Models of bed-load transport across scales: turbulence signature from grain motion to sediment flux

Abstract

Sediment transport controls the evolution of river channels, playing a fundamental role in physical, ecological, and biogeochemical processes across a wide range of spatial and temporal scales on the Earth surface. However, developing predictive transport models from first principles and understanding scale interactions on sediment fluxes remain as formidable research challenges in fluvial systems. Here we simulate the smallest scales of transport using direct numerical simulations (DNS) to explore the dynamics of bed-load and discover how turbulence and grain-scale processes influence transport rates, showing that their interplay gives rise to a critical regime dominated by fluctuations that propagate across scales. These connections are represented using a stochastic differential equation, and a statistical description through a path integral formulation and Feynman diagrams, thus providing a framework that incorporates nonlinear and turbulence effects to model the dynamics of bed-load across scales.

Diagrams for approximating moments

The diagrammatic expansion is used to simplify the calculation of moments of bed-load transport from the first terms of Eq. (12), which corresponds to a sum of “free” moments based on the PDF of the free action, $e^{-S_F}$. Here we follow the procedure outlined by Chow and Buice (2015), defining a set of rules to draw the diagrams that represent each term in the expansion. The moments are obtained by integrating the functions with respect to the free action, and expressing them in terms of the propagator or Green function of the linear first-order equation $G(t_2,t_1) =e^{-a(t_2-t_1)}=\langle q_{*}(t_1)\tilde{q}_{*}(t_2)\rangle _F$ for $t_2>t_1$, represented as an arrow in time, going from left to right. From Wick’s theorem (Chow and Buice 2015) and the symmetry of Gaussian processes, all the odd-numbered free-moments are zero, and the even moments are expressed as the sum of the moments of all the possible pairings of the variables at different instants in time. The Itō interpretation (Gardiner 2009; Särkkä and Solin 2019) or causality of the stochastic system, ensures that the variables of the model only depend on the past. Therefore, each time integral of the expansion in the interacting part of Eq. (12) is represented by a vertex in each diagram, with incoming arrows for each $q_{*}$ field, and outgoing arrows for each $\tilde{q}_{*}$ field. In this case we adopt the formulation and rules defined by Lera (2018), such that the diagrams that contain the initial condition (starting from a constant $\tilde{q}_{*0}$ value), the noise, and the nonlinear term are the following:

(A1)

(A2)

(A3)

These basic diagrams are the building blocks that are assembled to construct the expansion terms. Any general moment derived from the characteristic functional $Z[J,\tilde{J}]$ is defined as follows:

$$\begin{aligned} \left\langle \prod ^m_{i=1}\prod ^n_{j=1} q_{*}(t_i)\tilde{q}_{*}(t_j)\right\rangle&=\frac{1}{Z[0,0]}\prod ^m_{i=1}\prod ^n_{j=1} \frac{\delta }{\delta \tilde{J}(t_i)}\nonumber \\&\quad \times \frac{\delta }{\delta J(t_j)}Z[J,\tilde{J}] \end{aligned}$$

(A4)

which is approximated as a sum of diagrams with m incoming and n outgoing arrows. Therefore, we can intuitively derive diagrams that connect the terms, either for the characteristic functional or for the cumulant generating functional $W[J,\tilde{J}]$, which are easier to derive since only diagrams that connect all the nodes survive (Lera 2018). For example, for the first cumulant or mean of bed-load transport in Eq. (13), we sum the first three diagrams of the expansion with only one outgoing arrow, combining the diagrams in Eqs. (A1), (A2), and (A3). To calculate the numerical value of the moment approximation, each time integration is multiplied by a symmetry factor due to the number of repeated diagrams that appear in the expansion (Chow and Buice 2015), which corresponds to the number of ways arrows can be rearranged in each node of the diagram. The second and third diagrams in Eq. (13) have been multiplied by 2, since we have two ways of exchanging the incoming arrows in the same diagram. For the case of the second cumulant or variance in equation 16, we sum all connected diagrams with 2 outgoing arrows, where the number of nodes equal to the order of the term in the expansion.