A1. Equation of Motion
 Adopting a right-handed Cartesian coordinate system xi in which the subscript i denotes the three coordinate directions, and assuming that, because the particle Reynolds number Rp is large, the flow is effectively inviscid and hence forces due to fluid viscosity can be ignored, the momentum balance of a particle of mass mp is given by
where dVi/dt is the time derivative of the particle velocity Vi, taken following the particle, FSi is the solid contact force exerted by other particles or, in the case of the experimental test particle, by the stalk holding it stationary, gi is the acceleration due to gravity, P is the fluid pressure, and ni is the outward unit vector perpendicular to the element d of the particle surface .
 If the flow is irrotational as well as inviscid, the fluid velocity ui can be written as the gradient of a potential Φ, that is, ui = ∂Φ/∂xi. Substituting this into the Navier-Stokes equation for inviscid flow [Batchelor, 1967, p. 380], letting ρfu2 = ρfuiui, where ρf is the density of the fluid and terms with repeated subscripts follow the usual summation convention, writing gi in terms of the gravitational potential Γ, that is, gi = ∂Γ/∂xi, and moving the gradient operator ∂/∂xi out of all the terms shows that the expression P + ρfu2 + ρf∂Φ/∂t + ρfΓ has the same value everywhere in the flow [Batchelor, 1967, pp. 382–383]. If, as usually assumed, the particle affects only the nearby flow field, this expression would therefore also have the same value were the particle absent. Hence
where the subscript o denotes quantities in the flow with the particle absent, hereafter termed the reference flow.
 Substituting (A2) into (A1), letting 〈∂Po/∂xi〉 signify the average of ∂Po/∂xi over the particle volume in the reference flow, transforming the surface integral involving Po into a volume integral by means of
which is a form of Gauss's divergence theorem, eliminating ∂Po/∂xi by means of the Navier-Stokes equation for inviscid flow [Batchelor, 1967, p. 380], and rearranging the result leads to
where mf is the mass of fluid displaced by the particle. The second term on the right-hand side of (A4) is the buoyancy force FBi on the sediment particle.
 The quantity averaged over the volume in the third term is the acceleration following a fluid particle [Batchelor, 1967, pp. 72–73] in the reference flow. If the smallest length scale of the pressure gradient fluctuations is large compared to the size of the volume , which is not true of turbulent near-bed flows, in which bed particle wakes are important, the average can be approximated by DUi/Dt, which is Duoi/Dt evaluated at the center of the sediment particle. In the sediment transport literature, except, notably, the paper by Madsen , this term usually has been ambiguously, if not incorrectly, given as dUi/dt. Maxey and Riley  and Auton  have pointed out the importance of distinguishing between the derivative following the geometric center of the sediment particle and the derivative following the fluid at the position of the geometric center in the reference flow. Although for low particle Reynolds number Rp the error is not large, for high Rp it is substantial [Maxey and Riley, 1983; Mei et al., 1991].
 The fourth term on the right-hand side of (A4) is often called the added mass (or virtual mass) force FMi. It arises from the difference in acceleration of the reference fluid and the sediment particle. For weakly shearing flow, in which the reference velocity difference across the particle is small in comparison to the reference velocity relative to the particle, most authors have given for it the formula mfCMd(Ui − Vi)/dt. Auton et al.  noted, however, that the formula instead should be mfCM(DUi/Dt − dVi/dt). For spheres in weakly shearing flow the added mass coefficient CM = . For the strongly shearing turbulent near-bed flows studied in this investigation the formula , with CM empirically determined, will be adopted instead.
 Evaluation of the last term in (A4) requires solution for the flow around the sediment particle. The approach generally taken, and the one followed in this investigation, is to decompose the term into two forces that are treated separately: the drag FDi, which is in the direction of the reference velocity at the center of the particle, and the lift FLi, which is perpendicular to it.
 Inserting all these formulas into (A4) then finally leads to the schematic equation
which apart from the term FSi resembles virtually all equations of motion that have been used in the computation of saltation trajectories of sediment particles.
 Application of (A5) requires calculation of the drag and lift. Solution for the drag on a spherical particle using assumptions consistent with the development of (A5), namely inviscid, irrotational flow with pressure gradients only on scales large compared to the particle size, leads to the physically incorrect result FDi = 0 [Batchelor, 1967, p. 453]. A correct solution for large Rp requires treating the particle boundary layer and separation bubble, which means that the forces due to viscosity cannot be disregarded in the momentum balance for the fluid. The experiments reported in this paper show that a similar conclusion applies to the lift. Thus the process leading to (A4) and (A5) is faulty.
 To obtain closure, empirical formulas for drag and lift are used. These formulas, whose forms for simple flows can be obtained by dimensional analysis, take account of the viscous forces somewhat arbitrarily.
 An example derived from dimensional analysis that works adequately for numerous particle shapes is the formula
for the drag, where FD, U, and V are the vectors whose components are the previously defined FDi, Ui, and Vi, A⊥is the area of the projection of the particle on a plane perpendicular to U − V, and the drag coefficient CD is an empirically derived function of Rp. Coleman  has shown that this formula works well for a sphere resting on a bed of spheres when time-averaged FD and U are used.
 The lift on natural sediment particles near the wall in a turbulent boundary layer is less well understood than the drag. Some authors [Davies and Samad, 1978; Willets and Naddeh, 1986], for example, dispute even the direction of lift on particles near the bed, raising doubt as to whether a formula analogous to (A6) is appropriate.
 Lift arises in shear flow when the fluid velocity on one side of the particle is faster than on the other. For inviscid, weakly rotational flow around the particle, Auton et al.  obtained the formula
in which Ω = ∇ × U is evaluated at the position of the particle center and CLA is the Auton lift coefficient. This solution requires that the difference in velocity across the particle be much smaller than ∣U − V∣, whereas the two commonly are roughly equal for a fixed particle near a sediment bed. Unfortunately, it is the only solution available for large Rp.
 If for a uniform simple shear flow the particle and flow velocities are both in the x1 direction, taken horizontal, and the direction of increasing flow velocity and the perpendicular to the shear planes are both in the x3 direction, taken vertically upward, (A7) reduces to
where A∥ is the area of projection of the particle on the x1x2 plane, and uot1 and uob1 are the reference flow velocities at the top and bottom of the particle. For a spherical particle the analysis of Auton et al.  gives CLA = .
 Wiberg and Smith  adopted the same form for the lift but set the numerical coefficient to instead of , thereby defining a different lift coefficient CLVS. The analysis of Auton et al.  for a spherical particle in a uniform simple shear flow thus gives CLVS = . From the data of Chepil  for a hemisphere in a bed of evenly spaced hemispheres of the same size, Wiberg and Smith  obtained the empirical result CLVS = 0.2; reanalysis gives the range as 0.18–0.34. Analysis of the data of Chepil  for a sphere in a bed of evenly spaced spheres of the same size gives about 0.4–0.5. Anderson and Hallet  used Wiberg and Smith's  formula and Chepil's  data but set CLVS = 0.85CDMA, where CDMA is the drag coefficient read from the empirical curve of Morsi and Alexander ; for Chepil's [1958, 1961] data the range of the proportionality factor is 0.53–1.32.
 Other types of lift, such as spin lift and form lift will not be explored here. Spin lift has been investigated by Niño et al. , Niño and Garcia , and Lee and Hsu , who showed that it has a small but perceptible effect on the length and height of saltation trajectories. Form lift due to asymmetry in particle shape remains largely unstudied.
A4. Equation of Motion of Sediment Particle
 Using (A6), (A7), and the adopted formula for FMi, (A5) can be expanded to
for large particle Reynolds number Rp, where, by analogy with the drag coefficient CD, the lift coefficient CL is an empirically derived function of RP. This equation of motion applies whether or not the particle is fixed in position, touching the bed, or contacting other particles.
A5. Forces on Fixed Particle
 In the experimental setup the particle velocity V is fixed at zero and the force exerted by the stalk FS is (FB + FH), where FB is the buoyant force and FH is the hydrodynamic force exerted on the particle by the fluid. Substituting these relationships for V and FS in (A9) and rearranging gives the formula
which, despite the dubious assumptions and approximations in its derivation, of necessity serves as the starting point for the discussions in this paper. The horizontal and vertical components of the term on the left are the nominal drag and lift measured by the force transducer. The three terms on the right are the hydrodynamic drag, the hydrodynamic lift, and a force analogous to buoyancy, herein called the acceleration force, that arises from acceleration of the reference flow. In turbulent flows none of these three forces is always negligible in comparison to the others, nor are the hydrodynamic drag and lift usually in the directions of the nominal drag and lift; hence explicit distinction between the nominal and hydrodynamic forces is necessary.