Stochastic determination of entrainment risk in uniformly sized sediment beds at low transport stages: 1. Theory

Authors


Abstract

[1] The entrainment of sediments in rivers exhibits an intermittent behavior. Incipient motion should therefore be described as a random process requiring a stochastic predictive approach. The effect of near-bed turbulence on grain entrainment and the variation in bed particle stability due to local surface heterogeneity are included into a probabilistic framework. The intuitive evidence that hydrodynamic forces acting on the sediment bed and the resistance to motion of the bed particles are two mutually dependent aspects of a unique process is modeled by introducing a conditional independence hypothesis. Based on this concept, new insights into the stochastic aspects of incipient motion are obtained. For low ratios of the boundary shear stress to the critical shear stress, by including the mutual dependence of different processes the new theoretical development predicts up to 50% larger probabilities of grain removal from the bed surface compared to Grass' original formulation. This follows from the entrainment risk being not only dependent on the distributions of fluid forces and grain resistance, but also on the correlation that these distributions exhibit in relation to the geometrical configuration of the sediment bed. This complex interaction is neglected in existing probabilistic models of sediment transport. It is then demonstrated that such additional contribution explains better the influence of both flow turbulence and particle arrangement on key features of the overall grain entrainment process.

1. Introduction

[2] In the past decade a number of researchers have attempted to explore the probabilistic relationship between what is referred to as ‘pickup probability’ or ‘probability of entrainment’ and both turbulent flow conditions and bed texture properties that characterize the resistance to motion of sediments. Papanicolaou et al. [2002] explained that the initial development of stochastic models was primarily based on the assumption that hydrodynamic forces were statistically represented by a normal distribution. Other experimental findings [e.g., Wang and Larsen, 1994; Jiménez, 1998] had shown that the Reynolds normal stress in natural streams can be reasonably approximated by gamma or exponential distributions. Cheng and Chiew [1998] and Wu and Chou [2003]developed theoretical formulations of the entrainment probability, where the variability of instantaneous velocities was described adopting Gaussian and lognormal distributions, respectively. Higher-order distributions were presented byPapanicolaou et al. [2002] and Wu and Yang [2004]. Despite slightly different approaches presented in these studies, it is generally assumed that the near-bed velocityUf (or the fluid shear stress Tf) is represented by a cumulative distribution FUf with probability density function fUf, and that the probability of entrainment PE can be expressed as

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where ug denotes the threshold velocity or critical velocity for grain entrainment (equivalent to the critical shear stress τg for individual particles). Noting that Ug is a random variable represented by a cumulative distribution FUg with probability density function fUg, the mean entrainment probability is the expected value of equation (1) over the full range of Ug [e.g., McEwan et al., 2004]

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with FUg(uf) = P(Ug < ug = uf). Herein, capital letters indicate random variables, and lower-case letters indicate single occurrences of these variables.

[3] Equation (2) reflects the conclusions by Kirchner et al. [1990, p. 668] who suggested that “it may be possible to predict the bed-load transport rate from a ‘convolution’ of the probability distributions of the instantaneous local shear stress (…) and the local critical shear stress.” A. J. Grass first demonstrated that the entrainment of grains could be linked with turbulent flow features and combined with the probabilistic distribution of the critical shear stresses [Grass, 1970]. Solutions for the entrainment probability in the form of equation (2) are here referred to as following the ‘Grass approach’. A schematic representation of this method is reported in Figure 1, where the larger the overlapping of the shear stress distributions, the higher the probability that individual grains are entrained.

Figure 1.

Schematic representation of Grass joint probability. The joint probability of the critical shear stress, τg, lying within the range τ ± δτ/2 (having probability fτf δτ) and the fluid shear stress, τf, taking a value higher than τ(having probability 1-Fτf) gives the elemental probability of entrainment δPE = fτf[1-Fτf]δτ. Integrating δPE over the full range of τ leads to the probability of entrainment defined in equation (2).

[4] A crucial shortcoming of existing models based on equation (2) is that the underlying approach requires Tf and Tg (or Uf and Ug) to be statistically independent. This constraint is not generally satisfied as we suggest that both depend at least on the statistical distribution of the surface grain elevations, Zgand that this dependence has a significant impact on the probability of entrainment. Evidence indicates that the higher the exposure of a particle to the near-bed fluid, the higher the forces exerted by the flow (as the local mean velocity increases) and the weaker the resistance to motion of the particle (as the ratio of the lever arms of the submerged weight and fluid force decreases). Particles that rest in deep bed pockets are harder to move than particles in shallow pockets not only due to the higher forces required to begin motion, but also because the occurrence of the very high fluid forces is less frequent. A general form of the entrainment probability must account for such combined likelihood of occurrence of Tf, and Tg as follows

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where fTf,Tg is the joint probability density function of Tf and Tg, and the notation RE stands for risk of entrainment [e.g., Lopez and Garcia, 2001; McEwan et al., 2004]. The symbols τf and τg are used here to refer to the fluid and critical shear stress of a single grain, while the spatially averaged boundary shear stress and the threshold shear stress of the grain population are represented by τ0 and τ0c.

[5] This study is the first of two companion papers. This paper seeks to extend the original Grass approach by including the correlation between fluid forces and grain resistance and to make the problem tractable by introducing the hypothesis of conditional independence (Section 2). A general derivation of equation (3) is provided along with the analysis of the conditions that applies to the present model (Section 3). Results are used to illustrate the implications that the new theoretical development has for sediment transport (Section 4). The related companion paper [Tregnaghi et al., 2012] describes how detailed statistical information can be obtained to test the influence of the constituent sub-processes leading toequation (3) on sediment entrainment. This is achieved in a series of laboratory experiments using particle imaging velocimetry (PIV) techniques, and numerical simulations performed with discrete particles models (DPM).

2. Stochastic Model of Sediment Entrainment

2.1. Deterministic Criterion of Entrainment

[6] Bed load in open channel flows is commonly predicted as a monotonic function of the near-bed fluid shear stress. However, the instantaneous vertical exchange of momentum in the fluid is not necessarily correlated with the actual forces on the sediment bed, as stresses in a turbulent fluid are ‘apparent’ or Reynolds stresses [e.g.,Nelson et al., 1995; Schmeeckle and Nelson, 2003]. The pressure distribution on the particle resulting from the flow around its surface can be accounted for by a sufficient parameterization of the drag and lift terms, and as a consequence it is the turbulence fluctuations in the near-bed velocity that give rise to fluctuations in the forces on sediment grains. The grain entrainment mechanism is formulated in terms of force balance, where the fluid force is represented by the hydrodynamic drag force. The fluid shear stress for an individual grain can be obtained from its destabilizing forceFD as

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where ωfis the ratio of the exposed area to the cross-section of the particle,ρ the fluid density, and CD the drag coefficient. The shear stress that one calculates for a single grain is therefore equivalent to the ‘drag stress’ described by Buffington and Montgomery [1999], which is derived from the instantaneous streamwise velocity rather than from the instantaneous Reynolds stress.

[7] Equation (4) describes the simplest formulation of a static equilibrium of forces, which allows a statistical description by using univariate distributions. The vertical velocity, which may be important for lift force, is not accounted for as usually its effect on the entrainment of exposed particles has been neglected without significant error [e.g., Schmeeckle and Nelson, 2003]. The multivariate approach proposed by Papanicolaou et al. [2002] shows that the contribution of the vertical velocity fluctuations falls within a few percent of the total probability of entrainment, with the largest increments occurring for partially exposed particles. When a grain is surrounded by other particles forming a closely packed arrangement, hydrodynamic drag becomes less effective due to reduced exposure. At incipient transport conditions over relatively coarse sediments, it can be demonstrated that the lift force computed with the instantaneous velocity atop the mean bed surface is not of sufficient magnitude to trigger entrainment. Grains that are fully sheltered or surrounded by neighboring particles are effectively unavailable for transport. For the conditions under investigation rolling (or sliding) of fully and partially exposed particles drag forces is assumed to be the main mechanism driving entrainment, which is reasonably modeled by equation (4).

[8] Recently, Diplas et al. [2008] and Valyrakis et al. [2010] demonstrated that applied forces may last over such a short period of time that they do not accomplish full particle dislodgement. They concluded that an impulse approach is a more appropriate criterion for identifying entrainment conditions. This is particularly true for very low flow conditions, i.e., for τ0 less than about 0.1–0.2τ0c [Celik et al., 2010]. However, it is recognized that all forces above a given threshold are able to destabilize a particle within its pocket inducing particle vibrations or small displacements shorter than one grain size [e.g., Bottacin-Busolin et al., 2008]. In a sediment transport perspective, the two criteria may be then comparable if the hop length statistics account for the whole range of potential displacements, including such small movements. As argued by Valyrakis et al. [2011]a probabilistic entrainment model following a Grass-like approach should employ an impulse-based criterion. Currently, this is difficult to apply, as identifying both excess force magnitude and duration requires that the critical force that is exceeded and the geometry of the bed are obtained deterministically. In the present study instead these elements are modeled as interdependent stochastic variables.

2.2. Statistical Dependence of Fluid Forces and Grain Resistance

[9] The random nature of the entrainment process depends on many physical contributors including the temporal fluctuations of turbulent flows, heterogeneities in grain size, shape and density, local grain arrangement, exposure and sheltering effects, bed roughness. Some of these features pertain either to the fluid shear stress τf or to the critical shear stress τg, while other features may affect both τf and τg caused by the correlation between fluid forces and grain resistance. The effect of certain physical variables may be arbitrarily accounted for by a functional dependence of either τf or τg. For example, sheltering effects result in the modification of the local velocity downstream of a protruding grain and in the reduction of the exposed area, thus diminishing the action of the fluid exerted on the target particle. An equivalent effect may be regarded as an apparent enhancement of the critical entrainment force leading to the same probability of entrainment. Table 1 reports a schematic definition of the main physical factors involving the random variability of the grain entrainment process. In this study the dependence on the grain size variability is neglected and the derivations apply only to uniformly sized sediment. The grain elevation affects the fluid shear stress τf through the dependence on ωf = ωf(zg), μuf = μuf(zf) and σuf = σuf(zf), where μuf and σuf are the mean and standard deviation of the approach flow velocity uf, zf = zf(zg) is the elevation (from the reference level) of the application point of the drag force (see Figure 2).

Table 1. Definition of Variability Factors
Random ParameterPhysical EffectApparent EffectFunctional Dependence
Velocity fluctuations (flow turbulence)drag force fluctuations-fluid stress, τf
Grain sizeexposed area/submerged weight-fluid stress, τf critical stress, τg
Grain shapedrag coefficient (drag force)grain stabilitycritical stress, τg
Grain densitysubmerged weight-critical stress, τg
Target grain elevation (exposure)exposed area/fluid velocity-fluid stress, τf critical stress, τg
Upstream grain elevationfluid velocity/exposed areagrain stabilitycritical stress, τg
Downstream grain elevationgrain stability-critical stress, τg
Figure 2.

Geometrical definition of the fluid drag on particle at z = zg(longitudinal section). The fluid is assumed to flow over a flat bed of uniform-sized spherical particles, aligned with top elevation located atz = zbed, and mean bed elevation at z = μzg. The ‘target’ particle is located at z = zg, and the force application point is zf > zg. The reference level, zREF, is the elevation at which the mean velocity profile is zero.

[10] Cooper and Tait [2008]found that the spatial pattern of near-bed velocities had little coherence with bed surface topography at the grain-scale, and that local topography exerted less of an influence on the spatial organization of time-averaged velocities than relative submergence. This supports the assumption that, for ratios of the flow depth to the roughness layer larger than about 5, the structure of the near-bed velocity can be considered effectively as independent of the spatial grain arrangement. In other words, the characteristic distributions describing the variables exhibit no spatial or temporal dependence under uniform transport conditions. The analytical formulation offTf, fTg and FTf,Tgmay vary with time to account for the development of resilient bed structures occurring in water-worked sediment beds as suggested by findings ofPapanicolaou et al. [2001] and Marion et al. [2003], and illustrated by Valyrakis et al. [2008] using numerical simulations performed with discrete particle modeling (DPM).

2.3. Entrainment Risk Under the Conditional Independence Hypothesis

[11] In this stochastic approach to sediment entrainment turbulent stresses (or forces) are taken to be responsible for the initial dislodgement of sediment. The random nature of initial motion is caused by the time-varying near-bed flow field combined with the spatial variability of sediment bed properties. The key hypothesis is that although Tf and Tg are statistically independent, they can be regarded as conditionally independent random variables given the grain elevation Zg = zg [e.g., Shao, 2003], whence

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in which fTf(·|·) and fTg(·|·) are the conditional probability density functions of Tf and Tg, respectively; fTf,Tg(·|·) is the joint conditional probability density function of Tf, and Tg. Equation (3) can then be expressed as follows

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where fTf(τf|zg) represents the turbulent fluctuations of τf at the elevation zf(zg), which is stationary and applies uniformly to the surface under consideration. fTf(τf|zg) applies to all grains regardless of their position in the surface, thus any effect due to localized non-uniformity in the near-bed flow and sheltering effects are accounted for by the definition offTg(τg|zg). fTgrepresents the variation in critical entrainment shear stresses due to local surface heterogeneities, it is stationary in the sense that the grain exchange processes between the surface and bed load do not alter the statistics of bed grain stability. As a consequence of this quasi-steady assumption, the analysis is restricted to low-rate sediment transport. We believe that this methodology may apply also to conditions where transport is unsteady and non-uniform, as long as the time scales of the entrainment process and of bed modifications are different by at least an order of magnitude.

3. Theoretical Derivation of Entrainment Risk

3.1. Grain Exposure and Drag Force Position

[12] Figure 2 shows the geometrical definition of the fluid drag on a ‘target’ particle located at the elevation z = zg, where zg is a certain occurrence of the random variable Zg having cumulative distribution FZg and probability density function fZg. The fluid is assumed to flow over a flat surface composed of uniformly sized spherical particles, aligned with top elevation located at z = zbed. The reference level, zREF, is the elevation at which the mean velocity profile μuf(z) is zero.

[13] Bridge and Bennett [1992] reported on a number of studies where the reference level was commonly taken as 0.2 to 0.3 d below the tops of the bed grains, zbed. Laboratory studies by Kirchner et al. [1990], Sekine and Kikkawa [1992], and Nikora et al. [1998]found that the distribution of bed elevations in water-worked gravel beds was fairly close to Gaussian. The probability density functionsfZg of the target grain elevation is then estimated as

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where N(·) implies a Normal distribution, μzg = (zbed − d/2) is the mean particle elevation, and σzg is the standard deviation of the distribution. Sekine and Kikkawa [1992] indicated that, at three standard deviations above the mean zg = (μzg + 3σzg), the centroid of a grain is one grain diameter above the mean elevation, resulting in σzg = 0.33 d. Nikora et al. [2001a]suggested that the flow-sediment interface is comprised within individual roughness elements and occupies the flow region between roughness crests and troughs. The thickness of such interface for water-worked gravel beds was found to be approximately equal to 4σzg = 1.0 to 1.5 d, leading to σzg = 0.25 to 0.37 d.It is very rare for a particle in a natural bed to protrude entirely above the mean elevation, for most of the bed grains the fluid drag applies to an area smaller than the total cross-section of the particle. Grains located underneath the mean bed level experience negligible drag forces.

[14] The variation in fluid drag is accounted for by the target particle elevation and by the velocity fluctuations at the level zf. The mean velocity vertical distribution in the wall region of the turbulent flow over a hydraulically smooth bed can be described as

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where gi(·) stands for function of; the star notation is used to denote dimensionless quantities, and the scales that apply to fluid velocity and lengths are the shear velocity us and the particle size d, respectively. The fluid drag is computed as the integral of the elemental drag forces over the exposed grain surface Af = ωfπd2/4, and the elevation zf* = g2(zg*) of the application point of the drag force depends on the elevation of the target particle, zg*, on the elevation of the bed crest, zbed*, and on the vertical velocity profile g1(z). The elevation zf* of the reference drag force can be expressed as

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where dA(z*) is the differential area as illustrated in Figure 3; zmin* = zbed* for zg* < (zbed* + 1/2) and zmin* = (zg* − 1/2) for zg* ≥ (zbed* + 1/2). According to the transformation rules of probability distributions, the probability density function fZf(zf*) is given by

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where δ is the Dirac delta function; the notation g′ and g−1 stand for derivative of g and inverse function of g, respectively.

Figure 3.

Particle exposure to the fluid drag (cross section). The fluid drag is computed as the integral of the elemental drag forces over the exposed grain surface Af = ωfπd2/4.

[15] Schmeeckle and Nelson [2003] suggested that almost every particle is partially in the wake of protruding particles, as grains surrounding a target particle above the mean may be themselves located above the mean bed level, and it is therefore necessary to quantify the velocity reduction due to sheltering effects. This residual variability in the fluid drag distribution is accounted for in this work in the definition of FTg,Zg. Here, only the variation of grain exposure over the mean bed is included into the probabilistic description of the fluid stress. The exposed area of the target particle can be obtained as a function of zg* as follows (see Figure 3)

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Following the transformation rules of random variables, the probability density function of the particle exposure fΩf(ωf) is given as

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where the two impulse terms reflect the discontinuity of the cumulative distribution FΩf(ωf) at the lower and upper boundaries ωf = 0 and ωf = 1, respectively. In the next section equations (10a), (10b), (12a), and (12b) will be used to determine the conditional probability of exceedance P(Tf* < τf*|Zg* = zg*) and the relevant conditional probability density function fTf(τf*|zg*) for particles that are partly and fully exposed to the flow.

3.2. Fluid Shear Stress

[16] Under the assumption that a second-order probability density function describes satisfactorily well the streamwise velocity fluctuations in the turbulent boundary layer, and that the moments of the distribution are known functions of the vertical distance from the bed. Forzf* ≥ zbed* the conditional probability density function of Uf* given Zf* = zf* is

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in which D[∙] stands for an arbitrarily defined distribution and the mean and standard deviation of the distribution are deterministically described as function of zf*, which represents a specific distance from the reference level. Defining the variable yf* as a function of the fluid velocity

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where K = 0.5ρCD(us)2/(γΔd) is a dimensionless coefficient, γ is the specific weight of water, and Δ the relative submerged density of sediments. The dimensionless fluid drag τf* = τf/γΔd = ωf·yf* can be statistically defined by the joint density function of Yf* and Ωf. The joint probability density function fYf,Ωf can be expressed in terms of the conditional probability density function fUf(uf*|zf*) and the probability density function fZg(zg*), which are assumed to have known analytical solution.

[17] Mathematical derivations given in Appendix A lead to the following expression of the conditional density of Tf as a function of fUf(uf|zf):

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where

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is the Heaviside function of the variable zg* centered in zg* = (zbed* − 1/2). Here, the impulse term Γz accounts for the probability of fully sheltering conditions for particles located below the mean bed level, while the former term involves partial exposure conditions for 0 < g3(zg*) < 1, i.e., (zbed* − 1/2) < zg* < (zbed* + 1/2), and full exposure conditions for ωf g3(zg*) = 1, i.e., zg* ≥ (zbed* + 1/2).

3.3. Relation Between Fluid Shear Stress and Particle Elevation

[18] Following Dinavahi et al. [1995], the near-bed flow velocity at a given elevationz* is assumed now to follow a near-normal distribution, i.e.,fUf(uf*|z*) ∼ N(μuf*,σuf*), and the variation of the mean velocity with depth, μuf* = g1(z*), is sufficiently well represented by the logarithmic profile for hydraulically rough beds, resulting in

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where κ = 0.4 is the von Kármán constant, and the equivalent roughness height ks is replaced with the sediment size for uniformly sized beds. In equation (17) the coefficient C = 8.5 for the homogeneous sand roughness according to Nikuradse's formula, and C = 5.5 to 7.0 for natural gravel beds [e.g., Dittrich and Koll, 1997; Nikora et al., 2001a]. Estimates for the dimensionless near bed turbulence intensity σuf* = g(z/hu), where hu is the normal flow depth, were suggested by Nezu and Nakagawa [1993]. With reference to the measurements in a rough boundary open channel flow conducted by Kironoto and Graf [1994], assuming z < < hu leads the standard deviation of the fluid velocity distribution to be approximately twice the shear velocity, i.e., σuf ≈ 2us, regardless of the vertical position. Considering incipient flow conditions according to the Shields approach, i.e., τ0* = τ0c* ≈ 0.056. From the definition of critical shear velocity usc = (τ0c/ρ)0.5 the coefficient first introduced in equation (14) becomes here K = 0.5CDτ0c*. Assuming CD ≈ 0.5, Figure 4 shows the conditional probability density functions of the drag stress at the threshold of motion computed with equation (15) for different positions of the target particle along the vertical axis.

Figure 4.

Conditional pdf of drag stress at threshold of motion computed with equation (15) for different positions of the target particle zg along the vertical axis. The shape of the distribution follows a bell shaped distribution with positive skewness, for higher, less frequent zg (isolated flow regime). As zg reduces, the distribution loses its original bell shape and tends to become closer to exponential (higher packing density).

[19] The probability density fTf(τf*|zg*) is analytically defined as a continuous function for zg* > zbed*, while it reduces to the impulse term, Γz, otherwise. The distributions are strongly dependent on the value of zg*, although the actual variation of zg* is limited to a relatively narrow band, and even for the simplified case where the variation of the standard deviation with depth is neglected. These results compare favorably with the findings of Papanicolaou et al. [2002], who developed a probabilistic model based on a moment balance equation, and examined the different behaviors of three bed packing configurations, moving from low density (isolated flow conditions) to high density (skimming flow regime). Their analysis revealed that the shape of the distribution for the isolated flow regime follows a bell shaped distribution with positive skewness, which resembles the case discussed here of higher, less frequent zg* values. As the packing density increases, the distribution loses its original bell shape and tends to become closer to exponential. The change in the distribution shape is attributed to the effects of the bed grain-scale topography and appears to be accounted for in the present model by the variation of the bed particle elevations. These observations are in agreement with previous studies ofWang and Larsen [1994] reporting that the effects of a change in roughness were considered to be responsible for the change in shape of the normal stress distribution, thus supporting the hypothesis that one of the main source of variability of the forces exerted by the flow on the bed is to be attributed to the grain elevation.

3.4. Critical Shear Stress

[20] Schmeeckle and Nelson [2003]indicated that the high variability occurring when particle entrainment is observed during moderate bed load transport depends on (a) the turbulence of the near-bed flow field; (b) the large range of resisting forces to downstream motion resulting from the relative geometrical position of bed surface grains; (c) the local modification of the velocity field by upstream, protruding grains. There are difficulties arising in attempting to develop a proper probabilistic description of such variables and their mutual dependence. Empirical evidence on the mechanisms of grain detachment caused by the fluid suggests that, for uniformly sized sediment, the primary source of correlation between (a) and (b) is given by the particle elevation distribution. The processes involved in (c) can be modeled as an apparent increase of the grain stability.

[21] Schmeeckle and Nelson [2003]developed empirical relations of the time-averaged velocity reduction to account for how protruding bed particles modify the velocity incident on downstream grains. These results supported the spatial velocity reduction algorithm used for determining the force on individual particles in a direct numerical simulation of grain entrainment.McEwan and Heald [2001] proposed a comparable approach to calculate the form of the critical entrainment shear stress distribution for beds under the action of a constant, uniform flow. They identified the different contributions made by the supporting arrangement and the sheltering effects by recognizing that two grains may have the same critical entrainment shear stress but would actually be subjected to quite different fluid forces because of differences in the geometry of the bed upstream. These differences were then accounted for by empirically adjusting the incident velocity and so the critical entrainment force for individual particles. The computational approach used in discrete particle models appear rather consistent with the conceptual frame illustrated above. The distributions of the critical shear stress, fTg(τg*), for different grain sizes were found to fall onto the same curve and to a first approximation they could be considered as lognormal, i.e.

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where the mean and standard deviation of the distribution for the sheltering case were found to be μτg* = 0.87 and στg* = 2.77. The expected value of the random variable Tf* is different from the classical critical shear stress τ0c* that can be computed using the Shields diagram, as μτg* is associated with the instability of more bed particles than those implied by τ0c* (the corresponding fraction by weight entrained is roughly 0.5%). Equation (18) cannot be entered directly into equation (6) as it represents the marginal density function of Tg* and does not account for the conditional density function fTg(τg*|zg*), which is implicitly defined as

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Kirchner et al. [1990] reported that particles having larger projections also exhibit smaller mean friction angles, i.e., a weaker resistance to motion. They also indicated that the range of measured friction angles was larger as grain protrusion decreases. It can be argued that both the location and scale parameters of the distribution decrease with particle elevation. Relationships between the scale parameters and grain position are reported for beds of uniform spheres in the companion paper [Tregnaghi et al., 2012]. Equations (15) and (19) can then be introduced in equation (6) to obtain the risk of entrainment, RE. Figure 5 illustrates the new development of the Grass approach by comparison with the schematic of Figure 1: the entrainment risk is not only dependent on the distributions of fluid and critical stresses, but also on the correlation that such distributions exhibit depending on particle position. In turn, as particles protrude above the mean bed surface the entrainment risk increases because the fluid stress distribution moves toward higher values while the critical shear stress progressively reduces.

Figure 5.

New development of Grass approach: for particles protruding above the mean bed surface the entrainment risk increases as the fluid stress distribution moves toward higher values while the critical shear stress reduces. Inset boxes provide the relative frequency of particles located at elevation z = zg. Particles located at lower zg (Figure 5a) have larger probability of occurrence but lower probability of entrainment than particles located at higher zg (Figure 5c).

4. Implications for Sediment Entrainment

[22] In this section a comparison of the critical conditions for sediment motion is explored from a deterministic and stochastic perspective. The relationship between the marginal density functions fTf(τf*) and fTg(τg*) and their deterministic counterparts, namely τ0* and τ0c*, is explored. The outcome of the present model are then described to illustrate the implications that the joint description of Tf* and Tg* has for sediment entrainment prediction. This will highlight (a) the role of flow turbulence in determining the risk of a particle being entrained and (b) how spatial surface modifications of water-worked sediment beds may lead to more stable bed configurations.

4.1. Critical Conditions of Grain Motion

[23] By comparing the Grass probabilistic approach with the Shields deterministic criterion, it can be seen that for incipient flow conditions, the occurrence of values of the fluid stress exceeding the relevant critical stress is rare but not negligible. Deterministic sediment transport equations predict higher transport rates as the fluid shear stress increases beyond an apparent threshold of motion proportionally to (τ0*–τ0c*)n, where n is a calibration coefficient. From the probabilistic view, sediment transport may occur even when the expected value of the fluid stress distribution, μτf*, is well below the average critical stress, μτg*. Assuming, under the conditions described in section 3.3, the tops of the bed grains zbed* = 0.25 (with respect to the reference level), the mean particle elevation μzg* = (zbed* − 0.5) = −0.25, and the standard deviation of the distribution σzg* = 0.33, the mean and standard deviation of the drag stress can be computed as

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which results in μτf* ≈ 0.035 < <μτg* = 0.87 and στf* ≈ 0.071 for flows approaching deterministic critical boundary shear conditions, i.e., τ0* ≈ 0.056. The mean fluid stress acting on collections of particles on the bed is lower than what is commonly referred to as the threshold of motion as defined by Shields [1936]. It should not be surprising that μτf* < τ0c* ≈ 0.056, as the number of particles exposed to the flow and that can potentially be entrained is a fraction of the total number of bed surface particles. The mean fluid stress, μτf*, is lower compared to the average boundary shear stress for uniform flows, τ0*. However, the whole distribution of τf* spreads over a relatively wide range of values, thus demonstrating as inappropriate any deterministically based notion of incipient movement. This conceptual distinction between deterministic and probabilistic threshold conditions allows for different predictions of sediment transport, as similar average hydraulic conditions may result from dissimilar local flow field and bed topography characteristics. This evidence has direct implications for a large number of gravel bed streams, where most of the transport occurs at low transport stages, i.e., the ratios of the boundary shear stress to the critical boundary shear stress, less than about three. Barry et al. [2004] argued that sediment transport formulae containing a transport threshold typically exhibit poor performance as often they erroneously predict zero transport at low to moderate flows.

4.2. Role of Flow Turbulence

[24] As discussed in section 3.4, the entrainment risk is not only dependent on the distributions of Tf* and Tg*, but also on their positive correlation through the link with the particle position Zg*. Equation (6) will result in higher values of RE compared to the values of the pickup probability math formulaE estimated with equation (2), when the same distributions of fluid and critical stresses are used. The density function fTf(τf*) can be computed by multiplying equation (15) by fZg(zg*) and subsequently integrating with respect to zg* over the range (−∞, +∞). This can be introduced in equation (2) combined with the distribution of the critical shear stress fTg(τg*), as provided by equation (18), to obtain the pickup probability under the assumption of statistical independence of Tf* and Tg*.

[25] Figure 6aillustrates in a log-log plot the comparison between the entrainment risk obtained withequation (6) and the classical pickup probability obtained with equation (2), assuming that both the velocity mean and the standard deviation scale with the time-averaged shear stressτ0*. The difference between RE and math formulaE varies with τ0* up to about 50% for τ0* ≈ 1.5τ0c*, making the average particle displacement significantly more probable. The relative increment remains approximately constant for τ0* up to about 3–4τ0c*, while gradually decreasing for relatively higher shear values. In turn, for increased flow conditions the role of particle exposure becomes less significant as the relative sizes of the velocity fluctuations tend to reduce. The comparison may become ineffective for very large shear values as the model is intended for low transport stages. Similarly, as the shear stress moves toward the lower values, the number of potentially mobile particles is only a small fraction of the whole population, thus the variation of Zg* within this fraction is less significant. For these conditions, as the left tail of the conditional density function fTg(τg*|zg*) will closely approach the marginal density function fTg(τg*), the entrainment risk RE given by the joint description of Tf* and Tg* will tend to coincide with the pickup probability math formulaE.

Figure 6.

(a) Comparison between the entrainment risk RE (bold solid line) and the classical pickup probability math formulaE(dot-dashed line). The shaded area shows the increment of entrainment risk due to the correlation of drag stress and critical stress. (b) Comparison with the model byCheng et al. [2004] based on the statistical independence of drag and critical stresses. The variation of RE with the parameters σuf and Iτ (light solid lines) indicates that the effect of turbulence is amplified by the joint description of the stochastic variables of the problem.

[26] These results offer some new insights on the role of the flow turbulence in determining the irregular character of sediment transport. There exists evidence that over short time periods and for weakly intense flow conditions, sediment transport appears as an intermittent process [e.g., Ancey et al., 2006; Radice et al., 2009]. These observations suggest that long-term disturbances of the near-bed flow field may induce collective motion of particles resulting in the solid discharge made up of a series of pulses. Even though the effect of flow turbulence on grain entrainment has been extensively investigated [e.g.,Nelson et al., 1995; Papanicolaou et al., 2001] here the role of the correlation between destabilizing and resisting forces is analytically demonstrated to be of equal significance.

[27] Comparison can be made with the approach developed by Cheng et al. [2003], who carried out flume experiments to investigate directly the distribution of the distance between particles on a flat bed comprised of uniformly sized spheres. The variation of the critical shear stress, Tg*, was theoretically demonstrated to be approximately a narrow-banded random process with probability density function,fTg, following the Rayleigh distribution. The original analysis included no effects of the near-bed flow variations, implying that the proposed equations were only applicable for the case of steady uniform flows. Their results were subsequently extended byCheng et al. [2004], suggesting that for fluid shear stress, Tf*, varying with a known probability density function, fTf, the general solution could be expressed as equation (2) by means of the following density and distribution functions

display math
display math

where the mean fluid shear stress equals the time-averaged shear stress, i.e.,μτf* = τ0*, and Iτ is the relative fluctuation of τ0*, which decreases gradually with increasing Reynolds numbers [e.g., Alfredsson et al., 1988]. For τg* = μτg* the cumulative distribution FTg = 54%, indicating that slightly more than half of the bed particles are likely to be entrained by the flow. Assuming that the classical incipient condition for grain motion corresponds approximately to the probability FTg = 0.5÷1.0% leads to τ0c* ≈ 0.1 μτg*, similarly to the DPM results reported by McEwan and Heald [2001].

[28] Figure 6b shows the variation of math formulaEwith the time-averaged shear stressτ0* estimated by introducing equations (21a) and (21b) in equation (2) for different values of the relative fluctuation Iτ. The best correspondence between the present model and the model proposed by Cheng et al. [2004] was found for Iτ = 0.5. By comparison with Figure 6a, equation (6) exhibits a larger sensitivity to the variation of the turbulence intensity compared to models based on the assumption of statistical independence of Tf* and Tg*. For the two cases the turbulence intensity is varied by scaling σuf* and Iτ, respectively, with constant ratios ranging from 0.67 to 1.50. Within this range, for the mean shear stress τ0* ≈ τ0c*, the risk of entrainment RE is found to vary from 0.27% to 0.63%, while the pickup probability math formulaE ranges from 33% to 46%. This highlights the crucial role of the joint description of Tf* and Tg* for capturing the effect of the flow turbulence on the probability of entrainment at low transport stages. This is tested in the companion paper [Tregnaghi et al., 2012] against entrainment rate measurements collected with a flow field synchronized sediment-tracking system.Equation (6) also correctly predicts a reduction of the effect of the flow turbulence as mean shear stress τ0* increases, resulting in progressively less discontinuous particle motion.

4.3. Role of Particle Arrangement

[29] The present model relies on the assumption that the distribution of the critical shear stress is stationary. However, section 2.2pointed out that such distribution may vary with time to account for changes in water-worked sediment beds. These may result in particle arrangements that are more effective in partly dissipating the energy of turbulent flows, leading to the formation of more stable beds. As the new development of the Grass approach accounts for the correlation between destabilizing and resisting forces, it is expected thatequation (6) predicts more accurately the effect of the variation of the bed properties on the risk of entrainment compared to models based on equation (2).

[30] Wu and Yang [2004] proposed a probability model that explicitly incorporates the effect of random particle geometries on entrainment. They expressed the critical velocity Ug as a function of particle size D, grain elevation Zg, and friction height Hg (which is regarded as a surrogate of the friction angle), and computed the mean entrainment probability as the expected value of equation (1) over the full ranges of these random variables, i.e.

display math

where fD, fZg, and fHg are the probability density functions of D, Zg and Hg. It can be easily demonstrated that equation (22) falls within a particular formulation of equation (2). Figure 7 illustrates the surface plots representative of the risk of entrainment for different combinations of the bed statistics and for given bulk flow conditions, using equation (6) and equation (22), respectively. In both case the same distributions of the particle elevation, fZg, and of the near bed velocity, fUf, are used. Wu and Yang [2004] assumed Hg described by a uniform distribution. Sheltering effects are accounted for by selecting the hiding factor equal to 0.8 in equation (22), which makes the results from the two models similar for μzg* = −0.25 and σzg* = 0.25. A comparison of the surface plots reveals a weaker response of equation (22) to modifications of the bed arrangement as opposed to the results predicted with equation (6). Moving from surface bed configurations that are densely packed (small σzg*) to lower density bed packing geometries (large σzg*), the variation of entrainment risk illustrated in Figure 7a is on average twice than the variation represented in Figure 7b, depending on the position of mean bed level with respect to the zero velocity profile.

Figure 7.

Role of statistical properties of the bed in determining the entrainment risk. Comparison between (a) the present model based on a joint description of the variables and (b) the model by Wu and Yang [2004] based on the hypothesis of statistical independence. The latter shows a weaker response of two modifications of the bed arrangement as opposed to the results predicted with equation (6).

[31] These results have some significance in determining how the evolution of microscale bed features alter the resistance to motion of sediment particles over the bed surface. Nikora et al. [1998]found that bed elevation distributions in laboratory flumes and in natural gravel bed streams were close to Gaussian. Examination of experimental data showed that such distributions evolved from symmetrical for initial unworked beds to slightly positively skewed for water-worked beds.Hofland [2005] argued that the cause of the positive skewness was due to the large grain size ranges in natural rivers, as the larger grains have a tendency of protruding proportionately more above the average level. Nikora et al. [2001b] and Hofland [2005]used discrete particle models to replicate the behavior of water-worked beds comprised of well sorted and uniformly sized sediment, respectively. In both cases, calculations resulted in negatively skewed distributions of bed elevations, where the skewness was caused by the water-working. In these experiments, the removed particles were not replaced by upstream sediment supply, meaning that degrading beds was simulated. Here a similar approach is used to evaluate the contribution of the joint description of Tf* and Tg* to quantify of the stabilization of degrading sediment beds. Figure 8a shows the digital models of different uniformly sized sediment beds that were obtained progressively removing larger fractions of the most protruding particles. Figure 8b illustrates the statistical properties of the same beds moving from a Gaussian distribution (unworked beds) to distributions having negative skewness Szg(water-worked beds). The initial configuration may represent the case where net deposition has occurred due an excess incoming sediment supply with respect to local transport capacity of the flow, or during the recession of a flood event. Net erosion occurs when the sediment transport capacity exceeds the upstream supply, which is commonly observed downstream of dams.

Figure 8.

Representation of different uniformly sized sediment beds with (a) digital models of the bed and (b) density functions of grain elevation. From (1) to (4)beds move from Gaussian distributions (unworked beds) to negatively skewed distributions (water-worked beds).

[32] The statistics of the simulated beds are used to compute the entrainment risk using equations (6) and (22) for flow conditions corresponding to τ0* ≈ 1.5τ0c*. Initially, the most protruding and unstable particles will be gradually removed. If no sediment is supplied from upstream the bed tends toward an equilibrium. This is also associated with the hydraulic roughness of the bed slightly decreasing in time, which only partly contributes to the reduction of the transport rate. Consistent with laboratory and field observations, both models predict that the degree of sediment movement reduce as particle arrangement evolves to produce increasingly narrow bed elevation distributions. Figure 9 shows that for Szg = −0.4 the risk of entrainment reduces to about 30% compared to the initial configuration using equation (6), and to 50% using equation (22). The effect associated to the correlation of Tf* and Tg* introduced with equation (6) increases as the particle distribution tends to become more skewed, leading to almost one order of magnitude difference in the prediction of RE for Szg = −0.9 to −1.0.

Figure 9.

Effect of the joint description of drag and critical stresses on the mechanism of stabilization of water-worked degrading beds. ForSzg = −0.4 RE (bold solid line) reduces to about 30% compared to the initial configuration using equation (6), while math formulaE(dot-dashed line) reduce to 50% usingequation (22) as proposed by Wu and Yang [2004]. Such difference increases as the particle distribution tends to become more skewed, leading to almost one order of magnitude difference for Szg = −0.9 to −1.0.

5. Conclusions

[33] A probabilistically based framework for the commencement of sediment movement was proposed that jointly accounts for the role of flow field turbulent structures and bed topography for uniformly sized particles at low transport stages. By contrast with the original theory of Grass [1970], based on the assumption that the instantaneous fluid shear stress and the local critical shear stress are statistically independent, they are now considered as correlated random variables. The approach is founded on the assumption that the contributions of the near-bed turbulence and the variability of the resisting forces arising from water-worked grain arrangements can be jointly accounted for by means of a conditional independence hypothesis. It is also assumed that the effect of certain physical variables can be arbitrarily accounted for in the definition of either the fluid or critical stress, leading to a unique probability of entrainment. For example, the reduction of the approach velocity due to sheltering is regarded as an ‘apparent’ increase of the effective grain resistance.

[34] In the present analysis, the variability of the forces exerted by the flow on the bed is demonstrated to depend significantly on the grain position over the bed surface. This compares favorably with previous findings on the effect of different bed packing configurations on the shape of the force distribution. As a consequence, for low ratios of the boundary shear stress to the critical shear stress, the new theoretical development predicts up to 50% larger probabilities of grain removal from the bed surface compared to Grass' original formulation. This follows from the entrainment risk being not only dependent on the distributions of fluid and critical stresses, but also on the correlation that these distributions exhibit depending on particle arrangement. Although the present model is intended for low transport stages, it correctly predicts that the relative probability increase gradually reduces for relatively higher shear values, as the role of particle exposure becomes less important.

[35] The proposed approach has some important implications for sediment transport. First, it provides an additional explanation about the role of flow turbulence in determining the random character of grain entrainment. While long-term disturbances of the near-bed flow field may induce an intermittent and pulsating particle motion, existing models fail to capture this characteristic feature as they ignore any correlation effect between destabilizing and resisting forces. For comparable variations of the turbulence intensity at low flow rates, such correlation determines an almost three times wider variability of the entrainment risk. Second, we have analyzed how properties of sediment beds affect the resistance to motion as they evolve during the transition from uniform transport conditions to degrading water-worked conditions. Our investigation has shown that the stabilization mechanism is more effective if the correlation between fluid and critical shear stresses is accounted for, leading to almost one order of magnitude difference in the prediction of the entrainment risk for the most stable particle arrangements. This should motivate further research on bed stabilization effects as a result of the fluid action.

[36] The probabilistic method derived is general and all physical implications of the new theoretical development are independent of the selected probabilistic distributions. It can be applied in situations in which detailed statistical knowledge of the near-bed flow field, the bed geometry and resistance to motion are available. The present approach also overcomes any complexity arising from the combined modeling of correlated random variables, which is motivated by the evidence that the different nature of the random processes involved requires different and specifically developed tools of analysis and investigation. The related companion paper [Tregnaghi et al., 2012] shows that this can be achieved by using detailed descriptions of the near-bed turbulence obtained with Particle Image Velocimetry (PIV) combined with numerical simulations performed with discrete particle models (DPM) to provide detailed representations of random bed geometries and hence reliable distributions of the critical shear stress.

Appendix A:: Derivation of the Conditional Pdf of the Fluid Shear Stress

[37] The following is intended to provide the main relationships for the joint distributions and distribution functions of continuous random variables. Let X and Y be statistically dependent continuous random variables having joint density function fX,Y(x,y) and cumulative distribution FX,Y(x,y). Suppose that U and V are defined in terms of X and Y by the functions U = φ1(X,Y) and V = φ2(X,Y) where to each pair of values of X and Y there corresponds one and only one pair of values of U and V and conversely, so that the inverse system can be defined by the relations X = φ1−1(U,V) and Y = φ2−1(U,V). Then the fundamental theorem on transformation of random variables [e.g., Spiegel, 1975] states that

display math

in which det(·) denotes determinant; FU,V(u,v) is the cumulative distribution function of U and V; the Jacobian matrix is given by

display math

and the joint density function of U and V is then given by fU,V(u,v) where

display math

The gamma functions include impulse terms that account for the discontinuity of FU,V (if any) at the points (ui,v) and (u,vj), where the conventional derivative-based density functionhU,V(u,v) does not hold, and can be expressed as

display math

Let now X and Y be described by the marginal density function fX(x) and the marginal cumulative distribution FY(y), respectively. The conditional probability density function of X given (the occurrence of) the value y of Y, is linked to the joint density function X and Y and can be written as

display math

where it is required that dFY/d> 0 and dFY/dy gives the marginal density function for Y. Equations obtained next in this section are based on the properties defined by equations (A1)(A5).

[38] According to equations (4) and (14), the fluid drag τf* can be statistically described by the joint density function of Yf* and Ωf. Following derivations show that the analytical form of fYf,Ωf can be expressed in terms of the conditional probability density function fUf(uf*|zf*) defined by equation (13) and the probability density function fZg(zg*) as given by equation (7). Let us define the joint probability density function of Uf* and Zf* first. From equations (A1) and (A5), allowing for the existence of the geometrical discontinuity in zf = zbed yields the expression

display math

where

display math

Accounting for the definition of drag elevation given by (9) involves the latter limit in (A7) to be zero as the probability P(Uf* < uf*, Zf* < zbed*) ≤ P(Zf* < zbed*) = 0, while by recognizing that the condition zf zbed is statistically equivalent to the case zg < (zbedd/2) the former limit can be computed as follows

display math

where the equality holds P(Zf* = zbed*) = P(Zg* < zbed* − 1/2). Thus equation (A6) reduces to

display math

Accounting for (A5), the conditional probability density function of Yf* given Zg* = zg* can be expressed as

display math

then equation (A1) can be taken into account and equation (A10) becomes

display math

in which the Jacobian matrix has nonzero terms only on the main diagonal

display math

Taking into account equations (A6) and (A12), equation (A11) can be rewritten as follows

display math

with the last derivative in the right-hand side given by

display math

where the property is used that the derivative of the inverse function equals the inverse of the derivative. Equation (A13) then reduces to

display math

Let note now that the conditional probability density function of Yf* and Ωf can be expressed as

display math

Following a similar reasoning that led to equation (A8) and recognizing that the condition ωf = 0 is statistically equivalent to the case zg < (zbed-d/2), i.e., Pf = 0) = P(Zg* < zbed* − 1/2), it follows that

display math

where P(Yf* < yf*|Ωf = 0) ≤ P(Uf* < uf*(yf*)|Zf* = zbed*), resulting in

display math

Accounting for equation (A1), the second-order derivative in(A16) can be expressed as

display math

and introducing (A15) into the conditional density function in (A19) yields the following expression

display math

which finally leads to the conditional probability density function of Tf* given by

display math

Noting that

display math

Equation (A21) reduces to the expression given by (15), that follows from ωf g3(zg*)→0 as zf* = g2(zg*)→zbed*.

Notation
Af

particle area exposed to the fluid, [L2].

FD

drag force, [MLT−2].

FUf

cumulative distribution function of uf.

FUg

cumulative distribution function of ug.

FZf

cumulative distribution function of zf.

FZg

cumulative distribution function of zg.

FTf

cumulative distribution function of τf.

FTg

cumulative distribution function of τg.

FTg,Tg

joint cumulative distribution function of τf and τg.

FTf,Zg

joint cumulative distribution function of τf and zg.

FTf,Zg

joint cumulative distribution function of τg and zg.

PE

probability of entrainment.

RE

risk of entrainment.

Szg

skewness of zg.

d

representative grain size of the sediment, [L].

fUf

probability density function of uf.

fUg

probability density function of ug.

fZf

probability density function of zf.

fZg

probability density function of zg.

fTf

probability density function of τf.

fTg

probability density function of τg.

fTg,Tg

joint probability density function of τf and τg.

fTf,Zg

joint probability density function of τf and zg.

fTg,Zg

joint probability density function of τg and zg.

fΩf

probability density function of ωf.

hu

uniform flow depth, [L].

ks

equivalent bed roughness based on Nikuradse's formula, [L].

t

time, [T].

uf

near-bed fluid velocity, [LT−1].

ug

critical velocity, [LT−1].

us

shear velocity, [LT−1].

usc

critical shear velocity, [LT−1].

zbed

tops of the bed grains elevation, [L].

zf

fluid drag elevation, [L].

zg

target particle elevation, [L].

zREF

zero-velocity level elevation, [L].

Δ

relative submerged density of sediments.

γ

specific weight of water, [ML−2 T−2].

μuf

expected value of uf, [LT−1].

μzf

expected value of zf, [L].

μzg

expected value of zg, [L].

μτf

expected value of τf, [ML−1 T−2].

μτg

expected value of τg, [ML−1 T−2].

σuf

standard deviation of uf, [LT−1].

σzf

standard deviation of zf, [L].

σzg

standard deviation of zg, [L].

στf

standard deviation of τf, [ML−1 T−2].

στg

standard deviation of τg, [ML−1 T−2].

τ0

bed shear stress, [ML−1 T−2].

τ0c

critical bed shear stress, [ML−1 T−2].

τf

fluid shear tress, [ML−1 T−2].

τg

critical shear stress, [ML−1 T−2]

ωf

particle exposure to the fluid.

Acknowledgments

[39] Modeling and data analysis were carried out within the Project “PARTS: Probabilistic Assessment of the Retention and Transport of Sediments and Associated Pollutants in Rivers,” funded by the EU Research Executive Agency via a IntraEuropean Fellowship to M. Tregnaghi under a Marie Curie action funding scheme.