiency of transfer of flow momentum to the particle is assumed to be constant). On the basis of the above assumptions and using equation (20), the relationship between the two threshold flow conditions becomes b2 = b1ma.

[55] Similarly, for a given flow condition (or, equivalently, fixed values of the power law parameters), flow events of m times the magnitude of a reference impulse Iref appear maj times less frequently:

equation image

For example, considering that the range of flow conditions for the performed experiments is characterized by a relatively invariant negative exponent, a (=1.82), flow events of magnitude 2Iref have a rate of occurrence decreased by 3.5 times (for the same flow condition). This means that grains of the same diameter and about 52% greater density or, equivalently, the same density and 2.1-fold greater diameter are entrained downstream at a rate 3.5 times smaller compared to the case of the reference grain arrangement at the flow conditions examined here. In addition, when the bed surface is characterized by a range of particle sizes, the relative contribution of each size fraction to the total bed load transport may be estimated by measuring the average distance over which each particle will dislodge when an impulse above Icr is applied to it (e.g., using particle tracking and synchronized flow measurement techniques).

[56] Grass [1970] viewed incipient motion as a stochastic process and used the distributions of the applied hydrodynamic as well as resisting critical stresses to account for their variability. He suggested that the threshold conditions and, generally, any level of grain movement could be modeled by the overlap between the distributions of instantaneous bed shear stresses and the critical shear stresses required for the inception of motion of the bed surface grains. Following the same reasoning, the frequency distributions of flow impulses Ii and critical impulses Icr may be employed to characterize the flow-forcing and motion-resisting conditions (because of heterogeneities in grain and local microtopography parameters). Then the overlap between the two distributions of impulse (Figures 14a and 14b) may provide an alternative means of accurately accessing marginal bed load transport rates. The above approach demonstrates conceptually how the utility of an impulse model may be extended to characterize threshold conditions for a general bed configuration and variable grain parameters.

Figure 14.

Conceptual illustration of the utility of impulse concept in defining different levels of grain mobilization: (a) no entrainment, the distributions (f) of Ii and Icr are apart, and (b) threshold of motion characterized by a small but finite probability of grain mobilization or a certain degree of the distributions' overlap (higher degree of overlap will result in higher transport rates).

5.3. Implications for Bed Load Transport

[57] At near-threshold flow conditions and relatively low transport rates, grain entrainment becomes a highly intermittent process [Singh et al., 2009], which has been experimentally shown to affect mean bed load transport rates [Radice and Ballio, 2008]. The wide spatial and temporal fluctuations of bed load transport rates that have been observed in both laboratory experiments [e.g., Kuhnle and Southard, 1988; Ancey et al., 2008] and field studies [e.g., Drake et al., 1988] demonstrate the appropriateness of treating bed load movement as a probabilistic process. Einstein [1937] was among the first to recognize the stochastic nature of grain entrainment and develop distribution functions for the number of grains passing through a cross section. Recent probabilistic bed load transport studies model the intermittently occurring grain entrainment and distrainment as a Markov process [e.g., Lisle et al., 1998; Papanicolaou et al., 2002; Ancey et al., 2008; Turowski, 2009]. Particle instability may be triggered when the instantaneous hydrodynamic forces exerted to it exceed the resisting frictional and gravitational forces (u > uc).

[58] Here, under a similar context, flow events or impulses of sufficient magnitude, I > Icr, may impart enough momentum for particle removal to a downstream location. As shown in section 5.1, the distribution of impulse exceedances modeled by means of extreme value distributions, P(I > Icr), are directly linked to PE and consequently to the time average frequency of particle entrainment (1/nc). In addition, the frequency of impulse exceedances above critical (as predicted by equation (9)) exhibits an almost-linear relation (R2 = 0.93) to the average frequency of entrainment (modeled by the exponential distribution of time to entrainment, equation (18b)). Those observations strengthen the significance of impulse on particle entrainment.

[59] Herein, for low-mobility flow conditions the interarrival times between entrainments are modeled by an exponential distribution. This result offers an experimental validation of Einstein's [1937] assumption of exponentially distributed rest durations when no collective transport occurs. Furthermore, assuming an exponential distribution for the waiting time between entrainments, the rate of transport may be shown to follow a Poisson distribution [Ancey et al., 2008; Turowski, 2010]. The Poisson distribution for bed load is found to provide a good fit to high-resolution field data when the transport is not dominated by bed form motion [Turowski, 2010].

[60] In agreement with these studies, it is shown that at relatively low transport rates, particle entrainment (with Te following the exponential distribution) is a process without memory, implying that PE should remain constant over time (for the same flow conditions and bed geometry). As the transport rate increases, particles may be set into motion from bed material already entrained, which is not captured from a Poissonian representation of bed load [Ancey et al., 2008]. For such cases, it may be more appropriate to implement the Weibull distribution (equation (15)) to model the distribution of Te because of its flexibility to account for time dependence.

6. Conclusions

[61] The extremal character and episodic nature of the occurrence of high-magnitude impulse events and associated time to entrainments are considered here by employing stochastic measures and distributions from the extreme value theory for low-mobility flow conditions. The probability of particle entrainment is approximated by the probability of impulse exceedances above a theoretically defined critical level. Impulses and conditional impulse exceedances are treated as random occurrences of flow events of different magnitudes and durations.

[62] It is demonstrated that the distribution of impulses closely follows a Frechet distribution that is associated with a power law relation for the frequency and magnitude of impulses. The exponent of this relation did not show any significant trend for the range of examined flow conditions. The increase in flow rates was mainly demonstrated by an increase of the base coefficient. Such a description offers a useful tool for the prediction of particle entrainment for particular flow conditions.

[63] Additionally, the generalized extreme value distribution is shown to be an acceptable model for the tail of the distribution of impulses. The peaks over threshold method is implemented to extract the conditional excess impulses above a certain threshold. Guidelines for appropriate selection of the impulse threshold are provided, and the methods' sensitivity to this threshold is also assessed. Different methods are employed for the estimation of the model parameters. The robustness of the method is indicated by the satisfactory fit of the generalized Pareto distribution to the sample of conditional excess impulse data.

[64] The overall performance of the distributions is at least comparable to or better than the lognormal distribution, as assessed by direct comparison of the predicted and observed probabilities of entrainment for different flow conditions. In direct analogy to the statistical concept of Grass [1970], an extension of the utility of the proposed power law relation is offered by expressing the distribution of forces driving and resisting grain mobilization in terms of impulses rather than shear stresses.

[65] Further, the grain response is statistically described by employing concepts from reliability theory to model the time to full grain entrainment. The exponential distribution is a useful model providing mean time to entrainment and hazard rates for dislodgement, which efficiently characterize the intermittent nature of the phenomenon for low flow rates. The goodness of fit of the exponential model to the empirical distribution provides an experimental validation of the assumption employed by a number of bed load transport models.

[66] In addition to providing good statistical approximations to impulses and time to occurrence of grain entrainment, EVT models provide enhanced understanding and simulation abilities, which are required for the development of predictive equations for sediment entrainment.

equation image

bed slope.

equation image( )

Euler Gamma function.

equation image

shape parameter of the GEV distribution (equation (6)).

equation image

shape parameter of the GPD (equations (7) and (8)).

equation image

shape parameter of the Weibull model (equation (15)).

equation image

pivoting angle.

equation image

scale parameter of the Weibull model (equation (15)).

equation image

hazard rate of the exponential model (equation (17)).

equation image

location parameter of the GEV distribution (equation (6)).

equation image

conditional exceedance of impulse Ii above a threshold level.

equation image

density of the fluid.

equation image

density of the particle.

equation image

coefficient including effects of local grain arrangement and relative density of fluid and solid grain.

equation image

scale parameter of the GEV distribution (equation (6)).

equation image

scale parameter of the GPD (equations (7) and (8)).

equation image

dimensionless bed shear stress.


power law exponent (equation (9)) for flow conditions defined by index j.


base coefficient (equation (9)) for flow conditions defined by index j.


buoyancy force.


drag coefficient.


impulse coefficient.


added mass coefficient.


Weibull mean time to particle entrainment (equation (16)).


exponential mean time to particle entrainment equation image.


function of mean excess impulses over a threshold (Ithr).


total hydrodynamic force.


cumulative distribution function of the GEV distribution (equation (6)).


cumulative distribution function of the GPD.


critical force level.


hydrodynamic drag force.


hydrodynamic lift force.


mean force level.


mean rate of complete particle entrainments.


mean rate of particle entrainments.


probability density function of the GPD (equations (7) and (8)).


mean rate of occurrence of impulses.


hydrodynamic mass coefficient.


frequency of impulses in excess of Ii for flow conditions defined by index j.


gravitational acceleration.


hazard rate function (equation (17)).

equation image

impulse normalized with mean of distribution sample (Imean).


impulse event i (equation (1)).


critical impulse level (equation (5)).


threshold impulse level (defining the tail of distribution of impulses).


lever arm.


number of exceedances over a threshold (Ithr).


theoretical probability of exceeding the critical impulse level, P(Ii > Icr).


empirical estimation of probability of entrainment.


coefficient of determination.


particle Reynolds number.


survival function past time t.


time to particle entrainment.


duration of event i.


impulse duration (for entrainment by rolling).


time instance when event i occurs.


streamwise velocity component.


critical flow velocity.


local time averaged flow velocity.


volume of the particle.


weight of the particle.


submerged weight of the particle.


[67] The support of the National Science Foundation (grants EAR-0439663, EAR-0738759, and CBET-1033196) and the Army Research Office is gratefully acknowledged. The authors also thank A. Ozan Celik for providing access to experimental data used in this research.