Stochastic description of the bedload sediment flux
 ^{1}The University of British Columbia, Vancouver, Canada
 ^{2}Instituto Superior Técnico, Lisbon, Portugal
 ^{1}The University of British Columbia, Vancouver, Canada
 ^{2}Instituto Superior Técnico, Lisbon, Portugal
Abstract. We present a new formulation of the bedload sediment flux probability distribution. Individual particles obey Langevin equations which are switched on and off by particle entrainment and deposition. The flux is calculated as the rate of many such particles crossing a control surface within a specified observation time. Flux distributions inherit observationtime dependence from the onoff motions of particles. At the longest observation times, distributions converge to sharp peaks around classicallyexpected values, but at short times, fluctuations are erratic. We relate this scale dependence of bedload transport rates to the movement characteristics of individual grains. This work provides a statistical mechanics description for the fluctuations and observationscale dependence of sediment transport rates.
Kevin Pierce et al.
Status: final response (author comments only)

RC1: 'Comment on esurf20224', Anonymous Referee #1, 15 Feb 2022
This manuscript focuses on the statistical description of sediment transport, which is of great importance to numerous applications. Specifically, the authors have proposed a new formulation, combining the Langevin equation of the bedload particles in motion with a dichotomous noise for motionrest alternation. Then they analyze the sediment transport rate based on an approximation of this new model, which can be analytically treated by the Laplace transform and Fourier transform methods. As an overall evaluation, I recognize the contribution of this work in terms of devising some formal theoretical description for the particle transport consisting of both motions and rests, and can recommend eventual publication of this paper. However, I also have some concerns which need to be properly addressed during the revision, mainly focusing on how the authors can improve this work by clearly summarizing their contributions, demonstrating the limitations of their approach, and discussing existing results on this same topic.
1. The authors aim to propose a stochastic formulation based on grainscale mechanics, the idea of which is not new regarding the motion period of the transport alone; hence, the novelty of this paper lies in the consideration of the resting period under the same framework. Keep this in mind, the starting point of this work, Eq. (5), can only be considered as a “formal description”, because the entrainment and deposition of the grain are not formulated mechanically. That is, the start and end of the motions of a particle are not determined by the forces acting on it; thus no new information on the travel and resting times can be obtained based on incorporating this dichotomous Markov noise. From this perspective, the present formulation is intrincically the same as that used previously, for example, by Fan et al. (2014), who considered the motions mechanically, while simulated the transport process by switching the motions of the particle on and off (Fan et al., 2016). The authors may need to discuss this point explicitly. However, I must emphasize that through this “formal description” the authors can indeed obtain some new theoretical results based on Eq. (5), e.g. the master equation Eq. (11) with its solution Eq. (19), and corresponding transport characterisitics Eq. (20), flux rate Eq. (21), serving as key contributions of this work.
2. My second concern is on some fundamental assumption and approximation. During deducing the master equation Eq. (11), the authors assume Gaussian velocities for moving particles, and use the “overdamped" approximation.
 It seems that analytical solutions are only available for Gaussian velocities according to the authors; while for bedload transport at low transport rate, the exponential velocity distribution has also been observed in experiments, which can be even more popular than the Gaussian distribution (Fathel et al., 2015; Lajeunesse et al., 2010; Liu et al., 2019; Roseberry et al., 2012; Wu et al., 2020). Wu et al. (2020) provided an explanation for the existence of the two different distributions, by pointing out that the long trajectories contribute to the Gaussian velocities, and the mixture of both long and short trajectories results in the exponential distribution; the long and short trajectories are distinguished by the shift of the hop distancetime scaling. Resorting to this result I think is important for clarifying some key issues in this work, as will also be demonstrated in the following point. The authors are also suggested to discuss the effects of the velocity distributions on their deduced results.
 For the “overdamped" approximation, explained by the authors as “moving particles attain their steadystate velocities relatively quickly after entrainment”, which is only valid for the description of the long trajectories of particle motions. This is because only the long trajectories have a well defined mean velocity (e.g. the “steadystate velocity”); and the mean velocity for the short trajectories can on the average increase with their travel times (Wu et al., 2020). Since the short trajectories can cover over 80% of the total trajectories in experiments (Wu et al., 2021), applying this “overdamped” approximation may not be appropriate.
3. There are recent studies using different methods to theretically address the motion period of the bedload particle transport, for example, as discussed above (Wu et al., 2020; Wu et al., 2021), the results of which are compared with measured data. In other words, how the particle velocity changes with time was proposed and further determined based on experimental measurements (i.e. other means of specifying the external forces acting on the particle, F(u) in this work). The authors can compare the part of their formulation on the particle motions with different results.
Some minor point:
In section 3, the formulation of the sediment flux is based on N individual particles and then N is extended to infinity. The derivation is indeed complicated as indicator functions and delta functions are used. Could the derivation be started directly from the probability distribution function based on the continuum master equation (5)?
References
Fan, N., Zhong, D., Wu, B., FoufoulaGeorgiou, E., & Guala, M. (2014). A mechanisticstochastic formulation of bed load particle motions: From individual particle forces to the FokkerPlanck equation under low transport rates. Journal of Geophysical Research: Earth Surface, 119(3), 464482.
Fan, N., Singh, A., Guala, M., FoufoulaGeorgiou, E., & Wu, B. (2016). Exploring a semimechanistic episodic Langevin model for bed load transport: Emergence of normal and anomalous advection and diffusion regimes. Water Resources Research, 52(4), 27892801.
Fathel, S. L., Furbish, D. J., & Schmeeckle, M. W. (2015). Experimental evidence of statistical ensemble behavior in bed load sediment transport. Journal of Geophysical Research: Earth Surface, 120(11), 22982317.
Lajeunesse, E., Malverti, L., & Charru, F. (2010). Bed load transport in turbulent flow at the grain scale: Experiments and modeling. Journal of Geophysical Research: Earth Surface, 115(F4), F04001.
Liu, M., Pelosi, A., & Guala, M. (2019). A statistical description of particle motion and rest regimes in openchannel flows under low bedload transport. Journal of Geophysical Research: Earth Surface, 124(11), 26662688.
Roseberry, J. C., Schmeeckle, M. W., & Furbish, D. J. (2012). A probabilistic description of the bed load sediment flux: 2. Particle activity and motions. Journal of Geophysical Research: Earth Surface, 117, F03032.
Wu, Z., Furbish, D., & FoufoulaGeorgiou, E. (2020). Generalization of hop distancetime scaling and particle velocity distributions via a tworegime formalism of bedload particle motions. Water Resources Research, 56, e2019WR025116.
Wu, Z., Singh, A., FoufoulaGeorgiou, E., Guala, M., Fu, X., & Wang, G. (2021). A velocityvariationbased formulation for bedload particle hops in rivers. Journal of Fluid Mechanics, 912, A33.
 AC1: 'Reply on RC1', Kevin Pierce, 31 Mar 2022

RC2: 'Comment on esurf20224', Anonymous Referee #2, 21 Feb 2022
This paper present a theoretical analysis of the set of equation governing the motion of bedload particles subjected to noise (white noise for acceleration, and dichotomous noise for entrainment/deposition). The main contribution of the paper is in the simultaneous treatment of these 2 source of fluctuation, and there impact on the mean bedload flux.
The paper is well written, clear and concise. The approach is sound and standard mathematical tools are briefly introduced before or after they are used, which helps to understand the main ideas behind technical derivations. Main equations however miss a detailed physical explanation, term by term, to be understood by the readership.
The title should be more precise, several stochastic description of bedload having been already proposed.
A general concern is that the stochastic approach, although theoretically sound, is weakly linked to actual statistics of sediment transport by bedload, and thus the relevance of such complicated form of the bedload flux (eq 21!) is questionable for realistic transport conditions. In particular, there is no discussion on the actual values of Péclet number and the importance of considering both velocity fluctuations and entrainment/deposition as processes acting on similar time scales. There are considerable simplifications when decoupling both, so the authors should better point why such coupled approach is necessary. By doing so, the authors should also consider comparing their results with existing experimental or numerical data.
Other comments :
12 : drop “really”, and precise why /when fluctuations matter ?
16 : What is a “classic” description ? Deterministic ?
1719 : I do not get the point here. The approach followed by the authors is also mainly kinematic in that no discussion is made on the forces (gravitational, drag, friction, collision,…) acting on particles.
20 The original “probabilistic” description …
21 Later → replace by “more recently” (there were a lot a probabilistic studies between Einstein and Lisle)
22 “ by promoting his instantaneous steps to intervals of motion with constant velocity” I do not get the meaning of promoting hear.
75 – 85 No mention of Continuous Time Random Walks model is made. Authors should compare their approach with for instance (Schumer, R., Meerschaert, M. M., & Baeumer, B. (2009). Fractional advectionâdispersion equations for modeling transport at the Earth surface. Journal of Geophysical Research: Earth Surface, 114(F4))
l115 : Better explain how this equation can be physically understood, notably the presence of k and ke with time derivatives.
l137 Is the overdamped approximation similar to adiabatic elimination of the fast variable ? A deeper discussion is needed here, notably the validity of such approximation with respect to typical bedload transport scales.
l 143 : How does such expression compare with a spatiotemporal markov process, for instance eq 4.4 in Ancey & Heyman JFM 2014 ?
l217 : Why would velocity fluctuation during motion decrease diffusion at small time scales ? I would have imagined the reverse.
l230 Rewriting the Péclet in its usual form (diffusion time scale over advection time scale) would help understanding the transport process the authors are trying to characterize. In there definition of Peclet, the important length scale is the mean particle jump length. This should appear somewhere.
l264 : can you give an physical interpretation of why the flux is higher at the beginning ? Do we have a higher probability to sample particles in motion at short time scales ?
Figure 4a : why is there a plateau between 1100 s? Is the mean flux only dependent on Péclet and observation time ? If yes, can you make it appear clearly in eq 21. If not, what are the fixed parameter in this figure ? Can you compare with experimental/numerical (DEM) data ?
Figure 4b If the distribution is Poissonian, you should be able to rescale it by its mean and have a single timeindependent distribution. Could you plot this ?

AC2: 'Reply on RC2', Kevin Pierce, 01 Apr 2022
Thanks for the detailed reading and helpful comments on the manuscript. We were impressed by the engagement with the technical side of the paper. We have outlined our responses to your comments in the attached document. Thanks for your effort; we are sure it will improve the paper.

AC2: 'Reply on RC2', Kevin Pierce, 01 Apr 2022
Kevin Pierce et al.
Model code and software
Stochastic simulation of grainscale sediment transport Pierce, K. https://github.com/jkpierce/flippyflop
Kevin Pierce et al.
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