The physics of sediment transport initiation, cessation, and entrainment across aeolian and fluvial environments
Thomas Pähtz, Abram H. Clark, Manousos Valyrakis, Orencio Durán
CConfidential manuscript submitted to
Reviews of Geophysics
The physics of sediment transport initiation, cessation, andentrainment across aeolian and fluvial environments
Thomas Pähtz , , Abram H. Clark , Manousos Valyrakis , and Orencio Durán Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University, 866 Yu Hang Tang Road, 310058Hangzhou, China State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, 36 North BaochuRoad, 310012 Hangzhou, China Naval Postgraduate School, Department of Physics, Monterey, CA, 93943 USA Infrastructure and Environment Research Division, School of Engineering, University of Glasgow, Glasgow, UK Department of Ocean Engineering, Texas A&M University, College Station, Texas 77843-3136, USA
Key Points: • The physics of sediment transport initiation, cessation, and entrainment across aeolianand fluvial environments is reviewed • The focus lies on the simplest physical systems: mildly sloped, nearly monodispersesediment beds without complexities such as vegetation • A large part of the review concerns consensus-changing developments in the fieldwithin the last two decades
Corresponding author: Thomas Pähtz, –1– a r X i v : . [ phy s i c s . g e o - ph ] M a r onfidential manuscript submitted to Reviews of Geophysics
Abstract
Predicting the morphodynamics of sedimentary landscapes due to fluvial and aeolian flowsrequires answering the following questions: Is the flow strong enough to initiate sedimenttransport, is the flow strong enough to sustain sediment transport once initiated, and howmuch sediment is transported by the flow in the saturated state (i.e., what is the transport ca-pacity)? In the geomorphological and related literature, the widespread consensus has beenthat the initiation, cessation, and capacity of fluvial transport, and the initiation of aeoliantransport, are controlled by fluid entrainment of bed sediment caused by flow forces over-coming local resisting forces, whereas aeolian transport cessation and capacity are controlledby impact entrainment caused by the impacts of transported particles with the bed. Here thephysics of sediment transport initiation, cessation, and capacity is reviewed with emphasison recent consensus-challenging developments in sediment transport experiments, two-phaseflow modeling, and the incorporation of granular physics’ concepts. Highlighted are the sim-ilarities between dense granular flows and sediment transport, such as a superslow granu-lar motion known as creeping (which occurs for arbitrarily weak driving flows) and system-spanning force networks that resist bed sediment entrainment; the roles of the magnitude andduration of turbulent fluctuation events in fluid entrainment; the traditionally overlooked roleof particle-bed impacts in triggering entrainment events in fluvial transport; and the commonphysical underpinning of transport thresholds across aeolian and fluvial environments. Thissheds a new light on the well-known Shields diagram, where measurements of fluid entrain-ment thresholds could actually correspond to entrainment-independent cessation thresholds.
Plane Language Summary
Loose sediment grains can be transported by blowing wind (aeolian) or water flow-ing in a riverbed (fluvial). These processes are responsible for shaping much of the naturalworld, but they involve the combination of several very complex physical systems, like tur-bulent fluid flow near a rough boundary and the mechanical behavior of granular materials.Thus, there is no consensus about the minimum wind or water speeds required to initiate andsustain sediment transport. Additionally, wind and water-driven sediment transport are ob-viously similar, suggesting that it should be possible to capture both under one description.Recent advances in experiments and computer simulations have helped scientists to answersome key questions about why sediment transport is initiated and sustained. This article re-views many of these recent discoveries, focusing on three key topics: (1) the mechanical be-havior of granular materials; (2) how turbulence in the fluid helps to move grains; and (3) therole of inertia of mobile grains. We show that a deeper understanding of these topics helpsto resolve some major inconsistencies in our understanding of why sediment transport is ini-tiated and sustained and may help to unify sediment transport by wind and water under asingle theoretical description.
When an erodible sediment bed is subjected to a shearing flow of a Newtonian fluid,such as air or water, bed particles may be entrained (i.e., set into motion) by the action offlow forces and then transported by the flow, initiating a process known as sediment trans-port . The critical conditions that are required for the initiation of sediment transport havebeen studied for more than two centuries [e.g.,
Brahms , 1757]. Dating back to the pioneeringstudies for water-driven transport by
Shields [1936] and for wind-driven transport by
Bag-nold [1936, 1937, 1938] (summarized in his book [
Bagnold , 1941]), the initiation of sedi-ment transport in both cases has been commonly described by threshold values of the time-averaged shear stress τ that the flow applies onto the bed [see reviews by Durán et al. , 2011;
Kok et al. , 2012;
Merrison , 2012;
Dey and Ali , 2018, 2019;
Yang et al. , 2019, and referencestherein]. The idea of a threshold value of τ is natural, since a necessary condition for flow-driven entrainment (or fluid entrainment ) is that flow forces and/or flow-induced torques act- –2–onfidential manuscript submitted to Reviews of Geophysics ing on bed surface particles must overcome resisting forces and/or torques. Consistently, forwall-bounded flows (to which sediment transport belongs) at a given shear Reynolds numberRe ∗ ≡ u ∗ d / ν f , the shear velocity u ∗ ≡ (cid:112) τ / ρ f controls the near-surface profile of the stream-wise flow velocity when averaged over the entire spectrum of turbulent fluctuations [see re-view by Smits et al. , 2011, and references therein], where ρ f is the fluid density, ν f the kine-matic fluid viscosity, and d a particle diameter characteristic for bed particles. As forces re-sisting entrainment of a bed particle scale with the submerged gravity force ( ∝ ( ρ s − ρ f ) g d ),where ρ s is the particle density and g the gravity constant, it has been common among ge-omorphologists to nondimenionalize τ via Θ ≡ τ /[( ρ p − ρ f ) g d ] [ Shields , 1936], whichis known as the
Shields number or Shields parameter . In the aeolian research community,the threshold parameter √ Θ [ Bagnold , 1941, p. 86] is also often used.
Shields [1936] andnumerous researchers after him have measured transport thresholds for water-driven trans-port [see reviews by
Miller et al. , 1977;
Buffington and Montgomery , 1997;
Paphitis , 2001;
Dey and Papanicolaou , 2008;
Dey and Ali , 2019;
Yang et al. , 2019, and references therein].These measurements are usually summarized in a diagram showing the threshold Shieldsnumber Θ t as a function of Re ∗ (the Shields curve Θ t ( Re ∗ ) ), which is known as the Shieldsdiagram .However, the concept of a threshold shear stress for incipient motion (i.e., for the ini-tiation of sediment transport by fluid entrainment) has had several consistency problems.First, for wind-driven transport, the most widely used incipient motion models [
Iversen andWhite , 1982;
Shao and Lu , 2000], when applied to Martian atmospheric conditions, predictthreshold shear stresses for fine sand particles that are so large that transport should occuronly during rare strong Mars storms [
Sullivan and Kok , 2017]. However, this prediction iscontradicted by modern observations indicating widespread and persistent sediment activ-ity [
Bridges et al. , 2012a,b;
Silvestro et al. , 2013;
Chojnacki et al. , 2015], even of very coarsesand [
Baker et al. , 2018].A second inconsistency, which has long been known, concerns water-driven sedimenttransport and is tacitly acknowledged whenever the concept of an incipient motion shearstress is applied: the sediment transport rate Q (i.e., the average particle momentum perunit bed area) seems to never truly vanish for nearly any Θ > Θ t have relied either onindirect extrapolation methods or on vague criteria defining the value of Q (or a proxy of Q )at which transport is critical [ Buffington and Montgomery , 1997]. Such criteria had been in-troduced even before Shields [
Gilbert , 1914;
Kramer , 1935]. In particular, the experimentsby
Paintal [1971] suggest a power law relationship between Q , appropriately nondimension-alized, and Θ for weak flows over gravel beds: Q ∗ ≡ Q /[ ρ p d (cid:112) ( ρ p / ρ f − ) g d ] ∝ Θ (it wasnecessary to measure Q over tens of hours for the weakest flows), which describes a dramaticbut not infinitely rapid decrease of Q ∗ with decreasing Θ . Qualitatively similar observationswere reported by Helland-Hansen et al. [1974]. Largely because of Paintal’s experiments,
Lavelle and Mofjeld [1987] strongly argued in favor of stochastic sediment transport mod-els that do not contain a threshold shear stress [e.g.,
Einstein , 1950] in a highly cited paperwith the title, “Do Critical Stresses for Incipient Motion and Erosion Really Exist?” Despitethe fact that many researchers have been well aware of this inconsistency, the concept of athreshold shear stress has remained alive and never been truly questioned by the majority ofscientists working on water-driven sediment transport [
Dey and Ali , 2018, 2019;
Yang et al. ,2019]. There are two main reasons for the trust in this concept. First, above a value of Q ∗ that roughly coincides with typical criteria defining critical transport ( Q ∗ ≈ . Q ∗ and Θ turns into a much milder power law [ Paintal , 1971]: Q ∗ ∝ Θ . ,suggesting a clear physical meaning of the threshold Shields number associated with thistransition ( Θ t ≈ . Q ∗ ( Θ ) with Q ∗ ( Θ ≤ Θ t ) =
0) havebeen quite successful in reproducing transport rate measurements for well-controlled con-ditions when using very similar values of Θ t . For example, the scaling Q ∗ ∝ ( Θ − Θ t ) / –3–onfidential manuscript submitted to Reviews of Geophysics by Meyer-Peter and Müller [1948] with Θ t ≈ .
05 is one of the most widely used expres-sions in hydraulic engineering for gravel transport driven by water [
Wong and Parker , 2006].However, if this value of Θ t has a real physical meaning, what is it? Does it truly describeincipient motion, which has always been the predominant interpretation [see reviews by Miller et al. , 1977;
Buffington and Montgomery , 1997;
Paphitis , 2001;
Dey and Papanico-laou , 2008;
Dey and Ali , 2018, 2019;
Yang et al. , 2019, and references therein], despite thefact that Q ∗ ( Θ ≤ Θ t ) > Q ∗ > Θ ≈ . (cid:28) Θ t )?A third inconsistency in the concept of an incipient motion shear stress, which alsoconcerns water-driven sediment transport, is also old but much less well known, perhaps be-cause one of the key papers [ Graf and Pazis , 1977] is published in French language. Graf’sand Pazis’ measurements show that increasing the shear stress on the bed due to the wa-ter flow from zero up to a certain value τ (a transport initiation protocol) results in smallertransport rates Q than decreasing the shear stress from a larger value down to τ (a transportcessation protocol). This clearly indicates an important role of particle inertia in sustainingwater-driven sediment transport. Hence, any measurement of Θ t is affected by particle in-ertia because, regardless of whether an initiation or cessation protocol is used, particles arealready transported when Θ approaches Θ t (see the second inconsistency discussed above).Hence, Θ t is not, or at least not only, associated with fluid entrainment and thus incipientmotion. The importance of particle inertia was proposed and indirectly shown even earlier, ina largely ignored study (only eight citations indexed by Web of Science today, half a centuryafter publication) by Ward [1969]. In this study,
Ward [1969] measured smaller values of Θ t for a larger particle-fluid-density ratio s ≡ ρ p / ρ f (which is a measure for particle inertia) atthe same shear Reynolds number Re ∗ . A slight downward trend of Θ t with s even existed inthe pioneering experiments by Shields [1936]. Interestingly, a particle inertia effect in water-driven sediment transport has actually been studied. It is well known, although often not con-sidered to be crucial in the context of transport thresholds, that the flow strength at whicha transported particle can come to rest at the bed surface is weaker than the one at which itcan reenter transport [e.g.,
Francis , 1973;
Reid et al. , 1985;
Drake et al. , 1988;
Ancey et al. ,2002]. In contrast, another potentially important effect of particle inertia in water-driven sed-iment transport has not received the same attention: the interaction between particles that arealready in transport and particles of the bed surface (e.g., particle-bed impacts) may supportbed particle entrainment or even be predominantly responsible for it ( impact entrainment ).Particle inertia and particularly impact entrainment have been widely recognized ascrucial for sustaining wind-driven sediment transport since the pioneering studies by
Bag-nold [1941]. Yet, in contrast to water-driven transport, there seems to be a clear-cut shearstress threshold when applying an initiation protocol in wind tunnel experiments [e.g.,
Bag-nold , 1941]. This rather curious difference between wind-driven and water-driven transportis usually not discussed in the context of incipient motion. Why is it necessary to definecritical transport rates for measuring an incipient motion shear stress threshold in water-driven transport but not in wind-driven transport? A complete description of incipient mo-tion should be generally applicable and not limited to a subset of possible sediment trans-port conditions, since there is no reason to believe that the physical mechanisms involved inthe entrainment of a bed particle by a turbulent flow depend much on the nature of the flow.In fact, frameworks unifying sediment transport across driving fluids (not only in regard totransport thresholds) are scarce in general (e.g., apart from modern studies, only
Bagnold [1956, 1973] seems to have attempted unifying water-driven and wind-driven transport con-ditions).One of the most desired aspects of a general framework of sediment transport wouldbe its ability to reliably predict the general dependency of Q ∗ on Θ and other dimensionlessenvironmental parameters, such as the density ratio s . However, there is an obvious problem:since measured transport rates may depend on the experimental protocol for a given condi-tion, as was the case in the experiments by Graf and Pazis [1977] (see third inconsistency),does the concept of a general relationship even make sense? The consensus is, yes, it does –4–onfidential manuscript submitted to
Reviews of Geophysics make sense when referring to transport capacity (also known as transport saturation in ae-olian geomorphology), which loosely defines the maximal amount of sediment a given flowcan carry without causing net sediment deposition at the bed. However, a precise definitionof transport capacity is very tricky and controversial [see review by
Wainwright et al. , 2015,and references therein]. For example, the fact that equilibrium transport rates may depend onthe experimental protocol for a given condition implies that not every equilibrium transportcondition is equivalent to transport capacity and that transport capacity is in some way linkedto particle inertia. In fact, that the latter may be the case was recognized by no and García [1998], who numerically modeled water-driven sediment transport as a continuous motionof particles hopping along a flat wall. In particular, these authors mentioned that the capac-ity relation obtained from their numerical simulations contains a threshold Shields numberthat may not be associated with fluid entrainment, demonstrating the necessity for a goodunderstanding of transport capacity and its relationship to particle inertia in the context ofsediment transport thresholds.While this introduction has focused on introducing issues in our understanding of fluidentrainment, shear stress thresholds, particle inertia, transport capacity, and their mutual re-lationships from a historical perspective, there have been major developments in these topicsin the last two decades, largely because of the emergence of novel experimental designs andmodeling techniques. The purpose of this review is to draw the attention of the involved re-search communities to these developments that, if put together, resolve the above issues andprovide a largely improved conceptual understanding of sediment entrainment and transportthresholds.A large portion of recent developments in the field can be attributed to numerical stud-ies modeling the particle phase using the discrete element method (DEM). In comparison toother methods modeling the particle phase (e.g., continuum models), this method has the bigadvantage that it approximates the laws of physics at a very basic level, namely, at the levelof intergrain contacts. In fact, the force laws commonly used to model intergrain contactsare known to produce system results that match experiments extremely well [e.g.,
Stewartet al. , 2001;
Lätzel et al. , 2003;
Clark et al. , 2016]. Additionally, granular continuum modelsare formulated using DEM simulations [ da Cruz et al. , 2005] but reproduce complex exper-iments on granular flows often very accurately [
Jop et al. , 2006]. In the context of sedimenttransport, the main uncertainty of DEM-based models lies therefore in the modeling of thecoupling between the particle phase and the Newtonian fluid driving transport. However,many of the simulations that are described in this review show that the results are often in-sensitive to the details of how this coupling is treated. The authors of this review thus arguethat new physics uncovered by DEM-based numerical simulations are on a relatively solidfooting.To limit the scope of this review, it focuses on studies of mildly sloped beds of rela-tively uniform sediments unless mentioned otherwise. Also, because of the focus on physicalprocesses involving the bed surface, this review largely concerns nonsuspended sedimenttransport (i.e., the fluid turbulence is unable to support the submerged particle weight), inwhich transported particles remain in regular contact with the bed surface (typical for parti-cles of sand size and larger) and which is the relevant transport mode for the morphodynam-ics of planetary landscapes, riverscapes, and seascapes. In contrast, in suspended transport(typical for particles of silt or dust size and smaller), transported particles can remain out ofcontact with the bed surface for very long times (e.g., as atmospheric dust aerosols). In typ-ical nonsuspended wind-driven ( aeolian ) sediment transport, many particles move in largeballistic hops and the transport layer thickness h is therefore much larger than the particle di-ameter d . In the aeolian geomorphology community, such hopping particles are said to movein saltation and explicitly distinguished from particles rolling and sliding along the surface.However, this terminology is not used in this review. Instead, the term saltation transport isused for general transport regimes with h (cid:29) d , that is, it refers to all rather than a subset oftransported particles. In typical nonsuspended liquid-driven transport (henceforth referred to –5–onfidential manuscript submitted to Reviews of Geophysics as fluvial transport for simplicity although this mode is not limited to fluvial environments), h is of the order of d because the largest particle hops are small. Following the fluvial geo-morphology community, transport regimes with h ∼ d are termed bedload transport .This manuscript is organized into sections that focus on specific topics (sections 2–4)followed by a summary and outlook section (section 5) and a Notation section describingthe definitions of technical terms and mathematical symbols. It is noted that readers mayfind it useful to read section 5 first in order to organize the contents of the manuscript, andthen consult sections 2–4 for more detailed information on a particular topic. Section 2 re-views recent insights into the mechanics of beginning sediment motion and fluid entrainmentgained from studying sediment transport as a dense granular flow phenomenon. For example,it has become increasingly clear that granular material can flow even when a macroscopicmotion does not occur, such as for a collapsed pile of sand, because of a process known as creeping , which describes an irreversible superslow granular motion associated with spo-radic microscopic rearrangements. That is, it is crucial to clearly define what kind of motionone refers to when introducing sediment transport thresholds. Likewise, forces resisting theentrainment of a bed particle do not only depend on the local arrangement of bed particlesbut also on granular interactions with regions within the bed that are far away from the en-trainment location (i.e., sediment entrainment is a nonlocal phenomenon). This is because ofcollective granular structures that particles can form. Section 3 reviews insights gained fromrecent experimental and theoretical studies showing that the fluid shear stress applied ontothe bed surface alone only poorly characterizes the critical conditions required for fluid en-trainment by turbulent flows. These studies have provided more suitable criteria for sedimententrainment that take into account turbulent fluctuation events and, in particular, their dura-tions. However, section 3 also explains that a critical fluid shear stress for incipient motiondoes make sense when referring to the shear stress at which the fluid entrainment probabilityexceeds zero (which, for turbulent fluvial bedload transport, occurs much below the Shieldscurve [ Paintal , 1971]). For example, in wind tunnel studies (but not necessarily in the field),aeolian saltation transport is initiated at about this threshold. Finally, section 4 reviews stud-ies on the role of particle inertia in sediment transport, a topic that has very recently under-gone a dramatic change. In fact, while it is well established that impact entrainment is crucialfor aeolian saltation transport [see reviews by
Durán et al. , 2011;
Kok et al. , 2012;
Valanceet al. , 2015, and references therein], very recent experimental and theoretical studies revealedthat it is also crucial for sustaining fluvial bedload transport. Likewise, a very old argumentby
Bagnold [1941], which was forgotten or deemed unimportant, has recently been revived.
Bagnold [1941] pointed out that, for aeolian saltation transport, a predominant role of impactentrainment requires that the flow is able to sustain the motion of transported particles. Thisis only possible if the energy loss of transported particles rebounding with the bed is com-pensated by their energy gain during their trajectories via fluid drag acceleration. Modelsthat explicitly incorporate this requirement have been able to partially unify aeolian saltationand viscous and turbulent fluvial bedload transport. When combined, the insights from thestudies reviewed in sections 2–4 provide a conceptual picture free of inconsistencies, whichis described in section 5. For example, the shear stress threshold compiled in the Shieldsdiagram seems to characterize the cessation of sediment bulk motion and an appropriatelydefined transport capacity rather than incipient motion. Section 5 also summarizes importantopen problems and provides a brief outlook into related problems that have not been dis-cussed in this review, such as the effects of particle size heterogeneity on transport thresholdsand bed sediment entrainment.
In theoretical considerations of a problem as complex as the mechanics of beginningsediment motion, simplifying assumptions must be made. This often means that the gran-ular phase is treated extremely coarsely, as a continuum with a Coulomb-like friction coef-ficient [
Terzaghi , 1951;
Drucker and Prager , 1952], or very finely, where the pocket geom- –6–onfidential manuscript submitted to
Reviews of Geophysics etry of individual grains sets the bed strength [
Wiberg and Smith , 1987]. However, recentadvances in granular mechanics have shown that Coulomb-like behavior of granular materi-als is inherently nonlocal , so it must be treated on intermediate length scales. This is due tothe fact that the yielding condition, defined as the minimum shear stress required to achievepermanent granular flow, is set by emergent, collective networks of grains. These networkscan couple different sections of the material together over large distances. The purpose ofthis section is to provide an overview of recent work on yield and flow of dense granularmaterials in the context of sediment transport, with a particular focus on the nonlocal na-ture of granular yielding. To simplify the discussion, it is assumed throughout this sectionthat the granular bed is subjected to a constant bed shear stress (like for laminar flows), inwhich case the existence of a fluid entrainment threshold associated with bed failure doesmake sense. However, this is no longer true for turbulent flows, as reviewed in section 3. Formore information on dense granular flow, readers might consult recent reviews [
Forterre andPouliquen , 2008;
Jop , 2015;
Kamrin , 2018] devoted exclusively to the topic of dense gran-ular flow. For the connection between granular flow and sediment transport, the perspectiveand review by
Frey and Church [2009, 2011] are also recommended.
Surface grains sit in pockets on top of the bed, and the geometry of the pocket deter-mines the entrainment conditions for that particular grain via its protrusion (i.e., the grainheight above surrounding grains) and friction angle. When the downstream drag force fromthe fluid overcomes resistive forces from gravity and from contact forces with the pocket,the grain will begin to move. This conceptually simple scenario appears in many theoreti-cal studies [e.g.,
Wiberg and Smith , 1987;
Ling , 1995;
Dey , 1999;
Dey and Papanicolaou ,2008;
Ali and Dey , 2016]. However, this picture has several conceptual problems. For ex-ample, there are many different pocket geometries [
Kirchner et al. , 1990;
Buffington et al. ,1992] implying a distribution of entrainment thresholds.
Kirchner et al. [1990] made a sim-ilar argument, advocating for a statistical treatment of pocket geometries, where only thegrains with the smallest entrainment thresholds would be relevant. Additionally, when trans-port thresholds are discussed, one typically does not include transient behavior, after theflow has pushed grains from less stable to more stable pockets. For example, an entrainedgrain that then restabilizes in a nearby pocket would not constitute sediment transport. Aftersuch a rearrangement, the resulting bed would have a different intergrain force and contactstructure, which would be more suited to resisting the applied flow forces [
Masteller andFinnegan , 2017]. Thus, determining the fluid entrainment threshold amounts to determiningthe strongest bed that can be formed by the grains, subject to the flow forces and dynamics.This process necessarily involves transient behavior, as grains search for stable configura-tions, and spatial correlations, since information about each grain’s movement is transmittedthrough the intergrain force network.While this represents a very challenging problem, it is exactly the picture that hasemerged in recent years regarding the physical origin of frictional behavior in noncohe-sive soils or sediments. The yield criterion of granular materials is defined by the maxi-mum internal shear stress that a granular material can achieve, but grains must rearrange tofind this maximum stress, sometimes for a long time [
Clark et al. , 2018;
Srivastava et al. ,2019]. The yield criterion has the form of a friction coefficient, where flow occurs onlywhen µ ≡ τ p / P > µ s , where τ p and P are the granular shear stress and pressure P , re-spectively, that arise from intergrain contacts, and µ s is the static friction coefficient of thematerial. At first glance, this is not surprising, since the grains themselves have a surfacefriction coefficient µ g . However, µ s is only weakly dependent on µ g [ da Cruz et al. , 2005],as shown in Figure 1. Even frictionless spheres have µ s ≈ . Peyneau and Roux , 2008a,b],which arises from a preferred orientation for intergrain contacts that aligns with the com-pressive direction of the applied shear deformation. This effect is independent of whetherthe grains interact via linear spring forces [
Thompson and Clark , 2019] or more realisticHertzian interactions [
Peyneau and Roux , 2008a]. Similar behavior is observed for grains –7–onfidential manuscript submitted to
Reviews of Geophysics (a) (b) (a) (b)
Figure 1. (a) From
Peyneau and Roux [2008a], the normalized shear stress σ / P is plotted as a functionof strain γ . The shear stress builds up from zero, reaching its maximum value at γ ≈ .
1. Copyright 2008American Physical Society. (b) Data adapted from da Cruz et al. [2005] and
Kamrin and Koval [2014] show-ing a measurement of the bulk static friction coefficient µ s as a function of µ g , which is the static frictioncoefficient between the surfaces of two grains (simulated as two-dimensional disks). with surface friction and irregular shape [ Radjaï et al. , 1998;
Azéma and Radjaï , 2010, 2014;
Trulsson , 2018], but the maximum stress anisotropy is enhanced by these effects, since grain-grain contacts can have both normal and tangential components. This raises the yield stressslightly: frictional disks have µ s ≈ . − . da Cruz et al. , 2005] and frictional spheres have µ s ≈ . − . Jop et al. , 2006], with only a weak dependence on µ g for µ g > .
1. Addition-ally, µ s is nearly independent of polydispersity [ Voivret et al. , 2009]. This picture assumesgrains are slowly moving with persistent intergrain contacts, but µ s can be lowered signifi-cantly for more energetic kinds of driving, like vibration [ Gaudel and De Richter , 2019] orin aeolian saltation transport [
Pähtz et al. , 2019], probably because the tendency of the con-tact orientation to align with the compressive direction is somewhat suppressed [
Pähtz et al. ,2019]. Thus, frictional behavior in granular media arises primarily from the anisotropicstructure of force and contact networks, and grain-grain friction, shape, and polydispersityplay secondary roles.Here, µ is used to denote the local nondimensional shear stress in the granular materialitself, while the Shields number Θ is the dimensionless shear stress applied to the granularbed surface, so the two quantities are not equivalent but are closely related. At the surfaceof the bed, µ ≈ Θ if lift forces are neglected. The existence of a maximum shear stress thatcan be supported by a granular material (which is independent of grain size) suggests that,for noncohesive sediments, there should be a theoretical upper limit to the threshold Shieldsnumber Θ t , Θ max t ≈ µ s . This implies that the Shields curve must plateau at low values ofthe shear Reynolds number Re ∗ for laminar flows. This fact has been a subject of debate formany years, with some authors [ Shields , 1936;
Mantz , 1977;
Miller et al. , 1977;
Yalin andKarahan , 1979;
Govers , 1987;
Buffington and Montgomery , 1997;
Dey , 1999;
Hong et al. ,2015] showing a trend where Θ t continues to grow as Re ∗ gets smaller, while other stud-ies [ Wiberg and Smith , 1987;
Paphitis , 2001;
Pilotti and Menduni , 2001;
Ouriemi et al. ,2007] show a plateau at low Re ∗ . Recent work by the present authors [ Clark et al. , 2015a,2017;
Pähtz and Durán , 2018a] has investigated sediment transport thresholds over a widerange of Re ∗ and density ratio s using simulations based on the DEM to model noncohesivegrains that are coupled to fluid-driven shear forces. These studies all suggest that Θ t is a con-stant at low Re ∗ and s , corresponding to the strongest possible state of the bed. It is notedthat cohesive effects become important for very small grains, which can cause Θ t to continueto grow for smaller Re ∗ . –8–onfidential manuscript submitted to Reviews of Geophysics
Open Problem: Value of Viscous Yield Stress Θ max t Measured values of the viscous yield stress Θ max t vary substantially. For nearly monodis-perse beds of spherical particles, most studies reported Θ max t ≈ .
12 [
Charru et al. , 2004;
Loiseleux et al. , 2005;
Ouriemi et al. , 2007;
Seizilles et al. , 2014;
Houssais et al. , 2015], butlarger values of up to about 0 .
37 have also been reported [
Lobkovsky et al. , 2008;
Hong et al. ,2015]. Also, some measurements suggest that Θ max t depends on the median grain size [ Honget al. , 2015], in contradiction to the grain size independence of µ s , while other studies findno such dependence [ Ouriemi et al. , 2007]. To the authors’ knowledge, there is currentlyno convincing explanation for these contradicting observations. However, the scatter in thereported values for Θ max t (between 0.12 and 0.37) is within the range reported for the yieldstress of granular materials, ranging from low-friction spheres to rougher, more frictionalparticles. Thus, the yield stress of the bulk granular material may at least play some role insetting the scatter in Θ max t . In this context, it is worth noting that, for the entrainment of par-ticles resting on an idealized substrate by a laminar flow, threshold Shields numbers rangefrom zero to very large values depending on the packing arrangement [ Agudo et al. , 2017;
Deskos and Diplas , 2018;
Topic et al. , 2019;
Shih and Diplas , 2019].
The existence of a yield stress is one piece of a rheological description, which is a con-stitutive law that mathematically connects the strain rate to the local stress at each point ina material. For granular materials, dissipation implies that more force is required for fasterstrain rates, so µ will increase with strain rate (cid:219) γ . For the case of sediment transport, for-mulation of a constitutive law has obvious practical benefits, namely that it would allow ananalytical prediction of transport rates Q at varying Shields number Θ for transport condi-tions dominated by granular interactions. However, note that a bulk constitutive law maynot be able to capture certain cases, particularly very near to the onset or cessation of flu-vial bedload or aeolian saltation transport, where the transport layer is dominated by theisolated motion of a single grain along the bed (which is the typical situation in gravel-bedrivers [ Parker , 1978;
Phillips and Jerolmack , 2016]). Despite the fact that the force and con-tact networks discussed above are spatially extended, some progress has been made by con-sidering so-called local rheologies . Based on dimensional analysis, da Cruz et al. [2005]showed that µ for dry, uniform granular flows must depend on (cid:219) γ via a single dimensionlessnumber, I ≡ (cid:219) γ d / (cid:112) P / ρ p , where I is called the inertial number, similar to the Savage [ Sav-age , 1984] or Coulomb [
Ancey et al. , 1999] numbers. A functional form for µ ( I ) can thenbe measured from experiments or DEM simulations (a crude approximation is given by µ = µ s + c I I , where c I is a constant parameter). If one then assumes that a three-dimensional,tensorial generalization of this law is locally satisfied at each point in space in arbitrary ge-ometries, then the equations of motion are closed and one can predict (at least numerically)flow in any arbitrary geometry where the forces and boundary conditions are known. Exper-imental measurements of rapid, dense flow in several geometries show good agreement withthe local rheology [ MiDi , 2004;
Jop et al. , 2005, 2006].
Open Problem: Rheology of Nonsuspended Sediment Transport
There are many physical mechanisms that are relevant to nonsuspended sediment trans-port that are not included in the inertial number description, but recent work has suggestedthat appropriate dimensional analysis can be used to find a general rheological descriptionthat is relevant in all contexts. For example, viscous effects from the fluid can be included [
Boyeret al. , 2011;
Trulsson et al. , 2012;
Ness and Sun , 2015, 2016;
Houssais et al. , 2016;
Amarsidet al. , 2017;
Houssais and Jerolmack , 2017;
Guazzelli and Pouliquen , 2018] by replacing theinertial number I with the viscous number J ≡ ρ f ν f (cid:219) γ / P . This description is valid whenthe Stokes-like number I / J is small, and the standard µ ( I ) rheology again takes over forlarge I / J . This crossover can be heuristically written in terms of a viscoinertial number –9–onfidential manuscript submitted to Reviews of Geophysics K ≡ J + c K I , where c K is an order-unity fit parameter [ Trulsson et al. , 2012;
Ness and Sun ,2015, 2016;
Amarsid et al. , 2017], and the rheology takes the form µ ( K ) .The previous paragraph describes a unification of dry and wet, viscous granular flows,but some situations, like turbulent bedload or aeolian saltation transport, do not fit neatly intothis description. Maurin et al. [2016] showed that, for intense turbulent bedload transport,the inertial number I (used for dry flows) collapses the data best, but with a different µ ( I ) re-lation compared to dry flows. Additionally, the presence of more severe velocity fluctuationsand grain-grain collisions can weaken the material, giving a µ that is smaller than would bepredicted by a µ ( I ) or µ ( K ) rheology at a given shear rate [ Pähtz and Durán , 2018b]. An-other option is to build a rheological description that explicitly accounts for these fluctuationsand collisions via the Péclet number Pe ≡ (cid:219) γ d /√ T [ Pähtz et al. , 2019], where the granulartemperature T equals the mean square of kinetic particle velocity fluctuations. The advantageof Pe is that it is applicable to a wide range of different granular flows (e.g., it unifies intensefluvial bedload and aeolian saltation transport), whereas K is limited to relatively homoge-neous flows. The disadvantage is that Pe involves another granular property ( T ) that requiresmodeling. As discussed in section 1, some water flume experiments suggest that fluvial bedloadtransport never truly ceases for nearly any Θ >
0, which is usually attributed to turbulentfluctuations. However, as discussed in this section, the granular material itself may be par-tially responsible. In fact, it is well known that granular creep can be observed in a varietyof observational geophysical contexts [
Boulton and Hindmarsh , 1987;
Pierson et al. , 1987;
Ferdowsi et al. , 2018] as well as more idealized granular flows in a laboratory setting [
Roer-ing et al. , 2001;
Komatsu et al. , 2001;
Nichol et al. , 2010;
Moosavi et al. , 2013;
Amon et al. ,2013], including sediment transport explicitly [
Houssais et al. , 2015;
Allen and Kudrolli ,2018], as depicted in Figure 2. Generally, creeping refers to slow, typically intermittent flow(not limited to the bed surface) that occurs below a macroscopic yield criterion.One class of creeping flow involves systems where regions with µ > µ s and µ < µ s ex-ist nearby each other, which often occurs in systems with stress gradients (e.g., due to gravityor curvature). In this case, creeping flow is observed in regions with µ < µ s [ Fenistein andvan Hecke , 2003;
MiDi , 2004;
Crassous et al. , 2008;
Koval et al. , 2009]. This creeping flowis not steady or continuous, but occurs in a series of intermittent, avalanche-like slips, whichare triggered by the nearby steadily flowing region with µ > µ s . The time-averaged shearrate profiles decay quasi-exponentially with spatial distance to the steadily flowing region.Various nonlocal theories have been proposed [ Baran et al. , 2006;
Pouliquen and Forterre ,2009] that include a spatial length scale ξ over which flow can be triggered in this way. Themost successful theories [ Kamrin and Koval , 2012;
Henann and Kamrin , 2013;
Kamrin andHenann , 2015;
Bouzid et al. , 2013, 2015] suggest that the cooperative length scale ξ divergesat the yield stress (i.e., ξ ∝ | µ − µ s | − ν , where ν ≈ . Zhangand Kamrin , 2017] as well as how exactly to best mathematically formulate a nonlocal rheol-ogy [
Bouzid et al. , 2017;
Li and Henann , 2019] is still a subject of debate in the literature.The creeping flow captured by these nonlocal models is also apparent in laboratoryflumes used to model fluvial sediment transport.
Houssais et al. [2015, 2016] showed thatsediment transport involves the coexistence of three regimes: a dilute suspension above thebed surface, the bedload layer at the bed surface, and creeping behavior below the surface.These regions are depicted in Figure 2a. The shear rate profile in the creeping regime fol-lows an exponential decay, which is consistent with the predictions of nonlocal models. Sim-ilar behavior was also observed by
Allen and Kudrolli [2017], shown in Figure 2b, who alsostressed that the apparent agreement with nonlocal models formulated for dry granular ma- –10–onfidential manuscript submitted to
Reviews of Geophysics
I-Dilute II-Bed loadIII-Creeping J a mm ed E l e v a t i on z I = I c "J" Shear stress (cid:3) */ (cid:3) * c d C = C sat (a) (b) Figure 2. (a) From
Houssais et al. [2015] (Creative Commons Attribution 4.0 International License),a proposed phase diagram for granular flow behavior as a function of elevation z in the bed (vertical axis)and applied shear stress τ (cid:63) from the overlying fluid flow (horizontal axis). Bedload transport triggers slowcreeping flow below it, consistent with nonlocal rheological models that have recently been formulated fordry granular media, as described in the text. (b) From Allen and Kudrolli [2017], normalized velocity profiles u / u t for the fluid (blue squares) and grains (red circles) are plotted as a function of height z / d . Also plottedis the packing fraction of the grains φ g as a function of height. The top of the bed corresponds to the drop in φ g . Above the bed, grains move with the fluid. Below the bed, the grain velocity profile decays exponentially(a straight line on the semilogarithmic plot), which is a prediction of the nonlocal granular flow rheologiesdiscussed in the text. Copyright 2017 American Physical Society. terials implies that the fluid stress is not playing a major role in the observed creeping be-havior. In the creeping regime, µ < µ s , but flow events are triggered via the bedload trans-port regime at the top of the bed via spatial correlations in the force network. These creepingevents, although slow and intermittent, can lead to segregation effects over long times ( ∼ Ferdowsi et al. , 2017]. Thus, creep andnonlocal rheology may play a crucial role in armoring of gravel-bedded rivers, as opposedto size sorting in the transported layer. Additionally, recent computational work [
Pähtz andDurán , 2018b] has shown that sediment transport rheology is nonlocal even relatively farfrom the sediment transport threshold.There is a second class of creeping flow, which is currently not explained by any rhe-ological model. In the above discussion, creeping granular flow at µ < µ s was always in-duced by nearby regions with µ > µ s . In some cases, creeping flow can be observed at µ < µ s without any apparent granular flow nearby at µ > µ s [ Amon et al. , 2013]. Thisclass of creep is often accompanied by compaction of the bed. Slow shear and compactioninteract in a complex way that is not fully understood but can be crucial in regulating slow(e.g., millimeters to meters per day) geophysical flows [
Moore and Iverson , 2002]. Similarbehavior was also observed in laboratory sediment transport experiments by
Houssais et al. [2015] and further studied by
Allen and Kudrolli [2018], as shown in Figure 3. The latterauthors observed a granular bed with an overlying laminar shear flow and showed that slow(less than 0.1 grain diameters in 90 min) creeping flow persisted even for Θ (cid:28) Θ t (meaningthat µ < µ s everywhere in the granular bed). The grain motion in the direction of fluid flowfollowed an exponential decay with depth, similar to the creep described by nonlocal mod-els. However, it was not induced by granular flow but somehow by the laminar fluid flow.Streamwise creep was also accompanied by compaction of the bed, which can strengthenthe material and thus reduce creep. This second class of creep is therefore similar to com- –11–onfidential manuscript submitted to Reviews of Geophysics
GRANULAR BED CONSOLIDATION, CREEP, AND … (c) (d) ∆ x/d z / d z/d */ *c=0. 0*/ *c=0.25*/ *c=0. 5*/ *c=0. 8 γ x 10-3 z * c*/ Δ s d d (a) (b) z x FIG. 4. Movement of particles in the bed in the first 90 min of preshear at (a) no shear stress τ ∗ /τ ∗ c = . τ ∗ /τ ∗ c = .
8, where the color goes from dark to light with increasing particle movement. We see thatparticles move even at no shear, but there is greater movement at higher shear. Looking more closely in a shortsegment where we can track all the particles t = x and (d) gravity z directions. We see an exponential behavior inthe flow movement while the bed shifts down linearly with depth compacting uniformly. The inset of (d) is themeasure of the strain γ z from fitting the slopes of (d). IV. EVOLUTION OF BED STRUCTUREA. Rearrangements with depth
Figures 4(a) and 4(b) show examples of grain positions in a vertical slice in the bed recordedthrough 90 min, corresponding to shear stresses of τ ∗ /τ ∗ c = . y = ± . d are tracked and analyzed. The magnitudeof displacement of the individual grains s in the plane over this time interval is denoted using thecolormap to capture the bed evolution. One observes that both examples rearrange, including the onewith no applied shear, with greater motion occurring for τ ∗ /τ ∗ c = . t = z in Figs. 4(c) and 4(d), respectively. One observes that the bed creeps forward faster andsettles further with increasing shear stress. Moreover, the creep along the flow direction appearsto decay exponentially with depth as shown by the fits in Fig. 4(c). The decay length from theexponential fit in the case of the higher shear rates, where a meaningful variation occurs, is foundto be 2 . d ± . d . This decay is similar to the length scale over which grain speeds exponentiallydecay into the bed for τ ∗ > τ ∗ c [10] and was observed to be common to dry granular beds in gravitywhich are sheared horizontally at the top [27,28].At the same time, the linear compaction with depth at all shear rates implies that the bed settlesuniformly as grains rearrange in gravity. Such a linear increase would imply that the volume fractionof the bed increases uniformly into the bed, an issue we will examine more closely later in thediscussion. The strain gradient γ z = − (cid:2) z/z obtained from the linear fit is shown in the inset of Figure 3.
From
Allen and Kudrolli [2018], particle movement during 90 minutes with (a) no fluid flow and(b) fluid flow at 80% of the critical flow rate (i.e., τ ∗ / τ ∗ c = .
8) to initiate particle transport (brighter colorsindicate more particle movement). Movement is also plotted during times t = x )direction and (d) gravity ( z ) direction. There is exponential behavior in the x direction and a linear shift in the z direction. The strain γ z is shown in the inset to (d), by fitting the slopes of the data in (d). Copyright 2018American Physical Society. paction [ Knight et al. , 1995;
Ribière et al. , 2005] and creep [
Divoux et al. , 2008;
Candelierand Dauchot , 2009] that is induced by tapping or vibrations, despite the fact that no explicitvibrations were applied. The existence of this class of creep implies that sediment is likely always transported (albeit slowly) for arbitrarily small values of Θ , even in the absence of tur-bulence. Another recent experimental flume study [ Masteller and Finnegan , 2017] showeda similar result, where conditioning a bed by applying weak fluid flow led to zero net trans-port but a smoother bed profile with fewer protruding grains. Then, when the fluid flow ratewas increased to a value associated with significant transport for a conditioned bed, sedimenttransport rates were smaller when compared with an unconditioned bed.
Open Problem: Physical Origin of Creeping Below Macroscopic Yield
The physical mechanisms that lead to the second class of creep, where µ < µ s ev-erywhere in the system, are not known. One possible mechanism is contact aging [ Jia et al. ,2011], where the microscopic contact structure between two solid objects (i.e., grains) canevolve and weaken with time for reasons that are not fully understood [
Liu and Szlufarska ,2012]. Additionally,
Pons et al. [2016] showed that this second class of creep could be in-duced in dry granular flow by applying small pressure fluctuations to the interstitial air, withresulting shear rates of the order of 10 − . Similar fluctuations likely always exist in natu-ral systems. These two hypotheses are supported by the fact that, to the authors’ knowl-edge, this class of creep does not occur in DEM simulations, which use a Cundall-Strackmodel [ Cundall and Strack , 1979] or similar Coulomb-like yield criterion for the frictionalforces between grains, and fluctuating forces or slow variations in grain-grain friction are notincluded. Some DEM studies have observed creeping below a macroscopic yield criterionlike the angle of response [
Ferdowsi et al. , 2018], but the results from these studies seem toalways include some region of µ > µ s . –12–onfidential manuscript submitted to Reviews of Geophysics
Many experimental and computational studies [
Carneiro et al. , 2011;
Heyman et al. ,2013;
Houssais et al. , 2015] have observed that, near sediment transport thresholds (includ-ing the impact entrainment threshold, reviewed in section 4.1.3), the time t conv required forsome system measurement (e.g., the sediment transport rate Q ) to converge to its steady statevalue appears to grow very large. A common form [ Clark et al. , 2015a] to capture these longtime scales is t conv ∝ | Θ − Θ t | − β , where β is some positive exponent. A diverging timescale can arise in many ways, but one possibility is a critical phase transition . The study ofphase transitions, where a material abruptly changes as a control parameter is smoothly var-ied, originated in thermal physics (e.g., liquid-gas or ferromagnetic transition), but it has alsobeen successful in describing many other kinds of systems where thermal physics is not ap-plicable. The key feature of a critical phase transition is a diverging correlation length, suchthat small changes near the critical point can have system-spanning effects that last for arbi-trarily long times. The system is thus said to be scale-free at the critical point, since there isno largest length or time scale that is affected by a perturbation. Open Question: Is Flow-Induced Bed Failure a Critical Phenomenon?
Bed failure at the yield stress describes by definition a phase transition, but whetherthis transition is critical and how it arises from grain-grain and grain-fluid interactions re-main open questions. However, there is a growing body of work [
Clark et al. , 2018;
Srivas-tava et al. , 2019;
Thompson and Clark , 2019] suggesting that the yielding transition for gran-ular media is a critical transition. This is also suggested by the diverging correlation length ξ ∝ | µ − µ s | − ν that is present in the nonlocal models discussed above [ Kamrin and Koval ,2012;
Bouzid et al. , 2013]. In addition to describing creeping flow for µ < µ s , nonlocal the-ories are also able to correctly predict other size-dependent effects, like strengthening of thinlayers [ MiDi , 2004;
Kamrin and Henann , 2015]. The idea that yielding of granular media isa critical transition helps to explain certain experimentally observed behaviors in laboratoryand computational models of sediment transport. For example, using a laboratory flume nearthe viscous limit,
Houssais et al. [2015, 2016] found a diverging time scale near the criticalShields number that is “associated with the slowing down, and increasing variability, of theparticle dynamics; it is unrelated to hydrodynamics.” Evidence of scale-free channeling pat-terns [
Aussillous et al. , 2016] was also observed during erosion of granular beds, which wasattributed to the fact that the onset of erosion was behaving like a critical phase transition.When the physics controlling the onset of grain motion is no longer just the yield strengthof the granular material itself, then the picture changes somewhat. For example, once parti-cle inertia becomes important in sustaining nonsuspended sediment transport (see section 4),the granular phase may not have a frictional state µ that is close to µ s , and thus it may be farfrom the critical point. For viscous bedload transport (small Re ∗ ), when particle inertia is notimportant, computational studies typically show that t conv obeys system size dependence thatis consistent with a critical phase transition [ Yan et al. , 2016;
Clark et al. , 2018]. However,under steady driving conditions, when grain inertia starts to play a role (e.g., for larger Re ∗ ),then t conv still diverges, t conv ∝ | Θ − Θ t | − β , but systems of different sizes will have the same t conv [ Clark et al. , 2015a, 2017]. Thus, Θ t for inertial particles appears to be more similar toa dynamical instability rather than a true critical point.However, nonlocal effects still likely play a role in the initiation of permanent bed fail-ure. For example, if particle inertia plays a crucial role in sustaining sediment transport, asargued below in section 4, then a bed could be above the threshold needed to sustain motionbut not have any way to get started. Returning to the argument from Kirchner et al. [1990]discussed above, if only the grains with the lowest entrainment thresholds are susceptible tobeing moved by the fluid, then these grains might be thought of as weak links in the bed. Mo-tion that is initiated by these weak links could trigger flow elsewhere in the system, via the –13–onfidential manuscript submitted to
Reviews of Geophysics redistribution of forces or by collision.
Clark et al. [2015a, 2017] showed that the initiationof motion did indeed obey statistics consistent with a Weibullian weakest link scenario.
This section has described recent advances in the physics of sheared granular flows,with a focus on application to sediment transport. The main ideas are as follows. First, theyield condition for granular materials (e.g., a sediment bed) has the form of a static fric-tion coefficient µ s , but it is not set directly by grain-grain friction. Instead, µ s is an emer-gent property that arises from the maximum structural anisotropy that the grain-grain contactnetwork can support. Friction plays a minor role in determining this maximum anisotropy,and grain shape and polydispersity also play minor roles. Second, although these contact net-works are extended in space (and thus inherently nonlocal), local rheological descriptions(i.e., constitutive laws) can be very successful in many contexts. Recent advances suggestthat a unified, local rheological description might be within reach. This rule could be used tomodel any context of wet or dry granular flow with appropriate boundary conditions. Sucha description could be used to predict sediment transport rates and thresholds if the grainproperties (i.e., size distribution, friction coefficient, grain shape, etc.) were known, even ap-proximately. Third, the inherently nonlocal nature of yielding is dominant when the materialis near its yield condition. This causes creeping behavior in regions where a local rheologywould predict no flow, which complicates the search for a unified rheological description.However, the results described in Figures 2 and 3 showed that creeping is similar in wet anddry flows, since it very slow and thus dominated by grain rearrangements (not fluid). Thissuggests that the nonlocal descriptions for wet and dry flows might also be unified in a rela-tively simple way. The underlying physics behind this nonlocal behavior is not fully under-stood, but there is mounting evidence that yielding of granular materials represents a kindof critical transition, where different parts of the system can be correlated over arbitrary dis-tances. Remarkably, for sediment transport, creep seems to occur even much below the yieldtransition, that is, for seemingly arbitrarily small Shields numbers Θ .This section has considered only sediment beds sheared by nonfluctuating flows andusually neglected the effects of particle inertia in sustaining sediment transport. That is, ex-cept for the occurrence of creep, many of the results of this section do not apply to turbulentflows nor flows with significant particle inertia effects that are near the threshold for grainmotion (occurring for sufficiently large Re ∗ and/or s , see section 4). In particular, the averagefluid shear stress at which turbulent flows are able to entrain bed particles is usually muchbelow the yield stress of the granular phase. Nonetheless, both creep and the viscous yieldsstress Θ max t will play crucial roles in the new conceptual picture of sediment transport thresh-olds and sediment entrainment that is presented in section 5. This section reviews the state of the art on the entrainment of bed particles by a tur-bulent flow of Newtonian fluid. This process is not equivalent to the initiation of overallsediment motion, which occurs even in the absence of bed sediment entrainment becauseof creeping (see section 2.3). It is also not equivalent to the comparably simple physics offluid entrainment by a nonfluctuating flow. For example, when a laminar flow of a Newtonianfluid shears a target particle resting on the sediment bed, there are critical values of the fluidshear stress τ , which depend on the local bed arrangement, above which this particle beginsto roll and slide, respectively [ Agudo et al. , 2017;
Deskos and Diplas , 2018]. Once motionbegins, resisting forces weaken and, since the flow does not fluctuate, the particle will in-evitably leave its bed pocket (i.e., become entrained). The entrained particle will travel alongthe bed until it comes to rest in another pocket in which it can resist the flow, provided sucha pocket exists and is accessible (when the sediment bed has yielded, particles can no longerfind stable resting place, see section 2.1). In contrast, in turbulent flows, even though resist- –14–onfidential manuscript submitted to
Reviews of Geophysics ing forces weaken when a bed particle becomes mobilized, such a mobilized particle maynot find its way out of its initial bed pocket (i.e., incomplete entrainment). The prototype forthis situation is a turbulent fluctuation of the flow that exerts a large force on the particle, butthe fluctuation is too short-lived for it to become entrained. Hence, there are two importantingredients that need to be considered to accurately describe sediment entrainment by turbu-lent flows for a given pocket geometry: the magnitude and duration of turbulent fluctuations(evidence for this statement is briefly reviewed in section 3.1). Only entrainment criteria thataccount for both aspects are able to accurately describe fluid entrainment experiments (sec-tion 3.2). Shear stress-based criteria, in general, do not belong to this category. Yet one canstill define the critical shear stress τ In t above which the probability of fluid entrainment ex-ceeds zero. This and related thresholds have received a lot of attention in studies on aeolianand planetary transport (section 3.3). Turbulent fluctuations have been known to play a crucial role in fluid entrainment for along time. For example,
Einstein and El-Samni [1949], and later
Mollinger and Nieuwstadt [1996], measured large fluctuating lift forces on a fixed rough surface induced by pressuregradient fluctuations of the order of the mean pressure gradient. These authors concludedthat such pressure gradient fluctuations must be important also for the mobilization of bedsediment. In fact, numerous laboratory, field, and theoretical studies have advocated theviewpoint that the magnitude of peaks of the instantaneous flow force acting on a bed parti-cle, consisting of both lift and drag forces, is a key aspect of fluid entrainment [e.g.,
Kalinske ,1947;
Sutherland , 1967;
Paintal , 1971;
Heathershaw and Thorne , 1985;
Apperley and Raud-kivi , 1989;
Kirchner et al. , 1990;
Nelson et al. , 1995;
Papanicolaou et al. , 2001;
Sumer et al. ,2003;
Zanke , 2003;
Hofland et al. , 2005;
Schmeeckle et al. , 2007;
Vollmer and Kleinhans ,2007;
Giménez-Curto and Corniero , 2009;
Dwivedi et al. , 2010a,b;
Cameron et al. , 2019,2020]. However, while such force peaks explain certain observations, such as the episodiccharacter of very weak turbulent bedload transport [
Paintal , 1971;
Helland-Hansen et al. ,1974;
Hofland , 2005] or the strong increase of weak turbulent bedload transport in the pres-ence of vegetation [
Yager and Schmeeckle , 2013;
Yang and Nepf , 2018, 2019], they do notexplain all observations. In fact, experiments in which a target particle was placed on an ide-alized rough substrate and exposed to an electrodynamic force revealed that very high forcepulses do not lead to entrainment if their duration is too short [
Diplas et al. , 2008]. Likewise,moderate force pulses that only barely exceed resisting forces lead to entrainment if theirduration is sufficiently long. That the duration of force peaks is as important as their mag-nitude has also been experimentally confirmed both for particles resting on idealized, fixedbeds [
Diplas et al. , 2008;
Celik et al. , 2010, 2013, 2014;
Valyrakis et al. , 2010, 2011, 2013;
Valyrakis , 2013] and natural erodible sediment beds [
Salim et al. , 2017, 2018]. However,note that, for sediment transport along erodible beds (with the exception of viscous bedloadtransport), the vast majority of entrainment events are triggered by particle-bed impacts, ex-cept for very weak transport conditions (see sections 4.1.2 and 4.1.3). In the following, cri-teria are reviewed that account for both the magnitude and duration of turbulent fluctuationevents.
The initiation of movement of a target particle resting in a pocket of the bed surfacenecessarily requires that the instantaneous flow forces (or torques) F ( t ) acting on it at theinstant t of initial motion overcome resisting forces (or torques) F c : F ( t ) ≥ F c , (1) –15–onfidential manuscript submitted to Reviews of Geophysics
However, this criterion is not sufficient for entrainment to occur as the target particle maymerely move back to its initial resting place if F ( t ) becomes subcritical for times t too soonafter t o so that its gained kinetic energy is insufficient to overcome the potential barrier of itsbed pocket. For this reason, Diplas et al. [2008] proposed that the fluid impulse I f associatedwith larger-than-critical flow forces must exceed a critical value: I f ≡ t + T ∫ t F ( t ) d t ≥ I f c with F ( t ) ≥ F c for t ∈ ( t , t + T ) , (2)where T is the duration of the impulse event (i.e., the duration of the particle accelerationphase of a turbulent fluctuation event). Note that T can be much smaller than the time neededto leave the bed pocket as the latter also includes the particle deceleration phase. Diplas et al. [2008] confirmed their hypothesis with idealized experiments in which they subjected an iso-lated target particle with a constant electrodynamic, horizontal force F D for a given time T D ,for which I f = F D T D . In fact, their measured data of the force that is required for entrain-ment roughly obey the relation ˆ F D ≡ F D / F min D = T max D / T D ≡ ˆ T − D , where F min D is the minimalforce required for measurable particle motion (but not necessarily entrainment) and T max D theassociated time that is needed for F min D to cause entrainment (Figure 4). Figure 4.
From
Diplas et al. [2008] (M.V. is copyright holder), normalized magnitude ˆ F D of the electro-dynamic force pulse that is required for entrainment versus normalized duration ˆ T D of the force pulse. Datacorrespond to the entrainment experiments that were carried out for various particle arrangements and varyingsizes of the target ( d ) and base particles ( d ). The line corresponds to the prediction ˆ F D = ˆ T − D associatedwith a constant impulse threshold. In order to use equation (2) for predicting particle entrainment, one needs to know theimpulse threshold I f c . For entrainment into a rolling motion, Valyrakis et al. [2010] derivedan expression for the critical impulse I f c = F t T t ( F t is defined below) assuming a constant –16–onfidential manuscript submitted to Reviews of Geophysics pulse of a hydrodynamic force, separated into a horizontal drag and vertical lift component( F = ( F D , F L ) ), of short duration T t (so that the angular displacement ∆ ψ of the particleremains small for t ∈ ( t , t + T t ) ): I f c = F t g (cid:115) f ( ψ, α, s ) L arm g (cid:18) + C m s (cid:19)(cid:114) − m p g ρ ψ ( F n − F nc ) arsinh (cid:34) (cid:112) − ρ ψ ( F n − F nc )( m p g )( F t − F tc ) (cid:35) , (3)where F t = F D sin ψ + F L cos ψ and F n = − F D cos ψ + F L sin ψ are the tangential and normalcomponents, respectively, of the driving flow force at the rest position, m p = ρ p π d is theparticle mass, F tc = m p g cos ( ψ + α )/ sin ψ − ( m p g / s ) cot ψ the resisting force, L arm the leverarm length, C m = / f ( ψ, α, s ) = cos ( ψ + α ) sin α + [ − sin ( ψ + α )]( cos α − / s ) , with α the bed slope angle and ψ the pivoting angle (Figure 5). For L α F Fm gflow armL D FF n t m g/s p p Figure 5.
Sketch of pocket geometry. many conditions, this expression can be well approximated by [
Valyrakis et al. , 2010] I f c ≈ m p (cid:18) F t F t − F tc (cid:19) (cid:115) f ( ψ, α, s ) L arm g (cid:18) + C m s (cid:19) . (4) Lee et al. [2012] derived an alternate expression for short turbulent fluctuation events. In-stead of a pure rolling motion, they considered entrainment into a combined rolling and slid-ing motion (however, note that rolling is usually the preferred mode of entrainment) withoutbed slope ( α = µ C . Furthermore, instead of the pivoting angle, they de-scribed the pocket geometry by the horizontal ( ∆ X ) and vertical ( ∆ Z ) particle displacement(in units of d ) that is needed for the particle to escape (equivalent to ψ + α = π / Lee et al. [2012] reads I f c ≡ ( F e T e ) c = ( ∆ Z + µ C ∆ X ) m p (cid:114) F e F e − F ec (cid:115) g d (cid:18) + C m s (cid:19) (cid:18) + s (cid:19) , (5)where F e = F D ( sin ψ − µ cos ψ ) + F L ( cos ψ + µ C sin ψ ) is an effective hydrodynamic forceand F ec = m p g ( − / s )( sin ψ + µ C cos ψ ) its critical value. For entrainment into a hoppingmotion, defined as a lift force-induced particle uplift by a vertical distance ≥ d , Valyrakiset al. [2010] derived I f c ≡ ( F L T L ) c = m p (cid:114) F L F L − F Lc (cid:115) g d cos α (cid:18) + C m s (cid:19) (cid:18) + s (cid:19) , (6) –17–onfidential manuscript submitted to Reviews of Geophysics where the resistance force is given by F Lc = m p g ( − / s ) cos α . Note that equation (6) with α = ∆ Z + µ C ∆ X = F L ( c ) replaced by F e ( c ) .Equations (3)–(6) reveal that the impulse threshold I f c is constant only if the drivingflow force is very strong ( F ( t ) (cid:29) F c ). However, for near-critical fluctuation events ( F ( t ) → F c ), I f c diverges. This motivates the introduction of an energy-based entrainment criterion. The impulse criterion (equation (2)) accounts for the available momentum of the tur-bulent fluctuation event in comparison to the momentum required for entrainment. How-ever, close observation of near-bed turbulence reveals that fluctuation events are scarcely eversquare pulses or even single-peaked [
Valyrakis , 2013]. Instead, turbulent flows in nature ex-hibit a wide range of flow patterns and structures, some of which may be more efficient forparticle entrainment than others. For example, the transfer of energy from flow to particlesin turbulent fluctuation events with large driving flow forces ( F ( t ) (cid:29) F c ) is expected to bemuch more efficient than in fluctuation events with near-critical flow forces ( F ( t ) ∼ F c , seesection 3.2.1). This motivates the characterization of entrainment using the energy of thefluctuation event that is effectively transferred to the particle [ Valyrakis et al. , 2013]: C eff E f = C eff t + T ∫ t P f ( t ) d t ≥ W c , (7)where W c is the minimal amount of work required for complete particle entrainment and P f ( t ) = f [ u ( t ) ] the instantaneous flow power, parameterized by the cube of the local flowvelocity, and C eff is the coefficient of energy transfer efficiency of the turbulent fluctuationevent. The energy transfer coefficient C eff is expected to increase with (cid:104) F (cid:105)/ F c (see sec-tion 3.2.1), where (cid:104)·(cid:105) denotes the time average over the event. Water flume experiments onthe entrainment of a particle resting on an idealized substrate have confirmed that C eff tendsto increase with (cid:104) F (cid:105)/ F c (Figure 6). However, one has to keep in mind that C eff incorporatesalso other effects such as grain orientation and shape.In order to use equation (7) for predicting particle entrainment, one needs to know theenergy threshold W c . Valyrakis et al. [2013] derivedRolling: W c = m p cos α [ − sin ( ψ + α )]( − / s ) g L arm , (8)Hopping: W c = m p cos α ( − / s ) g d . (9)For typical sediment beds, the ratio between both energy thresholds ( [ − sin ( ψ + α )] L arm / d )is of the order of 0 .
1, demonstrating that a rolling motion is much more easily initiated uponentrainment than a hopping motion. Note that, in contrast to the expressions for the criticalimpulse for rolling (equations (3) and (4)), equation (8) does neither require the assumptionof a small angular particle displacement ∆ ψ during the acceleration phase of a turbulent fluc-tuation event nor the assumption of a short duration of this phase. The entrainment criteria reviewed in section 3.2 are able to predict whether a certainturbulent fluctuation event is capable of entraining a target particle, whereas a criterion basedon a critical shear stress would not suffice for this purpose. However, one can still define ashear stress threshold τ In t (the initiation threshold ) at which the fluid entrainment probabil-ity exceeds zero (i.e., below which entrainment never occurs). Such a threshold must existbecause the size of turbulent flow eddies is limited by the system dimensions, such as theboundary layer thickness δ . In fact, a limited size of turbulent flow eddies implies that also –18–onfidential manuscript submitted to Reviews of Geophysics ����� ����� ������������ � [ � ] � � ( � ) = � ( � � ( � )) [ �� � � / � � ] × �� - � ����� ����� ����� ������������ � [ � ] � � ( � ) = � ( � � ( � )) [ �� � � / � � ] × �� - � ◦◦◦◦ ◦◦◦ ◦ ◦◦ ◦ ◦ ◦◦◦◦ ◦ ◦◦◦◦ ◦◦ ◦ ◦◦◦ ◦ ◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦ ◦◦ ◦◦◦ ◦◦◦◦ ◦ ◦◦◦ ◦◦◦ ◦ ◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦ ◦◦◦◦◦ ◦ ◦◦◦◦◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦◦ ◦ ◦◦◦◦ ◦◦ ◦◦◦◦◦ ◦◦◦ ◦◦◦◦◦ ◦ ◦ ◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦ ◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦◦◦ ◦◦◦ ◦◦ ◦●●■◦■ □◦□ �������� ���� ���� ���� ���� �������������������������������� � [ � ] � � �� ���� ��� ���� ����������� � [ � ] � � ( � ) = � ( � � ( � )) [ �� � � / � � ] × �� - � (a) (b)(c) (d) Figure 6. (a–c) Flow power P f ( t ) versus time t for three different turbulent fluctuation events that lead toentrainment of a target particle resting on a prearranged substrate. The solid lines corresponds to experimentaldata [ Valyrakis et al. , 2013]. The dashed lines indicate the start of the respective fluctuation event. The dottedlines indicate the critical flow power that must be exceeded in order to overcome the resisting forces (i.e., u > u c ( t ) ), which depend on time because resisting forces weaken once the target particle starts to move.(d) Coefficient of energy transfer C eff versus duration of turbulent fluctuation event ( T ) for various recordedentrainment events (symbols). The green closed square corresponds to the event shown in (a), the blue opensquare to the event shown in (b), and the red closed circle to the event shown in (c). the magnitude of peaks of the flow force is limited. That is, one can always find a nonzeroshear stress below which even the largest fluctuation peaks do not exceed the resisting forcesacting on bed particles (however, note that the existence of sufficiently large flow force peaksdoes not guarantee a nonzero entrainment probability because their durations may always betoo short). Like for Θ max t , transient behavior associated with the flow temporarily pushingparticles from less stable to more stable pockets is excluded in the definition of τ In t , whichimplies Θ In t (cid:39) Θ max t for laminar flows at sufficiently low shear Reynolds number Re ∗ . Fur-thermore, surface inhomogeneities that can generate a lot of turbulence, such as vegeta-tion [ Yager and Schmeeckle , 2013;
Yang and Nepf , 2018, 2019], are also not considered inthe definition of τ In t . While τ In t is usually not measured for turbulent fluvial bedload trans-port (it is much below the Shields curve [ Paintal , 1971]), it has often been measured in windtunnel experiments (briefly reviewed in section 3.3.1), including those that sought to deter-mine the initiation threshold of aeolian saltation transport. The reason is that as soon as thefirst particles of the initially quiescent bed surface are entrained (i.e., begin to roll as rollingrequires the smallest flow forces), the flow is usually nearly sufficient to net accelerate themduring their downstream motion, resulting in larger and larger particle hops (i.e., the initia-tion threshold of aeolian saltation transport is only slightly larger than τ In t ) [ Bagnold , 1941;
Iversen et al. , 1987;
Burr et al. , 2015]. This occurs because, for typical wind tunnels, τ In t issignificantly above the cessation threshold of saltation transport (see section 4.3). However, it –19–onfidential manuscript submitted to Reviews of Geophysics will become clear that this statement may not apply to aeolian field conditions. Section 3.3.2briefly reviews models of τ In t derived from wind tunnel experiments, while section 3.3.3 re-views recent evidence that indicates that such models, in general, are unreliable, particularlywhen applied to field conditions. Two distinct experimental setups have been used to measure τ In t . In the first setup,small isolated patches of particles are placed at the bottom of a wind tunnel and then thefluid shear stress τ is increased until particles in such patches start to roll or detach [ Williamset al. , 1994;
Merrison et al. , 2007; de Vet et al. , 2014]. In the second setup, a complete bedof particles is prepared at the tunnel bottom and then the fluid shear stress τ is increased un-til saltation transport begins [e.g., Bagnold , 1937;
Chepil , 1945;
Lyles and Krauss , 1971;
Iversen et al. , 1976;
Greeley et al. , 1976, 1980, 1984;
Gillette et al. , 1980;
Greeley and Mar-shall , 1985;
Nickling , 1988;
Iversen and Rasmussen , 1994;
Dong et al. , 2003;
Cornelis andGabriels , 2004;
Burr et al. , 2015;
Carneiro et al. , 2015;
Swann et al. , 2020] (see also
Raf-faele et al. [2016, and references therein]). It is worth noting that, according to the definitionof τ In t , beginning saltation transport refers to the mere occurrence of saltation transport, evenif very sporadic, which is also the definition used by Bagnold [1937]. However, many exper-imental studies defined beginning saltation transport through a critical loosely defined salta-tion transport activity (similar to the definition of the fluvial transport thresholds compiled inthe Shields diagram), which yields slightly larger threshold values [
Nickling , 1988].
Open Problem: Qualitative Discrepancy Between Threshold Measurements
For cohesionless particles ( d (cid:38) µ m), existing threshold measurements based onthe second setup show that τ In t increases relatively strongly with the particle diameter d [ Raf-faele et al. , 2016]. In contrast, for the first setup, measurements indicate that τ In t remains con-stant with d for d (cid:38) µ m [ Merrison et al. , 2007; de Vet et al. , 2014]. The reason for thisqualitative inconsistency is not understood.
Merrison et al. [2007] suggested that the initi-ation of rolling (measured in their experiments) may be different to that of saltation trans-port. However, this suggestion is inconsistent with the observation that saltation transportin wind tunnels is preceded by rolling further upwind [
Bagnold , 1941;
Iversen et al. , 1987;
Burr et al. , 2015]. Furthermore, in contrast to standard wind tunnel experiments, for exper-iments in pressurized wind tunnels with Venusian air pressure, both an equilibrium rolling(lower initiation threshold) and an equilibrium saltation transport regime (higher initiationthreshold) exist, and both initiation thresholds strongly increase with d [ Greeley and Mar-shall , 1985].
Nearly all existing models of the initiation of aeolian rolling and saltation transport (in-cluding sand transport [
Bagnold , 1941;
Iversen et al. , 1976, 1987;
Iversen and White , 1982;
Shao and Lu , 2000;
Cornelis and Gabriels , 2004;
Lu et al. , 2005;
Claudin and Andreotti ,2006;
Kok and Renno , 2006;
Merrison et al. , 2007;
Durán et al. , 2011;
Duan et al. , 2013a; de Vet et al. , 2014;
Burr et al. , 2015;
Edwards and Namikas , 2015], drifting snow [
Schmidt ,1980;
Lehning et al. , 2000;
He and Ohara , 2017], and the transport of regolith dust by out-gassed ice on the comet 67P/Churyumov-Gerasimenko [
Jia et al. , 2017]) predict τ In t from thebalance between aerodynamic forces and/or torques and resisting forces and/or torques actingon a bed particle. Even though many of these models do not consider peaks of the aerody-namic force, and some of them do not treat τ In t as what it is (i.e., the threshold at which thefluid entrainment probability exceeds zero, see above), they are conceptually very similar andmainly differ in the empirical equations that they use for the aerodynamic and cohesive inter-particle forces. For this reason, only one of the most popular and simple models, the model –20–onfidential manuscript submitted to Reviews of Geophysics by Shao and Lu [2000], is discussed here. It reads Θ In t = A N (cid:18) + γ C ρ p g d (cid:19) , (10)where A N = . γ C = × − kg/s an empiricalconstant that accounts for cohesive interparticle forces. More complex models [e.g., Iversenand White , 1982;
Claudin and Andreotti , 2006;
Durán et al. , 2011] involve additional depen-dencies of Θ In t on the shear Reynolds number Re ∗ or, equivalently, on the Galileo number Ga ≡ (cid:112) ( s − ) g d / ν f ≡ Re ∗ /√ Θ (also called Yalin parameter [ Yalin , 1977]).
The size of turbulent flow eddies, and thus the duration of turbulent fluctuation events,is limited by the system dimensions, more specifically, the boundary layer thickness δ [seereview by Smits et al. , 2011, and references therein]. However, in most wind tunnel exper-iments and the field, the produced turbulent boundary layer should be so thick that any tur-bulent fluctuation has a nonzero probability to last sufficiently long for entrainment to oc-cur [
Pähtz et al. , 2018]. That is, the mere existence of aerodynamic force peaks that exceedresisting forces is sufficient for τ In t to be exceeded. However, this is no longer true when δ be-comes too small, at which point turbulent fluctuation events may cause particles to rock (i.e.,vibrate or wobble or oscillate) within their bed pockets but no fluctuation lasts long enoughfor the particles to completely leave them. Pähtz et al. [2018] physically modeled such situ-ations and derived an expression for the ratio between τ In t and the shear stress threshold τ In (cid:48) t of incipient rocking (equivalent to the Shields number ratio Θ In t / Θ In (cid:48) t ). These authors’ deriva-tion uses the impulse criterion of section 3.2.1 (even though Pähtz et al. [2018] start withthe energy criterion, their analysis is effectively equivalent to assuming a constant impulsethreshold) and the fact that the maximal duration T max of turbulent fluctuation events is con-trolled by δ and the local mean flow velocity u via T max ∝ δ / u [ Alhamdi and Bailey , 2017].The derived expression reads (cid:115) Θ In t Θ In (cid:48) t (cid:39) C < C if 1 ≤ C ≤ α f α f if C > α f (11) C ≡ α − f f ( G ) √ sd δ . where α f ≡ u m / u ≥ u m associatedwith the largest positive fluctuations and u , and f ( G ) is a factor that encodes informationabout particle shape, orientation, and the pocket geometry. Equation (11) encompasses threedifferent regimes. In one extreme, if there is a nonzero probability that turbulent fluctuationevents associated with the largest positive fluctuations last sufficiently long for particle en-trainment, then there will be a nonzero probability that incipient rocking evolves into incipi-ent rolling (i.e., Θ In t / Θ In (cid:48) t (cid:39) Θ In t / Θ In (cid:48) t (cid:39) α f ). In the intermediate regime between these two extremes, Θ In t / Θ In (cid:48) t is propor-tional to the square of the inverse dimensionless boundary layer thickness ( d / δ ) . Althoughweak logarithmic dependencies on δ / d are also incorporated in α f and Θ In (cid:48) t [ Lu et al. , 2005],they are dominated by this proportionality. In fact, Figure 7 shows that the prediction forthe intermediate regime is roughly consistent with the experimental data by
Williams et al. [1994] if one uses that the Shields number for incipient rocking ( Θ In (cid:48) t ) is approximately con-stant, neglecting the logarithmic dependency of Θ In (cid:48) t on δ (and further minor dependencieson Ga). Williams et al. [1994] set up their wind tunnel in a manner that produces a relativelythin developing turbulent boundary layer (i.e., δ increases with downstream distance). How-ever, once the intermediate regime is exceeded (i.e., Θ In t (cid:39) Θ In (cid:48) t ) because δ becomes too –21–onfidential manuscript submitted to Reviews of Geophysics
Figure 7.
From
Pähtz et al. [2018] (Creative Commons Attribution 4.0 International License), thresh-old parameter (cid:113) Θ In t versus dimensionless inverse boundary layer thickness √ sd / δ . Symbols correspond tothe measurements of incipient rolling by Williams et al. [1994], who set up their wind tunnel in a mannerthat produces a developing turbulent boundary layer, for four different sediments consisting of nearly uni-form, cohesionless particles. The solid line corresponds to equation (11) for the intermediate regime using (cid:113) Θ In (cid:48) t (cid:39) const (neglecting the weak logarithmic dependency of (cid:113) Θ In (cid:48) t on δ / d ). This regime turns into theextreme regime in which (cid:113) Θ In t (cid:39) (cid:113) Θ In (cid:48) t . This transition is shown by the dashed line assuming (cid:113) Θ In (cid:48) t = . (cid:113) Θ In (cid:48) t in the experiments by Williams et al. [1994] areunknown). It is suspected that the one extreme outlier for d = µ m may either have been a faulty mea-surement or be associated with the observation that the boundary layer for this particular sand sample was notalways fully turbulent [ Williams et al. , 1994]. large, as for most wind tunnel experiments with fully developed boundary layers, the loga-rithmic dependency of Θ In t on δ / d via Θ In (cid:48) t may become significant (Figure 8). For example,for the same Galileo number Ga, the threshold values measured by Burr et al. [2015] in Fig-ure 8, which were carried out in a pressurized wind tunnel with δ ≈ . Iversen et al. [1976], which were carried out in a wind tunnelwith δ ≈ . Open Problem: Unexpected Behavior of Saltation Transport Initiation Threshold forLarge Density Ratio
The very recent measurements by
Swann et al. [2020], who used a very-low pressurewind tunnel and three different beds of cohesionless particles ( d = [ , , ] µ m) tomimic Martian conditions, indicate that (cid:113) Θ In t unexpectedly increases substantially with Gaand thus d (Figure 8). A possible explanation could be that, because of the very large den-sity ratio s , some of the experimental conditions may have been in the intermediate regime(i.e., 1 ≤ C ≤ α f in equation (11)), in which (cid:113) Θ In t scales with d (Figure 7). In fact, 1 / C ∝ δ /(√ sd ) (cid:39) [ . , . , . ] for the three conditions, where only the largest value (correspond-ing to d = µ m) is larger than the critical value δ /(√ sd ) ≈ . Pähtz et al. [2018]associated with the end of the intermediate regime. In other words, the measurements for d = µ m and d = µ m may both have been in the intermediate regime, in which (cid:113) Θ In t roughly scales with d (cf. Figure 7). –22–onfidential manuscript submitted to Reviews of Geophysics
100 200 500 1000 2000 5000 10000
Figure 8.
Modified from
Pähtz et al. [2018] (Creative Commons Attribution 4.0 International License),compilation of measurements in wind tunnels with fully developed boundary layer of the initiation thresholdparameter of saltation transport ( ≈ (cid:113) Θ In t ) [ Iversen et al. , 1976;
Burr et al. , 2015;
Swann et al. , 2020] versusthe Galileo number Ga. The color indicates the thickness of the boundary layer δ relative to the particle diam-eter d , which controls the relative amplitude of turbulent fluid velocity fluctuations for a constant Ga. Circlescorrespond to threshold values obtained from the raw data by Swann et al. [2020]. The threshold values forthe experiments by
Iversen et al. [1976] are found in
Iversen and White [1982].
Controversy: Dependency of Saltation Transport Initiation Threshold on DensityRatio
Based on comparisons between experiments in pressurized wind tunnels with com-parably very thin boundary layers but larger-than-normal air density [
Greeley et al. , 1984;
Burr et al. , 2015] and nonpressurized wind tunnels with comparably very large boundarylayers [
Iversen et al. , 1976] (and normal air density),
Iversen et al. [1987] and
Burr et al. [2015] argued that there is an underlying decrease of the saltation transport initiation thresh-old (which is slightly larger than Θ In t for aeolian transport in typical wind tunnels, see above)with the density ratio s for a constant shear Reynolds number Re ∗ (equivalent to a constantGa). However, this dependency on s may be an artifact of huge differences in the dimension-less boundary layer thickness δ / d [ Pähtz et al. , 2018]. In fact, even though the dependencyof Θ In t on δ / d is logarithmic once the intermediate regime is exceeded (like for the measure-ments in question), such weak dependencies can still have significant effects once differ-ences in δ / d become very large. This point of view is supported by Figure 8, in which δ / d is color-coded. It can be seen that the yellow, open diamond (a measurement from a non-pressurized wind tunnel) exhibits a similar value of s as the blue symbols (measurementsfrom a pressurized wind tunnels), which was achieved by using a very light particle material( ρ p =
210 kg/m ). Nonetheless, the threshold (cid:113) Θ In t of the former is significantly smallerthan those of the latter. Also, the former measurement relatively smoothly connects to theother measurements carried out in the same nonpressurized wind tunnel, which exhibit muchlarger values of s . On the other hand, the measurements by Swann et al. [2020], for which s is comparably very large and δ / d of a similar size as for the measurements by Iversen et al. [1976], support the density ratio hypothesis because of comparably small values of (cid:113) Θ In t .Note that, for the discussion of threshold values, one has to keep in mind that threshold mea-surements are highly prone to measurement errors of various sources [ Raffaele et al. , 2016].Such errors are likely much larger than often reported because measurements of (cid:113) Θ In t can –23–onfidential manuscript submitted to Reviews of Geophysics vary by more than a factor of 2 for a given condition, even for cohesionless particles [
Raf-faele et al. , 2016].
Open Problem: Aeolian Bedload Transport in the Field
In wind tunnel experiments, rolling is being initiated at threshold values that are sig-nificantly above the cessation threshold of saltation transport (see section 4.3). This is whyrolling seems to always evolve into saltation transport (i.e., equilibrium rolling and thus ae-olian bedload transport does not seem to exist) [
Bagnold , 1941;
Iversen et al. , 1987;
Burret al. , 2015]. However, atmospheric boundary layers are several orders of magnitude thickerthan those of wind tunnels [
Lorenz et al. , 2010;
Petrosyan et al. , 2011;
Kok et al. , 2012;
Lebonnois et al. , 2018] and may therefore exhibit a significantly smaller rolling threshold.In contrast, the cessation threshold of saltation transport is predominantly a property of themean turbulent flow (see section 4.3) and therefore rather insensitive to the boundary layerthickness δ . Hence, for atmospheric boundary layers, it is possible that equilibrium rollingtransport exists. Note that equilibrium rolling transport has been observed in pressurizedwind tunnels with Venusian air pressure for a narrow range of Shields numbers Θ [e.g., Gree-ley and Marshall , 1985].
Open Problem: Reliable Models of the Initiation Threshold of Planetary SaltationTransport
The most widely used models for the initiation of aeolian saltation transport (see sec-tion 3.3.2), which have been adjusted to wind tunnel measurements, do not take into accountthe dependency of the relative magnitude of turbulent fluctuations on the dimensionlessboundary layer thickness δ / d . This may be the reason why these models, when applied toMartian atmospheric conditions, predict threshold shear stresses for fine sand particles thatare so large that transport should occur only during rare strong Mars storms [ Sullivan andKok , 2017], in contradiction to modern observations indicating widespread and persistentsediment activity [
Bridges et al. , 2012a,b;
Silvestro et al. , 2013;
Chojnacki et al. , 2015], evenof very coarse sand [
Baker et al. , 2018]. For example, for the Martian conditions reported by
Baker et al. [2018] ( ρ p = , ρ f = .
02 kg/m , g = .
71 m/s , d = . u In ∗ t ≡ (cid:113) Θ In t ( ρ p / ρ f − ) g d (cid:39) . Lu et al. [2005] proposed a model for the initiation of rolling that includes the ef-fect of δ / d . The authors of this review therefore recommend to use the model by Lu et al. [2005] in combination with models of the cessation threshold of saltation transport (seesection 4.3) for the estimation of the occurrence of saltation transport in real atmosphericboundary layers. However, it remains to be demonstrated that this approach yields reliablepredictions. In fact, in the field, atmospheric instability, topography gradients, and surfaceinhomogeneities, such as obstacles and vegetation, can dramatically enhance local turbulenceand thus fluid entrainment. Likewise, sublimation of subsurface ice in cold environments(the so-called solid-state greenhouse effect [ Kaufmann et al. , 2006]) can generate airborneparticles of carbon dioxide, methane, and nitrogen ice [
Hansen et al. , 1990;
Thomas et al. ,2015;
Jia et al. , 2017;
Telfer et al. , 2018]. Given that even a few entrained particle can re-sult in fully developed saltation transport provided that the fetch is sufficiently long [
Sullivanand Kok , 2017], it may well be that saltation transport in the field can almost always be initi-ated close to the cessation threshold [
Sullivan and Kok , 2017;
Pähtz et al. , 2018;
Telfer et al. ,2018]. Evidence for this hypothesis is seen on Pluto, where aeolian dunes and wind streakshave been observed even though saltation transport initiation had been thought to be virtuallyimpossible because of Pluto’s very thin atmosphere (pressure P = u max10m ≈
10 m/s) [
Telfer et al. , 2018]. –24–onfidential manuscript submitted to
Reviews of Geophysics
Open Problem: Lack of Direct Aeolian Sediment Transport Initiation Measurementsin the Field
The overarching problem associated with the rather poor current knowledge of aeo-lian sediment transport initiation in the field (see open problems above) is that, to the au-thors’ knowledge, there are no direct field measurements of the transport initiation thresh-old Θ In t . In fact, existing field experiments have focused on detecting aeolian saltation trans-port [ Barchyn and Hugenholtz , 2011, and references therein] rather than on how the fluidentrainment of individual bed particles, which usually starts out as a rolling motion, leads tosaltation transport. Hence, we currently do neither know the wind speeds that are requiredin the field to initiate rolling transport of individual bed particles nor whether such rollingtransport, like in wind tunnels, always evolves into saltation transport (see open problemsabove). What adds to the problem is that existing field studies either obtain saltation trans-port threshold estimates using methods that do not seek to distinguish saltation transportinitiation and cessation [
Barchyn and Hugenholtz , 2011, and references therein] or assumethat Θ In t coincides with the continuous saltation transport threshold [ Martin and Kok , 2018](which is a controversial assumption, see section 4.1.3).
As discussed in section 1, old experimental studies [e.g.,
Ward , 1969;
Graf and Pazis ,1977] strongly indicated that the fluvial transport threshold measurements that are com-piled in the Shields diagram are to a nonnegligible degree affected by particle inertia. As theShields diagram shows a rough data collapse of the threshold Shields number Θ t as a func-tion of the shear Reynolds number Re ∗ , this raises the question of whether Re ∗ is in someway associated with particle inertia. Indeed, while Re ∗ has usually been interpreted as theratio between the particle size and the size of the viscous sublayer of the turbulent boundarylayer, Clark et al. [2017] showed that it can also be interpreted as a number that compares theviscous damping time scale to the ballistic time scale between bed collisions. Importantly,these authors showed that the shape of the Shields curve can be partly explained by the factthat inertial particles at high Re ∗ are harder to stop.In general, the role of particle inertia in nonsuspended sediment transport can be twofold.On the one hand, entrainment by or supported by particle-bed impacts may be able to supplythe transport layer with bed particles and thus compensate captures of transported particlesby the bed (section 4.1). This mechanism gives rise to a shear stress threshold associatedwith impact entrainment. On the other hand, although the mean turbulent flow is usually tooweak to initiate transport (which instead usually requires turbulent fluctuation events, seesection 3), it may be able to sustain the motion of particles that are already in transport. Thismechanism gives rise to a physical process-based definition of transport capacity and a shearstress threshold, which has often been misidentified as an entrainment threshold by Shields [1936] and others (section 4.2). Various models for both shear stress thresholds that havebeen proposed in the literature are compared with one another in section 4.3.
Bagnold [1941] was the first to recognize that impact entrainment is crucial for sus-taining aeolian saltation transport. Based on his wind tunnel and field observations, he ex-plained [
Bagnold , 1941, p. 102], “In air, the grains, when once set in motion along the sur-face, strike other stationary grains, and either themselves bounce high (a distance measuredin hundreds if not thousands of grain diameters) into the relatively tenuous fluid, or ejectother grains upwards to a similar height.” Largely because of Bagnold’s observations, thestatistics of particle impacts onto a static granular packing have been subject of many exper-imental and theoretical investigations (section 4.1.1).
Bagnold [1941, p. 102] also believedthat impact entrainment is negligible for fluvial bedload transport: “If the physics of thisimpact-ejection mechanism is applied to sand in water, it is found that the impact momen- –25–onfidential manuscript submitted to
Reviews of Geophysics tum of the descending grains is insufficient to raise surface grains to a height greater than asmall fraction of one grain diameter.” However, Bagnold, and numerous researchers afterhim, did not consider that even a marginal uplift of a bed particle can make it much easierfor a turbulent fluctuation event to entrain it (section 4.1.2) and that, once bedload transportbecomes sufficiently strong, multiple particle-bed impacts occur in so short sequence thatthe bed can no longer be considered as static. In fact, for continuous transport, recent studiesrevealed that impact entrainment alone can sustain bedload transport (section 4.1.3).
The collision process between an incident particle and a static granular packing hasbeen investigated in many experimental [
Mitha et al. , 1986;
Werner , 1990;
Rioual et al. ,2000, 2003;
Tanaka et al. , 2002;
Nishida et al. , 2004;
Beladjine et al. , 2007;
Oger et al. ,2008;
Ammi et al. , 2009;
Clark et al. , 2012, 2015b, 2016;
Bachelet et al. , 2018;
Chen et al. ,2019] and theoretical [
Werner and Haff , 1988;
Anderson and Haff , 1988, 1991;
Haff andAnderson , 1993;
McElwaine et al. , 2004;
Oger et al. , 2005, 2008;
Zheng et al. , 2005, 2008;
Namikas , 2006;
Crassous et al. , 2007;
Bourrier et al. , 2008;
Kok and Renno , 2009;
Valanceand Crassous , 2009;
Ho et al. , 2012;
Duan et al. , 2013b;
Xing and He , 2013;
Comola andLehning , 2017;
Huang et al. , 2017;
Tanabe et al. , 2017;
Lämmel et al. , 2017] studies in orderto better understand aeolian saltation transport and other geophysical phenomena (e.g., rock-fall [
Bourrier et al. , 2008;
Bachelet et al. , 2018]); see also [
White and Schulz , 1977;
Willettsand Rice , 1986, 1989;
McEwan et al. , 1992;
Nalpanis et al. , 1993;
Rice et al. , 1995, 1996;
Dong et al. , 2002;
McElwaine et al. , 2004;
Gordon and McKenna Neuman , 2009, 2011] forcollision statistics during ongoing aeolian saltation transport. In typical experiments, a spher-ical incident particle of diameter d and mass m is shot (e.g., by an airgun) at a given speed v i and angle θ i onto a static packing of spheres of the same size. As shown in Figure 9 andsketched in Figure 10, as a result of its impact on the packing, the incident particle may re-bound (velocity v r , angles θ r , φ r ) and/or eject bed particles into motion (number N e , veloc-ity v e , angles θ e , φ e ), where a particle is typically counted as ejected if its center is liftedby more than d above the top of the bed surface. The statistics of this process has been the Figure 9.
From
Beladjine et al. [2007], high-speed images of the impact of an incident particle on a staticgranular packing. The time step between two successive images is 4 ms. Copyright 2007 American PhysicalSociety. subject of several recent experimental and numerical studies [e.g.,
Beladjine et al. , 2007;
Ammi et al. , 2009;
Tanabe et al. , 2017] (note that experimental studies that used only onecamera measured quantities projected into the incident plane: v Dr ( e ) ≡ (cid:113) v r ( e ) x + v r ( e ) z andtan θ Dr ( e ) ≡ tan θ r ( e ) / cos φ r ( e ) ). These studies have yielded the following insights: –26–onfidential manuscript submitted to Reviews of Geophysics zEjected V x � r O � i Vr Incident Vi � r θϕ y particleparticle eee Figure 10.
Sketch of collision process. (i) The incident particle loses much more energy in head-on than in grazing collisions.In fact, the average restitution coefficient and its two-dimensional projection obey the follow-ing empirical relationships for 10 ◦ ≤ θ i ≤ ◦ : e ≡ | v r |/| v i | = A − B sin θ i , (12a) e D ≡ v Dr /| v i | = A D − B D sin θ i , (12b)where the overbar denotes an ensemble average over collision experiments, and the A and B coefficients are empirical constants that vary slightly between the studies (e.g., A ≈ A D ≈ . B ≈ .
62 [
Ammi et al. , 2009], and B D ≈ .
72 [
Beladjine et al. , 2007]).(ii) The average vertical restitution coefficient exceeds unity at small impact angles andobeys the following empirical relationship for 10 ◦ ≤ θ i ≤ ◦ : e z ≡ v rz / v iz = A z / sin θ i − B z , (13)where A z ≈ . B z ≈ .
15 for the experiments by
Beladjine et al. [2007].
Pähtz et al. [2020] suggested the following modification of equation (13): e z = A D / (cid:112) sin θ i − B D . (14)This modification, which is also consistent with the experimental data, ensures the correctasymptotic behavior of the average rebound angle, θ r ∼ √ θ i [ Lämmel et al. , 2017], in thelimit θ i → ◦ ≤ θ i ≤ ◦ : θ r = θ + χθ i , (15a)sin θ Dr = e z sin θ i / e D , (15b)where θ ≈ ◦ and χ ≈ .
19 for the experiments by
Ammi et al. [2009].(iv) The average energy that the incident particle transfers to the bed is spent for theejection of bed particles. That is, it is proportional to the average of the sum of the kineticenergy of ejected particles ( E e = m v e and E De = m v D e ). In fact, the following empiri-cal relationships are obeyed for 10 ◦ ≤ θ i ≤ ◦ : N e E e = r ( − e ) E i , (16a) N e E De = r D ( − e D ) E i , (16b) –27–onfidential manuscript submitted to Reviews of Geophysics where r ≈ .
04 and r D ≈ .
038 for the experiments by
Ammi et al. [2009]. Note that r and r D decrease with the coordination number of the particle packing [ Rioual et al. , 2003].(v) The average number of ejected particles is a linear function of the incident speedfor 10 ◦ ≤ θ i ≤ ◦ : N e = n ( − e )[| v i |/( ζ (cid:112) g d ) − ] (cid:39) n ( − e D )[| v i |/( ζ (cid:112) g d ) − ] , (17)where n ≈
13 and ζ ≈
40 for the experiments by
Ammi et al. [2009]. Note that n decreaseswith the coordination number of the particle packing [ Rioual et al. , 2003].(vi) The average horizontal and lateral velocities of ejected particles are nearly inde-pendent of the incident velocity, but the average vertical velocity increases slightly with theincident velocity and is independent of the impact angle for 10 ◦ ≤ θ i ≤ ◦ [ Ammi et al. ,2009]: v rx ≈ v ry ≈ g d , (18a) v rz / (cid:112) g d ≈ . (| v i |/ (cid:112) g d ) / , (18b) v rz / g d ≈ . (| v i |/ (cid:112) g d ) / . (18c)(vii) The average ejection angle θ e is constant for 10 ◦ ≤ θ i ≤ ◦ [ Ammi et al. , 2009].However, its projection into the incident plane increases with the impact angle [
Beladjineet al. , 2007]: θ De ≈ π + . (cid:16) θ i − π (cid:17) . (19) Open Problem: Behavior of the Rebound Probability
Mitha et al. [1986] measured that about 94% of all impacting particles are not capturedby the bed (i.e., they successfully rebound). However, the range of impact velocities in theirexperiments was very narrow ( | v i | ∈ ( , )√ g d ). More systematic measurements of therebound probability P r are needed.Studies have attempted to physically describe both the rebound [ Zheng et al. , 2005,2008;
Namikas , 2006;
Lämmel et al. , 2017] and ejection dynamics [
McElwaine et al. , 2004;
Crassous et al. , 2007;
Kok and Renno , 2009;
Valance and Crassous , 2009;
Ho et al. , 2012;
Comola and Lehning , 2017;
Lämmel et al. , 2017]. For example, the rebound dynamics canbe analytically calculated for an idealized packing geometry and a given rebound location as-suming a binary collision between the incident particle and hit bed particle. From averagingover all possible rebound locations, one can then determine the rebound angle and restitutioncoefficient distributions. Using this procedure,
Lämmel et al. [2017] derived the followingexpressions for e D , e z , θ Dr , and P r in the limit of shallow impact angles ( θ i (cid:46) ◦ ): e D = β r − ( β r − α r ) θ i /( β r ) , (20) e z = − β r + ( / )( α r + β r ) (cid:112) / θ i , (21) θ Dr = e z θ i / e D ≈ ( / )( + α r / β r ) (cid:112) θ i − θ i , (22) P r = − + ln ξξ , with ξ ≡ max (cid:34) , √ ( + α r / β r ) θ i v i √ g d (cid:35) , (23)where α r and β r are the normal and tangential rebound restitution coefficients, respectively,in the impact plane, which depend on the binary normal and tangential restitution coeffi-cient (i.e., the ratio between the postcollisional and precollisional relative particle velocitycomponent normal and tangential, respectively, to the contact plane). Figure 11 comparesequations (20)–(22) with the experimental data by Beladjine et al. [2007] using the values α r = . β r = .
63, which
Lämmel et al. [2017] obtained from fitting the numeri-cal solution of the full problem (i.e., not limited to θ i (cid:46) ◦ ) to the experimental data. The –28–onfidential manuscript submitted to Reviews of Geophysics
Figure 11.
Test of the analytical expressions by
Lämmel et al. [2017] that describe the particle reboundof an impacting particle in the limit of shallow impact angles ( θ i (cid:46) ◦ ). (a) Average rebound restitutioncoefficient e D , (b) average vertical rebound restitution coefficient e z , and (c) average rebound angle θ r versusimpact angle θ i . Symbols correspond to experimental data by Beladjine et al. [2007]. Solid lines correspondto equations (20)-(22). agreement with the data with θ i (cid:46) ◦ is acceptable considering that the theory has beenderived mostly from first physical principles. Equation (23), which is the modified versionof equations (41) and (42) of Lämmel et al. [2017] that these authors describe in their text,cannot be tested because of the lack of systematic measurements of the rebound probability P r . A widely used alternative expression for P r was given by Anderson and Haff [1991]: P r ≈ . [ − exp (− γ r | v i |)] . However, this expression is empirical and contains the dimen-sional parameter γ r (note that Andreotti [2004] assumed γ r ∝ /√ g d ). Because e z (cid:38) Kok and Renno , 2009;
Comola and Lehning , 2017] and treating thecollision process as a sequence of binary collisions, in which the energy is split betweenthe collisional partners (i.e., incident and bed particle or two bed particles) [
McElwaineet al. , 2004;
Crassous et al. , 2007;
Valance and Crassous , 2009;
Ho et al. , 2012;
Lämmelet al. , 2017]. A minimal numerical model that is based on the latter approach has been ableto reproduce experimental data of both the rebound and ejection dynamics, including themeasured log-normal distribution of the vertical ejection velocity [
Crassous et al. , 2007].Furthermore, based on this approach and the derivation by
Ho et al. [2012],
Lämmel et al. [2017] derived the following analytical expression for the distribution of the ejection energy E e : P ( E e ) = √ πσ E e exp (cid:20) − ( ln E e − µ ) σ (cid:21) , with (24) σ = √ λ ln 2 ,µ = ln [( − e ) E i ] − λ ln 2 ,λ = [( − e ) E i /( m g d )] , –29–onfidential manuscript submitted to Reviews of Geophysics from which they further obtained expressions for N e , E e , and | v e | : N e = r ( − e ) E i E e erfc (cid:20) ln ( m g d ) − µ √ σ (cid:21) , (25) E e = m g d [( − e ) E i /( m g d )] −( − ln 2 ) ln 2 , (26) | v e | = erfc {[ ln ( m g d ) − µ − σ / ]/(√ σ )} erfc {[ ln ( m g d ) − µ ]/(√ σ )} √ ( µ / + σ / ) , (27)where r = .
06. Figure 12 shows that these expressions are roughly consistent with exper-imental data considering that they have been derived mostly from first physical principles.Note that equations (20)–(27), after some minor modifications, can also be applied to sit-
Figure 12.
Test of the analytical expressions by
Lämmel et al. [2017] that describe the ejection of bedsurface particles caused by the splash of an impacting particle. (a, c) Nondimensionalized average ejectionvelocity | v | and (b, d) average number of ejected particles N e versus (a, b) impact angle θ i and (c, d) nondi-mensionalized impact velocity | v i |/√ g d . Symbols correspond to experimental data by Beladjine et al. [2007].Solid lines correspond to equations (25) and (27) combined with the approximation | v e | (cid:39) | v | . uations in which the size of the impacting particle differs from the size of the particles ofthe granular packing [ Lämmel et al. , 2017]. Further note that equation (38) of
Lämmel et al. [2017], which is the equivalent of equation (27), contains a typo (a σ is missing in the de-nominator). Open Problem: Impacts Onto Mobile Beds
The findings from collision experiments with static beds are often applied to modelfluvial bedload [
Berzi et al. , 2016;
Pähtz et al. , 2020] and aeolian saltation transport [
An-dreotti , 2004;
Claudin and Andreotti , 2006;
Creyssels et al. , 2009;
Kok and Renno , 2009;
Kok , 2010a;
Jenkins et al. , 2010;
Lämmel et al. , 2012;
Pähtz et al. , 2012;
Huang et al. , 2014;
Jenkins and Valance , 2014, 2018;
Wang and Zheng , 2014, 2015;
Berzi et al. , 2016, 2017;
Boet al. , 2017;
Lämmel and Kroy , 2017;
Pähtz et al. , 2020]. However, if the time between suc-cessive particle-bed impacts is too short for a bed particle to fully recover from each impact,it can accumulate more and more kinetic energy with each impact. Hence, for a sufficientlylarge impact frequency and impact energy (both increase with the sediment transport rate Q ),the bed can no longer be treated as static and the findings from such collision experimentsmay no longer apply. For example, the simultaneous impact of two particles onto the bed –30–onfidential manuscript submitted to Reviews of Geophysics leads to a significantly different outcome compared with the situation in which each parti-cle impacts separately [
Duan et al. , 2013b]. For these reasons, future studies should try tosystematically investigate the effects of disturbances from the static bed on the outcome of aparticle-bed impact.
Open Problem: Effects of Particle Shape and Size Distribution
Chen et al. [2019] investigated the particle-bed collision process for natural sand par-ticles, which exhibit nonspherical shapes and nonuniform particle size distributions. Theyfound significant quantitative and qualitative deviations from the laws describing spherical,uniform particles. More systematic experimental studies are needed to pinpoint the exactmanner in which particle shape and size distribution affect the collision process.
Controversy: Effects of Viscous Damping
Binary collisions that occur within an ambient fluid can be significantly damped de-pending on the Stokes number St ≡ s | v r | d /( ν f ) [ Gondret et al. , 2002;
Yang and Hunt , 2006;
Schmeeckle , 2014;
Maurin et al. , 2015], where v r is the relative particle velocity just beforea collision. For example, experiments suggest that the effective normal restitution coefficient (cid:15) of a damped binary collision vanishes for St (cid:46)
10 [
Gondret et al. , 2002]. The questionthat then arises is how does viscous damping affect the rebound and ejection dynamics ofa particle-bed impact.
Berzi et al. [2016, 2017] assumed that the rebound restitution coeffi-cients e D and e z , like (cid:15) , also vanish when St falls below a critical value. In contrast, DEM-based simulations indicate that the dynamics of saltation [ Pähtz and Durán , 2018a] and par-ticularly bedload transport [
Drake and Calantoni , 2001;
Maurin et al. , 2015;
Elghannay andTafti , 2017;
Pähtz and Durán , 2017, 2018a,b] are not much affected by the value of (cid:15) , whichsuggests that the rebound and ejection dynamics of a particle-bed impact may not be muchaffected by viscous damping. A possible explanation for this unexpected behavior could bethat a nearly elastic particle-bed impact may be roughly equivalent to a sequence of binarycollisions between particles in contact at the instant of impact. In fact, a theoretical modelbased on this hypothesis reproduced experiments of the collision process [
Crassous et al. ,2007;
Valance and Crassous , 2009]. For the perfectly elastic case ( (cid:15) = (cid:15) = (cid:15) , which would imply that the momentum inthe direction tangential to the contact plane is what mainly matters. Collision experiments inan ambient viscous liquid could resolve this controversy. Open Problem: Effects of Cohesion
Cohesive interparticle forces, including van der Waals [
Castellanos , 2005], water ad-sorption [
Herminghaus , 2005], and electrostatic forces [
Lacks and Sankaran , 2011], becomesignificant in the collision process for sufficiently small particles (on Earth, for d (cid:46) µ m)because they scale with a lower power p in the particle diameter ( F coh ∼ d p ) than the gravityforce ( F g ∼ d ). However, collision experiments with so small particles have not been car-ried out because it is very difficult to detected their dynamics with cameras. Numerical stud-ies are also very scarce. To the authors’ knowledge, only the very recent study by Comolaet al. [2019a] studied cohesive forces, by implementing them in a numerical DEM-basedmodel of aeolian saltation transport. These authors investigated the impact of a particle ontothe bed for a large range of the strength of cohesive forces and found that cohesion decreases N e via solidifying the bed, while e slightly and | v e |/| v i | considerably increase. However,more systematic studies are needed to confirm these results and determine scaling laws de-scribing the effects of cohesion on the outcome of a particle-bed impact. –31–onfidential manuscript submitted to Reviews of Geophysics
To the authors’ knowledge, only a single study has resolved the effects of particle-bedimpacts on entrainment by turbulent fluctuation events in bedload transport [
Vowinckel et al. ,2016]. However, this study provided one of the largest, if not the largest, data sets of entrain-ment events associated with fluvial bedload transport with a very high resolution in spaceand time.
Vowinckel et al. [2016] coupled direct numerical simulations (DNS) for the fluidphase (i.e., the Navier-Stokes equations are directly solved without using turbulent closureassumptions) with DEM simulations for the particle phase (i.e., particles interact with eachother according to a contact model) using the immersed boundary method, which fully re-solves the geometry of particles (and thus the hydrodynamic forces acting on them) withoutremeshing the grid during their motion [
Vowinckel et al. , 2014]. Because of the sophistica-tion of this numerical method (i.e., resolving all relevant physical processes at very smallscale), the produced data can be considered to be very reliable. The simulated setup con-sisted of two layers of grains resting on the simulation bottom wall, the lower of which wasfixed, arranged in a hexagonal packing, and exposed to a unidirectional open channel flow ofthickness H = d (Reynolds number Re ≡ U b H / ν f = U b is the bulk flow ve-locity). The Shields number was at Θ = . Q ∗ was likely be-low the value associated with critical transport conditions (see section 1), which is consistentwith Vowinckel et al. [2016] reporting that only 3% of all particles were in motion on aver-age. For these conditions, it was found that, in the vast majority of cases (overall 96 . y p ) and (b) velocity of a bed surface particle ( u p ), whileFigure 14 shows the simulation domain and contour plots of the instantaneous flow field. It a b Figure 13.
From
Vowinckel et al. [2016], time evolution of a typical erosion event. At time instant A , a bedsurface particle is at rest. At time instant B , it is hit by an impacting transported particle. The impact causesa slight dislocation off its initial position. Once slightly lifted, the particle protrudes into the flow, enhancingthe flow forces acting on it. This enhancement in combination with much-larger-than-average flow velocitiesduring a turbulent fluctuation event (Figure 14) leads to entrainment (time instant C ), as indicated by thenondimensionalized (a) vertical displacement ( y p / H ) and (b) particle velocity ( u p / U b ) exceeding criticalvalues (dashed lines). Copyright 2016 Taylor & Francis Group. can be seen that, at the instant of entrainment, the instantaneous streamwise flow velocity ( u )exhibits larger-than-average values (Figure 14c). In fact, Vowinckel et al. [2016] reported that82% of the entrainment events were caused by sweep , characterized by positive fluctuations –32–onfidential manuscript submitted to
Reviews of Geophysics
A B C u / U b Figure 14.
From
Vowinckel et al. [2016], zoom into the simulation domain and contour plots of the instan-taneous streamwise flow field ( u / U b ) during a typical erosion event of a target particle (red). The color (gray,white, and black) corresponds to (nonerodible, resting, and transported) particles. Time instants A , B , and C are as in Figure 13. Copyright 2016 Taylor & Francis Group. of u and negative fluctuations of the flow velocity component in the direction normal to thebed. The results by Vowinckel et al. [2016] were obtained for an idealized hexagonal pack-ing and may not necessarily apply in their full extent to realistic sediment beds found in na-ture. While for a hexagonal packing, the vast majority of entrainment events are initiatedby particle-bed impacts, it remains unclear whether this holds true also for natural sedimentbeds, in which bed surface particles tend to protrude much more strongly into the flow. Onthe one hand, a larger protrusion makes it easier for a turbulent fluctuation event to entraina bed surface particle without a preceding particle-bed impact. On the other hand, particle-bed impacts can result in entrainment without the need of a turbulent fluctuation event (seesection 4.1.3).
Pähtz and Durán [2017] numerically studied the role of particle-bed impacts in sus-taining continuous nonsuspended sediment transport for transport conditions characterizedby a large range of the Shields number Θ , density ratio s , and Galileo number Ga. These au-thors coupled quasi-two-dimensional DEM simulations for the particle phase with a Reynolds-averaged description of the fluid hydrodynamics that neglects turbulent fluctuations aroundthe mean turbulent flow. While such simulations cannot resolve entrainment by turbulentfluctuation events, they are able to elucidate the importance of entrainment by particle-bedimpacts relative to entrainment by the mean turbulent flow. Also, the absence of turbulentfluctuations eliminates transport intermittency in the sense that transport in the simulationdomain is either continuous (i.e., periods of rest are absent) or it completely stops after a fi-nite time (except for potential creeping, see section 2.3). From their simulations, Pähtz andDurán [2017] determined an effective value of the local particle velocity averaged over ele-vations near the bed surface ( V b ) relative to the critical velocity that is needed to escape thepotential wells set by the pockets of the bed surface ( ∝ √ ˆ g d , where ˆ g = [ + /( s + C m )] is thevalue of the gravity constant reduced by the buoyancy and added mass force, with C m = / V b /√ ˆ g d exhibits a universal approximately con-stant value of order unity for continuous nonsuspended sediment transport if the following –33–onfidential manuscript submitted to Reviews of Geophysics constraint is obeyed: Im ≡ Ga (cid:112) s + C m (cid:38)
20 or Θ (cid:38) / Im . (28)The interpretation of V b /√ ˆ g d ≈ const is that particles located near the bed surface (whichincludes both particles of the bed and transported particles) are on average at the verge ofleaving it or being captured by its potential wells, consistent with a dynamic equilibrium thatis solely controlled by particle inertia. This implies that entrainment occurs solely due to theaction of particle-bed impacts. Consistently, Pähtz and Durán [2017] observed from visuallyinspecting simulations that obey equation (28) that every entrainment event is initiated by aparticle-bed impact, usually with a small time delay between the instant of impact and be-ginning visible motion. In contrast, for transport conditions that do not obey equation (28), V b /√ ˆ g d exhibits a smaller value, which means that the mean turbulent flow must assist parti-cles located near the bed surface in escaping the potential wells. For bedload transport, thefindings by Pähtz and Durán [2017] were independent of the effective normal restitutioncoefficient (cid:15) for a damped binary collision, which indicates that viscous damping does notsuppress impact entrainment (see also the discussion of viscous damping in section 4.1.1).The constraint set by equation (28) is obeyed by the vast majority of sediment trans-port regimes, including turbulent fluvial bedload transport. That is, for the absence of turbu-lent fluctuation events, only viscous fluvial bedload transport is significantly affected by theentrainment of bed sediment by the mean turbulent flow. The numerical prediction that im-pact entrainment dominates entrainment by the mean turbulent flow in turbulent fluvial bed-load transport is supported by experiments [
Heyman et al. , 2016;
Lee and Jerolmack , 2018].
Lee and Jerolmack [2018] studied bedload transport driven by a water flow in a quasi-two-dimensional flume (i.e., its lateral dimension was only slightly larger than the particle di-ameter d ). Because the size of turbulent structures, and thus turbulent fluctuation events,is strongly suppressed when the system dimensions are so strongly narrowed down, theirexperiments are somewhat comparable to the numerical simulations by Pähtz and Durán [2017] described above.
Lee and Jerolmack [2018] fixed the water discharge and fed parti-cles at the flume entrance with varying frequency f in (the tested range of f in was likely asso-ciated with a transport rate below capacity). In contrast to similar older experiments [ Böhmet al. , 2004;
Ancey et al. , 2008;
Heyman et al. , 2013], the bed was relatively deep, which en-sured the complete dissipation of shock waves associated with particle-bed impacts [
Rioualet al. , 2003].
Lee and Jerolmack [2018] reported that, for all tested conditions, every entrain-ment event is initiated by a particle-bed impact, exactly as numerically predicted, and thatthe number of transported particles roughly scales with the energy transferred to the bed byrebounding particles. The latter finding is remarkably similar to the scaling of the averageejected particle number N e in static bed experiments (e.g., see equation (25)). Lee and Jerol-mack [2018] also measured the frequency of particles passing an illuminated window nearthe flume exit ( f out ). They found that f out < f in for sufficiently small f in and that f out ≈ f in once f in exceeds a critical value.Similar observations were made by Heyman et al. [2016], who used a water flume witha narrow but larger width ( W = d ) than Lee and Jerolmack [2018] and who also used a rela-tively deep bed.
Heyman et al. [2016] measured that the entrainment rate was proportional tothe number of transported particles per unit bed area, which is indirect evidence supportingthat the majority of entrainment events is caused by particle-bed impacts. These authors alsoreported for all their tested feeding frequencies f in that the entrainment and deposition rateare equal to one another, in resemblance of the measurement f out ≈ f in for sufficiently large f in by Lee and Jerolmack [2018]. Note that one expects the approximate equality f out ≈ f in to break down for large f in (when the influx exceeds transport capacity) because increasingmomentum transfer from fluid to particles slows down the flow, which at some point can nolonger sustain the particle motion.The results by Heyman et al. [2016] and
Lee and Jerolmack [2018] suggest that mainly(but not solely) particles that were previously in motion are being entrained by particle-bedimpacts. Otherwise, there would be no reason to expect that the entrainment and deposition –34–onfidential manuscript submitted to
Reviews of Geophysics rate are relatively equal to one another for a large range of f in (instead, one would expect thatonly for transport capacity). This can be explained when assuming that particle-bed impactsare effective in mobilizing a bed particle almost only when the bed particle exceeds a criti-cal energy level just before the impact. On the one hand, this assumption would explain whybed particles that have never been transported only rarely become mobilized by particle-bedimpacts. On the other hand, this assumption is consistent with the fact that a transported par-ticle that has just been captured by a bed pocket exhibits a residual kinetic energy that takessome time to be completely dissipated, during which it can be remobilized by an impact froma particle coming from behind. It seems that, once f in exceeds a critical value, there is usu-ally a particle coming from behind in time and transported particles can only rarely settlecompletely even though they may temporarily stop. Temporary particle stops and reentrain-ment make transported particles tend to move in clusters near the flume exit even though theyare apart from one another at the flume entrance, which is exactly what Lee and Jerolmack [2018] reported and what can be observed in the numerical simulations by
Pähtz and Durán [2018a, Movie S2].There is evidence that the presumed impact entrainment mechanism described abovemay play an important role in nonsuspended sediment transport in general. In fact, in simula-tions of steady, homogenous sediment transport using DEM-based numerical models that ne-glect turbulent fluctuations around the mean turbulent flow , the steady state transport rate Q exhibits a discontinuous jump at a fluid shear stress τ ImE t [ Carneiro et al. , 2011, 2013;
Clarket al. , 2015a, 2017;
Pähtz and Durán , 2018a]. That is, for τ ≥ τ ImE t , transport is significantlylarger than zero ( Q >
0) and continuous, whereas Q (cid:39) τ < τ ImE t . Assuming thatonly impact entrainment took place in all these simulations (as the mean turbulent flow is tooweak for entrainment, see above), τ ImE t can be identified as the impact entrainment thresh-old. The discontinuous jump of Q thus means that, in order for impact entrainment to sus-tain transport, a critical transport rate must be exceeded. Like the critical feeding frequencyin the experiments by Lee and Jerolmack [2018], this critical transport rate may be inter-preted as the value above which most transported particles can be captured only temporarilyby bed pockets as they are usually hit in time and thus reentrained by an impact from a par-ticle coming from behind before dissipating too much of their kinetic energy. However, it iscrucial to point out that impact entrainment of bed particles that have never been transportedoccasionally occurs in DEM-based sediment transport simulations as well, which is why afurther interpretation of the physical origin of the discontinuous jump of Q has been pro-posed [ Pähtz and Durán , 2018a]. It states that, at a critical transport rate, bed surface parti-cles do no longer sufficiently recover between successive particle-bed impacts. They thus ac-cumulate energy between successive impacts until they are eventually entrained. In contrast,for subcritical transport rates, particles sufficiently recover between impacts so that impactentrainment is inefficient, causing transport to eventually stop. The two interpretations aboveare based only on the energy of bed particles or temporarily captured transported particles.In contrast, in the context of an idealized continuous rebound modeling framework (see sec-tion 4.2), an alternative mechanism based on the critical amount of energy E c that bed parti-cles need to acquire for entrainment (more precisely, for entering a quasi-continuous motion)can explain the discontinuous jump of Q without further assumptions (see section 4.2.1). Open Problem: Precise Mechanism of Impact Entrainment in Continuous Transport
The proposed impact entrainment mechanisms described above and in section 4.2.1 aremostly speculative and based on indirect experimental or theoretical evidence, or idealizedmodels. More direct investigations are therefore needed to uncover the precise nature of im-pact entrainment and the degree to which each of these mechanisms contributes. Such inves-tigations may also help to better understand fluctuations of nonsuspended sediment transport.For example, the longer the average time t conv it takes for transport to stop (in the absence ofturbulent fluctuations around the mean turbulent flow) when τ < τ ImE t ( t conv obeys a criticalscaling behavior at τ ImE t , see section 2.4), the larger are the transport autocorrelations, which –35–onfidential manuscript submitted to Reviews of Geophysics can be quite substantial in fluvial bedload transport [
Heathershaw and Thorne , 1985;
Drakeet al. , 1988;
Dinehart , 1999;
Ancey et al. , 2006, 2008, 2015;
Martin et al. , 2012].
Open Problem: Precise Definitions of Intermittent and Continuous Transport
As explained above, in simulations of steady, homogenous sediment transport usingDEM-based numerical models that neglect turbulent fluctuations, the steady transport rate Q (in a time-averaged sense) exhibits a discontinuous jump at the impact entrainment thresh-old τ ImE t . In contrast, for most natural conditions, fluid entrainment by turbulent events canreinitiate transport whenever it temporarily stops, meaning that Q remains significant be-low τ ImE t [ Carneiro et al. , 2011]. Hence, since turbulent events capable of fluid entrainmentoccur only at an intermittent basis (see section 3),
Pähtz and Durán [2018a] suggested that τ ImE t is equivalent to the continuous transport threshold for most natural conditions and thattransport becomes intermittent below τ ImE t . However, provided that fluid entrainment doesoccur, it is certain to find particles being in transport below τ ImE t at any given instant in timein the large-system limit, which renders the distinction between intermittent and continuoustransport somewhat ambiguous. For this reason, Pähtz and Durán [2018a] referred to inter-mittent conditions as those that deviate significantly from transport capacity (defined as insection 4.2.2). Consistently,
Martin and Kok [2018] and
Comola et al. [2019b] found fromaeolian field experiments that the long-term-averaged transport remains at capacity when thefraction f Q of active saltation transport is close to unity, that is, when transport quantifiedover a short but somewhat arbitrary time interval (2 s [ Martin and Kok , 2018] or 0 .
04 s [
Co-mola et al. , 2019b]) almost never stops. Interestingly,
Comola et al. [2019b] showed that thevalue of f Q can be indirectly estimated from the lowpass-filtered wind speed associated withlarge and very large scale turbulent structures (cutoff frequency Ω ≈ .
04 Hz). Alternatively,for their coupled DNS/DEM simulations of fluvial bedload transport,
González et al. [2017]fitted continuous functions to the distributions of the discrete transported particle number(defined as the number of particles faster than a somewhat arbitrary velocity threshold) atdifferent τ and identified the onset of continuous transport as the value of τ at which thesefitting functions predict a zero probability for a vanishing particle number. Future studiesshould investigate the compatibility of these and other definitions of continuous transport. Controversy: Threshold of Continuous Aeolian Saltation Transport
In the opinion of the authors, the evidence reviewed above for the hypothesis that con-tinuous transport occurs once impact entrainment alone is sufficient in compensating randomcaptures of transported particles is quite strong. (In other words, significant fluid entrainmentmay occur in continuous transport—and does so quite likely in aeolian saltation transportgiven that the turbulent intensity within the saltation transport layer increases with the sedi-ment transport rate [
Li and McKenna Neuman , 2012]—but it is not needed to sustain contin-uous transport.) However, it is worth pointing out that most aeolian researchers prefer a dif-ferent narrative for aeolian saltation transport. For example,
Martin and Kok [2018] assumedthat continuous aeolian saltation transport in the field occurs once the saltation transport ini-tiation threshold ( ≈ τ In t ) is exceeded, whereas the impact entrainment threshold describesthe cessation of intermittent saltation transport. This assumption is based on the idea thatfluid entrainment continuously provides the transport layer with bed particles. However, thisidea is problematic because turbulent events capable of fluid entrainment occur only at an in-termittent basis (see section 3). The interested reader is also referred to the commentary by Pähtz [2018], in which this controversy is extensively discussed.
In order for the mean turbulent flow to sustain the motion of particles that are alreadyin transport, it needs to compensate, on average, the energy dissipated in particle-bed re-bounds via drag acceleration during the particle trajectories. This mechanism, which is illus- –36–onfidential manuscript submitted to
Reviews of Geophysics trated in detail by means of a thought experiment in section 4.2.1, gives rise to a shear stressthreshold of sediment transport (henceforth termed rebound threshold ), as was already notedby
Bagnold [1941, p. 94] for aeolian saltation transport: “Physically [the rebound thresh-old] marks the critical stage at which the energy supplied to the saltating grains by the windbegins to balance the energy losses due to friction when the grains strike the ground [and re-bound].” It also suggests a clear-cut definition of transport capacity, which is otherwise diffi-cult to define [see review by
Wainwright et al. , 2015, and references therein], that leads to anexperimentally and numerically validated universal scaling of the transport load M (i.e., themass of transported sediment per unit bed area) with the fluid shear stress τ (section 4.2.2).From the appearance of the rebound threshold in this scaling of M , one can conclude that ata significant if not predominant portion of the threshold measurements by Shields [1936] andothers have been misidentified as measurements of the entrainment threshold (section 4.2.3).
To illustrate the concept of continuous particle rebounds, the motion of a particle alonga flat wall driven by a constant flow (e.g., the mean turbulent flow) is considered. This par-ticle shall never be captured and instead, for illustration purposes, always rebound with aconstant angle and lose a constant fraction of its impact energy (the core of the argument willnot significantly change if more sophisticated rebound laws, such as equations (20)–(22), areconsidered). For this idealized scenario, there are two extremes of possible particle trajecto-ries depending on the initial particle velocity v ↑ , which are sketched in Figure 15. First, if the Figure 15.
Sketch of continuous rebound mechanism. Depending on its initial kinetic energy E ↑ relative toa critical energy level E c that depends on the properties of the flow, a particle (yellow lines) either (a) gainssufficient energy in its hops along a flat wall (black lines) to approach a steady, periodic hopping motion or (b)net loses energy until it stops. corresponding initial kinetic energy E ↑ exceeds a critical value E c , the particle will spendsufficiently long within the flow so that it gains sufficient energy via fluid drag during itshops to approach a steady, periodic hopping motion, in which its energy gain via fluid dragis exactly balanced by its energy loss during its rebounds (Figure 15a). Henceforth, such par-ticles are termed continuous rebounders . Second, if E ↑ < E c , the particle loses net energy inits initial and all subsequent hops until it stops (Figure 15b). The critical energy E c dependson properties of the flow. Crucially, if the flow is too weak, all possible trajectories fall intothe second category (i.e., E c = ∞ ).There are a few takeaways from the this simple thought experiment for realistic sys-tems. First, as the mean turbulent flow is controlled by the fluid shear stress τ , it suggeststhe existence of a rebound threshold τ Rb t below which the energy losses in particle-bed re-bounds cannot be compensated by the flow on average [ Jenkins and Valance , 2014;
Berziet al. , 2016, 2017;
Pähtz and Durán , 2018a;
Pähtz et al. , 2020]. Second, the randomness –37–onfidential manuscript submitted to
Reviews of Geophysics introduced by inhomogeneities of the bed and turbulent fluctuations of the flow introducetrajectory fluctuations that can lead to random losses of continuous rebounders, particularlywhen the lift-off energy accidentally falls below E c [ Pähtz and Durán , 2018a]. Such lossesmust be compensated by the entrainment of bed particles into the continuous rebound layer.Hence, the mere mobilization of bed particles is not sufficient because the lift-off energy ofmobilized particles must also exceed E c . In particular, for rebound threshold models (seesection 4.3), it has been shown that E c becomes equal to the average rebound energy of con-tinuous rebounders in the limit τ → τ Rb t [ Pähtz et al. , 2020]. This implies that the impactentrainment threshold τ ImE t must be strictly larger than τ Rb t , since the energy of an entrainedparticle is much smaller than the energy of the particle that caused its entrainment (i.e., acontinuous rebounder) because of energy conservation. In particular, τ ImE t > τ Rb t auto-matically explains the discontinuous jump of the sediment transport rate Q at τ ImE t that hasbeen observed in the absence of fluid entrainment by turbulent fluctuation events (see sec-tion 4.1.3) because Q ( τ ImE t ) is controlled by the excess shear stress τ ImE t − τ Rb t > A third takeaway for realistic systems of the thought experiment described in sec-tion 4.2.1 involves the fact that, because of momentum transfer from flow to particles, theflow slows down with increasing transport load M . Hence, for a given τ > τ Rb t , providedthat there is an abundance of impact and/or fluid entrainment, the system tends to entrainbed material until the mean turbulent flow becomes so weak that it can barely sustain the av-erage motion of continuous rebounders [ Pähtz and Durán , 2018b]. Any further slowdownof the flow would then spike the deposition rate, leading to a decrease of M and subsequentincrease of the flow speed. That is, the system is at a dynamic equilibrium that may be inter-preted as transport capacity. Pähtz and Durán [2018b] analytically showed that this interpretation of transport ca-pacity leads to the capacity scaling M (cid:39) µ − b ˜ g − ( τ − τ Rb t ) , (29)where ˜ g = ( − / s ) g is the buoyancy-reduced value of the gravitational constant g and µ b = τ pb / P b an approximately constant bed friction coefficient (i.e., the ratio between theparticle shear stress τ pb and normal-bed particle pressure P b (cid:39) M ˜ g evaluated at the bedsurface). Note that the definitions of τ pb and P b (and thus µ b ), in contrast to the definitionsof τ p and P (and thus the yield stress ratio µ s , see section 2.1), include contributions fromstresses associated with the particle fluctuation motion in addition to contributions from in-tergranular contacts. The derivation of equation (29) by Pähtz and Durán [2018b] is basedon two main steps: showing the approximate constancy of µ b starting from a geometric con-straint on particle-bed rebounds in the steady state and assuming τ gb (cid:39) τ − τ Rb t , which ex-presses the aforementioned dynamic equilibrium condition associated with the continuousrebound motion. Interestingly, τ gb describes the momentum that is transferred from flow totransported particles per unit bed area per unit time, which implies that high-buoyant flu-ids (small ˜ g ), such as water, require a larger transport load M for a given rate of momentumtransfer (i.e., for a given M ˜ g ∝ τ gb ) than low-buoyant fluids (large ˜ g ), such as air. Pähtz andDurán [2018b] tested these derivation steps with numerical data from DEM-based simula-tions of nonsuspended sediment transport (the same as those by
Pähtz and Durán [2017],see section 4.1.3). It turned out that these steps, and thus equation (29), are obeyed acrossnonsuspended sediment transport conditions with Ga √ s (cid:38)
10 (all but relatively viscous bed-load transport) provided that the bed surface is defined as the effective elevation of energeticparticle-bed rebounds.The functional form of equation (29) is the foundation of the majority of theoreticaland experimental shear stress threshold-based expressions for the capacity transport rate, Q (cid:39) M v x (where v x is the average streamwise velocity of particles moving above the bed –38–onfidential manuscript submitted to Reviews of Geophysics surface), and goes back to the pioneering theoretical descriptions of nonsuspended sedi-ment transport by
Bagnold [1956, 1966, 1973]. However, Bagnold’s physical interpretationof the assumptions leading to this scaling was inaccurate: µ b is not equal to µ s and τ Rb t isnot an entrainment threshold, as Bagnold assumed [ Pähtz and Durán , 2018b]. In fact, equa-tion (29) has no association with sediment entrainment whatsoever, except for the fact thatsediment entrainment is a necessary requirement to keep transport at capacity [
Pähtz andDurán , 2018b].As explained in section 4.1.3, Q , and thus M , is significantly larger than zero at the im-pact entrainment threshold τ ImE t . In particular, transport becomes intermittent for τ < τ ImE t oreven stops in the absence of entrainment by turbulent fluctuation events (i.e., transport capac-ity cannot be sustained). Hence, equation (29) is, in general, valid only for τ ≥ τ ImE t and alsoconsistent with the rebound threshold model prediction τ ImE t > τ Rb t (see section 4.2.1). Notethat aeolian saltation transport experiments [ Carneiro et al. , 2015;
Martin and Kok , 2018]and coupled DNS/DEM fluvial bedload transport simulation [
González et al. , 2017], indeed,very roughly suggest τ ImE t ≈ . τ Rb t and τ ImE t ≈ τ Rb t , respectively. In order to extend thevalidity of equation (29), and thus of standard sediment transport rate relationships, to shearstresses τ with τ Rb t < τ < τ ImE t , one must abandon long-term averaging sediment transportdata. Instead, it is necessary to conditionally average M (or Q ) only over periods of near-capacity transport (on short-term average), but ignore periods with transport significantlybelow capacity or even at rest [ Bunte and Abt , 2005;
Singh et al. , 2009;
Shih and Diplas ,2018;
Comola et al. , 2019b]. Likewise, for realistic fluvial bedload transport, it is necessaryto exclude the turbulence-driven fluctuation motion (including turbulent entrainment events)when measuring M for equation (29) to remain valid; otherwise, transport does not vanishfor τ → τ Rb t . Salevan et al. [2017] demonstrated that implementing such constraints in theanalysis of experimental data is, in principle, possible. By separating the velocity distribu-tion of all measurable particles (including those that are visually perceived as resting) into aStudent’s t -distribution associated with the turbulence-driven fluctuation motion and an ex-ponential distribution associated with the bulk transport of particles (which automaticallyimplies conditional averaging as periods of rest do not affect this distribution), they obtaineda measure for the number of transported particles relative to the total number of bed surfaceparticles ( n tr / n tot ). This measure, indeed, vanishes within experimental precision below aShields number threshold (Figure 16a), which can be interpreted as Θ Rb t , whereas the num-ber of particles n v t that are faster than a certain velocity threshold v t remains nonzero for theentire range of Θ because of the turbulence-driven fluctuation motion (Figure 16b). The Shields diagram is a compilation of measurements of the threshold Shields num-ber Θ t as a function of the shear Reynolds number Re ∗ , which have been labeled as measure-ments of incipient sediment motion by numerous studies and reviews [e.g., Shields , 1936;
Miller et al. , 1977;
Yalin and Karahan , 1979;
Parker and Klingeman , 1982; van Rijn , 1984;
Wiberg and Smith , 1987;
Ling , 1995;
Buffington and Montgomery , 1997;
Dey , 1999;
Paphi-tis , 2001;
Cao et al. , 2006;
Dey and Papanicolaou , 2008;
Ali and Dey , 2016;
Dey and Ali ,2018, 2019;
Yang et al. , 2019, and references therein]. However, incipient motion of tur-bulent fluvial bedload transport is much better characterized by impulse and energy-basedcriteria (section 3.2), unless one refers to the Shields number Θ In t at which the fluid entrain-ment probability exceeds zero (section 3.3), which is much below the Shields curve [ Paintal ,1971]. Furthermore, in steady, homogenous turbulent fluvial bedload transport in which tur-bulence is suppressed (e.g., in narrow water flumes), the vast majority of entrainment eventsis caused by particle-bed impacts (see section 4.1.3). It is therefore here argued, based onthe results reviewed in section 4.2.2, that many of the threshold data compiled in the Shieldsdiagram are actually measurements of the rebound threshold Θ Rb t .The Shields diagram shows two kinds of threshold measurements obtained using twodifferent methods. The first method is the reference method , where one takes paired measure- –39–onfidential manuscript submitted to Reviews of Geophysics -3 -2 -1 -4 -3 -2 -1 Figure 16.
Measurements of particle activity by
Salevan et al. [2017]. (a) Number of transported particlesrelative to the total number of bed surface particles ( n tr / n tot ) and (b) number of particles n v t that are fasterthan a certain velocity threshold v t versus Shields number Θ . Error bars in (a) correspond to the standarderror computed from six experimental runs. ments of Θ and the nondimensionalized transport rate Q ∗ (or transport load M ∗ ≡ M /( ρ p d ) )and extrapolates them to the Shields number at which Q ∗ (or M ∗ ) either vanishes [e.g., Shields ,1936] (it is slightly controversial whether Shields really used this method [
Buffington , 1999])or equals a small reference value [e.g.,
Parker and Klingeman , 1982]. This method yieldsapproximately the rebound threshold Θ Rb t if an expression for Q ∗ (or M ∗ ) based on equa-tion (29) is used for the extrapolation and provided that the data used for the extrapolationare at capacity (i.e., Θ ≥ Θ ImE t ). For example, Lajeunesse et al. [2010] extrapolated theirmeasurements (many data points obeyed Θ ≥ Θ Rb t ≈ Θ ImE t ) to M ∗ = Θ Rb t . That the reference method yields the rebound threshold Θ Rb t is further supported by the fact that the values of Θ Rb t obtained from the DEM-based fluvialbedload transport simulations by Pähtz and Durán [2018a] are consistent with the compi-lation of reference method-based threshold measurements by
Buffington and Montgomery [1997].The second method is the visual method , where one increases Θ until criteria defin-ing what is considered critical transport are obeyed [e.g., Kramer , 1935] (see section 1).The threshold values obtained from this method depend significantly on the chosen crite-rion and are, on average, close to those obtained from the reference method [
Buffington andMontgomery , 1997]. For example, the transition point ( Θ , Q ∗ ) ≈ ( . , . ) at which thefunction Q ∗ ( Θ ) measured in the gravel-bed experiments by Paintal [1971] changed from Q ∗ ∝ Θ to Q ∗ ∝ Θ . (see section 1) is indistinguishable from the reference thresholdfor the same conditions within measurement uncertainty. In particular, a close examinationof Paintal’s and other gravel bed data has revealed that Paintal’s power-16 region can actu-ally be subdivided into two regions [ Dey and Ali , 2019, Figure 5] (see also [
Shih and Diplas ,2019, Figure 8b]): one region ( Θ (cid:46) .
04) with a milder power law and one with a strongerpower law (0 . (cid:46) Θ (cid:46) . Q ∗ by an order of magnitude at Θ (cid:39) .
04. Such a jump is consistent with exceeding the rebound threshold Θ Rb t becausetransported particles suddenly become able to move along the surface for comparably longtimes before being captured by the bed. Hence, it seems that also the visual method, at leastfor typical critical transport criteria, approximately yields the rebound threshold Θ Rb t ratherthan an entrainment threshold. –40–onfidential manuscript submitted to Reviews of Geophysics
The hypothesis that the Shields diagram shows measurements of the rebound thresh-old is further supported by the fact that certain rebound threshold models [
Pähtz and Durán ,2018a;
Pähtz et al. , 2020] reproduce the Shields curve without fitting to the experimentaldata compiled in the Shields diagram (see section 4.3), even when limited to only visuallymeasured data [
Pähtz et al. , 2020].
This section reviews theoretical models for both the rebound threshold Θ Rb t and impactentrainment threshold Θ ImE t . One of the early motivations for developing such models wasto better understand the hysteresis between the initiation and cessation of aeolian saltationtransport observed in wind tunnel experiments [e.g., Bagnold , 1941;
Chepil , 1945;
Iversenand Rasmussen , 1994;
Carneiro et al. , 2015]. While the difference between transport initi-ation and cessation is relatively small on Earth, wind tunnel experiments and observationssuggested a substantial difference on Mars, which needed to be explained [
Almeida et al. ,2008;
Kok , 2010b]. (However, note that extrapolating wind tunnel measurements of the ini-tiation threshold Θ In t to field conditions using standard initiation threshold models is actuallyinappropriate because Θ In t depends on the boundary layer thickness δ , as discussed in sec-tion 3.3.) Later on, cessation threshold models were developed with the purpose to unifyfluvial bedload and aeolian saltation transport in a single theoretical framework [ Berzi et al. ,2016;
Pähtz and Durán , 2018a;
Pähtz et al. , 2020].As cessation threshold models are associated with a sustained motion of transportedparticles, they require a physical description of the particle motion within the transport layerthat is coupled with boundary conditions that describe the interaction between transportedparticles and the bed surface. In general, there have been two approaches to describe thetransport layer and bed interactions. The first approach consists of representing the entireparticle motion by particles moving in identical periodic trajectories along a flat wall thatmimics the bed surface (section 4.3.1). The second approach consists of deriving generalcorrelations between transport layer-averaged physical quantities and obtain the correla-tion coefficients from numerical simulations (section 4.3.2). It will be shown that the latterapproach is probably a rough approximation of a variant of the former. Correlation-basedmodel equations elucidate the role that the density ratio s plays for the rebound threshold Θ Rb t in a simple manner and therefore provide a simple conceptual explanation for why Θ Rb t is smaller in aeolian saltation than in fluvial bedload transport (section 4.3.3). Open Problem: Effect of Cohesion on Transport Cessation Thresholds
Most of the sediment transport cessation threshold models reviewed here account forcohesive interparticle forces and do so in a similar manner as transport initiation thresholdmodels. However,
Comola et al. [2019a] recently revealed that the effects of cohesion ontransport cessation and initiation thresholds are actually fundamentally different from oneanother, which is why this section only considers versions of existing cessation thresholdmodels for cohesionless particles. The effect of cohesion on transport cessation thresholdsremains a major open problem.
Most studies proposing cessation threshold models start with the assumption that themotion of transported particles can be represented by a system in which all particles hop inthe same periodic trajectory, referred to as the average trajectory, driven by the mean tur-bulent flow along a flat wall, with which they interact according to certain boundary condi-tions [
Claudin and Andreotti , 2006;
Kok , 2010a;
Berzi et al. , 2016, 2017;
Pähtz et al. , 2020].(Note that, although
Kok [2010a] does not explicitly refer to identical periodic trajectories,his mathematical treatment of the problem is equivalent to IPTMs.) However, the assump-tion of identical periodic particle trajectories introduces a variety of potentially major weak- –41–onfidential manuscript submitted to
Reviews of Geophysics nesses, which has cast doubt on the reliability of IPTMs [
Andreotti , 2004;
Lämmel and Kroy ,2017;
Pähtz and Durán , 2017, 2018a]:1. In IPTMs, the particle concentration increases with elevation z and jumps to zerowhen z exceeds the hop height [ Anderson and Hallet , 1986]. In contrast, in real non-suspended sediment transport, it monotonously decreases with z , often exponen-tially [e.g., Durán et al. , 2012]. IPTMs that refer only to the motion of a well-definedspecies of particles (e.g., continuous rebounders) do not necessarily suffer from thisweakness because the concentration profile associated with this species may behavedifferently from that of the entire ensemble of transported particles.2. In IPTMs, the mean square of the vertical particle velocity ( (cid:104) v z (cid:105) ) decreases with z . Incontrast, in real nonsuspended sediment transport, it increases with z , except far fromthe bed surface [ Pähtz and Durán , 2017]. This behavior is a signature of the fact thatthe transport layer, in general, consists of different species of particles with differentcharacteristic velocities [e.g.,
Durán et al. , 2011, Figure 21]. That is, IPTMs that referonly to the motion of a well-defined species of particles (e.g., continuous rebounders)do not necessarily suffer from this weakness.3. Only particles that take off from the wall with an energy E ↑ that is larger than a crit-ical value E c can continue their motion after the initial few hops (Figure 15). Thatis, IPTMs that take into account the motion of entrained particles [ Claudin and An-dreotti , 2006;
Kok , 2010a] effectively assume that all entrained particles obey E ↑ ≥ E c even though most of them do not [ Pähtz and Durán , 2018a].4. IPTMs neglect particle motion via rolling and sliding, which is significant in bedloadtransport.Depending on the boundary conditions, three conceptually different kinds of IPTMs can bedistinguished:1. Models of the rebound threshold Θ Rb t consider only the dynamics of continuous re-bounders. Their rebounds are described, for example, by equations (12b) and (13),which link the streamwise ( x ) and normal-wall ( z ) components of the impact ve-locity v i to the streamwise and normal-wall components of the rebound velocity v r .Such models then look for the smallest Shields number that results in a periodic tra-jectory under the constraint that the hop height of particles exceeds one particle di-ameter ( z h ≥ d ). This constraint ensures consistency with the underlying modelassumption that continuous rebounders are never captured by the bed surface. Thethreshold resulting from this constraint is denoted as Θ Rb ∗∗ t . Pähtz et al. [2020] mod-ified this constraint to take into account that the near-surface flow can assist parti-cles in escaping the bed surface and is even predominantly responsible for the es-cape in the viscous bedload transport regime. These authors’ escape criterion reads Θ / Θ max t ≥ cot ψ / cot ψ Y , where Θ max t = .
12 is the viscous fluid entrainment threshold(see section 2.1), ψ Y = ◦ the pocket angle for particles resting within the deepestpockets of the bed surface, and sin ψ = sin ψ Y + v r /( g d ) . This criterion means thatthe rebound kinetic energy only needs to uplift a particle rebounding within the deep-est pocket to a point at which the near-surface flow is able to push it out of the pocket.The threshold resulting from this modified constraint is denoted as Θ Rb ∗ t .2. Models of the impact entrainment threshold Θ ImE t [ Claudin and Andreotti , 2006;
Kok ,2010a] do not neglect captures of continuous rebounders and therefore take into ac-count the entrainment of bed particles. One possible way to do this is by combiningrebound boundary conditions with an additional constraint that describes that oneparticle leaves the surface per impact on average (e.g., | v i | ∝ √ ˜ g d [ Claudin and An-dreotti , 2006]). However, the incorporation of entrained particles as part of the aver-age trajectory leads to consistency problems (see third point in the list above).3. Hybrids between continuous rebound and impact entrainment models [
Berzi et al. ,2016, 2017] look for the smallest Shields number (denoted as Θ Rb | ImE t ) that results in –42–onfidential manuscript submitted to Reviews of Geophysics a periodic trajectory under the constraint z h ≥ d (like before) and the additional con-straint that particle-bed impacts do not lead to entrainment. Berzi et al. [2016, 2017]modeled the latter constraint via | v i |/√ ˜ g d ≤ ζ / ≈
20 (cf. equation (17)), whichassumes that the fastest particles represented by the average trajectory of continuousrebounders do not exceed the value ζ of the nondimensionalized impact velocity thatis associated with the onset of entrainment (which can be roughly justified by assum-ing an even impact velocity distribution between 0 and ζ ). However, Pähtz and Durán [2018a] pointed out that this additional constraint is inconsistent with the experimen-tal and numerical evidence that impact entrainment to be effective requires that thetransport rate is significantly larger than zero (see section 4.1.3), which is never thecase at the rebound threshold Θ Rb t in the absence of entrainment by turbulent fluctu-ation events (see section 4.2.2 and equation (29)). Consistently, Pähtz et al. [2020]showed that, in the limit Θ → Θ Rb t , identical periodic trajectories of continuous re-bounders are unstable against trajectory fluctuations. That is, the energy that a parti-cle must acquire upon entrainment to become a continuous rebounder is equal to therebound energy of the continuous rebounder that has entrained it in this limit. Thisrequirement contradicts the fact that the entrainment energy is much smaller than therebound energy because of energy conservation, which implies that impact entrain-ment is impossible in this limit (see also discussion in section 4.2.1).Apart from these conceptual differences, existing IPTMs differ in several details (partly sum-marized in Table 1): the form of the fluid drag law, the consideration or neglect of verticaldrag forces on the particle motion, the form of the mean flow velocity profile (including thequestion of whether the viscous sublayer of the turbulent boundary layer is considered; formore details, see Appendix), and the bed boundary conditions (including the incorporationof viscous damping in the rebound laws). In this regard, it is reiterated that the effects of vis-Study Model Vertical drag Viscous sublayer Viscous damping Boundary ConditionsCA06 Θ ImE t yes yes no e D , θ Dr = constK10 Θ ImE t yes no no complexB16/17 Θ Rb | ImE t no no yes eqs. (12b) and (13)P19 Θ Rb ∗ t yes yes no eqs. (12b) and (14) Table 1.
Modeling details of the IPTMs by
Claudin and Andreotti [2006] (CA06),
Kok [2010a] (K10),
Berziet al. [2016, 2017] (B16/17), and
Pähtz et al. [2020] (P19). cous damping on the dynamics of particle-bed rebounds are probably negligible for bedloadtransport (for which viscous damping is deemed as potentially significant), even for condi-tions with strongly damped binary particle collisions (see section 4.1.1).In order to facilitate a comparison between the different model types that does not de-pend on modeling details but focuses only on conceptual differences, the same mean flowvelocity profile (equation (A1), which includes the viscous sublayer), boundary conditions(equations (20) and (21)), and fluid drag law (the drag law by
Camenen [2007]) are used forall model types. Following the trajectory calculation by
Pähtz et al. [2020], the impact veloc-ity v i as a function of the rebound velocity v r approximates asˆ v iz = ˆ v rz − ˆ t h , with ˆ t h = + ˆ v rz + W (cid:104) − ( + ˆ v rz ) e −( + ˆ v rz ) (cid:105) , (30a)ˆ v ix = ˆ v rx e − ˆ t h + V − s √ Θ f ( Ga √ Θ , V s s ˆ z ∗ + Z ∆ )( − e − ˆ t h ) , with ˆ z ∗ ≡ − ˆ v iz ( ˆ v rz + ) − ˆ v rz , (30b)where t h is the hop time, W the principal branch of the Lambert- W function, V s ≡ v s /√ s ˜ g d the dimensionless value of the settling velocity v s (defined in equation (31)), Z ∆ d = . d –43–onfidential manuscript submitted to Reviews of Geophysics the average elevation of the particles’ center during particle-bed rebounds (obtained fromexperiments [
Dey et al. , 2012;
Hong et al. , 2015]), and √ Θ f expresses the nonfluctuatingwall-bounded flow after Guo and Julien [2007], with f the function given in equation (A1).Furthermore, the hat denotes nondimensionalized quantities using combinations of ˜ g and v s ,which is given by v s = √ s ˜ g d µ b (cid:118)(cid:117)(cid:116) m (cid:115)(cid:18) C ∞ d Ga (cid:19) + m (cid:115) µ b C ∞ d − m (cid:115) C ∞ d Ga m , with µ b ≡ v ix − v rx v rz − v iz , (31)where C ∞ d = m = . Θ ( Ga , s , ˆ v rz ) . Then the thresholds are obtained from Θ Rb ∗ t ( Ga , s ) ≡ min ˆ v rz Θ (cid:40) Ga , s , ˆ v rz (cid:34) cot ψ Y Θ Θ max2 t ≥ (cid:18) sin ψ Y + v r g d (cid:19) − − (cid:35) (cid:41) , (32a) Θ Rb ∗∗ t ( Ga , s ) ≡ min ˆ v rz Θ [ Ga , s , ˆ v rz ( z h ≥ d )] , (32b) Θ ImE t ( Ga , s ) ≡ Θ (cid:20) Ga , s , ˆ v rz (cid:18) | v i | = ζ (cid:112) ˜ g d (cid:19)(cid:21) , (32c) Θ Rb | ImE t ( Ga , s ) ≡ min ˆ v rz Θ (cid:20) Ga , s , ˆ v rz (cid:18) z h ≥ d ∧ | v i | ≤ ζ (cid:112) ˜ g d (cid:19)(cid:21) , (32d)where the hop height is given by z h = [ v rz v s − v s ln ( + v rz / v s )]/ ˜ g (for small v rz / v s , z h (cid:39) v rz /( g ) ). In equations (32a)–(32d), the rebound threshold Θ Rb ∗ t is the only modeledcessation threshold that is linked to the viscous yield stress Θ max t and thus to dense gran-ular flow rheology (see section 2.1). In a complete model covering all transport regimes,such a connection must exist because Θ max t represents an upper limit to any kind of cohe-sionless sediment transport threshold. Also, a complete model of any kind of cohesionlesstransport threshold must reach this maximum value in the limit of vanishing particle in-ertia (i.e., when typical particle velocities during a trajectory become much smaller than √ ˜ g d ). The characteristic particle velocity scale in IPTMs is given by the settling velocity v s , which scales as v s ∝ Ga √ s ˜ g d in the viscous regime (Eq. (31) for small Ga). That is, acomplete model of any kind of cohesionless transport threshold must approach Θ max t in thelimit v s /√ ˜ g d ∝ Ga √ s →
0, where Ga √ s can be interpreted as a Stokes-like number [ Berziet al. , 2016, 2017;
Clark et al. , 2017;
Pähtz and Durán , 2018a].Figures 17a, 17b, and 18a show the thresholds calculated by equations (32a)-(32d) asa function of Ga √ s for five different density ratios s = (2.65, 40, 190, 2200, and 250000)corresponding to five different fluvial or aeolian conditions (Water, Venus, Titan, Earth, andMars). These figures also show cessation threshold measurements obtained for nearly co-hesionless conditions using different experimental methods. For turbulent bedload trans-port driven by water, the compilation of reference method-based measurements (measure-ment mean and its 95% confidence interval) by Buffington and Montgomery [1997], whichmake up a large portion of the Shields diagram, is shown. As explained in section 4.2.3,this method yields approximately the rebound threshold Θ Rb t . For viscous bedload transportdriven by water-oil mixtures, the visual incipient motion measurements by Yalin and Kara-han [1979] and
Loiseleux et al. [2005] and cessation threshold measurements by
Ouriemiet al. [2007] are shown (for viscous bedload transport, the differences between transport initi-ation, rebound, and impact entrainment threshold are very small [
Pähtz and Durán , 2018a]).For aeolian saltation transport, a few studies [e.g., Ho , 2012; Zhu et al. , 2019] carried out anindirect extrapolation to vanishing transport to obtain Θ Rb t using a proxy of Q : the surfaceroughness z o (see Appendix for its definition in the absence of transport), which undergoesa regime shift when saltation transport ceases. Furthermore, visual measurements of Θ Rb t by Bagnold [1937] and
Chepil [1945] are shown, obtained from successively decrementing Θ until intermittent saltation transport stops. Direct measurements of the intermittent salta-tion transport threshold (and thus Θ Rb t ), based on the so-called Time Frequency Equivalence –44–onfidential manuscript submitted to
Reviews of Geophysics -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 Figure 17.
Predictions of (a) the hybrid between rebound and impact entrainment threshold ( Θ Rb | ImE t )and (b) the impact entrainment threshold Θ ImE t from the IPTM as functions of the Stokes-like number Ga √ s (lines) for five different density ratios s = (2.65, 40, 190, 2200, and 250000) corresponding to five differentfluvial or aeolian conditions (Water, Venus, Titan, Earth, and Mars). Symbols correspond to threshold mea-surements (or measurement compilations) from various studies [ Bagnold , 1937;
Chepil , 1945;
Buffingtonand Montgomery , 1997;
Loiseleux et al. , 2005;
Ouriemi et al. , 2007; Ho , 2012; Martin and Kok , 2018;
Zhuet al. , 2019] and methods (see text).
Ouriemi et al. [2007] did not report single measurement values but aconstant threshold 0 . ± .
03 for a large range of viscous conditions, indicated by the dotted square. Errorbars correspond to 95%-confidence intervals of the compilation of reference method-based measurements by
Buffington and Montgomery [1997], which make up a large portion of the Shields diagram.
Method (TFEM) [
Wiggs et al. , 2004], by
Martin and Kok [2018] are also shown. Note that,although the evidence that the thresholds obtained from extrapolation to vanishing transportand from direct measurements of the cessation of intermittent saltation transport correspondto the rebound threshold Θ Rb t is quite strong (see section 4.1.3 and 4.2.2), many aeolian re-searchers believe that they correspond to the impact entrainment threshold Θ ImE t [e.g., Martinand Kok , 2018]. One of the reasons for this belief can be seen in Figure 17b: the predictionof Θ ImE t from equation (32c) is consistent with aeolian saltation transport data on Earth de-spite not containing fit parameters. In fact, for the range of conditions corresponding to thesedata, the predictions of Θ Rb ∗ t and Θ Rb ∗∗ t by equations (32a) and (32b) are equivalent and, co-incidentally, very close to the predictions of Θ ImE t and Θ Rb | ImE t by equations (32c) and (32d),which are also equivalent to each other. At this point, it is worth reiterating that differencesbetween the models caused by differences in the modeling details (e.g., those in Table 1)have been excluded here. Such detail differences cause the predictions of existing modelsto differ more strongly from one another than shown here.Figures 17a, 17b, and 18a show that the predictions of Θ Rb ∗∗ t , Θ ImE t , and Θ Rb | ImE t fromequations (32b)–(32d) overestimate threshold measurements for fluvial bedload transportby at least an order of magnitude. For Θ Rb ∗∗ t and Θ Rb | ImE t , this overestimation is caused bythe constraint in the minimization of Θ that the particle hop height z h must exceed one par-ticle diameter d to escape the bed surface (equations (32b) and (32d)), preventing solutionswith small particle velocities that would have a smaller threshold. However, the predictionof Θ Rb ∗ t from equation (32a), which is based on a modified escape condition that takes intoaccount the near-surface flow, is consistent with fluvial bedload transport conditions (Fig-ure 18a). The simultaneous agreement of the prediction of Θ Rb ∗ t from equation (32a) with –45–onfidential manuscript submitted to Reviews of Geophysics -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 Figure 18.
Predictions of the rebound threshold, Θ Rb ∗∗ t (dashed lines in (a)) and Θ Rb ∗ t (solid lines in (a)and (b) and dashed lines in (b)), from (a) the IPTM and (b) the correlation-based model by Pähtz and Durán [2018a] and its IPTM analogue as a function of the Stokes-like number Ga √ s for five different density ratios s = (2.65, 40, 190, 2200, and 250000) corresponding to five different fluvial or aeolian conditions (Water,Venus, Titan, Earth, and Mars). Symbols correspond to threshold measurements (or measurement compi-lations) from various studies [ Bagnold , 1937;
Chepil , 1945;
Buffington and Montgomery , 1997;
Loiseleuxet al. , 2005;
Ouriemi et al. , 2007; Ho , 2012; Martin and Kok , 2018;
Zhu et al. , 2019] and methods (see text).
Ouriemi et al. [2007] did not report single measurement values but a constant threshold 0 . ± .
03 for alarge range of viscous conditions, indicated by the dotted square. Error bars correspond to 95%-confidenceintervals of the compilation of reference method-based measurements by
Buffington and Montgomery [1997],which make up a large portion of the Shields diagram. For symbol legend, see Figure 17. The IPTM in (b)uses the modified boundary conditions µ b = .
63 and cot θ Dr = µ b [ /(√ c ) − ] (cid:39) .
4, and the modifiedviscous yield stress Θ max t = − µ b Z ∆ /( c ) + (cid:113) [ µ b Z ∆ /( c )] + µ b /[ ( − c ) c ] (cid:39) .
175 to mimic thepredictions from the correlation-based model by
Pähtz and Durán [2018a]. These modifications are explainedin the text. aeolian and fluvial transport regimes strongly supports modeling nonsuspended sedimenttransport within the continuous rebound framework.
Existing correlation-based cessation threshold models start with the assumption of aconstant bed friction coefficient µ b [ Pähtz et al. , 2012;
Pähtz and Durán , 2018a] ( µ b is theinverse of the parameter α in the model by Pähtz et al. [2012]). As discussed in section 4.2.2,the approximate constancy of µ b has been analytically linked to continuous rebounds [ Pähtzand Durán , 2018b]. However, in contrast to the purely kinematic meaning of µ b in IPTMs(equation (31)), for realistic nonsuspended sediment transport, µ b conveys information aboutboth the particle kinematics and interparticle contacts. Note that µ b (cid:39) const is also pre-dicted by IPTMs when vertical drag forces are small (i.e., the buoyancy-reduced gravity forcedominates the vertical motion) because this fixes e z (cid:39) e D , θ Dr , and µ b via therebound laws [ Pähtz et al. , 2020].A constant µ b links the average horizontal fluid drag acceleration a d x to the buoyancy-reduced gravity ˜ g via µ b (cid:39) a d x / ˜ g , where the overbar denotes a particle concentration- –46–onfidential manuscript submitted to Reviews of Geophysics weighted height average [
Pähtz and Durán , 2018a] (which is equal to the average over thehop time for IPTMs). This link subsequently fixes the value of the nondimensionalized av-erage velocity difference U x − V x ≡ ( u x − v x )/√ s ˜ g d = µ b v s /√ s ˜ g d [ Pähtz et al. , 2020] asa function of the Galileo number Ga via equation (31). In fact, equation (31) is not limitedto IPTMs but actually more general [
Pähtz and Durán , 2018a]. A further general correlationbetween U x and the nondimensionalized transport layer thickness Z ≡ z / d can be obtainedfrom approximating u x ( z ) (cid:39) u x ( z ) [ Pähtz and Durán , 2018a]. An analogous approxima-tion is also involved in some IPTMs, namely, in the right-hand side of equation (30b), sinceˆ z (cid:39) ˆ z ∗ (cid:39) ˆ v rz / v rz (i.e., when vertical drag forces are small). Up to thispoint, the two existing correlation-based models by Pähtz et al. [2012] and
Pähtz and Durán [2018a] are equivalent. From now on, only the latter model is reviewed as it constitutes asubstantial improvement of the former model in many regards.
Pähtz and Durán [2018a]derived the further correlation V z ≡ (cid:113) v z /( s ˜ g d ) = c µ − b V x , where c is a proportionality con-stant. This correlation with c = [√ ( cot θ Dr / µ b + )] − is also predicted by IPTMs in thelimit of small vertical drag forces. That is, up to here, the model by Pähtz and Durán [2018a]is effectively an IPTM that neglects vertical drag forces. The main differences between themodel by
Pähtz and Durán [2018a] and IPTMs lie in the latter two equations of the full set ofmodel equations: U x − V x = (cid:118)(cid:117)(cid:116) m (cid:115)(cid:18) C ∞ d Ga (cid:19) + m (cid:115) µ b C ∞ d − m (cid:115) C ∞ d Ga m , (33a) U x = (cid:113) Θ Rb ∗ t f [ Ga (cid:112) Θ t , ( Z + Z ∆ )] , (33b) V z = c µ − b V x , (33c) Z = c µ − b Θ Rb ∗ t + sV z , (33d) V x = (cid:113) Θ Rb ∗ t κ (cid:118)(cid:117)(cid:116) − exp (cid:34) − c κ (cid:18) U x / (cid:113) Θ Rb ∗ t (cid:19) (cid:35) , (33e)where µ b = . c = . c = . c = .
79, and Z ∆ = . Pähtz and Durán [2018a] obtained from adjusting equations (33a)–(33e) toDEM-based simulations of nonsuspended sediment transport (the same kind of simulationsas those by
Pähtz and Durán [2017], see section 4.1.3). Equation (33d) contains two terms:a term ( sV z ) that is associated with the vertical motion of particles (equivalent to ˆ z = ˆ v z in IPTMs) and a term ( µ − b Θ Rb ∗ t ) that is associated with particle collisions and particle-bedcontacts of particles moving above the bed surface level, which occur because of the surfacetexture [ Pähtz and Durán , 2018a]. A term analogous to the latter does not appear in existingIPTMs. Equation (33e) empirically merges two extremes. On the one hand, when the trans-port layer is completely submerged within the viscous sublayer of the turbulent boundarylayer (small U x / (cid:113) Θ Rb ∗ t ), it predicts V x = c U x . For viscous bedload transport (i.e., when thetransport layer is small: Z (cid:28) Z ∆ ), this correlation with c = − µ b /[ Θ max t ( c Θ max t / µ b + Z ∆ )] is also predicted by IPTMs that employ the constrained minimization principle in equa-tion (32a) to calculate the rebound threshold Θ Rb ∗ t . For viscous saltation transport (i.e., whenthe transport layer is large: Z (cid:29) Z ∆ ), IPTMs of the rebound threshold that consider verticaldrag forces also predict V x ∝ U x [ Pähtz and Durán , 2018a]. However, the proportionalityconstant exhibits a different value (but still near unity) that depends on µ b / cot θ Dr . On theother hand, when most transport occurs within the log-layer of the turbulent boundary layer(large U x / (cid:113) Θ Rb ∗ t ), equation (33e) predicts V x (cid:39) (cid:113) Θ Rb ∗ t / κ , which also follows from theminimization principle for turbulent saltation transport (i.e., Z (cid:29) Z ∆ ) [ Pähtz and Durán ,2018a]. For these reasons, equation (33e) can be interpreted as a rough approximation of theconstrained minimization in equation (32a) yielding Θ Rb ∗ t . In fact, Figure 18b shows that thepredictions of Θ Rb ∗ t from the model by Pähtz and Durán [2018a] are similar to those froman analogous IPTM and that they are also consistent with measurements across aeolian and –47–onfidential manuscript submitted to
Reviews of Geophysics fluvial environments. The predictions from these two models differ for turbulent bedloadtransport (large U x / (cid:113) Θ Rb ∗ t and small transport layer: Z ∼ Z ∆ ), mainly because the scal-ing V x (cid:39) (cid:113) Θ Rb ∗ t / κ that equation (33e) predicts for large U x / (cid:113) Θ Rb ∗ t does not capture theoutcome of the constrained minimization in equation (32a) for Z ∼ Z ∆ . The predictionsfrom these two models also differ for viscous saltation transport because the model by Pähtzand Durán [2018a] neglects vertical drag forces at various instances. However, note that thismodel does not completely neglect vertical drag forces because the scaling V x ∝ U x thatequation (33e) predicts for viscous saltation transport is associated with vertical drag [ Pähtzand Durán , 2018a], which is why deviations between this model and the analogous IPTM areonly moderate in this regime.
Open Problem: Reliable Models of the Impact Entrainment Threshold and Plane-tary Saltation Transport
Existing models of the impact entrainment threshold [
Claudin and Andreotti , 2006;
Kok , 2010a;
Pähtz et al. , 2012], which is arguably also the continuous transport threshold, donot take into account that the transport rate Q is significantly larger than zero at Θ ImE t , evenin the absence of entrainment by turbulent fluctuation events (see section 4.1.3). Instead, Q vanishes at the rebound threshold Θ Rb t , which is smaller than Θ ImE t (see section 4.2.2).Likewise, as mentioned before, existing models of Θ ImE t effectively assume that all entrainedparticles exhibit a kinetic energy that allows them to participate in the continuous reboundmotion even though most of them do not [ Pähtz and Durán , 2018a]. For these reasons, ex-isting impact entrainment threshold models seem to be missing important physics and needto be improved. This is problematic for modeling and predicting extraterrestrial sedimenttransport and associated bedform evolution [e.g.,
Almeida et al. , 2008;
Bourke et al. , 2010;
Kok , 2010b;
Ayoub et al. , 2014;
Lorenz , 2014;
Rasmussen et al. , 2015;
Jia et al. , 2017;
Telferet al. , 2018;
Durán Vinent et al. , 2019] because most predictions of the aeolian saltationtransport rate require that transport is continuous (i.e., at capacity).
The most important difference between aeolian saltation and fluvial bedload transportis the largely different density ratio s , which ranges from close to unity for oil and water tothe order of 10 for air on Mars. Equations (33a)-(33e) elucidate that s affects the modeledrebound threshold Θ Rb ∗ t in a relatively simple manner. In fact, it can be seen that s explic-itly appears only in Eq. (33d), which describes a monotonous increase of the dimensionlesstransport layer thickness Z with s . Subsequently, Z monotonously increases the dimension-less transport layer-averaged flow velocity U x via equation (33b). That is, given a certainsolution Θ Rb ∗ t ( Ga , s ) of equations (33a)–(33e), an increase of s leads to an increase of U x ,which must be compensated by a decrease of U x via a decrease of Θ Rb ∗ t to achieve a newsteady solution for the same Galileo number Ga. This mathematical fact expresses the phys-ical fact that particles that stay longer in the flow can feel a given effective flow forcing ata lower fluid shear stress, which is the ultimate reason for why Θ Rb ∗ t decreases with s for agiven Ga. Section 1 outlined five old, yet very significant, inconsistencies related to the conceptof a threshold shear stress for incipient motion. For the concept of a threshold shear stress tobe physically meaningful, these inconsistencies must be addressed and resolved. They can bebriefly summarized as follows:1. By design, existing models of incipient motion capture the conditions to which theyhave been adjusted (aeolian or fluvial transport on Earth). However, the predictions –48–onfidential manuscript submitted to
Reviews of Geophysics from standard models adjusted to aeolian transport on Earth [
Iversen and White , 1982;
Shao and Lu , 2000] are in stark disagreement with recent observations of aeoliantransport on Mars [e.g.,
Sullivan and Kok , 2017;
Baker et al. , 2018].2. Because of turbulent fluctuation events, fluid entrainment gives rise to fluvial bedloadtransport even for Shields numbers much below the Shields curve [
Paintal , 1971],which is the curve that is thought to describe incipient motion.3. Below the respective shear stress threshold associated with incipient motion, turbulentfluctuation events are able to initiate fluvial bedload transport but not aeolian saltationtransport. However, there is no reason to believe that the physics of incipient motionare different in aeolian and fluvial environments.4. Old experiments indicate a nonnegligible role of particle inertia in fluvial bedloadtransport [
Ward , 1969;
Graf and Pazis , 1977], which is problematic because criti-cal conditions are defined via nonzero transport rates (i.e., particles are in motion atthreshold conditions).5. In old numerical simulations of turbulent fluvial bedload transport [ no and García ,1998], it was recognized that the threshold shear stress obtained from extrapolatingthe simulated capacity transport rate to vanishing transport may not be associated withfluid entrainment. This is problematic because many of the threshold data compiledin the Shields diagram have been obtained from such or similar extrapolation meth-ods [e.g.,
Shields , 1936].As a result of the latest research reviewed in sections 2–4, a new conceptual picture hasemerged (section 5.1) that resolves these problems. However, it must be emphasized thatthis conceptual picture represents the authors’ synthesis of the current state of the art, andmany aeolian and fluvial geomorphologists may disagree. This is because, in some places,it stands in stark contrast to what has been a century old consensus. Likewise, there are stillmany open problems and controversies, summarized in section 5.2 (and highlighted in sec-tions 2–4), as well as a number of issues that have not been discussed in this review (e.g., theeffects of particle size heterogeneity on transport thresholds and sediment entrainment), intowhich section 5.3 presents a brief outlook.
Figure 19 summarizes the various shear stress thresholds of nonsuspended sedimenttransport and their relations to and effects on the transport characteristics. Details, with refer-ences to the research reviewed in sections 2–4, are described below. Θ > ) Creeping (see section 2.3) refers to a superslow granular motion, usually in the form ofintermittent local particle rearrangements within the sediment bed (not limited to the bed sur-face), that occurs below a macroscopic yield criterion (see section 2.1). One form of creep-ing is triggered by nearby regions above yield, while another form (the origin of which is notfully understood) occurs even in the absence of such regions. The existence of the latter formimplies that sediment likely is always transported (albeit slowly) for arbitrarily small valuesof the Shields number Θ , even in the absence of turbulence. Creeping of both kinds is veryimportant in determining the particle motion near transport initiation. It is fundamentallyrelated to the granular material, not a purely fluid-driven effect. Θ max t Apart from creeping, which affects the entire granular bed, bed surface particles canbe entrained directly by flow forces. When a sediment bed is subjected to a laminar flow ata sufficiently low shear Reynolds number Re ∗ , there is a critical Shields number, the yield –49–onfidential manuscript submitted to Reviews of Geophysics Θ Θ In t Probability of fluid entrainment from a static sediment bed
Initiation threshold
Impulse (energy) criterion for fluid entrainment of individual particles: I f ≥ I fc ( C eff E f ≥ W c ) Θ Fraction of capacity transport load ( M * / M e ) Θ Rb t Θ ImE t Rebound threshold Impact entrainment threshold do not depend much on turbulent fluctuations around the mean flow
Intermittent transport Continuous transport uncorrelated autocorrelated
Individual short transport events Predominantly bulk transport Long-lasting rolling, sliding, hoping motion M * = M e ∝ Θ − Θ Rb t Capacity transport load depends on the size of the largest turbulent eddies and thus on system dimensions, such as the boundary layer thickness
Summary of Transport Thresholds
Viscous bedload:Turbulent bedload: Aeolian transport (wind tunnel):Aeolian transport (field): unclear
Generalized Shields curve: Θ Rb t ≃ Θ max t ≃ Θ In t Θ Rb t ≫ Θ In t Θ Rb t < Θ In t Θ Rb t = f (Ga, s ) Creeping:
Θ > 0
Figure 19.
Sketch summarizing Shields number ( Θ ) thresholds of nonsuspended sediment transport. stress Θ max t , above which motion of bed surface particles is initiated and then never stops,whereas potential transient motion below Θ max t will inevitably come to an end (see section 2.1).The viscous yield stress Θ max t constitutes the upper limit for any kind of cohesionless sedi-ment transport threshold, including the Shields curve. The values of Θ max t reported in the lit-erature are somewhat scattered (between 0.1 and 0.4), but these numbers are within the rangeof the bulk friction coefficients for granular materials (ranging from low friction spheres tomore frictional, rough particles), suggesting that the granular material’s yield condition (seesection 2.1) is very important in determining the viscous yield stress. Θ In t While for laminar flows, the entrainment of individual bed surface particles is con-trolled by a critical Shields number, the entrainment of individual bed surface particles byturbulent flows is better described by an impulse (section 3.2.1) or energy (section 3.2.2)criterion. Nonetheless, one can still define an initiation threshold Shields number Θ In t (seesection 3.3) at which the probability of fluid entrainment exceeds zero (i.e., Θ In t (cid:39) Θ max t forlaminar flows at sufficiently low Re ∗ ). Because fluid entrainment is predominantly caused byturbulent fluctuation events, Θ In t depends not only on Re ∗ but also on properties that controlthe size of the largest turbulent flow eddies, such as the turbulent boundary layer thickness.This may be one of the reasons why aeolian incipient motion models adjusted to wind tunnelmeasurements fail when applied to atmospheric boundary layers (see first problem outlinedat the beginning of section 5). Further possible reasons include atmospheric instability, to-pography gradients, surface inhomogeneities, such as obstacles and vegetation, and sublima-tion of subsurface ice in natural atmospheres (see section 3.3.3). Θ Rb t (Generalized Shields Curve) The rebound threshold Θ Rb t (see section 4.2) is largely unrelated to the entrainment ofbed sediment (except for viscous bedload transport, for which Θ Rb t (cid:39) Θ max t ) but describesthe minimal dimensionless fluid shear stress that is needed for the mean turbulent flow tocompensate the average energy loss of rebounding particles by fluid drag acceleration duringtheir trajectories. Hence, for Θ ≥ Θ Rb t , transported particles rebound for comparably longer –50–onfidential manuscript submitted to Reviews of Geophysics periods before they deposit, whereas they deposit very quickly for Θ < Θ Rb t . The formertransport regime gives rise to transport autocorrelations, while the latter gives rise to individ-ual uncorrelated transport events. Hence, bulk sediment transport vanishes at Θ Rb t , which isdescribed by a general law for the dimensionless bulk transport load M ∗ at transport capacity(i.e., M ∗ = M e ∝ Θ − Θ Rb t , see equation (29)). In fact, fluvial incipient motion measure-ments compiled in the Shields diagram are actually measurements of Θ Rb t (see section 4.2.3),consistent with the fact that turbulent fluvial bedload transport does not vanish even muchbelow the Shields curve because of occasional strong turbulent fluctuation events [ Paintal ,1971]. The notion that Θ Rb t is largely unrelated to incipient motion and instead related toparticle inertia resolves the second, third, fourth, and fifth problems outlined at the begin-ning of section 5. There are relatively simple models (which neglect turbulent fluctuationsaround the mean turbulent flow) predicting Θ Rb t in agreement with measurements across ae-olian and fluvial environment without containing fitting parameters (models of Θ Rb ∗ t in sec-tion 4.3). Such models predict a generalized Shields curve of the form Θ Rb t ( Ga , s ) , whereGa ≡ (cid:112) ( s − ) g d / ν f is the Galileo number and s ≡ ρ p / ρ f the particle-fluid-density ratio,via modeling steady continuous particle trajectories. In fact, in aeolian environments, com-parably large values of s allow the flow to sustain comparably large steady trajectories at acomparably low Shields number Θ , causing Θ Rb t to be substantially smaller than in fluvialenvironments for a given Ga (see section 4.3.3). Θ ImE t Even for Θ > Θ Rb t , randomness introduced by inhomogeneities of the bed and turbulentfluctuations of the flow introduce trajectory fluctuations that can lead to random captures ofrebounding particles by the bed. To sustain transport capacity, these captures must be com-pensated by entrainment of bed sediment into the rebound layer by the action of the fluid(see section 3), by particle-bed impacts (see section 4.1.3), or a combination of both (seesection 4.1.2). Because entrainment involving the flow requires strong turbulent fluctuationevents (see sections 3 and 4.1.2), which occur only at an intermittent basis, transport remainsintermittent when impact entrainment alone is insufficient in providing the transport layerwith rebounders (i.e., for Θ < Θ ImE t ). However, once the impact entrainment threshold Θ ImE t (see section 4.1.3) is exceeded ( Θ ≥ Θ ImE t ), impact entrainment is sufficient to do so, evenwithout the assistance of fluid entrainment (i.e., significant fluid entrainment may occur, butis not needed). The impact entrainment threshold is strictly larger than the rebound thresh-old ( Θ ImE t > Θ Rb t ), which is associated with a nonzero bulk transport rate ( Q ( Θ ImE t ) > Θ ImE t are currently missing (see section 4.3). To avoid confusion, we reiterate that the terms bedload transport ( h ∼ d ) and salta-tion transport ( h (cid:29) d ) have been defined through the transport layer thickness h relative tothe particle diameter d (see notation and section 1). Depending on the relationship betweenthe initiation threshold Θ In t and the rebound threshold Θ Rb t , one observes different dynam-ics. For turbulent fluvial bedload transport, Θ In t (cid:28) Θ Rb t , which means that transport can beinitiated much below the Shields curve by occasional turbulent fluctuation events. However,whenever this happens, transport will very rapidly stop again. This is, indeed, the typical sit-uation for gravel-bed rivers, which adjust their shape so that they remain in a low-mobilitystate [ Parker , 1978;
Phillips and Jerolmack , 2016]. For aeolian transport in wind tunnels, Θ In t is significantly larger than Θ Rb t . This explains why aeolian bedload transport is usuallyvery short-lived. In fact, even though bed particles are usually entrained into a rolling mo-tion at Θ In t (i.e., h ∼ d ), this rolling motion rapidly evolves into saltation transport [ Bagnold ,1941;
Iversen et al. , 1987;
Burr et al. , 2015] as the flow is sufficiently strong to net accelerateparticles moving near the surface. By doing so, their hop height becomes larger and larger(i.e., h / d substantially increases) until a steady state is approached. For aeolian transport in –51–onfidential manuscript submitted to Reviews of Geophysics the field, the magnitude of Θ In t relative to Θ Rb t is unclear as Θ In t is smaller than in wind tun-nels because of a much larger boundary layer thickness δ , since δ controls the size of thelargest turbulent eddies and thus entrainment by turbulent fluctuation events (see section 3.3). The new conceptual picture described above has been derived nearly entirely from the-oretical and laboratory investigations. One may therefore wonder to what degree does thenotion of various transport thresholds have implications for natural field conditions, such asbedload transport in rivers and saltation transport driven by planetary winds. There are threemajor aspects in which the field differs from most laboratory experiments: much broaderparticle size distributions, much larger and more unstable boundary layers (mainly for aeo-lian transport), and various kinds of surface inhomogeneities, such as bedforms, obstacles,and vegetation. The effects of particle size heterogeneity have been excluded from this re-view (they are briefly discussed in the outlook, section 5.3.1). The remaining two aspectsare both associated with increasing turbulence and thus fluid entrainment (see section 3). Incontrast, the rebound threshold Θ Rb t and arguably the impact entrainment threshold Θ ImE t ,as well as the transport capacity scaling (which requires Θ ≥ Θ ImE t ), are mainly controlledby the mean turbulent flow and relatively insensitive to turbulent fluctuations around it andshould therefore be similar in laboratory and field (provided that the bed particle size dis-tributions are similar). That is, for Θ ≥ Θ ImE t ≈ ( . − ) Θ Rb t (typical for river floods andmany aeolian processes), one expects capacity relationships derived from laboratory exper-iments to reasonably work and laboratory and field to behave similar. This expectation isconsistent with observations reported in recent studies [ Recking , 2010;
Recking et al. , 2012;
Martin and Kok , 2017]. Even if transport is not at capacity, it is, in principle, possible to sep-arate the turbulence-induced random transport contribution from sediment transport rate datasets [
Salevan et al. , 2017] (see section 4.2.2) and to modify capacity relationships to accountfor noncapacity transport [
Comola et al. , 2019b].
Sections 2–4 have highlighted several important open questions and controversies thatneed to be addressed in future studies, which are summarized below. Section 2:1. Why do fluid-sheared surfaces creep below a macroscopic yield criterion? And whydo they do so even for seemingly arbitrarily small values of the Shields number Θ andin the absence of turbulence?2. What is responsible for the large spread of experimentally measured values of the vis-cous yield stress Θ max t ?3. Is flow-induced bed failure (i.e., yielding) a critical phenomenon?4. What is the rheology of nonsuspended sediment transport?Section 3 (although this section concerns both fluvial and aeolian transport conditions, openquestions and controversies in this section regard mainly aeolian transport):1. Why do different experimental designs for measuring the initiation threshold Θ In t ofaeolian rolling and saltation transport cause qualitative differences in the scaling of Θ In t with the particle diameter d ?2. Is the measured dependency of Θ In t on the density ratio s for constant Galileo num-ber Ga real or an artifact of differences in the boundary layer thickness of the windtunnels used to carry out the experiments?3. Is the measured strong increase of Θ In t with d for very large s in a wind tunnel withMartian pressure conditions real or an artifact of a limited boundary layer thickness ofthis wind tunnel? –52–onfidential manuscript submitted to Reviews of Geophysics
4. Is aeolian transport in the field on Earth and other planetary bodies, in contrast towind tunnels with similar atmospheric pressure conditions, always being initiatedclose to the rebound threshold Θ Rb t because of thick boundary layers, atmospheric in-stability, topography inhomogeneities, and subsurface ice sublimation? The answer tothis question is probably the most important one, since a positive answer would im-ply that a reliable model for Θ In t (i.e., answers to the previous three questions) is notrequired for predicting aeolian processes on such bodies.5. Does equilibrium aeolian bedload transport (i.e., h ∼ d ) exist in the field because ofthick boundary layers?6. Direct measurements of aeolian sediment transport initiation, which are currentlymissing, can help answering the questions above.Section 4:1. For a particle collision with a static sediment bed: how does the rebound probability P r depend on impact velocity and angle?2. How do particle shape and size distribution affect particle-bed collisions?3. Does viscous damping truly not much affect particle-bed collisions, as suggested bythe insensitivity of DEM-based sediment transport simulations to the normal restitu-tion coefficient (cid:15) of binary collisions? And if so, what is the physical reason?4. How do cohesive interparticle forces affect the collision process and thus sedimenttransport cessation?5. How do the laws describing a particle collision with a static bed change for a particlecollision with a mobile bed?6. It is straightforward to define intermittent and continuous sediment transport for theabsence of fluid entrainment because the sediment transport rate exhibits a discon-tinuous jump from nearly zero to a finite value at the continuous transport threshold.However, how does one universally define intermittent and continuous transport iffluid entrainment does occur?7. Is the transition from intermittent to continuous aeolian saltation transport associatedwith fluid entrainment (the current consensus) or with impact entrainment (the au-thors’ opinion, based on recent developments in the field)?8. What controls the impact entrainment threshold Θ ImE t and how does one model it? To limit the scope of this review, several important topics have been excluded. Two ofthem are briefly discussed below.
Perhaps the most important topic that has been excluded from this review is the ef-fects of the heterogeneity of the size of bed surface particles on sediment transport initiation,cessation, and entrainment. Naturally, sediment transport initiation and entrainment are size-selective. However, it is unclear whether this is also true for sediment transport cessation.While the continuous rebound mechanism (see section 4.2) is clearly a size-selective process(coarser particles are less accelerated during their trajectories), impact entrainment may notbe [
Martin and Kok , 2019;
Zhu et al. , 2019]. Furthermore, in heterogeneous sediment beds,relatively fine particles tend to be surrounded by coarser ones and their protrusion (i.e., theparticle height above surrounding sediment) is thus smaller than on average, whereas rela-tively coarse particles tend to have a larger-than-average protrusion. Because driving forcesdecrease and resisting forces increase with decreasing protrusion [
Yager et al. , 2018], rela-tively fine particles are more difficult to be entrained when compared with a bed made onlyof such fine particles. The ability of fine particles to continuously rebound is also suppressed –53–onfidential manuscript submitted to
Reviews of Geophysics by the presence of coarse particles [
Zhu et al. , 2019]. All these effects can make heteroge-neous sediment beds much less mobile than homogeneous ones of the same median particlesize. For example, for both fluvial bedload [
MacKenzie and Eaton , 2017;
MacKenzie et al. ,2018] and aeolian saltation transport [
Zhu et al. , 2019], it was found for certain heteroge-neous beds that the largest particles of the particle size distribution (larger than the 90th per-centile) have a very strong control on overall mobility. However, the manner and degree ofthe heterogeneousness seem to play an important role as not all kinds of heterogeneous bedsare so strongly affected by the presence of large particles [e.g.,
Wilcock , 1993;
Martin andKok , 2019]. In particular, in the early stages of bed armoring, the sediment transport rate canincrease because collisions between transported fine particles and coarse bed particles aremore elastic than collisions between particles of the same size [
Bagnold , 1973].
Another important topic that has been excluded from this review is the effects of steepbed slope angles on sediment transport initiation, cessation, and entrainment. For exam-ple, horizontal downslopes should, if everything else stays the same, increase bed mobilitybecause of the additional horizontal gravity force acting on particles [
Maurin et al. , 2018].However, in fluvial environments, steep slopes are usually accompanied by a very small wa-ter depth of the order of one particle diameter (or even lower), which strongly suppressesthe magnitude of hydrodynamic forces acting on particles, thus decreasing rather than in-creasing bed mobility [
Prancevic and Lamb , 2015]. Then again, an increasing downslopeangle α increases the bulk friction coefficient µ within the sediment bed (for turbulent flows, µ (cid:39) tan α [ + [( ρ p / ρ f − ) φ b ] − ] [ Maurin et al. , 2018], where φ b is the bed volume fraction).Once µ exceeds the static friction coefficient µ s associated with the yielding transition (seesection 2.1), the entire bed fails and a debris flow forms [ Takahashi , 1978;
Prancevic et al. ,2014;
Cheng et al. , 2018].
Appendix: Mean Flow Velocity Profile (Law of the Wall)
The mean flow velocity profile within the inner turbulent boundary layer above a flatwall (the law of the wall ) exhibits three regions: a log-layer for large nondimensionalizedelevations ( wall units ) Re ∗ z / d , a viscous sublayer for small Re ∗ z / d , and a buffer layer fortransitional Re ∗ z / d . For more details on turbulent wall-bounded flows, see the review by Smits et al. [2011]. In section 4.3, the following form of the law of the wall is used [
Guo andJulien , 2007]: u x (cid:112) ( s − ) g d = √ Θ f ( Re ∗ , z / d ) , f ( Re ∗ , z / d ) = (cid:18) Re ∗ zd (cid:19) +
73 arctan (cid:18) Re ∗ zd (cid:19) − .
52 arctan (cid:18) Re ∗ zd (cid:19) + ln (cid:34) + (cid:18) Re ∗ B zd (cid:19) ( / κ ) (cid:35) − κ ln (cid:26) + . ∗ (cid:20) − exp (cid:18) − Re ∗ (cid:19)(cid:21) (cid:27) , (A1)where κ = . B = exp ( . κ − ln 9 ) . Within the viscous sublayer of the turbulentboundary layer, u x / (cid:112) ( s − ) g d → Θ Ga z / d , whereas in the log-layer, u x / (cid:112) ( s − ) g d → κ − √ Θ ln ( z / z o ) . The roughness length z o equals d /( ∗ ) in the hydraulically smooth and d /
30 in the hydraulically rough regime [
Guo and Julien , 2007].
Acknowledgments
All data shown in the figures of this review can be found in the following references:
Peyneauand Roux [2008a], da Cruz et al. [2005],
Kamrin and Koval [2014],
Houssais et al. [2015], –54–onfidential manuscript submitted to
Reviews of Geophysics
Allen and Kudrolli [2017, 2018],
Diplas et al. [2008],
Valyrakis [2013],
Williams et al. [1994],
Iversen and White [1982],
Burr et al. [2015],
Swann et al. [2020],
Beladjine et al. [2007],
Vowinckel et al. [2016],
Salevan et al. [2017],
Buffington and Montgomery [1997],
Yalinand Karahan [1979],
Loiseleux et al. [2005],
Ouriemi et al. [2007],
Bagnold [1937],
Chepil [1945], Ho [2012], Martin and Kok [2018], and
Zhu et al. [2019]. We thank Michael Churchand three anonymous reviewers for their critical reviews and numerous insightful commentsand suggestions. T.P. acknowledges support from grant National Natural Science Foundationof China (No. 11750410687).
Notation τ Fluid shear stress [Pa] τ p Particle shear stress [Pa] P Particle pressure [Pa] ρ p Particle density [kg/m ] ρ f Fluid density [kg/m ] m p Particle mass [kg] u Instantaneous local flow velocity [m/s] U b Bulk flow velocity [m/s] u ∗ ≡ (cid:112) τ / ρ f Fluid shear velocity [m/s] ν f Kinematic fluid viscosity [m /s] δ Boundary layer thickness [m] H Flow thickness [m] W Flow width [m] d Characteristic particle diameter [m] h Transport layer thickness [m] (cid:219) γ Particle shear rate (strain rate) [1/s] T Granular temperature [m /s ] g Gravitational constant [m/s ] ˜ g ≡ ( − ρ f / ρ p ) g Buoyancy-reduced gravitational constant [m/s ] M Sediment transport load [kg/m ] Q Sediment transport rate [kg/(ms)] Θ ≡ τ /(( ρ p − ρ f ) gd ) Shields number or Shields parameter s ≡ ρ p / ρ f Particle-fluid-density ratio Re ≡ U b H / ν f Reynolds number Re ∗ ≡ u ∗ d / ν f Shear Reynolds number Ga ≡ (cid:112) ( s − ) gd / ν f Galileo number (also called Yalin parameter) St ≡ s | v r | d /( ν f ) Stokes number, where | v r | is the relative velocity between two particlesjust before they collide M ∗ ≡ M /( ρ p d ) Nondimensionalized sediment transport load Q ∗ ≡ Q /( ρ p d (cid:112) ( ρ p / ρ f − ) gd ) Nondimensionalized sediment transport rate I ≡ (cid:219) γ d / (cid:112) P / ρ p Inertial number J ≡ ρ f ν f (cid:219) γ / P Viscous number K ≡ J + c K I Viscoinertial number, where c K is an order-unity fit parameter Pe ≡ (cid:219) γ d /√ T Péclet number C m = / Added mass coefficient κ = . von Kármán constant ψ Pocket angle ψ Y Pocket angle for particles resting within the deepest pockets of the bed surface L arm Lever arm length [m] α Bed slope angle –55–onfidential manuscript submitted to
Reviews of Geophysics ∆ Z Critical dimensionless vertical particle displacement required for entrainment ∆ X Critical dimensionless horizontal particle displacement required for entrainment µ C Effective Coulomb friction coefficient encoding the combined effects of sliding androlling friction in entrainment µ ≡ − τ p / P Ratio between particle shear stress and particle pressure (bulk friction coeffi-cient) µ g Surface friction coefficient of granular particle µ s Static friction coefficient of granular bulk (yield stress ratio) µ b Bulk friction coefficient at the interface between bed and transport layer. In contrast to µ and µ s , µ b includes contributions from stresses associated with the particle fluctua-tion motion in addition to contributions from intergranular contacts. ξ ∝ | µ − µ s | − ν Correlation length associated with the yielding transition, where ν = . F , F D , F L , F t , F n , F e Instantaneous force applied by the fluid on a particle [kgm/s ]. Sub-script ( D , L , t , n , e ) refers to nature of force (drag, lift, tangential, normal, effective). T , T D , T L , T t , T n , T e Duration of turbulent fluctuation event [s]. Subscript ( D , L , t , n , e )refers to nature of applied fluid force (drag, lift, tangential, normal, effective). I f Impulse of turbulent fluctuation event [kgm/s] E f Energy of turbulent fluctuation event [kgm /s ] F c Force resisting initial particle motion [kgm/s ] u c Critical instantaneous local flow velocity associated with resisting forces [m/s] I f c Critical impulse required for fluid entrainment [kgm/s] W c Critical work done by flow event required for fluid entrainment [kgm /s ] C eff Energy transfer coefficient, describing the fraction of energy transferred from flow totarget particle during turbulent fluctuation event C ≡ α − f f ( G )√ sd / δ Inverse dimensionless boundary layer thickness α f ≡ u m / u Ratio between the characteristic flow velocity u m associated with the largestturbulent fluctuations and the local mean flow velocity u T max Maximal duration of turbulent fluctuation events [s] f ( G ) Factor that encodes information about particle shape, orientation, and the pocket ge-ometry v i Impact velocity [m/s] v r (v ) Rebound velocity (projected into incident plane) [m/s] v e (v ) Ejection velocity (projected into incident plane) [m/s] θ i Impact angle θ r ( θ Dr ) Rebound angle (projected into incident plane) θ e ( θ De ) Ejection angle (projected into incident plane) E i Impact energy [kgm /s ] E e (E ) Ejection energy (projected into incident plane) [kgm /s ] N e Average number of ejected particles P r Rebound probability ( e D ≡ | v |/| v i | ) e ≡ | v r |/| v i | (Projected) rebound restitution coefficient e z ≡ − v rz / v iz Vertical rebound restitution coefficient A , B , A D , B D , χ, r , r D , n , ζ Dimensionless parameters appearing in empirical or semi-empirical relations describing the collision process between an incident bead and agranular packing α r Normal rebound restitution coefficient in the impact plane β r Tangential rebound restitution coefficient in the impact plane (cid:15)
Restitution coefficient for binary particle collision V b Effective value of the local particle velocity averaged over elevations near the bed sur-face [m/s] f in Particle feeding frequency at flume entrance [1/s] –56–onfidential manuscript submitted to
Reviews of Geophysics f out Frequency of particles passing an illuminated window near the flume exit [1/s] f Q Fraction of active aeolian saltation transport v ↑ Initial particle velocity in thought experiment in section 4.2.1 [m/s] E ↑ Initial particle energy in thought experiment in section 4.2.1 [kgm /s ] E c Critical energy that E ↑ must exceed for particle to continuously rebound along the sur-face [kgm /s ] n tr / n tot Number of transported particles relative to the total number of bed surface particles n v t Number of particles that are faster than a certain velocity threshold v t z h Hop height [m] t h Hop time [s] f ( Re ∗ , z / d ) Function given by equation (A1) v s Settling velocity [m/s] U x ≡ u x /√ s ˜ gd Dimensionless transport layer-averaged fluid velocity V x ≡ v x /√ s ˜ gd Dimensionless transport layer-averaged horizontal particle velocity V z ≡ (cid:113) v z /√ s ˜ gd Dimensionless transport layer-averaged vertical particle velocity Z ≡ z / d Dimensionless transport layer thickness Z ∆ = . Dimensionless average elevation of the particles’ center during particle-bed re-bounds z o Surface roughness [m] c , c , c Model constants in equations (33c)-(33e)
Bed sediment entrainment
Mobilization of bed sediment
Fluid entrainment
Entrainment caused by the action of flow forces
Incipient motion
Initiation of sediment transport by fluid entrainment
Impact entrainment
Entrainment caused by the impacts of transported particles onto thebed
Sediment transport
Sediment motion caused by the shearing of an erodible sediment bedby flow of a Newtonian fluid
Aeolian sediment transport
Wind-driven sediment transport
Fluvial sediment transport
Liquid-driven sediment transport (despite its name, not limitedto fluvial environments)
Nonsuspended sediment transport
Sediment transport in which the fluid turbulence isunable to support the submerged particle weight
Saltation transport
Nonsuspended sediment transport with comparably large transport lay-ers ( h (cid:29) d ) Bedload transport
Nonsuspended sediment transport with comparably small transport lay-ers ( h ∼ d ) Transport capacity (or saturation)
Loosely, the maximum amount of sediment a givenflow can carry without causing net sediment deposition at the bed. More precisely, inthe context of nonsuspended transport of nearly monodisperse sediment, it is definedas a steady transport state at which any further net entrainment of bed sediment intothe transport layer would weaken the mean turbulent flow to a degree at which it is nolonger able to compensate the average energy loss of particles rebounding with thebed by their energy gain during their trajectories via fluid drag acceleration. The sodefined transport capacity obeys equation (29).
Creeping
A superslow granular motion, usually in the form of intermittent local particlerearrangements within the sediment bed (not limited to the bed surface), that occursbelow a macroscopic yield criterion Θ t ( τ t ) Shields number (fluid shear stress) at a nonspecified transport threshold. For specifi-cations, see below. t conv ∝ | Θ − Θ t | − β Time scale for transport property to converge in the steady state near Θ t ,where β is a positive exponent –57–onfidential manuscript submitted to Reviews of Geophysics
Shields diagram (Shields curve)
Diagram compiling measurements of Θ t as a function ofRe ∗ (the Shields curve Θ t ( Re ∗ ) ) for fluvial bedload transport conditions Θ max t ( τ max t ) Viscous yield stress. The upper limit of the threshold Shields number (fluidshear stress) in the Shields diagram, which is associated with viscous bedload trans-port. For Θ (cid:46) Θ max t , a sediment bed subjected to a laminar flow at low Shear Reynoldsnumber Re ∗ may temporarily fail but will eventually rearrange itself into a more stablepacking that resists the applied fluid shear stress. For Θ (cid:38) Θ max t , a sediment bed sub-jected to a laminar flow can no longer find packing geometries that are able to resistthe applied fluid shear stress. Θ In t ( τ In t ) Initiation threshold. Shields number (fluid shear stress) at which the probabilityof fluid entrainment of bed particles exceeds zero (which, for turbulent fluvial bed-load transport, occurs much below the Shields curve). For sediment beds subjected toturbulent flows, a critical fluid shear stress does no longer describe the fluid entrain-ment of individual particles. However, one can still define a Shields number ( Θ In t )below which fluid entrainment does never occur. Like for Θ max t , transient behavior as-sociated with the flow temporarily pushing particles from less stable to more stablepockets is excluded in the definition of Θ In t . Θ In (cid:48) t ( τ In (cid:48) t ) Rocking initiation threshold. Shields number (fluid shear stress) above (below)which there is a nonzero (zero) probability that peaks of flow forces associated withturbulent fluctuation events acting on bed particles exceed resisting forces. That is,there is a nonzero probability that particles rock (or wobble or oscillate) within theirbed pockets. Rocking may ( Θ In (cid:48) t = Θ In t ) or may not ( Θ In (cid:48) t < Θ In t ) lead to completeentrainment depending on the maximal duration of the strongest possible turbulentfluctuation events. Θ Rb t ( τ Rb t ) Rebound threshold. Shields number (fluid shear stress) above which the meanturbulent flow is able to compensate the average energy loss of transported particlesrebounding with the bed by their energy gain during their trajectories via fluid dragacceleration, giving rise to a long-lasting rebound motion. In general, this thresholdis unrelated to the entrainment of bed sediment. It is also the threshold that appears inmost threshold shear stress-based sediment transport expressions. Θ Rb ∗ t ( Θ Rb ∗∗ t ) Modeled rebound threshold. Values of Θ Rb t from models that consider (ne-glect) that the near-surface flow can assist rebounding particles in escaping the bedsurface. Θ ImE t ( τ ImE t ) Impact entrainment threshold. Shields number (fluid shear stress) above whichentrainment of bed sediment by impacts of transported particles onto the bed is ableto compensate captures of long-lasting rebounders (see Θ Rb t above) by the bed. Thisthreshold is arguably also the threshold of continuous nonsuspended sediment trans-port. Θ Rb | ImE t ( τ Rb | ImE t ) Modeled hybrid between rebound and impact entrainment threshold
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