### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Modeling of Impulses
- 3. Description of Setup and Experimental Process
- 4. Analysis and Results
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[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

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Modeling of Impulses
- 3. Description of Setup and Experimental Process
- 4. Analysis and Results
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[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 *I*_{i} 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 *T*_{i} for which the critical resisting force *F*_{cr} is exceeded (Figure 1):

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 *I*_{cr}, 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.

[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 (*T*_{i+}_{1} > *T*_{i}). 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 (*I*_{i+}_{1} > *I*_{i}), leading to a more pronounced response of the particle [*Valyrakis et al.*, 2010]. If this impulse exceeds the critical impulse level (*I*_{i+}_{1} > *I*_{cr}), 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 *u*_{cr}, used in extracting impulse events, as well as the theoretical impulse level for complete entrainment *I*_{cr} has to be provided.

#### 1.2. Detection of Impulse Events and Determination of *I*_{cr}

[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 (*F*_{D}), lift (*F*_{L}), and buoyancy (*B*_{f}) force components along the direction of particle displacement, exceed the corresponding component of particle's weight *W*:

where is the pivoting angle, formed between the horizontal and the lever arm (*L*_{arm} in Figure 2) and 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, , where *C*_{m} is the added mass coefficient (equal to 0.5 for water [*Auton*, 1988]), is the density of fluid, and is the particle's density, increases the effect of the submerged particle's weight, and equation (2) becomes

where , 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 (*F*_{cr} 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:

where *A* is the particle's projected area perpendicular to the flow direction and *C*_{D} 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 *f*_{h} = 1, by neglecting *C*_{m} 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 *T*_{roll} of the impulse event with the characteristic drag force *F*_{D} assumed constant over the duration of the flow event:

with

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 (*I*_{cr}), 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 *C*_{I}. For the typically encountered cases of water flows transporting grains of specific density ranging from 2.1 to 2.6, *f*_{h} is close to 1.4, and *C*_{I} 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

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Modeling of Impulses
- 3. Description of Setup and Experimental Process
- 4. Analysis and Results
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[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 *I*_{i} occurring at random instances in time (Figure 3b). Similarly, the sequence of conditional impulse exceedances above a certain threshold *I*_{thr} describes the point process of peak impulses. Here *I*_{thr} (not to be confused with *I*_{cr}, 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).

[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, *P*_{E} = P(*I*_{i} > *I*_{cr}). 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 *I*_{i} > *I*_{thr}) may be modeled separately. Then the probability of particle entrainment may be found from the conditional probability that the critical impulse level is surpassed, . 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.

#### 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 (, with *I*_{mean} being the sample's mean [*Celik et al.*, 2010, Figure 9]). However, particularly for , 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 (e.g., using *I*_{cr} = 0.0063 and *I*_{mean} 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 | *u*_{mean} (m s^{−1}) | | *Re*_{p} | *I*_{mean} (m^{2} s^{−2}) | *f*_{I} (events s^{−1}) | *f*_{E} (entrainment s^{−1}) |
---|

E1 | 0.248 | 0.007 | 424.18 | 0.0024 | 2.20 | 0.147 |

E2 | 0.243 | 0.007 | 412.75 | 0.0023 | 1.73 | 0.114 |

E3 | 0.238 | 0.006 | 397.51 | 0.0021 | 1.31 | 0.051 |

E4 | 0.230 | 0.006 | 384.81 | 0.0021 | 0.80 | 0.031 |

E5 | 0.228 | 0.005 | 377.19 | 0.0022 | 0.33 | 0.012 |

E6 | 0.218 | 0.005 | 364.49 | 0.0019 | 0.47 | 0.002 |

#### 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 *I*_{i} 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]:

where is the shape parameter (determines the type of the extreme value distributions), is the scale parameter, and is the location parameter. For , it corresponds to the Gumbel (type 1) distribution, for , it corresponds to the Frechet (type 2) unbounded distribution, and for 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 (*I*_{i} − *I*_{thr}), as it always fits asymptotically the tails of conditional distributions in excess of a sufficiently large threshold (*I*_{i} > *I*_{thr}) [*Pickands*, 1975]. The GPD is a right-skewed distribution parameterized by shape and scale parameters, with probability density function

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 and , respectively. For , GPD becomes the two-parameter exponential distribution:

While the shape parameter for the GEV and GPD has identical meaning and value, the scale parameters are interlinked with the threshold according to [*Coles*, 2001]. The relation between the cumulative distribution functions of the two distributions is *F*_{GPD} = 1 + ln(*F*_{GEV}). 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 (*I*_{thr} < *I*_{cr}). This renders GPD ideal for modeling the tail of distribution of impulses for low-mobility conditions since for such flow conditions, *I*_{cr} is relatively large, allowing for a sufficiently high choice of *I*_{thr} 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

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Modeling of Impulses
- 3. Description of Setup and Experimental Process
- 4. Analysis and Results
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[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 . 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 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].

[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 *F*_{D} = *f*(*u*^{2}).

[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 *f*_{E} 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, *Re*_{p}, 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 *u*_{mean}, dimensionless bed shear stress , and turbulent intensity (equal to 0.27). Contrary to the aforementioned flow parameters, which remain relatively invariant, the rate of occurrence of impulses, *f*_{I}, and *f*_{E}, change more than an order of magnitude (Table 1). Thus, estimation of the mean rate of particle mobilization is less sensitive if based on *f*_{I} 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.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Modeling of Impulses
- 3. Description of Setup and Experimental Process
- 4. Analysis and Results
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[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.