## 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 *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*for which the critical resisting force

_{i}*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*). 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*

_{i+}_{1}>

*I*), leading to a more pronounced response of the particle [

_{i}*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*), and buoyancy (

_{L}*B*) force components along the direction of particle displacement, exceed the corresponding component of particle's weight

_{f}*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*= 1, by neglecting

_{h}*C*at this stage) and identical to the critical level proposed by

_{m}*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*is close to 1.4, and

_{h}*C*is determined to have a value of approximately 0.5 (by matching the theoretical and experimentally defined critical impulse levels).

_{I}[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.